> Formex Configuration Processing | Hoshyar Nooshin and Peter Disney | Space Structures Research Centre, Department of Civil Engineering, University of Surrey, Guildford, Suey GU2 7XH, United Kingdom ABSTRACT: This is the frst paper in a series of papers that ae intended to cover the present state of knowledge in the field of formex configuration, processing. This field of knowledge has been developed during the last three decades and has now reached a level of maturity that makes it an ‘deal medium for configuration processing in many disciplines. In Particular, it provides a rich assortment of concepts that are of great Value to the engineers and architects involved in the design of space structures. 4.1 INTRODUCTION Formex algebra is a mathematical system. thet provides a convenient medium for configuration processing. Thé concepts are general and can be used in many fields, In particular, the ideas may be employed for generation of information about various aspects of structural systems such as element connectivity, nodal coordinates, loading details, joint numbers and support arrangements, The information generated may be used for various purposes, such as graphio visualisation or input data for structural analysis. ‘The rudimentiry ideas from which formex algebra has emerged were evolved in the early seventies. These embryonic concepts were published in 1975', Experience in using the early ideas resulted in a substantial revision which was presented at a short course in 1978 and subsequently published in 1981, ‘The first textbook on the subject appeared in 1984° This book provided a comprehensive account of the ideas of formex algebra as they stood at the time of the publication. The book remains a main source of information on formex algebra although new developments in the field have superseded some of the material of the book. These new developments have also enriched this field of knowledge extensively. International Journal of Space Structures Vol. 15 No. 12000 The objective of this paper is to introduce a skeleton of the ideas of formex algebra and the programming language Formizn which is a vehicle for the practical use of formex algebra. The material presented in this Paper will allow the reader to acquire a working knowledge of the elements of formex configuration processing. The term ‘configuration’ is used to mean an ‘arrangement of parts’. The elements of @ structure, for instance, constitute a configuration and so do the component parts of an electrical network and the atoms of a protein molecule. The most common usage of the term configuration is in reference to geometric compositions that consist of points, lines and surfaces. The term ‘configuration processing’ is used to mean the creation and manipulation of configurations and the term ‘formex configuration processing’ is used to mean configuration processing using formex algebra, 1.2 THE CONCEPT OF A FORMEX Consider the configuration shown in Fig 1.2.1 and let this represent the plan view of a diagonal grid. ‘The grid consists of two families of parallel beams that are rigidly connected together at the intersection points, This will result in 240 beam elements that are interconnected together at 136 nodes,(20 divisions @ 2.415 m) DOOOOOOOR Fig 1.2.1 Plan view of a diagonal grid <> 26.50 m . * (a2 divisions @ 2.375 m) Let it be required to produce data for the analysis of the grid of Fig 1.2.1. The complete set of data should include information about the constitution of the grid as well as the nodal coordinates, loading conditions, support arrangements, cross-sectional particulars of the elements and material properties. However, at the present, attention will be focused on the generation of information about the constitution of the grid. ‘The tem ‘compret’ is used to refer to the constitutional ‘aspects ofa configuration, To be specific, the compret of a configuration is ‘the arrangement of the component parts - of the configuration’. In order to describe the compret of a configuration it is necessary 10 list the component parts of the configuration and to specify the interconnections between these component parts. The terms ‘connectivity’ and ‘topology’ are sometimes used instead of compret. ‘Second direction o i 2-3 First direction Fig1.2.2 A nonmat 4.2.1 Normats “Fo specify the compret of the grid of Fig 1.2.1, one may begin by considering a small portion of the grid, ‘This portion is chosen to consist of the elements ij 2 Formex Configuration Processing I and jk at the bottom left comer of the grid. These elements are shown in Fig 1.2.2 together with e simple reference system that consists of two families of dotted lines. In terms of this reference system, element ij can be represented by the construct 11,0; 0,1) The first pair of numbers inside the square brackets represents node i, since this node is at the intersection of dotted line 1 of the first direction and dotted line 0 of the second direction. Similarly, the second pair of numbers inside the square brackets represents node j. ‘A graphical reference system that is used for the specification of the compret of a configuration is referred to as a ‘normat’. The reference system of Fig 1.2.2 is an example of a normat consisting of ‘“normat lines’ that intersect at ‘normat points’. 1.2.2 Uniples, Signets and Cantles Each number inside the square brackets in the construct 1.0; 0.1] represents a normat line and is referred to as a ‘uniple’, Bach uniple pair in the construct {1,05 0,1] represents a normat point and is referred to as a ‘signet’, Fig 1.2.3. The semicolon between the signets indicates a connection between the specified normat points. AAR oO [2,05 0,1] . ae Sena Smet Fig 1.2.3 Acentle ‘The construct (1,0; 0,1 isa simple example of a ‘formex’ and an equation of the form BL=[1,0; 0,1] implies that El is a ‘variable’ whose ‘value’ is, the formex on the right-hand side of the equation ‘Therefore, Bl represents the beam element ij of Fig 1.2.2, Similarly, the element jk in Fig 1.2.2 can be represented by E2= [0,15 12] International Journal of Spice Structures Vol. 15 No. 1 2000Hoshyar Nooshin and Peter Disney Also, the combination of elements ij and jk may be represented by {(1,05 0,1], [0,15 1,2]} This is a formex consisting of two ‘canties’, Fig 1.24. In general, a formex may have any number of cantles and the total number of these cantles is referred to as the ‘order’ of the forex. Curly brackeis are used to enclose the cantles of a former. However, in the case of a formex of the first order the curly brackets are optional. Thus, {{1,0;0,1]} and [1,0;0,1] are considered to be equivalent, In addition to its meening as a component of a formex, the term ‘cantle' js used to refer to a formex of the frst order. Typically, inthe context of a structural configuration ©” a signet represents a node, + acantle represents en element and © a formex represents & group of elements. cantle cantle signet signet signet signet 1,0; 0,1], [0,1; 1, {1405 9,1) [0.4 122 uniple uniple Fig1.24 A formex 1.2.3 Formex Composition The formex variable F= [1,05 0,1}, (0,1; 1,2]} that represents the combination of elements ij and jk in Fig 1.2.2 may also be given as FEI#E2 where EL=[1,0; 0,1] and E2= (0,1; 1,2] ‘The symbol # is referred to as the ‘duplus symbol’. This symbol -acts as an ‘operator’ indicaiing the ‘composition’ of formices represented by El and E2. ‘The term ‘formices’ is the plural of “formex’. Formex composition is a fundamental operation in formex algebra and is used to effect combination of entities represented by two or more formices. The concepts of formex algebra are normally used in conjunction with a computer software. In particular, the interactive. programming language ‘Formian’ provides s suitable medium for formex configuration International Journal of Space Structures Vol. 15 No. 1 2000 processing. An overview of the basic aspects of this, programming language is presented in the next Section and the continuation of the formex formulation of the grid of Fig 1.2.1 will be resumed in section 1.4, ‘The origins of Formian date back to the late seventies and various versions of the language have been in use since then’. The ideas of Formian have evolved and matured over the years and the material relating to Formian in the present work is based on the current state of the language. It is important for the reader to have access to a computer with Formian on it. The material presented in this paper may then be studied step by step with the examples being tried on the computer as progress is being made. Formian may be downloaded from the web site: 1.3 FORMIAN: THE FIRST ENCOUNTER In using Formien, one normally works with a computer ‘monitor’ together with a ‘keyboard’ and a ‘mouse’. These will be parts of a computing system that provides the required processing power as well as storage and inputloutput capabilities for the running of Formian. The term ‘system’ is used to refer to the entirety of the hardware and software components ttat are involved in running Formian on a computing system. The term ‘user’ is used to refer fo a person who uses the system and the term ‘session’ is used to refer to an occasion of the use of Formian that involves a user entering Formian, caying out some Formian operations and exiting Formian. 1.3.1 Formian Screen When Formian is entered, the screen of the monitor will display a number of items, as shown in Fig 13.1, The particular set-up of the screen that is displayed during a Formian session is referred to as the ‘Formian screen’, The main elements of the Formian screen consist of a number of ‘bars’ together with the editory and drawpad, as explained below: © The ‘title bar’ is a narrow rectangular region that appears at the top of the screen, * The ‘menu bar’ is another narrow rectangular region that is situated below the title bar and displays the names of the available ‘menus’,‘© The ‘tool bar’ is situated below the menu bar and displays a number of ‘tool butions’ each of which is for a specific activity, © The ‘editory” is a window that is situated on the left of the screen below the tool bar. The editory is used for the creation and editing of Formian instructions. © The ‘drawpad’ is a window that is situated on the right of the screen below the tool bar, The drawpad is used for graphical output. © The ‘status bar’ is situated under the editory and drawpad. The status bar is used by the system for the display of various kinds of information. © The ‘task bar’ is situated at the bottom of the screen and is used by the Windows operating system to display buttons for switching to various applications. (F Forman? ‘Gare falda foray Fig 1.3.1 Formian screen ‘The menus that are listed on the menu bar and the tool buttons that appear on the tool bar will be discussed at various points in the sequel. 1.3.2 Statements and Commands During a session, the user supplies the system with a sequence of ‘instructions’. These instructions are of two kinds, namely, statements and commands. ‘The first type of instructions are those that are given in textual form, An instruction of this kind is referred to as a ‘statement’, For exemple, EI-{1,0; 0,1); is a statement. The effect of this statement is to associate the formex [1,0; 0,1] with B1. That is, the ‘statement creates a ‘variable’ E1 whose ‘value’ is the given formex. Formex Configuration Processing I Instructions for performance of a number of activities are provided through menus, tool buttons and keyboard shortcuts. The tem ‘command’ is used to refer to an instruction of this kind, An example of a command is the ‘exit command’. This command is “issued” when the button on the top right-hand corner of the screen is ‘clicked’ using the mouse. This button has a cross appearing on it and is referred to as the ‘exit button’. To ‘click a button’, the ‘mouse pointer’ is placed on the button and then the ‘left- hand button’ of the mouse is pressed and released immediately. The effect of the exit command is to terminate the nmning of Formisn and bring the session to an end, The exit command may also be issued by typing ‘alt+X? from the keyboard. That is, by striking the X-key while the alt-eey is held down. This is ar: example of a ‘keyboard shortout’. Statements are entered into the editory by typing from the keyboard. The statements that are placed in the editory may be subjected to execution. By ‘execution’ of a statement it is meant carrying out, the actions that are necessary to put into effect what is specified by the statement. For example, the’ typing of the sequence of characters BL=[1,0; 0,1); E2=[0,1; 1,2]; F=EL#E2; will place the sequence in the editory. The sequence of letters in this particular case gives rise to a set of three ‘assignment statements’, 5 O ‘When an assignment statement is executed then the~ value of what is on the righthand side of the ‘assignment symbol’, that is , the symbol =, is ‘assigned’ to the name that is on the left-hand side of the symbol. In other words, the name on the left is ‘associated’ with the value on the right, The establishment of this association will tum the name into a ‘variable’. That is, a variable is a name which hhas been associated with a value, Thus, the execution of the above three assignment statements will result in the creation of three ‘formex variables” El, E2 and Fr Jn a programming language, as indeed in any mathematical context, a variable is simply a ‘name” that represents a ‘value’. However, it is a long established tradition that one talks about a variable as though it is the value itself. For instance, in relation to the formex varisble F discussed above, one may talk shout the “first cantle of F* where one really ‘means the ‘first cantle of the value of F’. Explicit International Journal of Space Structures Vol. 15 No. 1 2000Hoskyar Nooshin and Peter Disney reference to ‘value’ is sometimes made for ‘extra clarity’ but normally the direct reference to value is omitted, The name used for a variable is normally chosen to reflect what the variable represents so that the name can jog the memory regarding the significance of the variable, This name must be selected from a class of names that are referred to as identifiers. An ‘identifier’ is any sequence of up to eight letters and digits that starts with a letter. This rule provides over 2x10" possibilities for identifiers, The letters that are used in an identifier may be upper-case or lower-case without any distinction being made between them. In fact, in all Formian constructs upper-case and lower- case letters may be used interchangeably. That is, the ‘case’ of the letter is always ignored, ‘Note that the above discussed assignment statements El =[1,0; 0,1]; F2= (01; 1.2] FEL #E2; are simply the ‘formex equations’ discussed before except thet each equation is terminated by ‘a semicolon. In Formian, as a general rule, every statement must be terminated by a semicolon, Formex formulations in this work are presented in two different styles. To elaborate, when writing a formex formulation, without any direct Formian involvement, then the formalation is presented using the normal mathematical conventions, without any consideration regarding the Formien grammer. On the other hand, when a formex formulation is meant to be in terms of Formian statements, then the rules of the Formisn grammar are observed. For example, a formex equation such as FeEl #52 is not terminated by a semicolon unless it is meant to be a Formian statement. 1.3.3 Schemes A group of one or more statements that are to be ‘executed together’ is referred to es a ‘scheme’, To carry out the execution of a scheme, it is necessary to place the ‘cursor’ at a point of the scheme and to click the ‘execution button’ on the too! bar using the mouse. This will have the effect of issuing an ‘execution command’, Placing of the cursor on @ scheme is done using either. the mouse or the ‘cursor Keys’ on the keyboard. The execution button is the tool button with the leter E appearing on it, International Journal of Space Structures Vol. 15 No. 1 2000 While a scheme is being executed, the progress of execution is represented graphically by an advancing stip of colour at the bottom of the sereen on the status bar. This will provide a means of ‘seeing’ the Drogress of the execution as statements are executed ‘one after the other. The narrow rectangular area in Which the advancing strip of colour is shown is referred to as the ‘progress bar’, Fig 1.3.2. [Efenina cura arombay TT) [Fie Edt Uist Tianster Galery Help cisis) SESE) FS Sip OHS Career a g_[orowpas Ei=[06; 0. E2=[0.4 12) ‘Status bar Progress bar ue Sas a aay Fig 1.3.2 Execution of a scheme Fig 13.2 also displays. the above discussed assignment statements. These assignment statements are followed by the statement draw F; This is a ‘draw statement” Whose execution will place a plot of the value of the formex variable F in ‘the drawpad. The term ‘plot’ is used to refer to a graphical representation of a formex. The plot of Fis shown in the bottom left comer of the drawpad in Fig 1.3.2. ‘The effect of a draw statement may also be obtained by ‘highlighting’ a formex variable in the editory and clicking the ‘draw tool button’. This is the tool button withthe letter D appearing on it. The clicking of the draw tool button issues @ ‘draw command’ which will effect the display of the plot of the highlighted formex variable, The editory may contain more than one scheme, For instance, suppose that one wants to create a scheme for representation and plotting of the elements ‘im’ and ‘mk’, shown in Fig 1.3.3, The normat shown with dotted lines in this figure is identical to that shown in Fig 1.2.2. ‘The required scheme may be written as H= {(1,05 2,1), (2,1:1,2]}; clear;use vs(30); draw H; ‘The screen with this new scheme is shown in Fig 1.3.4, In addition to an assignment statement and a raw statement, the new scheme contains a ‘clear statemeat’ and a ‘use statement’. Second direction First direction Fig 1.9.3 A pert of the grid of Fig 1.2.1 F Foe ‘nen faery EI fie For Us Taraee Golery Holo SOoIsIseocirsocengcletal Edtry _ Curent: dap | Dravpad Bie [20603 Hae|oa: uz} FeEéiZ, daw? Ha{(a0: 2), as 2.2)) clear tse (80); raw Fi Fig 1.3.4 Formian screen with two schemes in the editory ‘The effect of a clear statement is to clear the drawpad. Thus, when the new scheme is executed, the plot of F, which was created by the first scheme, will be wiped out. ‘The effect of a clear statement may also be obtained by a ‘clear command’, This command is issued by clicking the ‘clear tool button’, .This is the tool bbutton with the letter C appearing on it. ‘The effect of the statement use vs(3 Farmex Configuration Processing is to alter the ‘plotting scale’. The letters ‘vs? stand for ‘view scale’ end the number 30 in parentheses is a measure of the plotting scale, The default value of the view scale is 10 and, therefore, the execution of the statement use vs(30); will make the scale three times larger. Once a plotting scale is set through a use statement, it will remain in force until the end of the session unless it is changed again by another use statement. ‘The plot of H with the increased scale is shown in Fig 1.3.4, Also shown in this figure is the symbol > separating the first scheme from the second one. This symbol is referred to as the “diamer’ symbol and is (~~) obtained by typing a ‘less than’ symbol followed by \-/ a ‘greater than’ symbol. ‘The diamer symbol mey also be obtained by clicking the ‘diamer tool button’. This is the tool button with a diamer symbol appearing on it. ‘The diamer symbol is the ‘scheme separator’ and every two consecutive scheme in the editory must be separated by one or more diamer symbols. 1.3.4 Examining the Value ofa Formex Variable The value of a formex variable may be examined through a ‘give statement’. For example, consider the.scheme oO E1=[1,0; 015; B2= (0,1; 1,2); F=E1#E2; give; ‘The last statement of the scheme is an example of a “give statement’. The execution of this statement will have the effect of displaying the value of the formex variable F in a ‘give box’ on the screen, Fig 1.3.5. Give statements provide a convenient mechanism for examining the values of formex variables. However, normally one would only use a give statement for displaying a formex that contains no more then a few cantles, Display of e large formex will give rise to a ‘huge volume of numbers that are difficult to follow and scrutinise. ‘The effect of a give statement may also be obtained by ‘highlighting’ @ variable in the editory an? clicking the ‘give tool bution’. This is the tool button with the letter G appearing on it. The clicking of the International Journal of Space Structures Vol. 15 No. 12000Hoshyar Nooshin and Peter Disney give tool button issues a ‘give command’ which will effect the display of the value of the highlighted variable, Frente — tare oan fe ta ba tame GEIS © RSIS EaE Ey Care a Bie[ttoah Epes Page ie Fig 1.8.5 Execution of a give statement [F Fenian Covent ar tony Be) [Fie Est Us Trnster Galery Hop DSS DORMS A Editory Curent te: ag |[ Draven ay Fig 1.3.6 Error indication 1.3.8 Diagnostic Information Consider’ the scheme in the editory shown in Fig 1.3.6. An attempt to execute this scheme will prove to be unsuccessful. The system will display an ‘error box’ with a message indicating that there is a ‘syntax error’ in the scheme. This implies that there is something wrong with the constitution of one of the statements and the statement that contains the error will be highlighted, On examination, it is found that the keyword ‘give? is misspelt, that is, itis spelt as ‘giive? and this is the error that is causing the problem International Journal of Space Structures Vol. 15 No. 1 2000 In general, once the system encounters an error in a statement, the execution of that statement and any remaining statements of the scheme will be abandoned and the nature of the problemi, with as ruch details as possible, will be reported through an error box. To proceed, one should * dismiss the error box by button’, conect the syntax error by editing and ‘try again to execute the scheme, clicking its ‘OK In general, a scheme may be modified, enlarged or reduced in any desired manner using the available editing facilities, 1.3.6 Editing Jn Formien, editing is performed using the normal Windows' editing conventions, The text in the editory may be edited using the edit mem, editing tool buttons and the keyboard, ‘The ‘edit menu’ is activated cither by clicking ‘Edit’ on the menu bar or by using the keyboard shortcut ‘alt+B’, Fig 1:3.7 shows the edit menu together with brief descriptions of the effects of its items, Undo | Undo the last editing action (one level aly), Cat. Cut the highlighted selection from the editory and put it on the clipboard. Copy | Copy the highlighted selection from the editory and put it on the clipboard. Faste | Insert the contents of the clipboard at the cursor position in the editory. » ind... | Find the specified text in the oditory, Replace...| Replace the specified text in the editory swith another text. Select ell | Select the entire contents of the editory, ee Fig 1.8.7 Editing commands ‘When a menu is active (that is, when it is displayed on the screen), then the effect of a ‘menu item’ may be obtained by: © clicking the item using the mouse or * touching the ‘underlined letter” of the item on the keyboard orhighlighting the item using the cursor keys and then touching the ‘return key’ on the keyboard, ‘The editing processes of ‘cutting’, ‘copying’ and ‘pasting’ make use of storage area. for temporary storage of material, This storage area is referred to as ‘he ‘clipboard’ and is symbolically represented by a clipboard. Find Figg what [) D1 Match whole word any Direction Di Match cave Oat Oup 0. Fig 1.3.8 Find box ‘When a menu item is followed by three dots, it implies that the item activates a ‘dialogue box’. For instance, the clicking of ‘Find’ on the edit menu will result in the appearance of the ‘find box’. This dialogue box is shown in Fig 1.3.8. To initiate the “find process’, it is necessary to specify the text to be found. There is a rectangular ‘field’ in the find box into which the required text should be entered. The selection of an option in a dialogue box is achieved by © clicking the comesponding ifem using the mouse or © ‘touching the ‘underlined letter’ of the corresponding item on the keyboard or © using the tab/cursor keys on the keyboard to ven up’ thé corresponding item and then touching the ‘return key’ on the keyboard. ‘The editing tool buttons are shown in Fig 1.3.9. Also shown in this figure is a ‘drop-down box’ that can be used to control the ‘font size’ of the text in the editory. Tr ee --- [¥][B)[Bl[e] (2 R]--- Fig 1.3.9 Editing tool buttons 4.4 BACK TO FORMEX ALGEBRA Returning to the formulation of the grid of Fig 1.2.1, one can produce a description of the compret of the configuration of the grid by writing a ‘longhand” formex. That is, a formex whose whole body is ‘written out explicitly in detail. Such a formex will have 240 cantles each of which will represent an 8 Formex Configuration Processing I element of the grid. The formex may be of the form {0,15 1,0}, [1.05 2,1}, (2,15 3,0), (3.0;4,1], ... * see (17,12; 18,11], [18,115 19,12}, (19,12, 20,117} However, if one actually tries to write such a formex, one is bound to find it an extremely tedious task, A much more convenient way of approaching the formulation is to make use of ‘formex functions” that allow the generation of compretic information taking advantage of the ‘regularities’ of te configuration, 1.4.1 Formex Functions The idea of formex functions is introduced through the simple configurations shown in Fig 1.4.1. In this _ figure, the combination of the elements ij and jk of the grid of Fig 1.2.1 is denoted by F’. The prime (hat is, the symbol "), signifies that the configuration ijk in Fig 1.4.1 is 2 ‘plot? of the formex variable F, where E> ({1,0; 0,1], (0,15 1,2)} Second direction Fig 1.4.1 Some formex plots Now, suppose that one wants to create a formex variable whose plot is the configuration denoted by FI" in Fig 1.4.1. This may be written as FL = {(7,0; 6,1], [6,15 7,21} Aliematively, the formex variable F1 may be given bby the equation Fi =tran(1,6) |F ‘The construct tran(1,6) isa ‘formex function’ that implies ‘translation’ in the first direction by 6 units, Fig 1.4.2. The former vatiable F in equation Fi =tran(1,6)|F International Journal of Space Structures Vol. 15 No. 1 2000Hosiyar Nooshin and Peter Disney is the ‘argument’ of the function and the symbol ‘|? is used to separate the function from its’ argument, This symbol is referred to as the ‘rallus symbol’ and is read as ‘of. The rallus symbol is sometimes represented as two short vertical lines with a gep in between, that is, as, The form of the rallus symbol with a single vertical line is used in this paper, but the other form is also acceptable, Fi = tran(1,6)|F r [Se ras symbol amount of tansation direction of translation abbreviation for translation Fig 1.4.2 Translation function The equation Fl = tran(1,6)| F may be read as: Fl is (equal to) translation, in the first direction, by 6 divisions, of F.- 4 formex function represents a ‘rule’ for transformation of a given formex into another formex. For instance, ia the above example the value of the argument F, that is, {11.05 0,1}, (0,15 1,29) is transformed into {17,05 6,11, (6,1; 7,2]} The rule for the transformation is to add the ‘amount, of translation’ to all the uniples that correspond to the ‘direction of translation’. For the example under consideration, the first uniple of every signet is increased by 6. Another formex function is introduced here in terms of the configuration labelled F2* in Fig 1.4.1. This configuration may be regarded as the plot of F2=ref{l,2)|F ‘The construct, ref(1,2) is a ‘reflection function’ implying reflection in the first direction with the plane of reflection being at normat line 2, Fig 1.4.3. The equation F2=ref(1,2)|F may be read as: F2 is (equal to) reflection, in the first direction, with reflection plane at 2, of F. International Journal of Space Structures Vol. 15 No. 1 2000 F2=ref(1,2)|F position of plane of reflection Airection of reflection abbreviation for reflection Fig 1.4.3 Reflection fuiction Formex fimetions may be used in combination, For instance, a formex variable representing the configuration labelled F3" in Fig 1.4.1 may be given by F3 = tran(2,2) | ref(1,1.5) | F It should be pointed out that the compret of a configuration can normeily be formulated in- many Afferent ways, For instance, a formex variable representing the configuration 4’ in Fig 1.4.1 may be given by F4=reff4) |F3 or | Fé=tran(2,2) | tran(1,5) | F or F4=ref{I,4.5)] tran(2,2) | F2 ... eto. A point regarding the usage of the term ‘compret” should be explained here. Namely, when a formex is said-to represent a configuration, it is meant that the formex represents the compret of the configuration, However, explicit reference to compret is often omitted, leaving it to be implied implicitly. 1.4.2 Functions that Allow Replication ‘The elements ij and jk that are represented by formex variable F are shown again in Fig 1.4.4, together with a number of other elements. U2 1 Fig 1.4.4 Translational replication The combination of all the elements shown in Fig 1.4.4 may be represented byHrin(1,4,2)|F The construct rin(1,4,2) is a ‘sindle function’ implying ‘translational replication’ in the first direction with the number of replications being 4 and the amount of translation at each step being 2, Fig 1.4.5, H=rin(1, ic DIE amount of translation ateach stop ‘number of replications - of replication abbreviation for rindle Fig 1.4.5 Rindle function Note that the number of replications is given as 4 rather than 3, That is, the number of replications includes the initial configuration which is represented by the argument. The term ‘rindle’ is an old English word meaning ‘water course’ and is adopted in formex algebra to imply translational replication. ‘The equation H=rin(1,4,2)|F may be read as: H is (equal to) rindle, in the first direction, with 4 replications, in steps of 2, of F. Incidentally, a new convention that is employed in Fig 1.4.4 is that the first and second directions of the normat are indicated by Ul and U2, respectively, where U1 implies ‘first direction uniples’ and U2 implies ‘second direction uniples’. This convention will be used henceforth. Another kind of replicational effect is obtained through the lambda function, To illustrate the effect, consider the configuration in Fig 1.4.6 consisting of four elements, This configuration may be represented by 2 aact R=lam(1,1)|F ‘The construct lam(1,1) is 2 ‘lambda function’ implying ‘reflectional replication’, Fig 1.4.7, ‘The effect is as though R is given by R=F #ref(1,1)|F ‘That is, the effect of a lambda fimction is to create a forthex that combines the value of the argument with 10 Formex Configuration Processing! that of a reflection of the argument. The reason for the adoption of the name ‘lambda’ is the shape of the upper-case Greek letter lambda. This letter is like an upside down V and may be regarded as consisting of a sloping line combined with its own reflection. Fig 1.4.6 Reflectional replication R=lam(i,1) IF position of ‘lane of reflection direction of reflection abbreviation for lambda Fig 1.4.7 Lambda function The equation Re=lam(1,1)|F ray be read as: R is (equal to) lambda, in the fi( ) direction, with refleotion plane at 1, ofF. Now, a formex variable deseribing the compret of the grid of Fig 1.2.1 may be given by G=rin@,6,2) | rin(1,10,2) | lam(1,1) |F Here, F represents the combination of elements ij and jk in Fig 1.2.1, Jam(1,1) | F "represents the combination of the four elements that forms the leftmost bottom rhombus in the configuration of the grid of Fig 1.2.1, rrin(1,10,2) | lam(1,1) |F represents all the elements forming the bottont row of thombuses and rin(2,6,2) | rin(1,10,2) | lam(1,1) |F represents all the elements of the grid, as shown i. Fig 148, International Journal of Space Structures Vol. 15 No. 1 2000Hoshyar Nooshin and Peter Disney ‘The translation, rindle, reflection and lambda functions belong to a family of formex functions that are referred to as ‘cardinal functions’, rin(2,6,2) {rin(1,10,2). HNam(1,2) |F XX SSS PRINS SOREL SSSR SSS QOOOSERE ES lam(1,1)1F SxS ¥ rin(1,10,2) lam(1,1) IF Fig 1.4.8 Stages of formulation 1.4.3 Functions that Allow Multiple Action ‘The above formulation of the compret of the grid of 1.2.1 may eltematively be written as G=rin(2,6,2) | rin(1,10,2) | lam(2,1) | Jamn(1,1) | [1,05 0,1) This equation can slso be given in the following ‘more convenient form G = rinid(10,6,2,2) | lamid(1,1) | [1,05 0,1) ‘The construct rinid(10,6,2,2) is referred to as a ‘rinid function’ and is equivalent to the ‘composite function’ ¥in(2,6,2) | rin(1,10,2) Also, the construct lamid(1,1) which is referred to as a ‘lamid function’ is equivalent to the composite function am(2,1) | tarn(1,1) The rinid function implies a ‘rindle effect” (that is, translational replication) in the first direction followed ‘by another rindle effect in the second direction, Fig 1.4.9, ‘The lamid function implies a ‘lambda effect’ (that is, reflectional replication) in the first direction followed by another lambda effect in the second direction, Fig International Journal of Space Structures Vol. 15 No, 1 2000 1.4.10. The suffix ‘id’ in “rinid” and ‘lamid? implies double action involving directions one and two. rinid(10,6,2,2) LO snows of tansation at each step in the second direction amount of translation at each step in the first direction ‘umber of replications in the second direction ‘number of replications in the first direction Fig 1.4.9 Rinid function lJamid(1,1) LL poston of plane of reiction for reflection in the second direction position of plane of reflection for ‘reflection in the first direction Fig 1.4.10 Lamid function The suffix ‘id’ may also be used with the translation and reflection functions. Thus, the composite translation function ‘tran(2,7)| tran(1,5) may be written as tranid(s,7) Similarly, the composite reflection function refl2,5) | ref(1,3) may be written as refid(3,5) ‘Trenid, refid, rinid and lamid fiumetions belong to a fail ‘sactions that are referred to as “tendial functions OR 1.4.4 Normic Properties and Retronormic Functions The complete specification of the eompret of the grid of Fig 1.2.1 is given by . G= rinid(L0,6,2,2) |lamid(1,1) | [1,0; 0,1] However, the information represented by G is in terms of ‘normat coordinates’. That is, the positions uof the nodal points are specified relative to the simple Ul-U2 normat shown in Fig 1.4.11. DOOOKKXOKOY RRR RLY FORRES BR ORRR RRS Bo OO 0S KKK KKKE KS o24e6 28.50 m (a2 divisions @ 2.375 m) Q Fig 1.4.11 Grid with U1-U2 normat and xy Cartesian coordinate system ‘A formex that contains the description of the ‘compret of the grid in terms of the x-y coordinate system may be represented by GI = bb(2.415, 2.375) 1G ‘The construct ‘bb(2.415, 2.375) is a function that changes the scales along the first and second directions. The scale factors are 2.415 and 2375 in the first and second directions, respectively. These factors are obtained from the actual dimensions of the grid, as given in Figs 1.2.1 and 1.4.11. GI represents 2 formex that conteins 240 cantles each of which represents an element of the grid of Fig 1.2.1. The uniples in this formex are the actual x and y Cartesian coordinates of the nodes of the grid. ‘The first seven cantles of formices G and G1 are shown below for comparison: G= {[1.05 0,1}, (1,0; 2,1}, (1,2; 0,1], [1,25 2.1], + [3,05 2,1, (3,05 4,1), 3,25 2,1), --=} Gl = {[2.415,0; 02.375], [2.415,0; 4.832.375], [2.415,4.75; 0,2.375], (2.415,4.75; 4.83,2.375], [7.245,0; 4.83,2.375], [7.245,0; 9.66,2.375), [7.245,4.75; 4.83,2.375], ....} Of course, normally, one is not concemed with the details of the numerical values inside a formex. One would work with formices as ‘bundles of information” and let the system deal with the details of numerical computation. The formex variable G may be transformed in various ways to obtain different geometric effects. For example, consider the equations: 2B Formex Configuration Processing 1 GA=00(1.5,0.8) |G and GB =09(0.75,15) |G Plots of GA and GB are shown in Figs 1.4.12 and 1.4.13, respectively. The configurations shown in Figs 1.4.11 (1.2.1), 1.4.12 and 1.4.13 have identical compretic properties but they have different ‘normic properties’. un First direction (radial) ? Fig 1.4.13 Plot of GB Normic properties are those particulars of O configuration that relate to the actual dimensions of the configuration. The normic properties of a configuration are normally specified in terms of the coordinates of the nodal points of the configuration. One way of distinguishing between the compretic and nomnic properties of a configuration is to imagine that the elements of the configuration are made from a freely deformable substance like rubber, The configuration may'then be distorted in any conceivable manner without affecting its compret, as long as no element is added or removed and as long as the connections between the elements remain unchanged. In contrast, any distortion of the configuration will alter its normic properties. Functions such as 1bb(1.5,0.8) and p(0.75,15) International Journal of Space Structures Vol. 15 No. 1 2000Hosiyar Nooshin and Peter Disney are referred to as ‘retronormic fanctions* or simply ‘retronorms’. The term ‘retronorm’ is used to refer to any function that transforms the normat coordinates ofa configuration into global Cartesian coordinates. The function bb(1.5,0.8) is a ‘basibifect retronorm’ effecting scaling in the first and second directions by factors 1.5 and 0.8, respectively. A basibifect retronorm may be used for the transformation of normat coordinates into the global x-y coordinates provided that the directions of the normat are the same as those of the global x-y axes. For the example under consideration, this condition is satisfied, The fiction 59(0.75,15) is 2 ‘basipolar retronorm’ that considers the uniples of its argument as ‘polar coordinates” and applies (linear) scaling of 0.75 in the first (radial) ditection and an angular scaling of 15* per division in the second (circumferential) direction. The resulting polar coordinates are then transformed into global Cartesian coordinates. The positions of the U1-U2 polar normat and the global x-y coordinate system are shown in Fig 1.4.13. A nonmat may be regarded as a ‘graphical retronorm’. For example, the Ul-U2 nomiat of Fig 14.11 may be considered to be the graphical equivalent of the basibifect retronorm b(2.415,2.375) and the UI-U2 nommat of Fig 1.4.13 may be considered to be the graphical equivalent of the basipolar retronorm bp(0.75,15) ‘The prefix “basi” in the word ‘basibifect’ implies ‘uniform sealing’ and the part ‘bifect” refers to the first and second directions. Thus, the © term ‘basibifect? implies uniform scaling in the first and second directions. The term ‘basipolar’ implies uniform scaling in the first and second directions of a polar normat followed by the transformation of the resulting polar coordinates into Cartesian coordinates. ‘The use of a normat in conjunction with a global Cartesian coordinate system is convenient in practice. The idea is that while describing the compretic aspects of a configuration, one works in terms of a normat, focusing the attention on the interconnection pattern without having to worry about the actual coordinates of the points. International Journal of Space Structures Vol. 15 No. 1 2000 Subsequently, the generated configuration may be subjected to suitable retronormic transformations to obtain the actual shape. However, it should be mentioned that there are cases when the globel Cartesian coordinate system itself is a suitable normat. u Fig 1.4.23 A three-way grid A generic formulation for the grid of Fig 1.4.23 may be written as: B= {(0,0; 2,0}, [0,05 1,1), 12.0; 1,1]) F=genid(m,n,2,1,1,-1) |E in(1,men,2) | [n,n; n+2,n) H= bb(d/2,¥3d/2) | (F #G) In this formulation © E represents the group of three elements that form the leftmost bottom triangle, © F represents all the elements of the grid other than the topmost horizontal elements, with m being the number of horizontal elements at the base in the first direction and’n being the number of rows of triangles in the second direction, 16 Formex Configuration Processing I 4 G represents the topmost horizontal elements ‘and +. Hrepresents all the elements of the grid, ‘The formex variables 8, F and G are relative to the ‘Ul1-U2 normat and H is relative to the x-y Cartesian coordinate system in Fig 1.4.23. The terms d/2 and ‘V3d/2 in the basibifect retronorm ‘bb(d/2,¥34/2) are the factors for scaling in the first and second directions. The above generic formulation makes use of the concept of a ‘genid function’, This is 2 mechanism for creation of non-rectangular arrays of objects. The particulars of the genid function are described in Fig 1.4.24. genid(m.n,2,1,1,-1) | L rapa amount of translation ateach stop in the second direction ‘amount of translation at ‘each step in the first direction umber of replicetions in the second direction initial number of replications in the first direction Fig 1.4.24 Genid function The first four parameters of the genid function a Fila to thse ofthe nid fancton, The remaining) two parameters may be desoribed as follows: ‘© ‘Bias’ represents the amount of translation in the first direction for every step in the second direction. Thus, if bias is denoted by ‘b” then the row of triangles indicated by R in Fig 1.4.23 will ‘undergo e translation by the amount b in the first irection, the row of triangles indicated by S will ‘undergo a translation by the amount 2b in the first direction and so on. © ‘Taper’ represeats the increment in the number of replications for every step in the second direction. Thus, if taper is denoted by ‘t’ thea the umber of triangles in the row indicated by R in Fig 1.4.23 will be m+t, the number of triangles in the row indicated by 8 will be m+2t and so on, The values for parameters in the above generic formulation that give rise to the grid of Fig 1.4.2° are: : m=10,n=7 and d=1 International Journal of Space Structures Vol. 15 No. 1 2000Hoskyar Nooskin and Peter Disney ‘The above generic formulation may be used to create a variety of trapezoidal three-way grids by using Gifferent values for parameters m, 2 and d. Three such examples are shown in Fig 1.4.25, with the comesponding values of the parameters shown alongside the configurations, Fig 1.4.25 Three-way grids created through generic formulation 1.4.7 Functions that Effect Rotation Consider the triangulated configuration in Fig 1.4.26. This is the grid of Fig 1.4.23 which is rotated by 90° about the point that is encircled in Fig 1.4.26. Also shown in this figure, is the boundary of the original position of the grid together with a curved arrow that indicates the rotation, ‘The rotation has been effected using the equation HI = ver(1,2,5, 343,90)! H ‘The construct ver(1,2,5, 3V3,90) is a ‘vertition function’ and ‘ver’ is an abbreviation for vertition, The term ‘vertition’ is a Latin based ‘word that means ‘rotation’. The parameters of the above vertition fimnction are described in Fig 1.4.27. ‘The first two parameters of the vertition function specify the directions that define the plane of rotation. The next two parameters specify the coordinates of the centre of rotation, In the present example, the Cartesian coordinates of the centre of rotation, from Fig 1.4.23, are x=5d and y=3V3d With the value of d taken as 1, the coordinates of the centre of rotation will be x=5 and y~3v3 International Journal of Space Structures Vol. 15 No. 1 2000 LVNZNZ\ PALL PAIN AAV AVAVAVAVN Fig 1.4.26 Effect of vertition function ver(1,2,5,313,90) L— angle of rotation ‘in degrees second coordinate of the centre of rotation of rotation first coordinate of the centre directions defining the plane of rotation abbreviation for vertition, Fig 1.4.27 Vertition function ‘The last parameter of the vertition function specifies the angle of rotation. Actually, the presence of the last parameter of the vertition function is optional, That is, it may or mey not be present. If this Parameter is not given then the angle of rotation is assumed to be 90°, ‘The vertition function ‘ver(1,2,5,313,90) may be written in the more convenient form verad(5,3"3,90) The suffix ‘ad’ implies an action involving the first and second directions. In contrast, the suflix ‘id? which was discussed before, implies an action in the first direction followed by a similar action in the second direction. The suffix ‘id’ implies a ‘double? action whereas ‘ad? implies a ‘single action, Fig 1.4.28 illustrates a rotational replication effect, ‘The configuration is obtained as a combination of the grid of Fig 1.4.23 with two rotations of itself, ‘The angle of the first rotation is 120° and that of the I7second rotation is 240°, The centre of rotation is shown encircled in Fig 1.4.28. ALS VANAN LOLLO2QEO. SOOLYYOYD,. LALLALALLAD, SSAA, EALYDALALADA, LIDPDLLAL BAL AALA LLL KLOAELLD, LEYLA PAVANANANANAVANAVAVAN Fig 1.4.28 Rotational replication 3,3,120) angle of rotation L angle of: foreach step number of replications second coordinate of the centre of rotation first coordinate of the centre of rotation. L_ diréctions defining the plane of rotation abbreviation for rosette Fig 1.4.29 Rosette function The formex that represents the configuration of Fig 1.4.28 may be written as H2 = pex | r0s(1,2,5,313,3,120) | H The construct ros(1,2,5,3V3,3,120) is a ‘rosette function’ and ‘ros’ is an abbreviation for rosette. The rosette function provides a mechanism for rotational replication with its parameters described in Fig 1.4.29. ‘The first four parameters of the rosette function are similar to those of the vertition function. The remaining two parameters specify, the required number of replications and the angle of rotation for ‘each step of replication The last two parameters of the rosette function are optional with the ‘default values’ being 4 and 90°, That is, if the last two parameters are not given then 18 Formex Configuration Processing I it will be assumed that the number of replications is 4 and the angle of rotation is 90°, ‘The equation that defines the configuration of Fig 1.4.28, that is, 2 = pex | ros(1,2,5,3¥3,3,120) | H involves a pexum function, This function was described earlier in relation to the examples of Figs 1.4.18 and 1.4.19. The role of the pexum function in the present cise is to eliminate the superffuous elements in the central region of the configuration of Fig 1.4.28. To elaborate, the three grids that constitute the configuration of Fig 1.4.28 have 2 number of overlapping elements. Therefore, without the effect of the pexum fumction, the central part of the configuration will have number of superfluous elements. a) SOLED SYRSIIIVVOD SALAD STAD SAID, LEADED. ARIIITLLLAELODA PAVANAAV NANA NVAVAVAVAVAVAVAN SSL DASA LRROY ALAALA CYS DAYANA AYAVAVN DAMS, PAY ANANANAVAY AV AN AVAVAVAVAVAN LADY AY WVAVAVAVAVAVAVAAVAVAVAVAVANAVAVAV/ AAP DAAAAAALYY AAA AAAAAAAYY I VAVAVANAVAVAVAVAVAV AVA AV VAVAVAVAVAVAVAVAVAVAVAYA LIDIA Oo Another rosette effect is illustrated in Fig 1.4.30, The three-way grid shown in this figure is obtained as a rotational replication of the grid of Fig 1.4.23. The configuration may be represented by HB = pex | r05(1,2,5,5%3,6,60) | The effect of the pexum function in this example is to remove the superfluous elements along the edges of the 6 trapezoidal parts that constitute the grid of Fig 1.4.30. Fig 1.4.80 A rosette effect Similar to the case of the vertition function, the rosette function os(1,2,5,573,6,60) ‘may be writien in the more convenient form rosad(5,5%3,6,60) where the suffix ‘ad’ implies an action involving the first and second directions, as described before. International Journal of Space Structures Vol. 15 No. 1 2000Hoshyar Noashin and Peter Disney 1.5 MORE ABOUT FORMIAN Fig. 1.5.1 shows the Formian soreen with two schemes in the editory and a formex-plot in the drawpad, [F Fontan? “Curent eer fomrbay EER Eft us Tawi Galey ep CISSS e breIsle)) £5 (0p:201 osrts}.20:220 epee [Gmtettarling ese Fae she ireey oars irra. 0; [0,0; 2,0}, [0.0; 1.1),[2, enid(.n,2,1,1,-1)| rin(1.m-n,2)|fnn; n+2,n} bb(d/2,(sqrt|3)*d/2)| (F#G); use v2}: clear; draw H; <> HA=rosed(5,0,3,120) |; HB=pex|HA; clear; draw HB; Fig 1.5.2 An editory display The contents of the editory are also shown in a frame, with a double line at the top, in Fig 1.5.2. An amangement of the form shown in Fig 15.2 is referred to as an ‘editory display’. An editory display is used for displaying the whole or a part of the contents of the editory. The termi ‘record’ is used to refer to the textual material that constitutes the entire contents of the editory. A record normally consists of a sequence of Formian statements that may have been divided into schemes. International Journal of Space Structures Vol. 15 No. 1 2000 1.5.1 Setting Values for Generic Parameters Consider the first scheme in the editory display of Fig 15.2. This scheme contains the generic formulation for the grid of Fig 1.4.23, as discussed in section 1.4.6. The scheme starts with the assignment statements m=10; These assignment statements set values of the parameters m, n and d for use in the subsequent statements. This is the usual style of setting values for parameters of a generic formulation. That is, a sequence of aésignment statements for’ setting “parameter values is placed at the top of the scheme. ‘Thereafter, the values of the parameters are adjusted by editing as required, 1.5.2 Automatic Control of Plot Sizes The use statement in the first scheme of the record in the editory display of Fig 1.5.2 involves a new ‘use- item’, This use-item is of the form vm(2) where vm stands for ‘view mode’, The effect af this use-item ig te let the sizes of formex plots in the drawpad be determined avtomaneaiy: To elaborate, ‘wnen ‘view mode 2? is ‘current’, then every formex plot is photographically scaled such that it just fits into the drawpad, Onee the system is put into the automatic sizing mode it will remain in this mode until the end of the session unless the mode is changed again by the user. The system may be put back into the default non-automatic sizing mode through a use statement of the form use vm(1); 1.5.3 Numeric Expressions and Functions ‘The record in the editory display of Fig 15.2 contains numeric expressions such as mn and (sqrt | 3)*a/2 19In general, a ‘numeric expression’ may appear at any position in a Formaian statement where a ‘number’ can appear. ‘A numeric expression in Formian is any meaningful combination of mumbers, mumeric variables, arithmetic operators, numeric functions and parentheses, The arithmetic operators are + = * / and 4 (for exponentiation) ‘The available set of numeric functions in Formian is, similar to the usual set of standard numeric functions in common programming languages. These functions are sign, ‘abs (absolute value), sqrt (square root), sin (sine), cos (cosine), tan (tangent), asin (arcsine), acos (arecosine), atan (arctangent), exp (exponential), In (natural logarithm), ric (rounded integer conversion), tic (truncated integer conversion), floc (Floatal conversion) and ran (random number). The rallus symbol is used to separete a numeric function from its argument. For example, sinx is written as sin [x ‘Thus, the assignment statement y =008 | 60; will create a numeric variable y with a value of 0.5, (that is, cosine of 60°). In Formian, angles are always given in degrees, ‘The functions ric, tic and floc are ‘conversion’ functions, For example, if x = 1.76 then ric |x is equal to 2, ric | is equal to-2, tic|x is equal to 1 and tic |-x is equal to-1. Also, the assignment statement y= floc | 6; will result in the value of 6 in “floating point form’ to be assigned to y. 20 Formex Configuration Processing I Before this section is ended, attention is drawm to the differences in the styles of presentation in ‘free formulations’ and their corresponding ‘Formian formulations’. To exemplify the point in mind, consider the formex equation H=bb(d/2,13d/2) | (F # G) and its corresponding Formian statement H-= bb(d/2,(sqrt | 3)*/2) | F #G); ‘The above formex equation is the last line of the formulation for the three-way grid configuration of Fig 1.4.23. In the case of this equation, a free mathematical style of presentation is used. However, the style of the comesponding Formian statement is compatible with the Formian grammar. In particuler, the expression B42 of the formex equation is given as (sqrt | 3)*d/2 in the Formian statement, Fig 1.5.2. 1.5.4 Saving Records Configuration processing activities are performed by executing sequences of Formian statements that are placed in the editory. itis often required to save the material in the editory for future use and further development, The current record in the editory may be saved in a “text ile’ using the ‘file menu’, ‘The file menu is activated by either clicking the item. “File? on the menu bar or by using the keyboard shortcut ‘alt+F’. The file menu together with brief descriptions of the effects of its items are shown in Fig 1.53. ‘When the file mem is active (that is, when it is displayed on the screen), the current record (that is, ‘the contents of the editory) may be saved through the ‘save? or ‘save as’ item of the file menu, If the current record has already been saved in a text file, then the name of this text file appears on the title bar of the editory. Subsequently, if the record is modified in any way then the contents of its text file may be updated using the ‘save? menu item. ‘The ‘save as’ menu item is used © when the current record has been saved in a text file with the name of the text file appearing on the title bar of the editory and it is required to save the current record in another text file, or © when the current record has not yet been saved. and itis required to save it in a text file. International Journal of Space Structures Vol..15 No. 1 2000Hoshyar Noashin and Peter Disney New Clear the contents of the editory, Open... Place the contents of a text file c in the editory, Save Save the contents of the editory in the currently open text file. Save as. Save the contents of the editory ina textile, Print text... | Print the contents of the editory. Print picture... | Print the picture displayed on the drawpad, Print dust... | Print the visible part of the editory together with the picture displayed on the drawpad. 1 tring > Open this file. 2 diag Exit Exit Formian, Fig 1.5.3 File menu In cither of the above cases, the current record may be saved through the ‘save as’ menu item. The ‘save as" men item activates the standard Windows ‘save dialogue box’ that may be used to specify a file name. It is not necessary to save the current record in a text file in every occasion, It frequently happens that the objectives of a configuration processing task are achieved through a few simple statements and there is no real need for saving the statements, Indeed, if one tries to create a new text file for every little activity, then the accumulation of the saved files will soon become a nuisance. However, if there is any chance that the record in the editory may be of use later, then it is wise to save it, A practical approach in this respect is as follows: © For material that is needed for fisture use and farther development, the related schemes may be grouped together and saved as a single record in a file (rather than saving the schemes separately in different files), eA file may be designated for temporary storage of material (called, for instance, ‘workfile’), ‘Then, any record that is needed for only a short while can be saved in this file which will be International Journal of Space Structures Vol. 15 No. 1 2000 fequcatly overwritten with the latest required record. If necessary, one may have more than one file for temporary storage of material (called, for instance, temp1, temp2, .. etc) The effect of the ‘new’ item of the file memu is to clear the editory for creation of a new record. The use of the ‘new’ menu item causes the entire contents of the editory to be deleted. But, before deleting, the user is asked whether it is required to save the record in the editory, The effect of the ‘open’ item of the file mem is to open a previously created ‘text file’ and place its contents in the editory. ‘The ‘open’ menu item activates the standard Windows ‘open dialogue box’ that displays the list of all the previously created files in the current ‘folder’. The required file may then be selected and ‘opened. This will cause the contents of the file to replace the current record in the editory, In Formian, only one text file can be open at any given time. Therefore, the opening of a new file will result in the loss of the current material in the editory unless this material is saved before opening the new file, The user is appropristely prompted by the system in this regard, ‘The last few text files that have been opened are listed in the lower part of the file menu, For instance, in Fig 1.5.3 the latest files opened are given as ‘triag" and ‘diag’, Each of these files may be opened through its item on the file menu. This route for opening a file is more convenient than that through the ‘open’ menu item and the ‘open dialogue box’. Itis good practice to have a ‘folde:* thet contains all the text files created in Formian, This folder may then be used as the ‘current folder’ in Formien sessions. The name of the current folder is always displayed on the title bar of the Formian screen. In the examples given, the name of the current folder is ‘formbay’. A folder may be created using standard Windows procedures, The last item on the file menu is ‘exit’. This mena item may be used to exit Formian, The file menu bas another three items that have not been referred to yet. These menu items are for printing operations and will be discussed in the next section. The effect of some of the items on the file menu may also be obtained through tool buttons. There are four ‘such tool buttons and theses are shown in Fig 1.5.4. 2IFig 1.5.4 File tool buttons 1.5.5 Printing The information generated through configuration processing activities in Formian can be outputted in ‘two main ways, namely, © export to structural analysis programs, Graughting packages, graphies systems, numerically controlled machines, ... eto, as will be discussed in the second paper in this series, and ‘+ printing in textual and/or graphical form. Printing of information may be effected through the file menu. There are three items on the file menu that relate 10 printing operations, namely, ‘print text’, ‘print picture’ and ‘print duet’, Fig 1.5.3, The ‘print text’ menu item may be used to print 2 part or the whole of the current record in the editory. The ‘print text” menu item activates the ‘print text box’ shown in Fig 1.5.5. The entries in this box may be set as required and then the printing operation say be initiated through the OK button. Formex Configuration Processing T instance, ifthe printing of a duet is initiated when the screen has the arrangement shown in Fig 1.5.1, then the printed output will be'as shown in Fig |. (Cotati otelazingels aay eck tevagh Se ew oats BIE, Hire 7 CAAA Bonin CARY AKER KAA? Fig 1.5.6 A duet 1.5.6 Properties of Variables ‘The list of current variables and their properties may be inspected at any time during a Formian session. ‘This list may be displayed through the ‘list menu’ by © clicking the item ‘List’ on the menu bar or using the keyboard shortcut ‘alt+L? and © selecting the item ‘variables’ from a choice of three options that will be displayed. For example, suppose that the above steps are taken following the execution of the schemes shown in Fig 1.5.1 (which are also shown in the editory display of Fig 1.5.2 and the duet of Fig 1.5.6). This will result in the display of the ‘variables box’ shown in Fig Pintiod 157. Print: [ Varabies Number of copies: [1 Variable Type ‘Order Plextude Grade Size d FT 4D © NTFMK 3 2 2 a 1 oo WTENK «Mr 22 2D g | INTFMX 3 2 2 4 Fig 1.5.5 Print text box ho FLTFMX 500-2 22k 7 ha RLTFMK 400-227 | : hb RLITFMK 35-22 8D ‘The ‘print picture” menu item can be used to print a mont * copy of the contents of the drawpad, That is, all the now a graphic effects that are visible on the drawpad. The ‘print picture’ menu item activates the ‘print picture ~Teancel box’ which has a form similar to the ‘print text box’, shown in Fig 1.5.5. The ‘print picture box’ may then be used to initiate the printing operation. ‘A copy of the combination of the visible material of the editory and the graphic effects on the drawpad is referred to as a ‘duet’. A duet may be produced through the ‘print duet” item on the file menu or the ‘print duet’ tool button. This is the tool button with 2 ‘pritter symbol appearing on it, Fig 1.5.4, For 22 Fig 1.5.7 Variables box ‘The information presented in the ‘variables box’ is arranged in a number of columns. The first column lists all the current variables in the alphabetical order. The variables in this column are always given in lower-case letters. International Journal of Space Structures Vol. 15 No. 1 2000 oaHoshyar Nooshin and Peter Disney The second column of the ‘variables box’ shows the ‘types’ of the variables. The abbreviations used ere as follows: FLT stands for. floatal, FMX stands for formex and INT © stands for integer. To elaborate, two kinds of numerical values are represented in computing systems, namely, ‘integer’ and ‘flostal’. An ‘integer value’ is ‘stored in 2 manner ‘that the full precision of the value is preserved. A “floatal value’ is a numerical value that is stored in a ‘floating point form’. For instance, the assignment statement y= 725.328; will result in the number 725.328 to be assigned to y. ‘The value will be stored in a ‘flosting point form’. This ‘floatal number’ may be represented by 0.725328 x 10° This value may also be represented by 0.725328E3 or 72532882 or 72.5328E1 ... ete, The letter E in. the above ‘floating point representation’ stands for ‘exponent’ and may also ‘be given in lower-case as ‘e’, The range of values that can be represented by ‘flostal numbers’ is enormously larger than the range that can be represented by ‘integer numbers’ However, unlike the ‘integer form of storage’, the ‘floatal form of storage’ does not necessarily preserve the full accuracy of the values, The above statement is not meant to imply that the available range of integer values is too restrictive, since integer numbers with up to about 7 digits can be used. Also, the accuracy of floatal numbers is quite adequate for most practical purposes, since one can normally rely on an accuracy of 7 decimal places. This represents an accuracy better than a millimetre in a mile, There are situations when only integer numbers can be used meaningfully, like specifying the number of nodes in 2 configuration. Obviously, it does not make sense to have 270.34 nodes in a configuration! On the other hand, if one wants to specify the length of a bar element, then either an integer number or a floatal number can be used. It is meaningful to refer to a '2 metre long bar element” or a ‘2.15 metre long bar element’. [In relation to formices, a formex is said to be an ‘integer formex” provided that all of its uniples are International Journal of Space Structures Vol. 15 No. 1 2006 integers, In contrast, a formex is said to be a ‘noninteger formex’, provided that one or more of its uniples are noninteger. For example, {04.25 3,11, -6,4; 4,2} is an integer formex and {(4,2; 3,1], [-6,4; -4,2.5]} and {{4.72,2.14; 3,1), [-6.67,4; 4.2]} are noninteger formices, In Formian, the uniples of an integer formex are stored as integers and the uniples of a noninteger formex are stored as flostal numbers. Hence, a noninteger formex in the context of' Formian’ is referred to as a ‘floatal formex’, For instance, consider the assignment statements El = {[4,25 3,1], [-6,4 -4,2]}; and B2= {(4,2; 3,1], [-6,45 -4,2.5]}; The only difference between the two formices Concems the last uniple which is the integer number 2 in the first formex and the noninteger number 2.5 in the second one. Let the execution of the above two assignment statements be followed by the execution of the give statement sive BLLE2, This will result in the display of the give box shown in Fig 1.5.8. FI= 4,2 3, 1, [6, 4-429 E2={[ 4.000000E+000, 2.0000006+000; 3,000000E+000, 1:000000€+000}, [-6.000000E+000, 4.000000E+000, -4.000000E+000, 2'500000E+000)} Echo to edtory Cancel Fig 1.5.8 Integer and floatal formices It is seen that the value of formex variable El is given with integer uniples but the value of E2 is given in terms of flostal uniples, Incidentally, the give box has a button for ‘echoing to editory’. The clicking of this button has the effect of placing a copy of the contents of the give box at the end of the editory. 23In the early days of formex algebra, one was very conscious of the ‘integemess’ and ‘nonintegemess’ of formices. This was mainly due to the fact that, with the available computing facilities of those days, work with integer formices was appreciably faster than that with noninteger formices. However, the attitudes in this regard have changed. The. present attitude is to work with both integer and noninteger formices as the situation may demand and let the system take cate of the rest. Returning to the discussion of the ‘variables box’ of Fig 1.5.7, the information given in the second column indicates that dis‘a ‘floatal variable’, ¢, fand g are ‘integer formex variables’, hh, ha and bb are ‘floatal formex variables" and m and n are ‘integer variables’. ‘The third column of the ‘variables box’ of Fig 1.5.7 is only relevant to formices and, therefore, it does not have any entries for the ‘numeric variables? dm and n, An entry in the third column indicates the ‘order’ of a formex. That is, the number of cantles of the former, ‘The fourth end fifth columns of the ‘variables box’ of Fig 1.5.7 give information about ‘plexitude’” and ‘grade’ that are relevant to formices only. The term ‘plexitude’ means the number of signets in a cantle and the term ‘grade’ means the number of uniples in a signet. Each one of the formex variables listed in Fig 1.5.7 represents an assembly of two-noded beam elements ‘with each cantle of the value of the variable having two signets corresponding to the two nodes of an clement. Thus, the plexitude of the formex variables in Fig 1.5.7 is given as 2. Also, the formex variables in Fig 1.5.7 are either relative to a two directional normat or a two dimensional Cartesian ‘coordinate system with each signet consisting of two uniples, Thus, the formex variables are of grade 2, as indicated in Fig 1.5.7. ‘The last column of the ‘variables box’ gives the size of the storage area used by each of the variables. The size is given in terms of bytes, where ‘b” stands for “byte? and ‘Kb” stands for ‘kilobyte’ (1000 bytes). 4.5.7 Concept of Tolerance A usefil piece of information that may be deduced ftom the third column of the ‘variables box’ of Fig 1.5.7 concerns the formex variables HA (ha) and HB (hb). To elaborate, both of these formex variables represent the three-way grid shown on the drawpad 24 Formex Configuration Processing 1 in Fig 1.5.1 (and Fig 1.5.6). However, the grid represented by HA contains a number of superfluous clements whereas HB represents the grid with the superfluous elements removed, as discussed in relation to Fig 1428 in section 14.7. The climination of the superfluous elements is achieved through the pexum fimction, es shown in the second scheme of the record in the editory display of Fig 1.5.2, Now, from the third column of the ‘variables box’ in Fig 1.5.7, it is seen that formex HA has 450 cantles whereas formex HB has 315 cantles. This shows that there have been 135 superfluous elements that were deleted by the pexum function. ‘A question that may be raised regarding the working of the pexum function is as follows: Considering the imate nature of floatal numbers, how could the equality of uniples in 2 floatal formex for the ( ‘operation of the pexum fumetion be checked? : To elaborate, consider the arrangenient shown in Fig 1.5.9. The elements 1 and 2 in the figure are supposed to represent two coincident elements in a ‘grid represented by a floatel formex. The elements are shown with greatly exaggerated inaccuracies in the coordinates of their end nodes. The theoretically correct position of both elements 1 and 2 is indicated by the dotted line. sesseves+ Correct position, Element 1 —— — Hlement 2 tT Fig 1.5.9 Tolerance range The question is: With the inevitable presence of inaccuracies, as indicated in Fig 1.5.9, how can the pexum fonction detect the (theoretical) coincidence of the two elements. “The answer is that the coincidence of the elements is detected with the aid of the concept of “tolerance”. To elaborate, in Formien, whenever two numbers m. and n are to be compared for equality, then the following inequality is considered |mnjst : where, a typical value for t is'0.00001. International Journal of Space Structures Vol. 15 No. 1 2000Hoshyar Nooskin and Peter Disney The above inequality compares the value of t with the absolute value of the difference between m and n, If the inequality is satisfied then m and n aro regarded as equal. For instance, with t= 0.00001, 4.329123 and 4.329114 are regarded as equal since their difference is 0.000009, which is less than t, but 2.825631. and 2.825642 are regarded as unequal since their difference is 0.000011, which is greater than t, The term ‘t’ is referred to as ‘tolerance’. The default value of t is 0.00001 but this value may be changed by the user through a ‘use-item’ of the form tol(r) where ‘tol stands for ‘tolerance’ and r is the required tolerance. For instance, the statement use tol(0,000005);, will make the tolerance equal to 0.000005. ‘The effect of tolerance is illustrated graphically in Fig 1.5.9; where each of the two squares shown represents the ‘tolerance range’, The elements 1 and 2 will be regarded as ‘coincident’ provided thet their corresponding end points remain within the tolerance range at both ends. 1.5.8 Arrangement of Statements and Inclusion of Comments Consider the duet shown in Fig 1.5.10, The record in the editory consists of a single scheme which is obiained by combining the two schemes of Fig 1.5.1 and changing the values of the parameters m and n as well as the coordinates of the centre of rotation in the rosad function. The scheme is also shown in the editory display of Fig 1.5.11. ‘The: scheme illustrates the use of ‘comments’. In Formian, a ‘comment’ is a sequence of characters that is enclosed between two ‘comment brackets" The ‘comment bracket’ js the compound symbol oO This compound symbol is referred to as the ‘floret symbol’. The ‘body’ of the comment that lies between the initial an terminal floret symbols may be any ‘sequence of characters with the only restriction that it must not include a floset symbol. A flotet symbol may be either typed from the keyboard or obtained by clicking the ‘floret tool ‘button’ on the tool bar. This is the tool button with a floret symbol appearing on it. International Journal of Space Structures Vol. 15 No. I 2000 CHEB pent he fm") ‘tts ew Fig 1.5.10 Inclusion of comments (*) Hexagonal Arrangement (*) m=8; (*) Width of base unit (*) a= (°) Depth of base unit (*) 4=1.0; (+) Element length (*) E={[0,0; 2,0), 10,0; 1,1], (2,0; 3,1]}; F=genid(m,n,2,1,1,-1}|E; G=rin(1,m-n,2)| {n,n; n+2,n); He=bb(d/2 (sqrt) 3)*d/2) | F#C); HA=rosad(4,-2"sqrt|3,6,60)|H: (*) Hrepresents the base unit (*) HB=pex|HA; (*) HB represents the final form (*) clear; use vm(2); draw HB; | SBEnRycasnnnnesettGnseeeemssnssnsanil| Fig 1.5.11 A scheme with comments in=5;d=1.0;E= {[0,0;2,0] [0,0;4,1],{2 O;1,1]}F=gonid(m,n,2,1,1,-1)|E;G=rin(t mrn,2)] {n,nn4+-2,n};H=bb(d/2,(sqrt[3)*d /2)| (P#G) HA=rosad(4,-2*sqrt|3,6,60}|H; HB=pex|HA;clearyuse vm(2);draw HB; Fig 1.5.12 A ‘squashed’ scheme A comiment may appear at any point of a scheme where a ‘white space’ can eppear. The term ‘white space’ refers to an ‘empty space’ that is obtained by typing a space, newline or tab character from the Keyboard. White spaces and comments may appear at any point of a scheme which is not an intermediate point of a number, an identifier, a keyword or a compound symbol. It is good practice to include comments in the schemes. A comment may be used to give a title to a scheme, describe the purpose of the scheme, indicate the significance of a variable, ... etc. 25White spaces and comments do not affect the execution of a scheme. As far as the system is concerned, the scheme of Fig 1.5.11 is equivalent to the ‘squashed’ scheme shown in the editory displey of Fig 1.5.12. This scheme is obtained by removing all the white spaces and comments from the scheme of Fig 1.5.11, with two exceptions. Namely, the spaces after the keywords ‘use’ and ‘draw’ ere not removed, This is because keyword must always be followed by one or more spaces. It is good practice to arrange the statements in a scheme in a manner that the material can be easily read. There is no syntactic difference between the schemes in Figs 1.5.11 and 1.5.12. However, it is much easier for a human to follow the material as given in Fig 1.5.11. 1.5.9 Arrangement of Schemes When a scheme is subjected to execution, then the statements that constitute its body are executed one after the other, starting ftom the first stetement. The scope of the execution will be confined to the body of the scheme and will not involve any statement outside the scheme. In this sense, a scheme is an independent unit and the position of the scheme wwitbin the record has no particular significance. However, the variables created by a scheme are not ‘private’ to that scheme. All the variables created are held in a ‘common’ storage area without any reference to the schemes that have created them. If 2 scheme creates a variable F and if the scheme that is, ‘executed next also creates a variable F, then the new value of F overwrites the value created by the previous scheme with no trace of the old value of F remaining. Also, if a scheme creates a variable, then ‘a subsequently executed scheme can make use of the value of this variable without any restriction, ‘The general availability of the variables for use by different schemes imply that a schenie can provide ‘variables for use in other schemes. Therefore, if desired, the schemes may be made to depend on one another, In this case, the order of execution of the scheme’ should be conformable with the required order for the availability of variables. However, even, in this case, the actual positions of schemes within the record remain unimportant. Although, it will be convenient to arrange the schemes in the same ne order as they are to be executed, ‘As an example, consider the editory display of Fig 1.5.13. The record shown in this figure consists of three schemes. The first scheme creates a formex variable H representing a trapezoidal three-way grid, 26 Forme: Configuration Processing I based on the generic formulation that has been used in the provious examples. The second scheme uses the variable Hi to create formex variable HEX1 ‘whose plot is shown in Fig 1.5.14, The third scheme again uses the variable H to create a formex variable HEX2 whose plot is shown in Fig 1.5.15. o Trspeasidal grid (*) ¥egenid(m,n2,1, Gerin(1,m-=n,2)) fa. ‘H=bb(¢/2,sq"d/2) | (F#G); clear; use vm(2); drew Hi <>(*) Hexagonal grid 1 (*) N=rosad(8,0,4,-60) {{3,0}, (7,01) HEX1=pex_|lam(2,0) ux(N)|H; lost; draw HEX1; <>(*) Hexagonal grid 2 (*) Ne=rosad(8,0,3,-60)| {(4,sa), [5.6], (4.5,1.5%sq)): ‘HEX2=pexlam(2,0)|lux(N) |H; clear; draw HEX2; Fig 1.5.13 Asecord with three schemes A&X LLLS. LANEY ROOD AIDED CO OHOAD, AY LY YY YS LSD. LLB ALY SVAVAVAVANAV AN AVAY AN AVAV AN WWAVAVAVAVAVAVAVAVAVAVANAVAVA SVAVAVAVAV AAV AV AVAVANAVAY BY DAY WY A AMA LLY DAY SVAVAVAVAVAVAVAVAN Fig 1.5.14 Plot of HEX1 In this example, the second and third schemes are both dependent on the first scheme, Therefore, itis necessary to execute the first, scheme to create the formex variable H before the second or third scheme can be executed. However, there is no link between International Journal of Space Structures Vol. 15 No. 1 2000Hoshyar Nooshin and Peter Disney the second and third schemes and they can be executed independently. LLY, LLOLLALISA, ELLY YADA Ly NVAVAVAVAVAVAVAVAVAVAVAVAVAVAVA/ WX YRAY ALY VAVANAN LY CA LY WVAVAVAVAN A DALLY WVAVAVAVAN LLY WVAVAVAVAVAVAVAVAVAY WVAVAVAVAVAVAVAVAVA Fig 1.5.15 Plot of HEX2 Incidentally, it is a convention to use three consecutive diamer symbols to indicate the end of a record. This is shown at the end of the record in the editory display of Fig 1.5.13. 1.5.10 Ending a Session A session may be brought to ari end by issuing an ‘exit command’. This is done by * clicking the exit button, that is, the button on the top right comer of the screen with a cross appearing on it, or using the keyboard shorteut ‘alt+X? or © using the ‘exit’ item on the file memu, © te ending of the session wilt cause any material that may be in the editory as well as all the values of variables created during the session to be wiped out, However, before closing the session, if the editory contains any unsaved material, then the system will ask whether the material is required to be saved. On the other band, as far as the variables are concerned, their values will disappear at the end of the session. This should not raise any alarms since, usually, there is no point in holding on to the value of a formex variable beyond the end of a session. The value of a typical formex variable is a huge bundle of numbers. In contrast, the formex formulation that creates such a value is normally a few lines of text that can be incorporated in a scheme and saved in a text file. The execution of the scheme will then generate the value when required, So, what should be saved is the rule for generation of a formex value rather than the value itsel® : ‘International Journal of Space Structures Vol. 15 No. 1 2000 The session may also be ended through a different mechanism. To elaborate, sometimes an error in a statement may cause the system to enter into an endless chain of operations. The user will: become aware of this by the failure of the system to complete the execution of a scheme within a reasonable period of time, To get out of a situation of this kind, one may ‘terminate the session by using the key combination ‘control+alt+delete’ from the keyboard. 1.6 SPACE STRUCTURES The term ‘space structure’ refers to -a. structural system in which the load transfer mechanism involves three dimensions. This is in contrast with a “plane structure’, such as a plane truss, in which the Joad transfer mechanism involves no more than two dimensions. The above definition is the ‘formal’ definition of a Space structure. However, in practice, the tem ‘space structure’ is simply used to refer to a number of families of structures that includes grids, barrel vaults, domes, towers, foldable systems and tension structures, There are numerous examples of space structures built ell over the world for sports stadiums, ‘gymnasiums, cultural centres, auditoriums, shopping malls, railway stations, sircraft hangars, leisure centres, radio telescopes and many other purposes, A number of single layer grid configurations were considered in the previous sections. These configurations were used as examples for explaining some basic concepts of formex configuration processing. Examples of a mimber of other kinds of Space structure configiations will be considered in the sequel. These examples will again be used as vehicles for describing basic ideas and procedures in formex configuration processing, 1.7 DOUBLE LAYER GRIDS A perspective view of a double layer grid is shown in Fig 1.7.1. In general, a double layer grid consisis of two parallel layers of elements that are connected together by ‘web elements. In the double layer grid of Fig 1.7.1, the top layer has a ‘square pattem’ and consists of 112 elements. The bottom layer also hes a square pattern with 84 elements, The number of web elements is 196. The top layer elements are shown by thick lines and the bottom layer elements as well as the web elements are showa by thin lines. The plan and elevation of the.grid of Fig 1.7.1 are shown 27in Fig 1.7.2. Also shown in this figure are the dimensions of the structure together with the global Cartesian coordinate system x-y-z. 7 Trop layer Bottom layer Web Fig 1.7.1 Perspective view of a double layer grid Be Elevation: 1 25.76 m (14 divistons @ 1.54 m) @ we Ke NI ® z 8 3 4 be z 2 2 aa “ a: x epee Seen Plan Fig 1.7.2 Plan and elevation of the double layer grid of Fig 1.7.1 The configurations in the examples of the previous sections were formulated using simple two directional normats. The same basic approach can be used for the formulation of the doubie layer grid of Fig" 1.7.1. However, in this case it is necessary to 28 Formex Configuration Processing 1 work in terms of a ‘three directional normat’. Such a normat is shown in Fig 1.7.2, with the first, second and third directions being indicated by Ul, U2 and U3, respectively. ‘A formex formulation for the double layer grid of Fig 17.1 relative to the U1-U2-U3 nomat of Fig 1.7.2 may be written as ‘TOP =rinid(7,8,2,2) | [0,0,15 2,0,1] #. rinid(8,7,2,2) | {0,0,13 0.2,1] BOT = rinid(6,7,2,2) | [1,1,0; 3,1,0] # inid(7,6,2,2) |[1,1,03 1,30) ‘WEB = rinid(7,7,2,2) | rosad(1,1) | (0,0,15 1,1,0) GRID = TOP # BOT # WEB In this formulation © rinid(7,8,2,2) | (0,0,15 2,0,1] - represents all the top layer elements that are in! / the frst direction, # rinid(8,7.2,2) | [0,0,13 0,2,1] represents all the top layer elements that are in the second direction, © rinid(6,7,2,2) | [1,1,05 3,1,0] represents all the bottora layer elements that are in the first direction, @rinid(7,6,2,2) | (1,1,05 1,3,0] represents all the bottom layer elements that are in the second direction, © rinid(7,7,2,2) |rosad(1,1) | [0,0,1; 1,1,0] represents all the web elements and © TOP#BOT# WEB represents all the elements of the grid, It should be noted that the rosad function used in the above formulation for the gencration of the we”) elements, that is, ns rosad(1,1) is an abridged form of rosad(1,1,4,90) ‘This represents 4 rotational replications by steps of 90° with the centre of rotation at point (1,1). The abridged form of the rosad function can be used whenever the number of replications is 4 and the rotation at each step is 90°, as explained in section 1.4.7. Of course, tae use of this abridged form is not compulsory and one can always include the third and fourth parameters of the rosad function for the sake of clarity. Another point to be noticed is that all the formices involved in the above formulation are of ‘grade 3” ‘That is, all the signets of the formiccs consist o” three uniples, This is a consequence of the fact tha, International Journal of Space Structures Vol. 15 No. 1 2000Hoshyar Nooshin and Peter Disney all the formices involved are relative to a three directional normat, ‘The formex variable GRID in the above formulation represents the compret of the configuration of the double layer grid of Fig 1.7.1 (1.7.2) relative to the ‘UI-U2-U3 normat. A formex variable that describes the compret of the grid of Fig 1.7.1 (1.7.2) in terms of the global x-y-z coordinates may be written as GRIDX = bt(1.84,1.84,1.65) | GRID The construct be(1.84,1.84,1.65) is a ‘basitrifect’ retronorm. This function is similar to the ‘basibifect’ retronorm that was used for scaling two directional configurations in the previous sections. A description of the basitrifect retronorm is given in Fig 1.7.3. bt(1.84,1.84,1.65) factor for scaling in the third direction factor for scaling in the second direction, Lez ltstt for Scaling inthe fst direction abbreviation for basitrifect, Fig 1.7.3 Basitrifact retronorm The effect of the basitrifect retrononm is to change the proportions of a three directional configuration by using scale factors. Another two examples of double layer grids are shown in Figs 1.7.4 and 1.7.5. A formex representing the configuration of the grid of Fig 1.7.4 relative to the U1-U2-U3 normat of Fig 1.7.2 may be obtained as follows: e N=rinid(3,3,4,4) | 3,3,0] GRIDA = Iux(N) | GRID In this formulation * Nrepresents nine points, specifying the positions for the removal of elements, * the luxum function lux(N) effects the removal of all the clements that have a connection to the points represented by N and © GRID represents the compret of the grid of Fig 1.7.1 {1.7.2), as formulated before. A formex representing the grid of Fig 1.7.5 relative to the Ul-U2-U3 normat of Fig 1.7.2 may be obtained as follows: GRID # tranid(6,6) | GRID N= {(7,7,0, (13,13,0]} GRIDB = lux(N) | pex | F International Journal of Space Structures Vol. 15 No. 12000 In this formulation °F represents an overlapped combination of the arid of Fig 1.7.1 (1.7.2) with a displaced version of itself, * the pexum function has the effect of ‘pruning? the configuration by removing all the overlapped superfluous elements and * lux(N) effects the creation of the two openings, Fig 1.7.4 Perspective view of the double layer grid represented by the formex variable GRIDA Fig 1.7.5 Perspective view of the double layer grid represented by the formex variable GRIDS As another example, consider the double layer grid whose plan is shown in Fig 1.7.6. The top layer elements together with the web elements in this grid constitute (inverted) tetrahedral units, Also, all the triangular units in the top and bottom layers of the grid are equilateral, A generic formulation for the grid of Fig 1.7.6 may be written as s= 13/3, TOP = genid(m,m,2,3s,1,-1) | rosad(1,s,3,120) | [0,0,1; 2, 29BOT = genid(m-1m-1,2,35,1,-1)| rosad(2,2s,3,120) |[1,8,0; 38,0] ‘WEB = genid(mn,2,35,1,-1) | osad(I,s,3,120) |[0,0,3; 1,0} TRIAN = TOP # BOT # WEB The ‘parameter _m in this generic formulation represents the number of tetrahedral units along each side of the grid. For the grid in Fig 1,7.6 the parameter m is equal to 6. Top layer va rane LISTSLIN KOO Ra DOK ON EASES ESE SS OX OK i ° DPE BEES 04.2345 6 —*UI Fig 1.7.6 Plan view of a double layer grid with a ‘riangle-on-triangle pattern ‘The nirmat in Fig 1.7.6 involves the factor 3= 13/3 which is used to obtain the correct scaling in the second direction. This example shows that normat coordinates can be noniinteger. Indeed, the fact that all the normat coordinates used so far were integer ‘numbers is incidental rether than essential, Another point to be noticed in Fig 1.7.6 is that the actual dimensions of the grid and the x-y-z coordinate system are not included. The reason for this omission is that the objective of the exercise which is the formulation of the compret of the grid is independent of the actual dimensions. And, once the compret of the grid is formulated, the actual dimensions may be taken into account easily. For this reason the actual dimensions will be omitted in most of the configurations that are considered henceforth. ‘As the last example in this section, the above generic formulation is extended to allow curtailing of the comers of the double layer grid of Fig 1.7.6. The extension involves two additional equations as follows: 30 Formex Configuration Processing 1 C= rosad(nn,m 53,120) | genidn,n,2,38,1,-1) | {10,0.11, [1.s,0]} ‘TRIANA = lux(C) | TRIAN where TRIAN is the formex variable representing the configuration of the double layer grid of Fig 1.7.6, as formulated above. Fig 1.7.7 Positions for removal of elements for m=5 and n=2 LAA LE CBB CBE \B BOR \EBBOX/ LL ELRSLNS PROX ON LRRD KREIKA SRI \ OK MOK / ESERLSEREAS \ RRR ESERERSES ES LESESE 3s n=3 Fig 1.7.8 Three examples of corner curtailment ‘The formex variable C represents the points for the removal of elements, where © m represents the number of tetrahedral units along each side of the grid of Fig 1.7.6, © n indicates the extent of comer curtailment and 2 s=133. International Journal of Space Structures Vol. 15 No. 12000Oo Hoshyar Nooshin and Peter Disney For instance, for m=5 and n=2, the points represented by the formex variable C are indicated by little circles in Fig 1.7.7. ‘The above generic formulation can be used to, represent a variety of different double layer grids with @ pattem similar to that in Fig 1.7.6 and with comer curtailment. Three such examples are shown in Fig 1.7.8 with the corresponding values of m and n shown for each case, Consider egain the formex equation C= rosed(m,ms,3,120) | genid(a,n,2,38,1,-1) | {[0,0,1), [1,s,0]} which is a part of the above generic formulation. In ‘tracing’ the effects of a formex equation such as this, one would normally work from right to left ‘Thus, one would first identify the points represented by {[0,0,1], [1.8,0]} then; one would work out what is represented by genid(,n,2,3s,1,-1) | {{0,0,1], [1.3,0]} and finally, one would find out the effects of rosad(m,ms,3,120) | ‘genid(n.n,2,35,1,-1) | {[0,0,1}, [1,3,0]} An example of auch a tracing process is shown in Fig 1.48. Double layer grids are one of the most popular forms of space structures and there are many impressive double layer grids built all over the world. The examples of double layer grids considered in this section cover a few basic forms. However, there are many other patterns that are commonly used for double layer grids in practice. 1.7.1 Perspective views To acquire a “feel” for the overall visual effect of @ space structure configuration, it is often required to produce one or more perspective views of the configuration, as exemplified in Figs 1.7.1, 1.7.4 and 1.7.5. The manner in which perspective views are created in Formian is described in this section, using double layer grids as examples. The notion of a perspective view in Formian is explained using the double layer grid of Fig 1.7.1. This grid, together with the global x-y-z coordinate system, is shown in Fig 1.7.9. The required perspective view is specified in terms of a ‘view helm’, To elaborate, it is ‘imagined’ that the grid is viewed from a point that is referred to as International Journal of Space Structures Vol. 15 No.1 2000 the ‘view point’. The line of vision is ‘imagined’ to be directed from the view ‘point towards a point that is referred to as the ‘view centre’. A ‘vector’ is ‘imagined’ to emanate from the view centre, This vector is referred to as the ‘view rise” and its role is to specify the direction that is to become the ‘vertical direction’ in the required perspective view. The “broken vector" that is shown by thick line in Fig 1.7.9 is referred to-as the ‘view helm’. The view helm consists of the line from the view point to the view. centre and the view rise. <—— View point “ View helm View rise 4 Fig 1.7.9 View helm (*) Perspective view (*) ‘TOP=rinid(7,8,2,2)|10,0,1; 2,0,1]# Hinid(8,7,2,2)| {0,0,45 0,2,4); rimid(6,7,2,2)|[1,1,0; 3,1,0]# Finid(7,6,2,2)|[1,1,0; 1,3,0}; WEB=rinid(7,7,2,2)|rosad{(1,1)| {0,0,3; 1,1,0}5 GRID=TOP#BOT#WEB; use vin(2),vt(2), vb(7,-14,42, 7,7,0, 7,7,1); clear; draw GRID; <><><> BOT: Fig 1.7.10 A scheme for obtaining a ™* perspective view A view helm is specified through a use-item. An ‘example of such a use-item is given in the scheme in the editery display of Fig 1.7.10. The execution of this scheme. will produce a perspective view similar to the one shown in Fig 1.7.1. The assignment statements in the scheme of Fig 1.7.10 follow the formulation of the double layer grid of Fig 1.7.1 (1.7.2) as given in section 1.7. ‘The ‘use statement? in the scheme of Fig 1.7.10, that is, suse vma(2),vt(2),vh(7,-14,42, 7,7,0, 7.7.1); 31contains three use-items, The first use-item, that is, ‘ym(2), has the effect of putting the system in the ‘automatic scaling mode’, as discussed in section 152. ‘The second use-item, that is, vi(2) indicates that a ‘perspective view’ is required. The term ‘vt? is an abbreviation for ‘view type’. The number in parentheses following ‘vt’ can be either 1 or 2. The ‘usecitem vi{1) causes the view to be ‘isometric’ and the use-item vi(2) causes the view to be ‘perspective’. The default setting for the view type use-item is vt(1). ‘The third use-item is ‘vh(7,-14,42, 7,7,0, 7,7.) ‘where ‘vh’ is an abbreviation for ‘view helm’. The details of this use-item are described in Fig 1.7.11. wh(7,-14,42, 77,0, 7.7.1) coordinates of the arrow end of the view rise coordinates of the view centre (end the starting point of the view rise) coordinates of the view point abbreviation for view helm Fig 1.7.11 View helm use-item ‘The view helm use-item specifies ‘¢ the point from which the ‘object is viewed (that is, the view point), * the point towards which the line of vision is irected (that is, the view centre), * the direction that is to become the vertical direction (that is , the direction of the view rise) and © the ‘up’ and ‘down’, with the view rise regarded as pointing ‘upwards’. ‘An important point that needs explanation concems the coordinate system with respect to which the view helm is to be specified. To elaborate, the coordinates of the view helm should be given relative to the coordinate «system in terms of which the configuration to be viewed is given, In the present example, the formex variable GRID that represents the configuration to be viewed is relative to the U1- U2-U3 normat of Fig 1.7.2, Therefore, the coordinates of the view helm in the scheme of Fig 1.7.10 are given relative to this normat. 32 Formex Configuration Processing I The coordinates of a view helm must always be relative to a Cartesian-type reference system. This may be the global x-y-z coordinate system or a normat such as U1-U2-U3 in Fig 1.7.2. The term ‘Cartesian-type reference system’ implies a reference system that has orthogonal linear axes. Thus, a curvilinear reference system is not of a Cartesian- type, Tae term ‘curvilinear reference system” is used to imply a reference system that involves one or more curved axes (surfaces). Examples of such a system are cylindrical and spherical coordinate systems. ‘A minor point to be noticed in Fig 1.7.11, and elsewhere in the paper, is that the coordinates in the view helm usé-items are grouped together with spaces in between. This is @ useful convention that helps to separate groups of related items and is used“) in the specification of view helms and other entities." d\, N/ VY VAN gr, (mS WP aN \/ Ny VM AVA YY Be Wy RX y J\ \/ és NAN LQOQ LYXYDQACY LEAYDBAC LLYN OSM LRP BAYAAAY LIAAA DON DR LS Oo Fig 1.7.12 A perspective view with vh(-15.-15.50, 7,70, 7.7.1) To further exemplify the effects of the view helm, another two perspective views of the double layer grid of Fig 1.7.1 are shown in Figs 1.7.12 and 1.7.13, ‘The view helm used for Fig 1.7.12 is vh(-15,-15,50, 7,7.0, 7.7.1) and the view helm for Fig 1.7.13 is vh(7,-10,-10, 7,7,0, 7,7,1) ‘The default setting for the view helm use-item is ‘vh(0,0,1B4, 0,0,0, 0,1,0) That is, the view point is high up on the positive side of the third axis (z-axis), © the view centre is at the origin of the coordinate system and International Journal of Space Structures Vol. 15 No. 1 2000Hoshyar Nooshin and Peter Disney ‘+ the view rise is along the positive direction of the second axis (y-axis). ‘The default setting of the view helm will normally give rise to the plan view of an object. VAAL AA VARMA (ZNZRR Fig 1.7.13 A perspective view with vh(7,-10,-10, 7,7,0, 7,7,1) It is not always casy to produce 2 good perspective view of an object and it is normal to try @ number of different view helms until a satisfactory view is found. To obtain a general view of an object, it is not always necessary to work in the ‘perspective mode’, that is, under the currency of the use-item vi(2). One can also obiain an ‘isometric’ view of an object, under the currency of vi(1), using a view helm use. item in the usual manner. A perspective view is closer to a human view of an object, as compared with an isometric view. Nevertheless, isometric views could also be quite effective and, for some purposes, they may be more appropriate than perspective views, 1.7.2 Line Width, Style and Colour A feature of the examples considered in section 1.7 is the use of different line thicknesses in the plots of double layer grids for ‘layer identification’. ‘The manner in Which line thicknesses may be specified in Formian is exemplified in terms of a double layer grid in the scheme showa in the editory display of Fig 17.14. ‘The assignment statements .in the scheme of Fig 1.7.14 are based on the generic formulation for the triangular double layer grid of Fig 1.7.6, as given in section 1.7. The assignment statements are followed by 2 use statement that includes the use-item bv(0.6) This is.a ‘line width’-use-item, where ‘Iw’ stands for ‘line width’ and the number in parentheses specifies the required line width jn millimetres. The effect of the use-item is to change the ‘current setting? for the line width to 0.6. As a result, every line that is drawn International Journal of Space Structures Vol. 15 No. 1 2000 will have a thickness of 0.6 mm, until the setting for the line width is changed. The default setting for the Tine width use-item is 1w(0.3). (*) Triangular grid (*) m s=(sart|3)/3; TOP=genid(m,m,2,3*s,1,-1)] rosad(1,s,3,220)|[0,0,1; 2.0,4}; BOT=genid(in-1,m-1,2,3*6,1,-1)| Tosad(2,2*5,3,120) [1,8,0; 3,8,0}; = genidm,m,2,3*9,1,-1)] rosad{1,s,3,120)|[0,0,1; 1,5,0]; clears_use & vs(30),1(0:6 draw TOP; use lw(0.4); draw BOT; use 1w(0.2}; draw WEB; <><><> WEI Fig 1.7.14 Line width specification Retuming to the scheme of Fig 1.7.14, the first use statement is followed by a draw statement for plotting the top layer elements, These elements will then be drawn with lines that are 0.6 mm thick. The next statement in the scheme is a use statement that changes the current setting of line width to 0.4, This is followed by a draw statement for plotting the bottom layer elements, which will be drawn with a line width of 0.4mm. The following two statements Tepeat the process for plotting the web clements using a line width of 0.2 mm. The plotting results are shown in the duet of Fig 1.7.15. (Brewpad KK KK Fig 1.7.15 Line width variation Specification of different ‘line styles’ is achieved using the same approach as described for the line ‘width. The use-item for ‘line style’ is of the form Is(n) 33‘The term ‘ls? stands for ‘line style’ and n is a ‘code umber’ indicating a style of line, where © n= 1 indicates “full line’, *, n=2indicates ‘dashed line’ and « n=3 indicates ‘dotted line’. ‘The default setting for the line style use-item is Is(1). ‘An effective way of ‘distinguishing’ between different parts of 2 configuration. is to use colour. The approach in the scheme of Fig 1.7.14 for specifying line width is also used for specifying line colour. The ‘colour’ use-item is of the form (Ln) where ‘e" stands for ‘colour’ and where the first parameter which is given as 1 indicates that the specification is for a line. The second parameter of the colour use-item (thet is, n) is 2 ‘code number’ for colour. There are fifty available colours 2s listed in the ‘palette’ (colour table). The palette may be displayed by clicking the ‘palette tool bution” which has a ‘coloured window’ appearing on it. A sketch of the palette is shown in Fig 1.7.16. Palette (Colour Table) Colours and colour numbers ” a4 val_] 2 a3|_] 28] 14L_] 2a] 1s{_| 25] reL_] 2 a7] 27 1s[_] 28 ia[_ | 29] 20[_] 30] Fig 1.7.16 Palette (colour table) Colour samples in little squares are displayed on the palette with the ‘code mumber’ for each colour appearing next to it, The first column on the palette displays the ‘grey band’ starting with black (colour 34 Formex Configuration Processing I ‘code 1) at the top and going down to white (colour code 10) with eight shades of grey in between, The second column on the palette displays the ‘red band’ starting with dack red (colour code 11) at the top, followed by lighter sheds of red and with some colours in the brown and yellow ranges further down, The third, fourth and fifth columns on the palette display the ‘purple band’, ‘green band’ and “phue band’, respectively. The default setting for the line colour use-item is (1,23). In general, when different sections of a configuration are to be plotied separately with different line thicknesses, styles and/or colours, then the system should not be in the ‘automatic scaling mode’. Otherwise, each section will be scaled independently to fit the drawpad and consequently, the plots of different sections will not necessarily ‘fit together’) properly. It is for this reason that the drawing of different parts, represented by formex variables TOP, BOT and WEB, in the scheme of Fig 1.7.14 is carried out ‘with a non-automatic scaling mode, However, the fact that the drawing operations in the scheme of Fig 1.7.14 are effected under the non- automatic mode, that is under the currency of vm(1), is not immediately obvious. To elaborate, the use statement in the scheme of Fig 1.7.14 is of the form use &,¥8(30),lw(0.6); The second and third use-items here are, a view scale ‘use-item and a line width use-item, respectively. The effects of these useitems have been explained before. However, the symbol & thet appears as the first item in the ebove use statement has not bec.) discussed yet. As an item in a use statement, the symbol & is referred to as the ‘ampersand use-item? and has the effect of making the default setings of all the use-items current, Returning to the use statement in the scheme of Fig 1.7.14, since non-automatic scaling is the default for the view mode, the effect of the ampersand use-item is to put the system in the non-automatic mode and this is the mode in which the drawing of TOP, BOT and WEB ere produced, 1.8 BARREL VAULTS Consider the configuration shown in Fig 1.8.1. This is a ‘curved grid’ with a cylindrical form. ‘The structure consists of 178 ‘straight’ beam elements that are connected together at 99 nodes. The node: lie on the surface of a circular cylinder. The ‘span’ of the structure is 28.75 m, the ‘rise’ (that i, the height International Journal of Space Structures Vol. 15 No. 12000Hoshyar Nooshin and Peter Disney at the centre) is 5.45 m and the ‘length’ is 39,50 m. This is an example of a structural form that is referred to as a ‘barrel vault’, where the circular oylindrical surface that contains all the nodal points is referred to as the ‘ciroumcylinder’ of the barrel vault, | =28.75m_ (span) | Fig 1.8.1 A barrel vault ‘A convenient reference system for the formulation of the configuration of the barrel vault of Fig 1.8.1 is a ‘cylindrical normat’, as shown in Fig 1.8.2. Tn this figure, the barrel vault is shown with its nodes lying on the circumeylinder. The ‘origin’ of the normat is at the centre of a cross-section of the circumeylinder, The first direction of the normat is along 2 “radius” of the cross-section. The second direction of the normat is along the ‘circumference’ of the cross- section. The third direction of the normat is slong the ‘longitudinal axis’ of the circumeylinder. The first and third directions of the normat are “linear” and the second direction is ‘angular’, ‘The Cartesian coordinate system corresponding to the nonmat is also shown in Fig 1.8.2. The xaxis is collinear with the Ul-exis and the z-axis is collinear with the U3-axis, The y-axis lies in the cross-section of the circumeylinder and is perpendicular to the x-, axis. A comer of the barrel vault, together with the normat, is shown in Fig 1.8.3. In this normat, the length of the radius of the cross-section of the circumeylinder is chosen to be 1 and the divisions along the second and third dizections are chosen to suit the positions of the nodal points of the barrel. vault International Journal of Space Structures Vol. 15 No. 12000 ut i u2 Circumterefaat Fig1.8.3 Comer of barrel vault of Fig 1.8.1 All the nonnats considered so far were of Cartesian- type (except for the polar nommat discussed in section 1.4.4). However, curvilinear normats are of frequent use in formex configuration processing and the cylindrical normat is one of the commonly used curvilinear normats, Now, focusing on the formulation of the configuration of the barrel vault of Fig 1.8.1, the compret of the configuration relative to the normat of Fig 1.8.3 may be represented by = rinit(8,11,1,1) | [1,0,0; 1,1,0] # Finit(9,10,1,1) |[1,0,0; 1,0,1] In this formulation * 1,0,0; 1,1,0] represents the element ‘if’ in Fig 1.8.3, indicated by 35© rinit(8,11,1,1) | [1,0,0; 1,1,0] represents all the elements that are in the circumferential direction (second direction), + [1,0,0; 1,0,1] represents the element indicted by ‘Sik’ in Fig 1.8.3 and # rinit(9,10,1,1) | [1,0,0; 1,0,1} represents all the elements that are in the longitudinal direction (third direction). In the above formulation, the first uniples of all the signets are equal to 1. This is a consequence of the fact that the length of the radius of the cross-section of the circumeylinder is chosen to be 1. ‘The functions rinit(8,11,1,1) and rinit(9,10,1,1) in the above formulation are analogous to the ‘rinid” function with the suffix ‘id’ replaced by ‘it. The suffix ‘i? implies a double action in directions 2 and 3 (whereas, the suffix ‘id? implies a double action in directions 1 and 2), ‘The function rinit(8,11,1,1) implies 8 translational replications in the second direction with steps of 1 followed by 11 translational replications in the third direction with steps of 1, as shown in Fig 1.8.4, rinit(8,11,1,1) amount of translation at each step in the third direction amount of translation at each step in the second direction number of replications in the third direction \__ number of replications in the second direction Fig 1.8.4 Rinit function ‘All the fanctions with ‘id’ suffix have their equivalent ‘i’ versions, as will be seen in various examples henceforth, The formex variable B in the above formulation represents the configuration of the barrel vault of Fig 1.8.1 relative to the nomat of Fig 1.8.3. A formex representing the configuration relative to the x-y-z coordinate system of Fig 1.8.2 may be written as F=bo(R,A/4,L/10) |B 36 Formex Configuration Processing I where © Ris the radius of the cross-section of the circumeylinder, as shown in Fig 1.8.5, © Ais the ‘sweep angle’ of the barrel vault, that is, half the central angle of the barrel vault, as shown in Fig 1.8.5 and © Lis the length of the barrel vault, as shown in Fig 1.8.1. fete Fig 1.8.8 Cross-section of circumeylinder The construct defR,N/4,L/10) is a ‘basicylindrical retronorm’ that transforms the cylindrical normat coordinates into Carlesian coordinates, The general form of this function is shown in Fig 1.8.6, be(b1,b2,b3) LL tector for scaling in the thd dizection (linear scale factor) factor for scaling in the second direction (engular scale factor) factor for scaling in the frst direction (Oinear scale factor) abbreviation for besicylindsical Fig 1.8.6 Basicylindricel retronorm ‘The first parameter of the basicylindrical retronorm is a ‘linear’ scale factor for scaling of the normat coordinates in the first direction. This scale factor, in ‘the present example, should be R since the length of ‘the radius of the circumeylinder in the normat was taken as 1. The second parameter of the basicylindrical retronorm is an ‘angular’ scale factor for scaling in the second direction, This parameter specifies the angle (in degrees) for every division along the second direction. The third parameter of the basicylindrieal retronorm is a ‘linear’ scale factor for scaling in the third direction, The first and third parameters of the basicylindrical retronorm: should International Journal of Space Structures Vol. 15 No. 1 2000 ©Hoshyar Nooshin and Peter Disney be in a ‘length unit? such as metre or millimetre, as appropriate, The values of R, A and L in be(R,A/4,L/10) may be obtained from the information given in Fig 18.1. The length L of the barel vault is given directly in Fig 1.8.1 and the values of A and R may be obtained in terms of the span $ and rise H of the barrel vault, as follows: Itmay be seen form Fig 1.8.5 that tana =2H/S, Aw2a and sin A= S/R Therefore A=2 arctan 2H/S and R=S/(2 sin A) (*) Barrel vault of Fig 1.8.1 (*) $=28,75; (*) span (*) Hes! (*) rise (*) 39.50; (+) length (+) A=2*atan|(2*H/S); (*) sweep angle (*) ReS/(2*sin|A); | (*) radius (*) m=8; —_(*) units in U2 direction (*) (*) units in U3 direction (*) () view adjuster (*) rinit(mm.n-+3,1,1)|[1,0,0; 1,1,0]# rinit(m+1,0,1,1)][1,0,0; 1.0.1); Febe(R,2*A/m,L/n)|E; Beverad(0,0,90-A)|B use &.vm(2).vi(2), vh(v2"R-R, 0,0,R, 0,1,R); leer; draw B; <><><> BSE ee ee eee eee EE} Fig 1.8.7 A generic scheme for barrel vault of Fig 1.6.1 v=; A Formian scheme for the barrel vault of Fig 1.8.1 is shown in the editory display of Fig 1.8.7. This scheme has a generic form. That is, it is written in terms of a number of parameters. These ‘parameters are: © spans, + riseH, © length, * number of units in the second direction, denoted bym, - International Journal of Space Structures Vol. 15 No, 1 2000 * number of units in the third direction, denoted by nand * 8 parameter ‘v’ that is referred to as the ‘view adjuster’ and will be discussed later. In the scheme of Fig 1.8.7, the formex variable F Tepresents the configuration of the barrel vault of Fig 1.8.1 relative to the x-y-z coordinate system of Fig 1.8.2, as discussed before, In Fig 1.8.7, the assignment statement B= verad(0,0,90-A) | F; effects a rotation of the configuration in the x-y plane. This rotation will place the configuration in a convenient position relative to the x-y-z coordinate system, Where the x-axis is horizontal and the y-axis is vertical, as shown in Fig 1.8.8, / Cross-section of barrel vault of Fig 1.8.1 Fig 1.8.8 Rotated position of the barrel vault When working with a cylindrical normat, a ‘translation’ in the second direction is equivalent to a ‘rotation’ in the x-y plane, Therefore, the effect of the statement B =verad(0,0,90-A) | F; in the scheme of Fig 1.8.7 may also be achieved through a translation in the second direction (before the application of the basicylindrical retronorm). In this case, the statements F=be(R,2*A/m,Lin) | B = verad(0,0,90-A) | F; can be replaced by E = tran(2,m*(90-AV(2*A)) [E; B=bo(R,2*A/m,L/n) |B; ‘The reader may like to Prove that the second Parameter of the above translation function should indeed be m(90-A)/2A The setting of the view helm in the scheme of Fig 1.8.7 ensures that the y-axis remains vertical in all the views with the value of the ‘view adjuster’ v determining the ‘sway’ of the body of the barrel a7‘vault, Three views of the barrel vault of Fig 1.8.1 for different values of v are shown in Fig 1.8.9. Fig 1.8.9 Effect of view adjuster Now, suppose that the formex variable B int the scheme of Fig 187 is subjected to the ‘transformation EBI = bt(1,1.5,1)|B where, the construct bt(1,1.5,1) is a basitrifect retronorm, as described in section 1.7. ‘The effect of the transformation is that all the y- coordinates of the nodes of the barrel of Fig 1.8.1 are multiplied by 1.5. The result is the barrel vault shown in Fig 1.8.102. The span of this barrel vault is the sare as the original berrel vault but the rise is increased by a factor of 1.5. Therefore, the cross- section of the barrel vault will assume an elliptic form. That is, the nodes of the barrel vault will lie on the surface of a cylinder with an elliptic cross section, An ellipse that indicates the new proportions of the cross-section is shown under the barrel vault of Fig 1.8.10a, © Fig 1.8.10 Elliptic barrel vaults (b) Another example involving the scaling of the barrel vault of Fig 1.8.1 is shown in Fig 1.8.10b. The transformation that gives rise to this configuration is given by EB2 = bt(1,2,1) |B Here again the barrel vault has an elliptic form with the span being the seme as the original barrel vault and the rise being increased by a factor of 2. An ellipse indicating the new proportions of the cross- section is shown under the barrel vault in Fig 1.8.10b. The configuration in Fig 1.8.10c represents 38 Formex Configuration Processing 1 the original barrel vault which is shown for comparison. ‘The above examples demonstrate the fact that a barrel vault configuration which is based on a circular cylinder can be easily transformed into an lliptic form. ‘Therefore, when it is required to formulate a barrel vault configuration whose nodes lie on an elliptic cylinder, to begin with, the configuration may be formulated relative to a simple cylindrical normat, The result may then be transformed into an elliptic barrel vault, In relation to the use of the basitrifect retronorm for obtaining the elliptic barrel vaults of Fig 1.8.10, the following points are to be noted: The basic role of a retronorm is to transform the normat coordinates into global Cartesian coordinates. However, as far as the” basitrifect retronorm is concemed, it involves. nothing other than simple scaling. Therefore, it may also be employed as a ‘scaling function’ in any formulation where simple scaling is required. An example of this type of usage is the creation of elliptic barrel vaults, as discussed - above. Incidentally, to obtain the elliptic barrel vaults of Fig 1.8.10, one can also'use the basibifect retronorm as a “scaling function’. In this case, the formex variables BI and EB2 can be obtained as EBI=bb(I,1.5) |B and EB2=bb(1,2)|B Now, consider the configuration shown in Fig 1.8.11. This illustrates a group of four barrel vaults that are connected together along their sides. A. structure of this type is referred to as a ‘compout barrel vault’. The barrel vaults that constitute th compound barrel vault of Fig 1.8.11 are all identical to the barrel vault of Fig 1.8.1. Fig 1.8.11 A compound barrel vault ‘A formex representing the compound barrel vault of Fig 1.8.11 may be written as CB = pex | rin(1,4,28.75) | B where B is the formex variable in the scheme of Fig 1.8.7 representing the barrel vault of Fig 1.8.1. Thr effect of the pexum function is to remove the superfluous overlapping ‘valley elements’, International Journal of Space Structures Vol. 15 No. 1 2000¢ > Hoshyar Nooshin and Peter Disney Compound barrel vaults represent @ popular structural form and are frequently used in practice. The individual barrel vaults that constitute a compound barrel vault need not necessarily be identical or at the same level. For instance, the: structure shown in Fig 1.812 is a ‘stepping’ compound barrel vault in which the constituent barrel vaults are at different levels. A scheme for the configuration of the stepping barrel vault of Fig 1.8.12 is shown in the editory display of Fig 1.8.13. In this scheme, the formex variable G represents the part whose boundary is shown by thick lines and the formex variable SB represents the whole configuration. Also, the variable C represents the central angle of the part enclosed in thick lines and the variables R, D and T represent the dimensions indicated in Fig 1.8.12. Fig 1.8.12 A stepping compound barrel vault (*) Barrel vault of Fig 1.8.12 (*) (*) Central angle (*) ‘init{6,11,1,2)] [1,0,0; 1,1,0]# rinit(7,10,1,1)|[3.0,0; 10,1]; Febo(R,C/6,2)|Es G=verad(0,0,90-C)[F; ex (lam(1,0)| (G#tranid(D-T)|C)# tranid(2"D,-2"T) |G); use &vm(2).vt(2), vb(0,3°R,-3°R, 0,0.R, 0,1,R); clear; draw SB; <><><> Fig 1.8.13 A scheme for the stepping barrel vault of Fig 1.8.12 The examples of barrel vaults considered so far have 2 ‘two-way’ pattern of elements. However, a barrel vault may have many éther patterns, For instance, a barrel vault with a diagonal pattem is shown in Fig International Journal of Space Structures Vol. 15 No. 1 2000 1.8.14, A barrel vault with this type of pattem is Teferred to as a ‘lamella’ barrel vault. Fig 1.8.14 A lamella barrel vault A formex variable representing the compret of the lametla barrel vault of Fig 1.8.14, relative to the indicated U1-U2-U3 cylindrical normat, may be written as E = rinit(8,10,2,2) | lamit(1,1) | (1,0,0; 1,1,1) # rinit(8,2,2,20) | [1,0,05 1,2,0] # rinit(2,10,16,2) | [1,0,0; 1,0,2] In this formulation, rinit(8,10,2,2) | Iamit(1,1) | [1,0,0; 1,1,1] Tepresents all the diagonal elements, rinit(8,2,2,20) | [1,0,0; 1,2,0] . represents the edge elements that are in the second direction, that is, inthe direction of U2, and rinit(2,10,16,2) | [1,0,05 1,0,2] Tepresents the edge elements that are in the third direction, In the above formulation, Jarnit(1,1) | (1,0,0; 1,1,1] represents the four elements that constitute the ‘cross’ in the front left comer of the-barrel vault, shown by thick lines in Fig 1.8.14, Here, the construct lamit(1,1) is a ‘lamit’ fimction. The effects of a amit function are similar to those of a lamid function but in the second and third directions (rather than the first and second directions). To be specific, a lamit function effects a double ‘lambda action’ in the second and third directions, There are many examples of actual barrel vaults that are constructed using the ‘lamella’ pattem. A number of other commonly used patterns for barrel vaults are 39shown in Fig 1.8.15. The formex formulations for these barrel vaults are left as exercise to be cartied ‘out by the reader, Fig 1.8.15 Some common types of barrel vault configurations All the examples of barrel vaults considered so far have a rectangular boundary in plan. However, there are ng restrictions regarding the general shape of the boundary of a barrel vault. For instance, the configuration shown in Fig 1.8.16 represents a barrel ‘vault with a triangular boundary. Fig 1.8.16 A finite element mesh In addition to the shape of the boundary, the example of Fig 1.8.16 has another new feature, Namely, the configuration in Fig 1.8.16 represents a ‘finite element mesh’ consisting of ‘triangular elements’ with comer nodes. This is in'contrast with all the 40 Formex Configuration Processing I configurations considered so far in which the elements were ‘two-noded linear elements’. Therefore, unlike all the previous examples; the line segments in the configuration of Fig 1.8.16 represent the ‘edges’ of the triangular finite elements rather than individual line elements. The configuration of Fig 1.8.16 represents a ‘cylindrically curved shell” which may altematively be thought of as a “continuous barrel vault’. One may now proceed to produce a formex formulation for the finite element mesh of Fig 18.16. However, before attempting this, it should be made clear that the understanding of the formalation ofthe mesh of Fig 1.8.16 does not depend on a knowledge of the ‘finite element -method’ of structural analysis, In this relation, all that one needs to know is that a ‘finite element mesh” for a shel”? structure is obtained by dividing the shell into a~ umber of ‘tiles’ each of which is referred to as an ‘clement’. Actually, subdivision of a ‘surface’ into an array of ‘tiles’ provides an effective way of graphically visualising the surface. Therefore, for a reader who is unfamiliar -with the finite element method of structural analysis, the finite element mesh of Fig 1.8.16 may be simply regarded as an ‘array of triangular tiles” that defines a cylindrically curved surface. Fig 1.8.17 Comer of finite element mesh of Fig 1.8.16 A fonmex formulation for the finite clement mesh of Fig 1.8.16 may be written as El = [1,0,0; 1,2,0; 1,1,1] E2= [1,205 13,1; 111] FI = genit(12,12,2,1,1,-1) | El F2= genit(11,11,2,1,1,-1) | B2 FFL#F2 In this formulation, the equation EL = [1,0,0; 1,2,0; 1,1,1] defines a formex variable .E1 that represents the triangular element denoted-by TI in Fig 1.8.16. The International Journal of Space Structures Vol. 15 No. 1 2000Hoshyar Nooshin and Peter Disney element TI may also be seen in Fig 1.8.17, This figure shows an enlargement of the front lef comer of the mesh of Fig 1.8.16, ‘The cantle representing element T1, that is, 11,0,0; 1,2,05 1,1,1] has three signets. The first signet, that is, 10,0 represents node i in Fig 1.8.17, where the radius in the cylindrical normat is assumed to be 1 and, therefore, the first uniple is equal to 1. Also, the second and third signets of the above cantle represent nodes j and k, respectively. Similarly, the element T2 is represented by E2=[1,2,05 13,15 11,1] ‘The equation FL = genit(12,12,2,1,1,-1) [EL in the above formulation defines a formex variable FI that represents all the elements of the mesh of Fig 1.8.16 that are similar to TI. Also, the equation F2 = genit(11,11,2,1,1,-1) | E2 defines a formex variable F2 that represents all the elements of the mesh that are similar to T2. The constructs gonit(12,12,2,1,1,-1) and genit(11,11,2,1,1-1) are ‘genit” functions, The effects of a genit function are similar to those of a ‘genid’ fmotion, as described in section 1.4.6. However, a genit function operates in the second and third directions, in contrast with the genid fiinction that operates in the first and second directions. ‘The composition of formex variables F1 and F2, that is, F=Fl#F2 represents the entire mesh of Fig 1.8.16 relative to the indicated cylindrical normat. ‘The formex variable F may be transformed into a fommex variable thet represents the mesh of Fig 1.8.16 relative to the x-y-z coordinate system using the procedure described before in relation to the barrel vault of Fig 1 Another example of a cylindrically curved shell together with a finite clement mesh is shown in Fig 1.8.18; Also, a scheme for the generation of this mesh is shown in the editory display of Fig 1.8.19. ‘The normat used for the formulation of the mesh is the same as that shown in Fig 1.8.16. Iniernational Journal of Space Structures Vol. 15 No. 1 2000 Fig 1.8.18 A finite element mesh for a cylindrically curved shell (*) FE-mesh of Fig 1.8.18 (*) (2) sweep angle (*) (*) radius (*) ) enit(12,8,2,1,1,-1)|E1e genit(11,6,2,1,1,-1)|E2; lam(3,0) Es 0(R,A/12,1/26)|F ‘MESH=verad(0,0,0-A) |G; use & vm(2),vt(2}, vh(0,2°R,-2°R, 0,0,R, 0,1.R); clear; draw MESH; <><><> Fig 1.8.19 A scheme for finite element, mesh of Fig 1.8.18 In the scheme of Fig 1.8.19, * El represents the triangular element TI in Fig 18.18, + E2 represents the triangular element T2 in Fig 1.8.18, © E represents the trapezoidal part whose bx is shown by thiok lines in Fig 1.8.18 and + F represonts the entire mesh relative to the Ul- U2-U3 normat (shown in Fig 1.8.16). The formex variable F is obtained from E through a lambda function with the plane of reflection intersecting the circumeylinder along the curve indicated by ‘ab’ in Fig 1.8.18. Thus, the mesh is obtained by putting together two trapezoidal parts along the curve ab. However, this does not give rise to any ‘overlapping’ along the curve ab. The edges Of the elements on the two sides of the curve ab are 41‘touching’ each other but no part of any element ‘overlaps? any other element, ‘The formex varieble G, in the scheme of Fig 1.8.19, represents the mesh of Fig 1.8.18 relative to the global x-y-z coordinate system (shown in Fig 1.8.16), The basicylindrical retronoim used is be(R,ANI2,L/16) where, R is the radius of the cireumcylinder, A is the sweep angle of the mesh and L is the length of the mesh in the z-direction. The statement MESH.= verad(0,0,90-A) | G in the scheme of Fig 1.8.19 creates a formex variable MESH. This represents a ‘rotated’ version of the mesh with the y-axis assuming a vertical position, as, discussed before. After the execution of the scheme of Fig 1.8.19, one may want to check the properties of the variables created by the scheme. This may be done by displaying the ‘variables box’, as explained in section 1.5.6, In the present exarnple, the variables box will be as shown in Fig 1.8.20. ‘Variables Py Variable Type Order Plextude Grede_ Size @ NT “> © oNTFMK 1280 3g 4m. | ef INTFMX 3 3 38D | 2 INTFMK 33 am | 1 IWTFWX 28-33 oz | g FITFMX «2580-38 mb | Tost | mesh FLTFMK 258 33 gab | rT 2 | Caneel Fig 1.8.20 Variables box after the execution of the scheme of Fig 1.8.19 Focusing on variables F, G and MESH, the following points may be noted: © "The ‘type’ of F is’ given as INT FMX (integer formex) and the type of G and MESH is given as FLT FMX (floatal formex). This is a reflection of the fact that F is relative to the cylindrical nommat U1-U2-U3_ with integer coordinates ‘whereas G and MESH are relative to the global x-y-z coordinate system with noninteger coordinates, The ‘order’ of F, G and MESH is given as 256. * This indicates that the formex represented by 2 Formiex Conjiguration Processing 1 each of these variables has 256 cantles, where each cantle represents an element of the mesh, ‘Thus, the mesh bes 256 elements. ‘The ‘plexitude’ of F, G and MESH is given as 3. This shows that each cantle in the formices represented by F, Gand MESH has three signeis. This is a reflection of the fact that, in the present ‘example, each cantle represents a ‘three-noded’ finite element (triangular tle), The ‘grade’ of F, G and MESH is given as 3, ‘This indicates that each signet in the formices represented by F, G and MESH hes three uniples, This is a consequence of the fact that the cylindrical normat used has ‘three directions’ and the Cartesian coordinate system used has ‘three dimensions”. All the barrel vaults considered so far consisted of a -) single layer of elements, However, in practice, many barrel vaults are constructed with elements arranged in two or more layers. For example, consider the barrel vault shown in Fig 1.8.21. Web Bottomlayer _f 7 Top layer Oo Fig 1.8.21 Perspective view of a double layer barrel vault This barrel vault consists of © a layer of elements that forms the ‘top layer’, © layer of elements that forms the ‘bottom layer’ and © the ‘web’ elements that interconnect the top and bottom layers, The top layer elements of the barrel vault in Fig 1.8.21 are shown by thick lines and the bottom layer elements as well as the web elements are shown by thin lines. All the top layer nodes of the barrel vauit lie on a cylindrical surface. This surface is the "top circumcylinder’ of the barrel vault. Also, all the bottom layer nodes lie on a cylindrical surface, This is the ‘bottom circumeylinder’ of the barrel vault. The top and bottom circumcylindefs share the same International Journal of Space Structures Vol. 15 No. 1 2000Hoshyar Nooshin and Peter Disney longitudinal axis, The plan and elevation of the barrel vault of Fig 1.8.21 are shown in Fig 1.8.22, Also shown in this figure are the dimensions of the barrel vault together with a cylindrical normat for the formulation of the compret of the configuration of the barre! vault. watz un 7 3 4 g g 8 i Mh 4A pees 5 0 2 4 6 ——u S=24.62 m (Span) | Plan H=5.12 m (Rise) (Depth) U3, 2 (into paper) Elevation Fig 1.8.22 Plan and elevation of the double layer barrel vault of Fig 1.8.21 ‘The ‘span’ of the barrel vault is 24.82 m, its ‘rise’ is 5.12 mand its ‘length’ is 28,64 m. The ‘depth’ of the barrel vault is 1.35 m. The depth is the distance between ‘the top and bottom layers. More specifically, the depth is the difference between the radii of the top and bottom circumeylinders of the ‘barrel vault International Journal of Space Structures Vol. 15 No. 12000 ‘The compret of the barrel vault relative to the Ul- ‘U2-U3 normat of Fig 1.8.22 may be represented by the following formex formulation: ‘TOP = rinit(6,8,2,2) | [Rt,0,0; Rt,2,0] # Finit(7,7,2,2) | [Rt0,0; Rt,0,2] BOT = rinit(5,7,2,2) | [Rb,1,1; Rb,3,1) # tinit(6,6,2,2) | [Rb,1,1; Rb,1,3) WEB = rinit(6,7,2,2) | lamit(1,1) | {Rt,0,0; Rb,1,1] B=TOP#BOT# WEB Inthis formulation. © rinit(6,8,2,2) | [Rt0,0; Rt,2;0] represents all the top layer elements that are in the second direction, * rinit(7,7,2,2) | [R0,0; Rt,0,2] represents all the top layer elements that. are in the third direction, + tinit(S,7,2,2) | [Rb,1,1; Rb3,1) represents all the bottom layer elements that are in the second direction, @ rinit(6,6,2,2) | [Rb,1,1; Rb,1,3] represents all the bottom layer elements that are in the third direction, + rinit(6,7,2,2) | lamit(1,1) | [Rt,0,0; Rb,1,1] Tepresents all the web elements and ° TOP#BOT# WEB represents ali the elements of the barrel vault, All the formex fimctions used in the above formulation are as introduced and discussed previously, However, the above fommuletion does involve a new aspect regarding the use of a eylindrical normat. To elaborate, in using the cylindrical normats so far, all the configurations formulated involved only @ single cylindrical surface. However, in the present example, the configuration involves two cylindrical surfaces. Actually, there is no limit to the number of cylinders of a normat that may be involved in a formulation. A cylindrical normat has infinitely many coexial cylinders whose radii range from 0 to e, Fig 1.8.23, un Us, Fig 1.8.23 Coaxial cylinders of a oylindrical normat 43‘A Formian scheme for the barrel vault of Fig 1.8.21 is shown in the editory display of Fig 1.8.24. The scheme is generic. That is, it is written in terms of a number of parameters, where «© Mis the mumber of top layer modules in the U2 direction, © Nis the number of top layer modules in the U3 Girection, Sis the span, His therise, Dis the depth, Lis the length and vis the view adjuster. pe (*) Double layer berrel vault of Fig 1.8.21 (*) (*) top modules along U2 (*) (*) top modules along U3 (*) $=24.62; (*) span (*) (*) rise (*) (4) dopth (*) (4) length (*) (*) view adjuster (*) atan|(2*H/S);(*) sweep angle (*) s((2*sin| A); (*) top radius (*) RED; (*) bottom radius {*) TOPsrinit(MN+1,2,2)} IRt,0,0; Rt2,0]# rinit(M+1,N,2,2) | [Rt,0,0; Rt,0,2}; BOT=rinit(M-1,N,2,2}|[Rb,1,1; Rb,3,1]# rinit(M,N-1,2,2)|[Rb,1,1; Rb.1.3} }=rinit(M,N,2,2)|lamit(1,1)| {Rt0,0; 8b,4,4); \P#BOT# WEB; Bi=be(1,A/M,1/(2°N)[B: BV=verad(0,0,90-A)|B4; use &vm(2),vt(2), vh{v.2.75*Rt-Rt, 0,0,Rt, 0,1,Rt}; clear; draw BV; <> wa Fig 1.8.24 A generic scheme for the double layer barrel vault of Fig 1.8.21 The variable A in the scheme of Fig 1.8.24 represents the sweep angle of the barrel vault. The sweep angle is given by A=2 arctan (24/8) as derived previously and shown in Fig 1.8.5. Also, using the information given in Fig 1.8.5, the radius of the top circumeylinder of the barrel vault of Fig 1.8.21 is found to be Rt=S/(2 sin A) ‘The radius of the bottom cireumcylinder of the barrel vault is then given by 4 Formex Configuration Processing Rb=Rt-D ‘The formulation of the compret of the barrel vault of Fig 1.8.21, as given in the scheme of Fig 1.8.24, is a generic version of the formex formulation given above. The last statement in this formulation creates the formex variable B that represents the compret of the barrel vault relative to the cylindrical normat U1 U2-US of Fig 1.8.22. It should be noted that in this normat the coordinates in the Ul direction are based ‘on the ‘true’ dimensions rather than a simplified graduation Graduations along normat directions are normally chosen for convenience in the formulation of ‘configurations. However, in some situations the actual dimensions provide the most convenient graduation along normat direction. And, this/~ happens to be the case for the Ul direction in the example under consideration, In the scheme of Fig 1.8.24, the configuration of the barrel vault relative to the global Cartesian x-y-2 coordinate system of Fig 1.8.22 is given by Bl =be(1, A/M, L(2*N)) | B; The construct be(1, AMM, LA2*N)) is a basicylindrical retronorm, as discussed before (Fig 1.8.6). «The first parameter of the retronorm, which is given as 1, is the scale factor in the Ul direction, The value is 1 since the normat graduation in this direction corresponds to the actual dimensions. 6 © The second parameter is the (angular) scald. factor in the U2 direction and is given as A/M. This has the effect of keeping the sweep angle equal to A for all values of M. © The third parameter of the retronorm is the scale factor in the U3 direction and is given as LiQ*N). This has the effect of keeping the length of the barrel vault equal to L for all values ofN. The. statement following the basicylindrical transformation in the scheme of Fig 1.8.24 is BY = verad(0,0,90-A) |B1; The effect of this statement is to rotate the whole barrel vault around the z axis by (90-A)®. As aresult, with respect to the new position of the barrel vauit, the x and y axes assume the directions indicated by (x) and (y) in Fig 1.8.22. This rotational transformation wes previously discussed in relation to the barrel vault of Fig 1.8.1, as shown in Fig 1.8.8, International Tournal of Space Structures Vol. 15 No. 1 2000 >) )Hoshyar Nooshin and Peter Disney A generic scheme, such as the one in Fig 1.8.24, is a useful aid for a designer since it allows vatious possibilities to be examined conveniently by using different values for the parameters, For instance, the double layer barrel vault of Fig 1.8.25 is produced by the scheme of Fig 1.8.24 using the parameter values ‘M=10, N=10, S=32, H=10, D=1.6 and L=40, Fig 1.8.25 A double layor barrel veult generated by the scheme of Fig 1.8.24 with M=10, N=10, S=32, H=10, D=1.6 and 1.9 DOMES An example of a dome configuration is shown in Fig 1.9.1. The dome has 400 elements that are connected together at 144 nodes. All of these nodes are situated on 2 sphere that is referred to as the ‘circumsphere* of the dome, A cross-section of the circumsphere of the dome is shown in Fig 1.9.2. Also shown in this figure is the cross-section of the dome in thick lines. ‘The particulars of the dome are as follows: © The ‘span’ is $= 36 m: © The ‘tise’ is H=7 m, © The ‘sweep angle’ is A= 2arctan(2H/S) = 42.501° ® The radius of the circumsphere of the dome is R= S/2 sin A) = 26.643 m * The ‘central angle’ of the dome is twice the sweep angle and is equal to 85.002°. ‘The formulae for the sweep angle A and radius R, as given above, are the same as for a cylindrical barrel vault derived previously and shown in Fig 1.8.5, International Journal of Space Structures Vol. 15 No. 1 2000 ¥ig 1.9.2 Cross-section of the circumsphere of the dome of Fig 1.9.1 A dome with a pattem of elements as shown in Fig 1.9.1 is referred to as a ‘Schwodler dome’ (after jhe German Engineer J W Schwedler who builly a number of domes of this type in the nineteenth century). In a Schwedler dome, each group of elements that forms a horizontal polygon is referred to asa ‘ring’ and cach individual element of a ring is referred to as a ‘ring element’, Also, cach group of elements that lie along a meridional line between the crown and the base ring is referred to as a ‘rib’ and exch individual clement of a rib is referred-to as a ‘rib element’. Finally, there are the diagonally disposed elements that are referred to as ‘diagonal elements’. ‘The most convenient reference system for the formulation of the compret of the configuration of the Schwedler dome of Fig 1.9.1 is a ‘spherical 45normat’. A spherical normat may be imagined as consisting of an infinite number of concentric spheres. The circumsphere of the dome may then be considered to be coincident with the normat sphere of radius R, Fig 1.9.3. This sphere is imagined to have a number of ‘parallels’ and ‘meridians’ to suit the configuration of the dome. These parallels and meridians, together with a part of the dome of Fig 1.9.1, are shown in Fig 1.9.4 LES © Fig 1.9.3 Concentric spheres of a spherical normet UL First direction Parallel (radial {us in Third direction Second direction (meridional) —_(cizcumferential) Fig 1.9.4 Parallels and meridians of a spherical normat ‘The normat of Fig 1.9.4 has three directions: © ‘radial’ direction, denoted by U1, representing the radii of the spheres of the normat, ‘circumferential? direction, denoted by U2, sepresenting angles along the parallels and 46 Formex Configuration Processing 1 + ‘meridional’ direction, denoted by U3, representing angles along the meridians. ‘A close-up of the top part of the normat of Fig 1.9.4 is shown in Fig 1.9.5, Also shown in this figure are a few elements of the dome and graduations along the second and third directions. NAD oe Fig 1.9.5 Normat graduation for the foimulation of the compret of the dome of Fig 1.9.1 A formex formulation for the compret of the configuration of the Schwedler dome of Fig 1.9.1, relative to the normat of Fig 1.9.5, may be written as follows: B= rinit(16,8,1,1) | {{R.0,1; R11), [R.0,1; R,0,2], [R,0,15 R123} B=rin(2,16,1) | [R,0,9; R,1,9] D=E4B Oo In this formulation © {(R.0,1; R,1,1), [R,0,15 R,0,2}, (R,0,15 R,L2]} represents the elements ab, ac and ad of the dome, as shown in Fig 1.9.5, © rinit(16,8,1,1) | {IR,0,1; R11], [R.0,1; R,0,2], [R,0,1; R121} represents all the elements of the dome other than those along the base ring, © rin@,16,1) | [R.0,9; R,1,9] represents all the elements along the base ring and « E#B represents all the elements of the dome. ‘A generic Formian scheme for the formulation of the compret of the Schwedler dome of Fig 19.1 is shown in the editory display of Fig 1.9.6. In this scheme © Mrepresents the number of elements on a ring, International Journal of Space Structures Vol. 15 No. 1 2000© Hoshyar Nooshin and Peter Disney N represents the number of elements on a rib, © S represents the span and © Hrepreseats the rise, ‘The formex formulation in the scheme of Fig 1.9.6 is @ generic version of the forniulation given above, with the difference that the first uniples of the formices in the scheme are given as 1 rather than R. This will be discussed later. (7) Schwedler dome of Fig 1.9. (*) M=16; (*) No of elements on a ring (*) N=; (+) No of elements on a rib (*) C2) span (*) ‘ (C) rise (*) *atan|(2*H/S); (*) sweep angle (*) (2rsin|A); (+) radius (*) (MAN 1, 2)|{(2,0,45 1,1,4 (2.0,4; 1,0.2}, (2,0,45 1,1.2] Berin(2,M,1)|[1,0,.N+1; 1,1,N+1]; D=E#B; DD=bs(R,360/M,A/(N+1)}[D; use & vm(2),vt(2}, vh(0,2°R4°R, 0,0,0, 0,0,1); clear; draw DD; <> Fig 1.8.6 A generic scheme for the Schwedler dome of Fig 1.9.1 bs(b1,b2,b3) Li ‘actor for scaling nth tind irection (angular scale factor) fector for scaling in the second direction (angular scale factor) factor for scaling in the fret direction (near scale factor) abbreviation for basispherical Fig 1.8.7 Besispherical retronorm The statement. following the formulation of the ‘compret in the scheme of Fig 1.9.6 is DD =bs(R, 360/M, A((N+1)) | ‘The construct, bs(R, 360/M, A/ON+1)) is a ‘basispherical” retronorm. The effect of this function is to transform the spherical normat coordinates into global x-y-z Cartesian coordinates, ‘The basispherical retronorm has three parameters that act as scale factors, as explained in Fig 1.9.7. International Journal of Space Structures Vol. 15 No.1 2000 The first parameter bl is a scale factor in the radial direction along U1, the second parameter b2 is an angular scale factor for the circumferential direction along U2 and the third parameter b3 is an angular seale factor for the meridional direction along U3, Inthe scheme of Fig 1.9.6, the first parameter of the retronomm is given as R. This implies that every first uniple of the value of the formex variable D will be multiplied by R. Thus, all the nodal points of the dome will assume their correct positions on the normat sphere of radius R. Alfemnatively, the formex variables E and B in the scheme could have been given as E=rinit(M.N,1,1) | {1R0,1; R11), 10,1; R,0,2), [R,0,1; 1,23}; B= rin@M,1) | [ONH; RLN+}; Jn this case, the first parameter b1 of the retronorm jn the scheme should be given as I rather than R. The second parameter b2 of the basispherical Tetronorm in the scheme of Fig 1.9.6 is given as 360/M_ This implies that every second uniple of the value of the formex variable D will be multiplied by 360/M. The effect is that, for any value of M, the elements ‘on a ring cover 360° and create a closed polygon, ‘The third parameter b3 of the retrononm in the scheme is given as AMN+1) This implies that every third uniple of the value of the formex variable D will be multiplied by A(N+1). Consequently, for any value of N, the sweep angle of the dome will remain equal to A. ‘Meridional direction us | UL, Redial direction Circumferential direction y U2 Fig 1.9.8. Relationship between the U1-U2-U3 spherical normat and the global x-y-z Cartesian coordinate system a7In the scheme of Fig 1.9.6, both formex variables D and DD represent the compret of the Schwedler dome of Fig 1.9.1. However, D represents the compret of the dome relative to the spherical normat of Fig 1.9.5, whereas DD represents the compret of the dome relative to the global x-y-z Cartesian coordinate system, The relationship between these ‘two reference systems is shown in Fig 1.9.8. If a typical: signet of the formex variable D is represented by {U1,02,U3] and if the corresponding signet of DD is represented by xyz] then the basispherical retronorm bs(b1,b2,b3) will transform [U1,U2,U3] into [xy.z] using the equations. x= b1U1xc08(b2+U2)xsin(b3xU3) y= blxULxsin(b2-U2)xsin(b3xU3) = blUlxcos(b3xU3) ‘These equations are based on the standard formulae for transformation of spherical coordinates into Cartesian coordinates. To conclude the discussion of the scheme of Fig 1.9.6, reference should be made to the ‘use’, ‘clear’ and ‘drew’ statements at the end of the scheme. ‘These statements effect the setting up of the viewing particulars and the drawing of the dome. The details Of these statements have been discussed in the previous sections. Another example of a Schwedler dome is shown in Fig 1.9.9. This dome has 10 rings and 24 ribs. However, some of the ribs have been ‘trimmed’ back at the top and the arrangement of the elements in the central part has been altered. An operation of this kind on the configuration of a dome is referred to as ‘trimming? and the resulting dome is referred to as a ‘trimmed’ dome. Trimming may be carried out in many different ways and applied to different types of domes. ‘The reason for trimming is to avoid ‘element cluttering’ in the central region of a dome, To elaborate, consider the Schwedler dome shown in Fig 1.9.10. This configuration is generated using the scheme of Fig 1.9.6 with M = 24 (oumber of elements on a ring) and N = 9 (number of elements onarib), Formex Configuration Processing I Zz S KZA Mie SSW eg COIS MEER ESRI. WF RRSEREGAN Wy ERK LEN \\ Fig 1.9.10 A Schwedler dome generated by the scheme of Fig 1.9.6 with M=24 and N=9 ‘The main bodies of the domes in Figs 1.9.9 and Oo 19.10 are similar but their central regions have different patterns. In the central region of the dome of Fig 1.9.10 there is, evidently, an undesirable cluttering of the elements. However, this problem has been overcome by trimming in the dome of Fig 19.9, ‘A. Formian scheme for the generation of the configuration of the dome of Fig 1.9.9 is shown in Fig 1.9.11. In this scheme «the formex variable Bo represents the elements in the central region of the dome, - © the formex variable Eb represents the elements in the main body of the dome excluding the ones oon the base ring, + the formex variable B represents the elements on ‘the base ring and © the formex variable D represents all the elements of the dome. International Journal of Space Structures Vol: 15 No. 1 2000Hoshyar Nooskin and Peter Disney (*) Trimmed Schwedler 4), 13,0.4: 1,0,5], (1,0,45 11,5]; B=rin(2,24,1)|[1,0,1 i D=Ec#Eb#B; DD=bs(25,360/24,3.5)|D; suse &,vm(2),vt(2), vh(0,50,100, 0, clear; draw DD; <><><> Fig 1.9.11 A scheme for the trimmed Schwedler dome of Fig 1.9.9 In the scheme of Fig 1.9.11, the staternent DD =bs(25, 360/24, 3.5) |D; has the effect of transforming D, representing the compret of the dome relative to a spherical normat, into DD, representing the compret of the dome relative to the global x-y-z coordinate system, In the basispherical retronorm bs(25, 360/24, 3.5) + The first parameter sets the circumradius of the dome equal to 25 units (the term ‘cireumradius* means the ‘radius of the circumsphere’), * The second parameter sets the angular scale factor for the circumferential direction equal to (360/24), so that, with 24 circumferential ivisions, the rings will close. * the third parameter sets the angular scale factor in the meridional direction equal to 3.5°, This will result in the sweep angle of the dome being equal to 35° (since there are 10 divisions on each meridian from the crown to the base of the dome). It is to be noted that in the case of the scheme of Fig 1.9.11, the cireumradius and the sweep angle of the dome are given directly, rather than being calculated in terms of the span and the rise (as in the previous example). However, if required, the span and the rise of the dome can be obtained using the relations given in Fig 1.9.2. To be specific, the span is obtained as S=2R sin A= 28.679 units and the rise is obtained as International Journal of Space Structures Vol. 15 No. 12000 H=(S tan(A/2))2 = 4.521 onits Note that here the values for the span $ and rise H are given in terms of ‘units (of length)’, rather than a specific unit such metre or millimetre, The reason is thet, in the present context, the discussion concems the arrangement and the proportions of the elements Tather than the actual sizes..In such a context, one may choose to use the general term ‘unit? instead of 2 specific unit of length. This emphasises the fact that the discussion is independent of the actual sizes, Fundamentally, the information contained in a formex is in tenms of an arrangement of ‘pure’ numbers without any inherent association ‘with specific units of length, angle, time, force, ... ete, ‘The association of units with the information in a formex is effected by a ‘human’ to suit the context in which the formex is used. RAK} SERA ee aN ese Wesey Fig 1.9.12 Some varieties of Schwedler dome Fig 1.9.13 Examples of ribbed domes Further examples of dome configurations are shown in Figs 1.9.12 to 1.9.14, The domes in Fig 1.9.12 ste examples of various types of Schwedler domes. Examples of a different family of dome configurations are shown in Fig 1.9.13. These are 49referred to as ‘ribbed’ domes, with the one on the right being a ‘trimmed ribbed’ dome. ‘The basic characteristic of a Schwedler dome configuration is that it consists of rib, ring and diagonal elements, In contrast, a ribbed dome configuration (mainly) consists of rib and ring elements, There is another major family of dome configurations in which the elements are (mainly) ring and diagonal elements, Examples of this family of domes are shown in Fig 1.9.14. These are referred to as ‘lamella’ domes. Also, the bottom right dome in Fig 1.9.14 is a ‘trimmed lamella’ dome. Fig 1.9.14 Examples of lamella domes A lamella dome configuration may contain a few rib elements, as in the bottom dome configurations in Fig 1.9.14. Also, ribbed domes may sometimes involve a few diagonal elements. ‘The formex formulations for the domes in. Figs 1,9,12 to 1.9.14 are left for the reader to carry out as exercise. Spherical dome configurations may be transformed into ellipsoidal domes using simple scaling in a ‘manner similar to that discussed in relation to barrel ‘yaults in section 1.8. For example, let D be a formex variable representing a trimmed Schwedler dome of the type shown in Fig 1.9.9 relative to the global coordinate system. A plan view of this dome is shown in Fig 1.9.15(@), The formex variable DD = bb(1,1.25)|D will then represent an ellipsoidal dome whose dimensions in the y direction are 1.25 times greater than those in the original dome. The plan view of the dome represented by DD is shown in Fig 1.9.15(b). 50 Formex Configuration Processing 1 SEs LEI LIER ZEEE VERRAN WER DEY aa BAR Eh MRR ACO Y SEE ad WERE] INNA RR URE WRENS WO @ Weer Wey TERESI VARY) Mee RY eKee/ KSISEX7 WSS wy Fig 1.9.15 Examples of ellipsoidal and ‘ovate domes (7) Ovate dome of Fig 1 Ee=rin(2,3,4)|{[1.0,0; [1,0,2; 1.0.3], [1.0, Fin(2,6,2) |{(2.0,35 1,2,3), 1,0,4), [1,0,3; 1,4,4), [14, Bb=rinit(12,3,4,1) |{[1,0,4 1,4,4), [2,0,4; 1.0.5}, (1,0,45 1.1.5 -rin(2,12,1) | [1,0,7; 11,7); #ED#B; DD=bs(25,360/24,3.5)|D; DD=DD#bb(1,1.35) frefid(o0,0)|DD; use &,vmi(2); clear; draw DD; <><><> Fig 1.9.16 A scheme for the ovate dome of Fig 1.9.15(c) Simple scaling may also be used to create ‘ovate’ (cee-shaped) domes of various forms. For example, Fig 1.9.15(c) shows the plan view of an ovate dome obtained by putting together half of the spherical dome of Fig 1.9.15(a) with half of an ellipsoidal dome. A scheme for the generation of the compret of this ovate dome in shown in the editory display of International Journal of Space Structures Vol. 15 No. 1 2000Hoshyar Nooshin and Peter Disney Fig 1.9.16. Fig 1.9.15(4) shows the plan view of another ovate dome that is obtained by putting together two half ellipsoidal domes, As the last example in this section, consider the double layer dome of Fig 1.9.17. This is a circular canopy structure with a large opening at the middle, ‘The dome has 324 top layer elements, 252 bottom layer elements and 576 web elements, In Fig 1.9.17, the top layer elements of the dome are shown by thick lines and the bottom layer elements as well as the web elements are shown by thin lines. A section through the centre of the dome is shown in Fig 1.9.18. ISSR ee Eee NESS ReRep ey RSIS < Top layer Bottoni layer Fig 1.9.17 Porspective view of a double layer canopy dome S=75 m (Span) P=45 m (Gap) | Rb =48.953 m este Tink ercsin =P =27.363" Greresin -B=27.083 Fig 1.9.18 A section through the centre of the double layer dome of Fig 1.9.17 International Journal of Space Structures Vol. 15 No. 1 2000 The span of the dome, that is, the diameter of the base ring of the top layer, is S= 75 m and the ‘gap’ at the middle, that is, the diameter of the central ring of the top layer is P= 45 m, The position of the circumsphere of the top layer of the dome is indicated by a dotted circle in Fig 1.9.18, ‘Also shown in this figure is the ‘depth’ of the dome which is equal to 1.5 m. This is the difference between the circumadii of the top and bottom layers of the dome. The sweep angle of the dome is given as 50°, as shown in Fig 1.9.18. The circumradius of the top layer is then obtained es Rt=S/(2 sin A) = 48.953 m where S is the span and A is the sweep angle. The derivation of this equation bas been discussed previously Fig 1.8.5). To obtain the ‘gap angle’ G, one may write (@ayRi= sin G or G = aresin(P/(2 RY) = 27.363° (*) Double layer dome of Fig 1.9.17 (*) 3 (*) No of elements on a ring (*) “) No of elements on a top rib (*) (2) span. (*) (7) gap (*) (*) depth (*) (2) sweep angle (*) S=75; $/(2*sin|A); (*) top radius (+) Rb=RtD; (*) bottom radius (*) asin |(P/(2*RO); (*) gap engle (*) i=(A-GV2"N); | (*) increment (*) ‘TOPsrinit(M,N4+1,2,2") |[R}0,G; Rtz,C}# init(M.N,2,2*i)[[RLO.G; Ri0,G42" BOT=rinit(M.N,2,2"%) |[Rb,1,G+iRb,3,C-45]# rinit(M,N-4,2,2°H)|[Rb,1,G-+i; Rbjt,G+3"i}; WEB =rinit(M.N,2,2*i) lamit(a,C-+i)| [Rt0,G;Rba,.G+i}, DD=TOP#BOT#WEB; DD=bs(1,360/(2*M),1) |DD; use &.vm{2),vi(2), vh(0,2*Rt,4*Rt, 0,0,0, 0,0,1); clear; draw DD; <><><> Fig 1.9.19 A generic scheme for the double layer dome of Fig 1.9.17 A generic scheme for the dome of Fig 1.9.17 is given in Fig 1.9.19, The initial data, namely, 8 = 75, P= 45, D = 1.5 and A = 50 are given as parameters in the scheme and the values of the circumradii Rt and SIRb and the gap angle G are then obtained in terms of the parameters. Also given as parameters in the scheme are © M, denoting the number of elements on a top or bottom layer ring, and +N, denoting the number of elements on a top layer rib. ‘A spherical normat for the formulation of the compret of the dome is shown in Fig 1.9.20. In this figure, the normat coordinates in the third direction (meridional direction) are given in terms of the gap angle G and an increment (A-GVQN) ‘The normat coordinates in the third direction are actual angular values, starting from the gap angle G at the central ring of the top layer, incrementing towards the base ring of the top layer, reaching the sweep angle A. uz Fig 1.9.20 Spherical normat for the formulation of the compret of the double layer dome of Fig 1.9.17 ‘The formulation of the formex variables TOP, BOT. and WEB in the scheme of Fig 1.9.19 is in terms of the actual dimensions in the first direction and actual angular values in the third direction. However, in the second direction the normat coordinates are given in terms of the simple graduation shown in Fig 1.9.20. Corisequently, in the basispherical retronorm given in the scheme of Fig 1.9,19, namely, bs(1, 360/(2*M), 1) only the normat coordinates in the second direction need scaling. The (angular) scale factor in this direction is 52 Formex Configuration Processing I 360/2M Thus, with M members on a ring and with each member involving two divisions, the angles in the second direction will add up to 360° and the rings will close. ACKNOWLEDGEMENTS ‘The early work in formex configuration processing was greatly helped by substantial donations from 2 group of Iranian Engineers. These are A. Sarshar, A. Jahanshahi, C, G. Abkatian, G. A. Mirzareza, M, S. Yazdani and J..Hassanein and their contributions are gratefully acknowledged. In the early nineties, the Taiyo Kogyo Corporation of Japan played a crucial role in supporting research in formex configuration processing and, more recently, the Tomoe Corporation of Japan has been instrumental in supporting research in this field. Their generous help is gratefully acknowledged. REFERENCES 1. Nooshin, H. Algebraic Representation and Processing of Structural Configurations, International Journal of Computers and Structures, Vol. 5, 1975, 119-130. 2. Nooshin, H. Formex Formulation of Double Layer Grids, This work was presented at a short course on “analysis, Design and Construction of Double Layer Grids at the University of Surrey, UK, in September 1978 and subsequently publishied as chapter four in: Analysis, Design and Construction of Double Layer Grids, Edited by Z. 8. Makowski, Applied Science Publishers, London, 1981, 119-183, 3. Nooshin, H. Formex Configuration Processing Structural Engineering, Elsevier Applied Science Publishers, London, 1984, 4, Disney, P. and Etabbar, O. An Introduction to Formian, Proceedings of the 3" Intemational Conference on Space Structures, Edited ty H. ‘Nooshin, Elsevier Applied Science Publishers, 1984, Disney, P. Formian: The Programming Language of Formex Algebra, Proceedings of the [ASS Symposium on Membrane Structures and Spece Frames, Edited by K. Heki, Elsevier Science Publishers, 1986, Nooshin, H. and Disney, P. Elements of Forman, International Journal of Computers and Structures, Vol. 41, No. 6, 1991, 1183-1215. 7. Nooshia, H., Disney, P. and Yamamoto, C. Formian, ‘Multi-Science Publishing Co, Ltd., 1993, 6 International Journal of Space Structures Vol. 15 No. 1 2000Formex Configuration Processing II Hoshyar Nooshin and Peter Disney Space Structures Research Centre, Department of Civil Engineering, University of Surrey, Guildford, Suey GU2 TXH, United Kingdom ABSTRACT: This is the second paper in a series of papers that are intended to provide ‘2 comprehensive coverage of the concepis of formex configuration processing and their applications in reletion to structural configurations. In the present paper, attention is focused on the configuration processing for a number of femilies of space structures, namely, pyramidal forms, towers, foldable systems and diamatic domes. Also included is 4 section on information export as well as en Appendix on basic formex functions. The section on information export describes the manner in which the information about the details of a configuration, generated by the programming language Formian, can be exported to graphics, draughting and structural analysis packages, 2.1 INTRODUCTION Formex configuration processing provides’ a powerfil medium for the processing of configurations of all kinds, Formex configuration processing uses the concepts of formex algebra through the programming language Formian to generate and process configurations. The preliminary concepts and ideas of formex configuration processing are described in the first instalment in this series of papers, Ref 1. The material in the present paper is highly dependent on that of Ref 1 and, therefore, the reader should be thoroughly familiar with the material of Ref 1 before attempting to study the present paper. For further information and downloading of Formian visit the web site: 2.2 PYRAMIDAL FORMS Consider the pyramidal structure shown in Fig 2.2.1. ‘The structure consists of 171 beam elements that are connected together forming. a pyramid. The base of the pyramid is an equilateral triangle with each side being equal to L. The position of the global x-y-z coordinate system is assumed to be as shown in Fig 2.2.1. The base of the pyramid lies in the x-y plane with the origin of the coordinate system being at the centre of the base. International Journal of Space Structures Vol, 16 No. 1 2001 Fig 2.2.1. A pyramidal structure ‘The height (altitude) of the pyramid is equal to H with the z axis passing through the apex. The pyramid has three identical triangular faces each of which consists of a planar arrangement of beam elements, The distance denoted by $ in Fig 2.2.1a isthe distance between the apex of the pyramid and the midpoint of the base line of a face. The distance denoted by C in Fig 2.2.1c is the perpendicular distance from the centre to a side of the base of the pytamid. This distance is equal to the projection of S on the x-y plane. y ce s x a0 es (@) Step 1 () Step 2 () Step 3 (@) Step 4 Fig 2.2.2 Procedure for the generation of the configuration of a face of the pyramid ‘The configuration of a face of the pyramid may be formulated using the following four steps: STEP1 ‘The configuration of 2 face of the pyramid is formulated in the x-y plane in terms of a convenient system of graduations along x and y axes, Fig 2.2.22, STEP2 ‘The configuration of Fig 2.2.2a is scaled in the x and y directions such that the face assumes its correct dimensions, Fig 2.2.2b. STEP3 ‘The configuration of Fig 2.2.2b is translated in the y direction such that the distance between the base of the face and the x axis becomes equal to C, Fig 22.20, STEP 4 The configuration of Fig 2.2.2c is rotated about the base line of the face such that the face assumes its correct spatial position, Fig 2.2.24, Formex Configuration Processing It ‘The formulation of the configuration of the face shown in Fig 22.2a in terms of the indicated ‘graduations along the x and y axes may be written as El = genid(1,6,2,1,-1,1) | {[0,0,0; -1,1,0), 10,0,0; 11,0}, (-1,1,0; 1,1,0}} In this equation, the part genid(1,6,2,1,-1,1) is a genid fonction that generates the triangulated arrangement of elements in Fig 2.2.2a, as described in Section 1.4.6 of Ref 1 and Section 2.4.4 of the Appendix. The argument of the genid function in the above equation, that is, {{0,0,0; -1,1,0}, (0.0.0; 1,1,0}, [-1,1.05 1,1,0]} represents the three elements that are shown by thick ines in Fig 2.2.2a. ‘The configuration of Fig 2.2.2b is obtained by scaling the configuration of Fig 2.2.24 along the x ‘. and y axes. This may be achieved by writing E2=bb(L/12, $/6) | El, ‘The construct bb(L/12, 8/6) is a basibifect retronorm that effects scaling by L/12 and 8/6 in the x and y directions, respectively, as, discussed in Section 1.4.4 of Ref 1 and Section 2.A.11 of the Appendix. ‘The configuration of Fig 2.2.2c may be obtained by writing 3 =tran(2, C-S) | E2 In this equation, the part tran(2, C-S) is a translation fimetion that effects a translation in) the y direction by C—S, see Section 1.4.1 of Ref 1 and Section 2.A.3 of the Appendix. The rotation of the face about its bese line (Fig 2.2.24) may be achieved by writing B4 = verat(C,0,-B) | E3 The construct vverat(C,0,-B) is a verat function that effects a rotation in the y-z plane about the base line of the face, see Section 1.4.7 of Ref 1 and Section 2.A.5 of the Appendix. ‘The distances C and S and the angle-B may be obtained in terms of L and H as follows: From Fig 2.2.3 that shows the base of the pyramid, L/2C = tan 60° and C=L/(2 tan 60°) International Journal of Space Structures Vol. 16 No. 1 20010 Hoshyar Nooshin and Peter Disney L ef \ le Centre of, the base Fig 2.2.3 Base of the pyramidal structure of Fig 2.2.1 Also, from Fig 2.2.24, s=(C +H)? and tan B=H/C B=arctan(H/C) A Formian scheme for the generation of the configuration of the pyramidal structure of Fig 2.2.1 is showa in the editory display of Fig 2.2.4 (Formian schemes and editory displays are introduced in Ref 3, in particular, sce Sections 1.3.3, 1.5 and 1.5.8). The scheme of Fig 2.2.4 contains Formian instructions that are based on the formulations preseated above. However, the formulations in the scheme of Fig 2.2.4 are more general involving two additional paranieters m and n, where © m denotes the number of elements along an edge ofa face of the pyramid and © n denotes the number of sides of the base of the pyramid. ‘The number of elements along an edge-of a face of the pyramid (that is, m) is referred to as the ‘frequency’. This is a measure of the ‘density’ of the pattern of the configuration of the pyramid, ‘The scheme of Fig 2.2.4 is a ‘generic scheme’. The term ‘generic’ implies that the formulation is carried out in terms of parameters. This would allow the scheme to be used for exploring a variety of configurations rather than being restricted to a single configuration, sce Sections 1.4.6 and 1.5.1 of Ref 1. In the scheme of Fig 2.2.4, the formex variable E4 represents the configuration of a face of the pyramid as shown in Fig 2.2.2d. The configuration of: the ‘whole pyramid is then obtained by writing = pex | rosad(0,0,n,360/n) | E4; In this Formian statement, the construct rosad(0,0,n,360/n) International Journal of Space Structures Vol. 16 No. 1 2001 is 2 rosad furiction that generates the configuration of the entire pyramid by composing n rotations of the face represented by 4, see Section 1.4.7 of Ref 1 and Section 2.A.5 of the Appendix. ‘The part ‘pex’ in the above Formian statement is the pexum function that effects the removal of the superfluous overlapping elements along the edges of the pyramid, see Section 1.4.5 of Ref I and Section 2.4.1 of the Appendix. (*) Pyramidal structure of Fig 2.2.1. (*) L=10; ‘(*) length of each side of the base (*) H=10; (*) height of pyramid (*) () frequency (*) n=3; — (*) number of sides of the base (*) C=L/(2"tan|(180/n)); S=sqrt|(C*C+H"H); B=atan| (HC); E1=genid(1,m,2,1,-1,1) | {[0,0,0; -1,1,0], {[0,0,0; 1,4,0}, [-1,2,0; 1,1,0]}5 E2=bb(l/(2*m), Sim} |E1; ES=tran(2, C-S)|E2: Fasverat{C,0,-B) E3; P=pex|rosad(0,0,n,360/n) |E4; use & vm(2),vt(2),vh(0,-2*1,16*H, 0,0,0, 0,0,1); clear; draw P; <><><> Fig 2.2.4 A generic schome for the pyramidal structure of Fig 2.2.1 ‘The use statement in the scheme of Fig 2.2.4, that is, use &,vm(2),vt(2),vh(0,2*L, 16*H, 0,0,0, 0,0,1); has the effect of setting the viewing particulars for ‘the perspective view of. the pyramidal structure shown in Fig 2.2.1a, see Sections 1.5.2, 1.7.1 and 1.7.2 of Ref 1. With the choice of the parameter values: L=10,H~10,m=6 anda=3 the scheme of Fig 2.2.4 will generate the pyramidal structure of Fig 2.2.1. However, the scheme of Fig 2.2.4 may be used to generate a variety of other pyramidal forms by simply changing the values of the parameters. Four such examples are shown in Fig 2.2.5 with the corresponding values of the parameters shown for each case. In the case of the configuration in Fig 2.2.5d, the height of the pyramid is equal to zero, This then represents a ‘dogenerate’ pyramidal form which is a ‘flat’ grid. A flat grid of this type is referred to as a-‘sectorate grid’ with each one of the triangular parts that corresponds to a pyramidal face being called a ‘sector’, The grid of Fig 2.2.5d consists of 12 sectors,L210, H=0, m=5, n=12 Fig 2.2.5 Examples of pyramidal forms generated by the scheme of Fig 2.2.4 Incidentally, to obtain the views shown in Fig 2.2.5, the use statement in the scheme of Fig 2.2.4 should be ruse &,vm(2),v1(2).vh(0,-2*L,,6*H, 0,0,0, 0,01); for Figs 2.2.5a, 2.2.5b and 2.2.5¢ and suse &vn(2); for Fig 2.2.54. FEISS Z HINSS STAN RS THEN TAINS HE HY “ x * CH CRANKY ete BNI NY BONY ¥ig 2.2.6 Further examples of pyramidal forms Some further examples of pyramidal configurations are shown in Fig 2.2.6. The formulation of these Formex Configuration Processing II configurations is left to be carried out by the reader asexercise. A pyramidal structure may have two or more ayers’. An example of a double layer pyramidal structure is shown in Fig 2.2.7. In this structure, the length of the base of the top layer of each face is L=20 unit length, the height of the apex of the top layer is. H=7 unit length and the perpendicular distance between the two layers is 5 unit length, SLIT RDO SEA MOORAS SENN INAS SIO) BEBO SLOPES SERED Plan view of top layer Plan view of web Plan view of bottom layer (~) Fig2.2.7 A double leyer pyramidal structure ‘A major consideration in the design of pyramidal structures is the manner in which the elemeats are connected together. In the case of a ‘single layer’ pyramidal structure (like the ones in Figs 2.2.1, 2.2.5 and 2.2.6) the elements are, in general, under the effects of benditig moments, shear forces, torques and axial forces. The connections should, therefore, be designed such that there are adequate rigidities in different directions to allow the transfer of the components of force and moment. The situation is rather different for a. double layer ‘pyramidal structure. In this case, the axial forces are usually the dominant effects-in the members. Therefore, it is normally acceptable to use a connector that behaves, ‘more or less like a pin-joint, The above comments also apply in relation to single and double layer domes and barrel vaults. International Journal of Space Structures Vol. 16 No. 1 2001Hoshyar Nooshin and Peter Disney (*) Double layer pyremidal structure of ig 2.2.7 (*) ; (*) lenigth of each side of the base (*) (0) height of pyramid (*) () distance betwoon two layers (*) (*) frequency of top layer (*) () number of sides of the base (*) C=L/(2*tan|(180/n)); S=sqrt](C*C+H*H); Beatan |(H/C); ET=genid(1,m,2,3,-1,1)| {{0,0,0; -1,3,0}, 10,00; 1,3,0}, [1,2,0; 1,8,0)}; E2=bt(Li(2*m),S/(3*m),D) [B1; E3=tran{2,C-S)| 32; Ee=verat{C,0,-B)|E3; 2,1); Risverat(C,0-B)| tran(2,0-S) [bt(L/(2*m),S/(3*m),D)| Qi; Ri=verad(0,0,360/n) [Ris Ki=tig| (RAR); Keszinex(0,0,E, -L/(2*xa),C/m,(m-t)*H/m, m2) |K1; P=pex|rosad{0,0,n,360/n) |(E4#K2); iase &,vm(2),vt(2),vh(0,-2*L,6"H, 0,0,0, 0,0,1)s clear; draw P; <><><> Fig 2.2.8 A generic scheme for the double layer pyramidal structure of Fig 2.2.7 i y We MAAR eee WAAAY for top layer forbottom layer YOO o WW Fig 2.2.9 Graduations along x, y and z axes for the formulation of a face of the pyramidal structure of Fig 2.2.7 Retuming to the example of the double layer oyramidal structure of Fig 2.2.7, a scheme for the generation of the configuration of the structure is given in the editory display of Fig 2.2.8. The International Journal of Space Structures Vol. 16 No.1 2001 formulations in this scheme follow the same four- step procedure described for the single layer Pyramidal structures, see Figs 2.2.2 and 2.2.4. The step 1 of the fortmulation for the example of Fig 2.2.7 is carried out in terms of the graduations along the x, y and 2 axes as shown in Fig 2.2.9. Here, the top layer of the face is assumed to be in the x-y plane and the bottom layer is assumed to be in the plane zed In the scheme of Fig 2.2.8, the formex variables ET, EB and EW represent the top layer elements, the bottom layer elements and the web elements of the face in Fig 2.2.9, respectively. The only part of the scheme of Fig 2.2.8 that needs further explanation relates to the formulation for the elements that link the’ bottom layers of the neighbouring faces of the pyramid. One of these ‘link elements’ is denoted by ij in Fig 2.2.9. The position of node i relative to the reference system of Fig 2.2.9 is given by the signet G= 102-1] JE Ri is a signet representing the final position of node i in the pyramid, then Ri is obtained from Qi through the following transformations: sealing by bi(L/(2*m),S/(3*m),D), corresponding to step 2 of the procedure described in Fig 2.2.2 (the function bi(L/(2*m),S/3*m),D) is a basitrfect retronorm, as described in Section 1.7 of Ref I and Section 2.4.11 of the Appendix), © translation by tran(2,C-S), corresponding to step 3 of the procedure described in Fig 2.2.2 and © rotation by verat(C,0,-B), corresponding to step 4 of the procedure, ‘Therefore, Ri =verat(C,0,-B) | ‘tran(2,C-S) | bYL/(2*m),S/(3*m),D) | Qi Also, if Rj is a signet representing the final position of node j in the pyramid, then Rj is obtained as Rj = verad(0,0,360/n) | Ri The link element ij in its final position may be represented by K1=tig| @i#R) ‘The term ‘tig’ is the imprint of the ‘tignum function’ and the argument @i#R)is equal to the ingot * {Giyizi), iii} where xi, yi, zi, xj, yj and zj are the coordinates of nodes i and j in their final positions in the pyramid relative to the x-y-z coordinate system. The role of the tigaum function is to create a cantle from the signets of an ingot, see Section 2.A.16 of the Appendix. Therefore, the construct tig | (RI# RI) is equivalent to the cantle [xiyizis xiyi.2i] This cantle represents the link element jj in its final position in the pyramidal structure of Fig 2.2.7, Now, consider all the link elements in Fig 2.2.9. These. elements in their final positions in the pyramidal structure of Fig 2.2.7 are given by K2 = rinax(0,0.H, -L/Q*m),C/m,(oa-1)*T/m, m) | KL The construct stinax(0,0.H, -L/(2*m),C/m,(m-1)*H/m, m) is a ‘riiax function’ which is a generalisation of the rindle “function. A rinax function effects @ translational replication in the direction given by @ ‘diroction vector’. The direction vector is specified by the coordinates of its end points, as shown in Fig 2.2.10. Also, the amount of translation at each step of replication is given by the ‘length’ of the direction vector. Further information about the rinex function is given in Section 2.4.5 of the Appendix. rinax(X1,Y1,Z1, X2,Y2,Z2, m) number of replications coordinates of the end point of the direction vector ‘coordinates of the starting point of the direction vector abbreviation for rindle proviax Fig 2.2.10 Rinex function In the case of the example under consideration, the soordinates of the starting point of the direction vector are’ specified as 00H and the co-ordinates of the end point of the direction vector are specified as =L/(2*m),Clmn,(m—1)*Hra Formex Configuration Processing IT This direction vector will be coincident withthe final position of the top layer element indicated by ab in Fig229. Fig2.2.11 A sectorate double layer grid generated by the scheme of Fig 2.2.8 (°) Double layer grid of Fig 2.2.11 (*) L=20; (*) length of each side (*) ; (*) distance between two layers (*) (*) frequency of top layer (*) (*) number of sides (*) S=I/(2*tan|(180/n)); ET=genid(1,m,2,3,-1,: ADL Kio. 0, (0.0.0; 1,3, EB=genid(1,m-1,2,3,-1, [0,2,-4; 1,5,-1], [-1,5,- EW=sgenid(1.m,2,3,-1,1)[{10,0,0; [-1,3,0; 0,2,-1], [1,3,0; 0,2,-1)}: E=bt(L/(2*m), $/(3*m),D)|{ET#EB#EW); Qi=[0,2,-2]; U2), SI8°20)D}1 Ks jerad(0,0,360/n)|Ri; ig| (RAR) -rinad(0,0, -I/(2"m),S/m, m)|K1; G=pex|sosad{(0,0,n,980/n) | (E#K2); use &,vm{2); clear; draw G; <><><> Fig 2.2.12 A generic scheme for the sectorate double layer grid of Fig 2.2.11 The scheme of Fig 2.2.8 may be used to generate a variety of double layer pyramids. It can also be used to create ‘sectorate double layer grids’, These are ‘degenerate’ cases of double layer pyramids obtained International Journal of Space Structures Vol. 16 No, 1 2001Hoshyar Nooshin and Peter Disney by letting the height of the pyramid equal to zero. An example of a sectorate double layer grid is shown in Fig 2.2.11. This is generated by the scheme of Fig 2.2.8 using the parameter values L=20, H=0, D=2.15, m=6 and n=8 A simplified version of the scheme of Fig 2.28 for the generation of sectorate double layer grids is shown in Fig 2.2.12, The simplification results from the fact that H=0, S=C and steps 3 and 4 of the procedure of Fig 2.2.2 are not required for a seotorate double layer grid. Also, note that the rinax function of the scheme of Fig 2.2.8 is replaced by the simpler ‘rinad function’ in the scheme of Fig 2.2.12, see Section 2.A.5 of the Appendix. 2.3 TOWERS ‘The formulation of a number of lattice tower configurations is discussed in this section, As the first case, consider the tower a perspective view of which is shown in Fig 2.3.1. The tower consists of 144 elements that are connocted together at 54 nodes, The nodes lie on a cylindrical surface that is referred to as the ‘circumoylinder’ of the tower. The height of the tower is H=25 unit length and the radius of the cross-section of the circumeylinder is ReS unit length. : Fig 2.3.1 A cylindrical tower ‘The most convenient reference system for the formulation of the compret of the configuration of the tower of Fig 2.3.1 is a cylindrical normat, A cylindrical normat that suits the tower of Fig 2.3.1 is shown in Fig 23.2 (Cylindrical normats are discussed in Section 1.8 of Ref 1). ‘The three normat directions Ul, U2 and U3 in Fig 2.3.2 are shown together with the 1-s-z cylindrical International Journal of Space Structures Vol. 16 No.1 2001 coordinate system as well as the x-y-z global Cartesian coordinate system. Also shown in Fig 2.3.2 is a rhombic unit of the tower. This is the part of the tower that is indicated by the letters A, B, C and D in Fig 2.3.1. Ashombic unit U3) of the tower 2 \e 4 U1 (7) 1 2 > 2 Fig2.3.2 Cylindrical normat for the formulation of the tower of Fig 2.3.1 A formex F representing the compret of the configuration of the tower of Fig 2.3.1 relative to the nommat of Fig 2.3.2 may be written as F =rinit(6,4,2,2) | Camit(0,1) j[1,0,05 1,-1,1]}¢ {11,15 11,1), 01.0,2; 1,229) Ti this formulation ¢ Femnit(,1) | (1,0,05 1,-1.1] represents elements AB, AC, BD and CD and @ {O-1b 11,1), (1,0,2; 1,2,2)) represents elements BC and DE. ‘The constructs rinit(6,4,2,2) and. Tamit(0,1) are rinit and lamit functions, see Section 1.8 of Ref 1 and Section 2.4.4 of the Appendix. The configuration of the tower of Fig 2.3:1 relative to the global x-y-2 coordinate system can be obtained as T= be(5,360/12,25/8) | F ‘The construct bo(5,360/12,25/8) is a basicylindrical retronorm that transforms the Ul~ U2-U3 nomnat coordinates into the equivalent x-y-z global coordinates, as discussed in Section 1.8 of Ref Land Section 2.A.11 of the Appendix.In the above basicytindrical retronorm. © the first canonic parameter, that is, 5, specifies the length corresponding to every division in the Ul direction of the normat (this is the radius of the circumeylinder in the present example), © the second canonic parameter, that is, 360/12, specifies the angle in degrees that corresponds to every division in the U2 direction of the normat and © the third canonic parameter, that is, 25/8, specifies the length corresponding to every ivision in the U3 direction of the normat, The reason for giving the second canonic parameter as 360/12, rather than 30, is to clarify the logic behind the specification. Namely, there are 12 subdivisions along the U2 direction (2 per rhombic, unit) and the angle per subdivision must be such that these 12 subdivisions cover 360° (so that the rings of the tower close). Also, the third canonio parameter is given as 25/8 because there are 8 subdivisions along the U3 direction (2 per rhombic unit) and these 8 subdivisions should correspond to the full height of the tower which is 25'mnit length. ‘A. generic scheme for the creation of the configuration of the tower of Fig 2.3.1 is shown in the editory display of Fig 2.3.3. In this scheme, the formulations for the formex variables F and'T are the parametric versions of the formulations given above. ‘The parameters in the scheme are: © H, denoting the height of the tower, eR, denoting the radius of the circumeylinder of the tower, © m, denoting the number of sides of a ring of elements in the tower and © 1, denoting the muinber of rhombic units in the vertical direction, ‘The statement following the formulations of the formex variables F and T in the scheme of Fig 2.3.3 is the use statement ruse &,vm(2),vt(2),vh(14*R,14°R,2, 0,0,0, 0,0,1) ‘The role of this use statement is to set the viewing particulars for the perspective view of the tower in Fig 2.3.1, see Sections 1.5.2, 1.7.1 and 1.7.2 of Ref 1 The scheme of Fig 2.3.3 may be used to generate a variety of cylindrical tower configurations by choosing different values for the parameters H, R, m and n, Three such examples are shown in Fig 2.3.4, Three further examples of cylindrical tower configurations are shown in Fig 2.3.5. These may be obiained by slight modifications of the formulation Formex Configuration Processing II of the formnex variable F in the scheme of Fig 2.3.3. ‘These modifications are left for the reader to carry out as exercise. (*) Tower of Fig 2.3.1 (*) H=25; (*) height of tower (*) () radius of circumeylinder (*) {*) number of sides of a ring (*) (*) number of rhombic units in the vertical direction (*) Ferinit(m,n,2,2)] lamit(0,1)]{1,0,0; 1,-1,1]# {(1e1, A], (1,0,25 1,2,2]}); ‘Tebe(R,360/(2*m),2/(2*n)) | use & vin(2),vt(2),vh(14*R,14*R,2, 0,0,0, 0,0, clear; drew Fig 2.3.3 A gonoric scheme for the creation of i configuration of the tower of Fig 2.3.1 O Fig 2.3.4 Examples of cylindrical towers generateu by the scheme of Fig 2.3.3 (in all the three examples H=25) Fig2.3.5 further examples of cylindrical tower. International Journal of Space Structures Vol. 16 No. 1 2001Hoskyar Nooskin and Peter Disney Now, consider the tower shown in Fig 2.3.6, This is a ‘tapered’ tower with the radius of its cross-section reducing gradually from the base radius of ReS unit length to the top radius of R1=2,2 unit length. Since the tower has seven ‘levels’, the reduction in the radius of the tower per level is t=(-RIV7 =0.4 unit length ‘The height of the tower is H=25 unit length, 2R1, Fig 2.3.6 A tapered tower ‘The nodal points of the tower of Fig 2.3.6 lie on a ‘conical’ surface a vertical section of which is shown on the left of the tower in Fig 2.3.6. This surface is referred to as the ‘circumsurface’ of the tower. Actually, the term ‘circumsurface” is a general term ‘that can be used to refer to a surface that contains all ‘the nodal points of a configuration, For example, one may sey that the circumsurface of the tower of Fig 2.3.1 is a cylinder or, in relation to a dome, one may. say that the dome has a- spherical (elliptical, conical...) circumsurface. ‘The formulation’of the configuration of the tapered tower of Fig 2.3.6 may be carried out using the normat of Fig 2.3.2: In doing this, the tower is imagined to be positioned such that the base of the tower is in the x-y plane with the cenive of the base being at the origin and with node A of the base being ‘onthe x axis, One may then write P=rinQ,6,2) | lib =0, 6) | {Roi i; RAG) iL], iis RAGH) LT], PRAGH+) I-41; RAG) T= be(1,360/12,25/7) |F Jn the above formulation, the formex variable F represents the configuration of the tower of Fig 2,3.6 relative to the cylindrical normat of Fig 2.3.2 and T represents the configuration of the tower relative to the global x-y-z coordinate system, International Journal of Space Structures Vol. 16 No.1. 2001 The above formulation involves a formex fimetion that had not been encountered so far in the discourse, namely, libG=0, 6) This is a ‘libra function’ with the term ‘i? being referred to as the ‘libra index’ or ‘libra variable’, Fig 23.7. lib( 0,6) L. final value of libra index initial value of libra index bra index (libra variable) abbreviation for libra Fig 2.3.7 Libra function ‘The libra index assumes a sequence of integer values starting from the ‘initial’ value of O through to the ‘final’ value of 6, in steps of 1. To explain the effect of the libre fiction, let © Go denote the value of the argument of the above libra function for i+0, that is, Go= ((R,0,0; R4-1,1], [R,0,0; Rt 1,1), R+-LE R411), * Gi denote the value of the argument of the function for i=1, that is, Gi (R411; R-2t,0,2], [R+t1,1; R-2,2,2), [R-24,0,2; R-24,2.2) © and soon, ‘The libra function will then produce the formex Go# Gt # G2# G3 # G4 # Gs # Go ‘The formex Go represents the elements AB, AC and BC of the tower in Fig 23.6, the formex Gi Tepresents the elements CD, CE and DE of the tower and so on. The libra designator libG=0, 6] {Ri RAG), Ri, i+1,itl), HEH PHL} Tepresents the ‘spiral’ of elements that is shown on the right of the tower in Fig 2.3.6. ‘The ‘looping mechanism’ of the libra function is similar to that of the ‘sigma operator’ in scalar algebra. However, the’ libra famction is concemed . with composition of fonnices whereas the sigma operator is concerned with surnmation of numbers, ‘The libra function is a valuable concept and provides a versatile mechanism for déaling with .manyproblems in formex configuration processing. In the case of the’ present example of the tapered tower, none of the previously introduced replicating functions could have dealt with the problem in a convenient mamer. A more detailed description of the libra function is given in Section 2.A.12 of the Appendix. A generic scheme which is based on the above given formulation for the tapered tower of Fig 2.3.6 is shown in the editory display of Fig 2.3.8. Also, three farther examples of tapered towers are shown in Fig 2.3.9, These towers are generated using the scheme of Fig 2.3.8 with different choices of parameters as indicated in Fig 2.3.9, Formex Configuration Processing II circumsurface is surface of revolution with a parabolic generator, 26 shown in Fig 2.3.10. The ‘basic dimensions of the tower are as follows: The overall height is H=25 unit length, The height at the ‘neck’ of the tower is unit length (The ‘neck’ of the tower is thé position of the ‘smallest cross-section of the circumsurface of the tower). * The radius of the cross-section of the circumsurface at the base of the tower is R=10 ‘unit length. * The radius of the cross-section of the circumsurface at the neck of the tower is RI=S unit length. (*) Tapered tower of Fig 2.3.6 (*) H=25; (*) height of tower (*) R=5; (+) radius of the base of tower (*) ‘Ri=2.2; (*) radius at the top of tower (*) (*) number of sides of the base (*) (1) number of levels (") AI) LL4a), (+2)i+1 i441), IRA'G+1), 1,140; RE GH Ii+4i41)); T=bo(1,360/(2*m),H/n) | sé &,vm{2),vt(2);vh{10°R,20°R,2, 0,0,0, 0,0,4); clear; draw T; <><><> Fig 2.3.8 A generic scheme for the tapered tower of Fig 2.3.6 R=8,Ri=4.4, m=10, a: Re4, R115, m=6, n=5 Fig 2.3.9 Examples of tapered towers generated by the scheme of Fig 2.3.8 fin all the three examples As the next example, consider the tower shown in Fig 2.3.10. This is a ‘doubly curved’ tower whose 10 Fig 2.3.10 A parabolic tower ‘The equation of the generator of the circumsurface ig obtained from the general parabolic equation x=ad+bte with the following three conditions: © when 2-0 then x-R, © when z-HI then x=R1 and ® when z-HI then the derivative of x with respect to zis equal to zero, that is, (@vdz) = 22 +b=0, From these conditions, the coefficients a, b and ¢ are found to be a=(RRIJHV, b=-2aH1 and c=R ) ‘The configuration of the parabolic tower of Fig 2.3.10 may be formulated in terms of the normat of Fig 2.32 using an approach identical to that explained for the tapered tower of Fig 23.6. A generic scheme which is based on this approach is shown in the editory display of Fig 2.3.11, The only new point here is that the U1 coordinates in the argument of the libra function are given as a(diy’ +bdi+e International Journal of Space Structures Vol. 16 No. 1 2001Haskyar Nooshin and Peter Disney or a(dG+l))? + baG+1) +0 where d=Hn and di and d¢+1) represent z coordinates. Therefore, the expression a(di’ +bdi+e or (d+)? + bd) + represents a +bzte and this specifies a coordinate of a node of the tower in the radial ditection of the cylindrical nommat of Fig 2.3.2. poe ec (*) Parabolic tower of Fig 2.3.20 (*) (*) height of tower (*) 18; (*) height at the neck of tower (*) (2) radius of the base of tower (*) (¢) radius at the neck of tower (*) (+) number of sides of the base (*) (1) number of levels (+) a=(RRIVHI%2; b=-2"a"Hi; o=R; F=tin(2m,2) |libi=0,n-1)| {le (@*i) *24b*d*it+eii; a*(a°G+1))*2+b'E 4a) 4oh4i+1), [et(@"i) “24d*d*it Gi; ar(d"(itt))*24b*d* ita) toit1i41], (e*(d*G+1)) *24b'aG+a)toen iti; a*(a'(i41))*24b*d" (41) 4oi+Li41)); ‘T=be(1,380/(2*m),4)|F, ‘use &.vm{2),vt(2),vh(10*R,10*R,2, 0,0,0, 0.0.1); clear; draw Ty <><><> Fig2.9.11 A generic scheme for the parabolic tower of Fig 2.3.10 Three further examples of parabolic towers are shown in Fig 23.12. These are generated by the scheme of Fig 2.3.11 with parameter values H=30, R=8, R1=3, m=10 and a=8 for all the three cases and with varying values for the parameter Hl. In the case of the example on the left, the neck of the tower is at the height of HI=20 unit length, In the case of the example at the middle, the neck of the tower is at the height of HI=30 unit Tength. That is, the neck is at the very top of the tower, In the caso of the example on the right, the aeck is at the height of Hi=50 unit length which is higher than the top of the tower. It is permissible for International Journal of Space Structures Vol. 16 No. I 2001 the position of the neck to be ‘outside’ the tower since the neck is defined with respect to” the ciroumsurface of the tower and the circumsurface extends beyond the actual body of the tower. Fig 2.3.12 Examples of parabolic towers generated by the scheme of Fig 2.3.11 (in all the three examples H=30, R=8, R1=3, m=10, n=8) Plan view ‘Fig 2.3.13 A saddle shaped barrel vault Perspective view Side view The approach used in this section for the generation of tapered and doubly curved towers may also be employed to generate tapered (conical) and doubly curved barrel vaults. For instance, the doubly curved bane! vault of Fig 2.3.13 is generated using the scheme shown in the editory display of Fig 2.3.14. ‘The barrel vault of Fig 2.3.13 bas a saddle shaped body. The particulars of the barrel vault are as follows: * It consists of 430 beam elements that are connected together at 231 nodes. + Its sweep angle is 70°, see Section 1.8 of Ref I. * Its Jength is 40 unit length, i© Its circumsurface is a surface of revolution with fa parabolic generator where the radius of the cross-section of the circumsurface at the ends is 12 unit length and where the minimum radius is at the middle and is equal to 8 unit length. + Tho ‘froquency’ (that is, the mmber of elements along an edge) in the circumferential direction is equal to 10. ©: The frequency in the longitudinal direction is equal t0 20. (*) Doubly curved barrel vault of Fig 2.3.13 (*) Pe! ; (*) sweep angle of barrel vault (*) (+) length of barrel vault (*) (+) radius at the ends (*) (*) redius at the middle (*) *) frequency in U2 direction (*} 2=20; (*) frequency in U3 direction (*) *(RRI/L~2; b=-a"l; c=R; d=La; rin(2,m,1) 1ib(i=0.n)] [ar(a**24b*Gti+0,0,5; ata) 24 b*di+c,1i)F rin(@,m+1,1)|b@=0,n-1)| far(d*i) *24b*d*i+c,0,5 a*{a*(i43)) *24b"d*(41)+0,0,i+1); B=verad(0,0,90-P) [bo(1,2*P/m,d)|F ‘use & vm(2),vt(2),vh(0,3*L,-2*L, 0.0, clear draw B; e>e>e> 1D; Fig 2.3.14 A generic scheme for the doubly curved barrel vault of Fig 2,3.13 Perspective view Plan view Ann Side view Fig 2.9.15 A convex barrel vault generated by the scheme of Fig 2.3.14 (with Ri=16) ‘The scheme of Fig 2.3.14 works in a manner similar to the scheme of Fig 2.3.11 and it is written to suit the barrel vault of Fig 2.3.13. DR Formex Configuration Processing IT In relation to the scheme of Fig 2.3.14, the following points are worth noting: © The rotational function verad(0,0,90-P) appearing before the basicylindrical retronorm in the scheme has the effect of making the y axis vertical, see Section 1.8 of Ref 1. © Theuse statement ‘use &,vm(2),vi(2),vb(0,3*L,-2"L, 0,0,L, 0,1,L) in the scheme is for the perspective view of the barrel vault in Fig 2.3.13, see Section 1.7.1 of Ref 1. © The plan view of the barrel vault can be obtained using the use statement use &vm(2),vi(1),vh(,1,0, 0,0,0, 0,0,1) The plan view may also be obiained, with reasonable accuracy, by simply enlarging the y coordinate of the view point in the original use ~ statement, that is, use &,ym(2),vi(2), ‘yh(0,10000*L,-2*L, 0,0,L, 0,1,L) © The side view of the barrel vault of Fig 2.3.13 can be obtained using the use statement, use &vm(2),vi(1),vh(1,0,0, 0,0,0, 0,1,0) This side view may also be obtained, with reasonable accuracy, by simply enlarging the x coordinate of the view point in the original use statement, that is, use & vm(2),t(2), vh(10000*L,3*L,-2*L, 0,0,L, 0,1,L) The scheme of Fig 2.3.14 may be used to generate a variety of doubly curved barrel vaults. Included in these are the ‘convex’ barre! vaults an example of” which is shown in Fig 2.3.15, \ In this case, the radius of the circumsurface of the barrel vault at the middle is RI=16 unit length which is larger than the radius at the ends, namely, R=12 ‘unit length. 2.4 FOLDABLE SYSTEMS A number of foldable configurations are considered in this section, These relate to the type of ‘foldable structures? (deployable structure) that consist of seissors-like units’, To begin with, consider an example of a very simple foldable system, namely, the ‘lazy-tongs’ configuration shown in Fig 2.4.12. The configuration consists of four scissors-like units. ‘The comers of one of these units are indicated by letters i, j, k and 1. The elements ik and jl arc ‘pivoted’ together at the middle such that the uni can be opened and closed like a pair of scissors, A International Journal of Space Structures Vol. 16 No. 1 2001Hoskyar Nooshin and Peter Disney unit of this kind is referred to as a ‘“duplet” and each one of the parts ik or jl is referred to as a ‘uniplet’. The “duplet ijkl is ‘pinned (hinged) to its neighbouring duplets at points i,j k and I, so that the whole configuration of Fig 2.4.1a can be opened and closed like @ pair of scissors. The extent of the folding of the configuration of Fig 2.4.12 may be specified by a single parameter, This parameter is chosen to be the angle t, as shown in the figure. ‘Three folded states of the lazy-tongs configuration are shown in Fig 2.4.1. These correspond to =90", 4=120° and 160°, as shown in the figure. The angle t is referred to as the ‘control angle’. j k t=60" Fig 2.4.1 Three folded states of lazy-tongs configuration To work out the geometry of a folding duplet, consider the arrangement shown in Fig 2.4.2. The duplet shown by full lites in this figure has the same horizontal and vertical dimensions D, The control angle for this duplet is equal.to 90°, For any other value of the control angle the horizontal dimension H of the duplet will be different from its vertical dimension V. This is illustrated for a folded state of ‘the duplet shown by dotted lines in Fig 2.4.2. ey PE CeeecoeeeeHe Fig 2.4.2 A folding duplet International Journal of Space Structures Vol. 16 No. 1 2001 The horizontal and vertical dimensions of the dotted duplet are found to be: H= 12D sin(t/2) and V = 2D cos(t/2) ‘The two-directional extension of the idea of lazy- tongs is a ‘foldable (double layer) grid” an example of which is shown in Fig 2. Fig 2.4.3 A foldable grid The grid of Fig 2.4.3 consists 6f 71 identical duplets that are pinned together creating an assembly of duplets that can be folded into a ‘bundle’. The dimension D (as shown in Fig 2.4.2) for the duplets in the grid of Fig 2.4.3 is equal to 0.8 unit Jeng‘h and the control angle of the duplets is equal to 100°. In practice, a foldable grid of the kind shown in Fig 2.4.3 is stabilised (and stiffened) by adding a number of elements and/or by constraining the grid at the support points when the grid js in the required final unfolded state, A formex formulation for the configuration of the foldable grid of Fig 2.4.3 may be written as follows: E=rinid(6,6,1,1) | {10,0,0; 1,0,1, [0,0,1; 1,0.0)}¢ rinid(7,5,1,1) | {[0,0,0; 0,1,1], [0,0,1; 0,1,0}} F=btGLH,V) |B In this formulation: © the formex £{[0,0,0; 1,0,1], (0,0,1; 1,0,0), represents the duplet in the left comer of the g7id in the U1 direction (shown by thick lines), the formex {{0,0,0; 0,1,1], [0,0,1; 0,1,0]} represents the duplet in the left comer of the grid in the U2 direction (shown by thick lines), © E represents the configuration of the grid relative to the given U1-U2-U3 normat and © F represents the configuration of the grid relative to the global x-y-z coordinate system, where H and V are as described for the duplet of Fig 24.2. B(*) Foldable grid of Fig 2.4.3 (*) t=100; (*) control angle (*) D=08; (*) dimensions ofa duplet for t=00 (*) m=6; | (*) frequency in the x direction (1) (*) frequency in the y direction (*) H=sqrt|2*D*sin|(v/2); ‘Vesqri|2*D*c0s (1/2); Ferinid(m.n+1,1,1)|{[0,0,0; 1,0,V], [0,0,¥; H,0,0}} #rinid(m+1,0,2,1)] {{0,0,0; 0,H,V}, [0,0,V5 0,H,0]}; ‘use &,vm/(2),vt(2),vh(2*m*H,-n"H,15*H, m*H.n*H,0, m*H,0"H,1); clear; draw F <><><> ‘ig 2.4.4 A generic scheme for the foldable grid of Fig 2.4.3 A generic scheme for the foldable grid of Fig 2.4.3 is shown in the editory display of Fig 2.4.4. The formex formulation in this scheme is based on the formulation given above with two differences, Firstly, the duplet frequencies in the x and y directions are given by parameters m and a. Secondly, to show an alternative approach in dealing with ihe problem, the formulation has been carried out directly in terms of the global x-y-z coordinate system without the involvement of an intermediate Fig 2.4.5 Some folded states of the grid of Fig 2.4.3 generated by the scheme of Fig 2.4.4 The scheme of Fig 2.4.4 may be used to generate foldable grids of the kind shown in Fig 2.4.3 with “4 Fores Configuration Processing II different values for the frequencies m and n and the size of the duplets represented by D. Also, the scheme can generate the configurations of different folded states of a grid by changing the value of the control angle t. For example, four different folded states of the grid of Fig 2.4.3 are shown in Fig 2.4.5. ‘Those are genetated by the scheme of Fig 2.4.4 with the control angles 120°, t=90°, 60° and 1-30" Now, consider the foldable barrel vault shown in Fig 246. The barrel vault has 42 duplets in the circumferential direction. These create 6 circular ‘arches’. The arches are connected together by 40 duplets in the longitudinal direction, The duplets in the circumferential direction create arches because they have a “trapezial’ shape, as shown in Fig 2.4.7. Ina ‘traperial duplet’ the length Lt of the upper part-~ of the uniplets is different from the length L2 of the lower parts. In the present example, Li-0.8 unit Jength, L2=0.7 unit length and the control angle t is equal to 140°. Details of one of the arches of the barrel vault of Fig 2.4.6 are shown in Fig 2.4.8, The duplets in the longitudinal direction of the barrel vault of Fig 2.4.6 are ‘rectangular duplets’, In this case the uniplets are pivoted together at the middle, as shown in Fig 2.4.7 (The duplets in the lazy-iongs of Fig 2.4.1 and the grid of Fig 2.4.3 are rectangular duplets) Rectangular duplet Fig 2.4.6 A foldable barrel vault In order to work out the geometry of the foldable barrel vault of Fig 2.4.6, it is necessary to obtain the radius R of the top citcumcylinder, the depth D of the barrel vault, the central angle C of a duplet in the circumferential: direction and the distance Di ‘between the arches in terms of Li, L2 and the control angle t. International Journal of Space Structures Vol. 16 No.1 2001Eoshyar Nooshin and Peter Disney Cylindrical coordinate directions W2 Di Rectangular Guplet (for longitudinal direction) ‘Trepezial duplet (2) or etreumferential rection) Fig 2.4.7 ‘Trapezial and rectangular duplets Span=2R sin P “Top circumeylinder Bottom circumcylinder ~~ Radius of top ( eee Sweep angle R Fig 2.4.8 Details of one of the arches of the barrel vault of Fig 2.4.6 oO For any triangle with side lengths a, b, cand the corresponding opposite angles a, 9, y, the following rales apply: @ _ a 2 SSN a eb Sina “sinp siny a =b'+c'—2be cosa Fig 2.4.9 Two general rules for a triangle From triangle 124 in Fig 2.4.7 and the second rale in Fig 2.4.9, D= (Li? +12 ~ 20112 c08 tx)? Also, from the rectangular duplet in Fig 2.4.7, Di=@?-p)? International Journal of Space Structures Vol. 15 No. 1 2001 From triangle 124 in Fig 2.4.7 and the first rule in A= arcsin(L2 sin ti/D) Again, from triangle 124 in Fig 2.4.7, Bi = 180°-A-ti = 180°-A~ (1801) =-A +t and B= 180°-B1= 180°+ At From triangle 135 in Fig 2.4.7 and the first rule in Fig 2.4.9, Risin B= (R-Dy/sin A or R= (sin BY(sin B-sin A) Also, Lsin C= R/sin B or C= arcsin(L sin BIR) A formex formulation for the barrel vault of Fig 2.4.6 may now be written as Bi =rinit(7,6,C,D1) | {(R-DL0,0;2,C,0], [R,0.0; R-D,C,0]} E2=rinit(8,5,C,D1) | {(R-D,0,0; R,0,D1}, [R,0,0; R-D,0,D1}} F=bo(1,1,1) | 61 # Ez) P=7Ci2 BY = verad(0,0,90-P) | F Tn the above formalation © El represents the duplets in the circumferential direction relative to the -rs-z cylindrical coordinate system, © Ez represents the duplets in the longitudinal direction relative to the 1-s-z coordinate systom, © F represents the whole configuration relative to the global x-y-z coordinate system, where, be(1,1,1) is a basioylindrical retronorm, as described in Section 1.8 of Ref 1 and Section 2.A.11 of the Appendix, © P is the ‘sweep angle’ of the barrel vault, as described in Section 1.8 of Ref 1 and © BV is a rotation of F representing the barrel vault, of Fig 2.4.6 with the y axis in the vertical Position, as discussed in Section 1.8 of Ref 1. A generic Formian scheme which is based on the above formulation is shown in the editory display of Fig 2.4.10. This scheme is used to generate three folded states of the barrel vault of Fig 2.4.6. These are shown in Fig 24.11 together with the corresponding values of the control angle. 15(¢) Foldable barrel vault of Fig 2.4.8 (*) (*) control angle (*) (¢) length of upper part of uniplet (*) (+) longth of lower part of uatplet (+) () circumferential frequency (*) (*) longitudinal frequency (*) +L2; t1=160-; D=sqpt|(L1*2+L2% 2-2*L1*L2*cos|t1); Di=sqrt|(L*2-D%2) asin|(L2*sin|t1/D)}; 0+ At; in |B/{sin|B-sin|A); in| (L*sin| BR); Eszinit(m.a+1,C,D1}| {(R-D,0,0; R.C.0}, [R,0,0; R-D,Cop# rinit(mn+4,0,C,.D1) | | {(R-D,0,0; R,0,D3}, [R,0,0; R-D,0,D1}}; Febe(1,1,1)[E; P=m*C/2; BY=verad(0,0,90-P) |, | ‘use &vm(2)}.vt(2).vh(6,8°R,12*R, 0,0.R, 0,1,R) ‘ear; dre Side=n*D1; Swoop=P; Spi Riso=R*(1-cos|P); ive Radius, Depth, Side, Sweep Span Ris <><><> Fig 2.4.10 A ganeric scheme for the foldable barrel vault of Fig 2.4.8 ‘A-usefil feature of the scheme of Fig 2.4.10 is that it ‘effets the display of information in a give box (see Section 1.3.4 of Ref 1). An example of a give box for the barrel vault of Fig 2.46 is shown in Fig 2.4.12. This provides a convenient tool for the design of a foldable system since the effects of changes in the parameters t, Li, L2, m and n can be easily observed and the parameters can be adjusted to suit the design requirements. NA SRG aA Fig 2.4.11 Three folded states of the barrel vault of Fig 2.4.6 generated by the scheme of Fig 2.4.10 16 Formex Configuration Processing II 5=4.172521E+000 pth=5.215652E-001 Side=7.082016E +000 ‘Sweep=7.265675E+001 Span=7,965644E+000 Rise=2.9287116+000 Echo to itony Fig 2.4.12 Give box displaying the values of the radius of the top circumcylinder, depth, side Jength, sweep angle, span and rise of the barrel vault of Fig 2.4.6 In the above. example, the initial parameters are ‘chosen to be Lt, L2, t,m and n. However, if require’ the problem may be formulated in terms of <><> Fig 2.5.8 A generic scheme for the configuration of the diamatic dome of Fig 2.5.1 With the choice of R= 30, m= 6,n=6 and A=36%. ) the scheme of Fig 2.5.8 generates the diamatic dome” of Fig 2.5.1. However, many other diamatic dome configurations may be generated through the scheme of Fig 2.5.8 by simply changing the values of the parameters. Four such examples are shown in Fig 25.9. These are generated by the scheme of Fig 2.5.8 with different values for the parameters m and 1, as indicated in Fig 2.5.9. SIRES ESSN RSSEY Fig 2.5.9 Examples of diamatic domes generated by the scheme of Fig 2.5.8 Jncideatally, diamatic pattems of the type shown in Figs 2.5.1 and 2.5.9 may also be employed for finite element meshes. In this case, the component pats of the configurations will be ‘tile’ elements with three ‘or more nodal points rather than ‘linear’ elements with two nodes at the ends. Such finite elemer meshes are used for the analysis of spherical shells, International Journal of Space Structures Vol.16 No. 1 2001) Hoshyar Nooshin and Peter Disney re (*) Diamatic finite element mesh (+) Rab; (*) radius of circumsphere (+) (2) frequency {*) (*) number of sectors (*) (*)- sweep angle (*) E=genit(2,m,1,1,0,1)|[1,0,0; 40,2; 1,1,2}¢ genit{t.m-1,1,1,0,2)|[2,0,25 1,25 1.4,1)5 '=bd(R,360/n,A/m) |B; G=rosad(0,0.n,360/n)|, use &vm{2),0t(2),0(3,40), vh(1.732"RR,3°R, 0,0,0, 0,0,1); clear; draw G; — Fig 2.5.10 A generic scheme for creation of diamatic finite element meshes A Formien scheme for the generation of diamatic finite element meshes is shown in the editory display of Fig 2.5.10. The schemes of Figs 2.5.8 and 2.5.10 produce ‘similar looking’ configurations. However, the configuration produced by the scheme of Fig 2.5.8 will consist of two-noded linear elements whereas the configuration produced by the scheme of Fig 2.5.10 will consist of triangular elements with three comer nodes, The main difference between the schemes of Figs 2.5.8 and 2.5.10 is in the formulation of formex variable E representing the configuration of the first sector relative to the nonmat of Fig 2.5.4. Another difference between the two schemes is that the formulation for the formex variable G in the scheme of Fig 2.5.10 does not involve the ‘pexurn function’. This is due to the fact that in the finite element version of the configuration the neighbouring sectors do not have any overlapping parts, see Section 1.8 of Ref 1. ‘The schemes of Figs 2.5.8 and 2.5.10 have a further ‘two minor differences. Firstly, the values given for the radii of-circumsphere in the two schemes are different. Secondly, the use statement in the scheme of Fig 2.5.10 has an extra use-item, namely, (3,40) ‘The effect’of this use-item is that the finite-clements will be plotted with an infill colour. The infill colour will be the colour whose code number is 40, sce Section 1.7,2 of Ref 1. Returning to the discussion of ‘lattice’ diamatic domes, it should be mentioned that the examples considered so far (Figs 2.5.1 and 2.5.9) belong to a particular family of diamatic domes that ace referred to as ‘parallel lamella domes’. The distinguishing characteristic of this family of diamatic domes is thet International Journal of Space Structures Vol. 16 No, 1 2001 each sector is fully triangulated with the ‘element lines’ being approximately ‘parallel’ to the edges of the sector. However, there are many ‘other “diamatic pattems? that can be used for domes. In general, a ‘diamatic pattern’ is defined as any pattern that can be obtained 25 a combination of elements whose ‘nodal points’ are on the ‘normat points’ of a diamatic. norinat. For example, consider the dome a perspective view of which is shown in Fig 2.5.11. This is a diamatic dome with a honeycomb patter, ‘Tor the above dome: m=10, n=7 and the sweep angle A=45° Fig 2.5.11 A honeycomb diamatic dome togethar with a generic formulation Included in Fig 2.5.11 is a generic formulation for the dome together with the diamatic normat with respect to which the formulation is carried out, The elements that are shown on the normat correspond to those elements that are shown by thick lines at the top of the first sector of the dome in Fig 2.5.11. To create a scheme for the generation of domes of the form shown in Fig 2.5.11, the formulations for the formex variables E, F and G in the scheme of Fig 2.5.8 should be replaced by those given in Fig 25.11 Using the same style of presentation as in Fig 2.5.11, smother example of a diamatic dome is shown in Fig 2.5.12. A new feature in this example is the use of Giamatic normat coordinates ‘beyond’ the borders of ‘the first sector. 21Exgenit(1,mit,3,3,0,1)| 1(1,2,45 14,2], (1,24; 11.5], 1124s 14.5))# rin(S.m+1,3)|[1,1,25 11,2] Febd(R,360/at,Ai/3m) sad(0,0,n,360/n) |F 74 ‘t For the above dome: Peete m=8,n=6 and the “1 U2 sweep angle A=36° Fig2.5,12 An alternative honeycomb diamatic dome together with a generic formulation To elaborate, the element indicated by ¢ in Fig 2.5.12 crosses the left border of the first sector. One may thea wonder how to specify the coordinates of the ‘outside’ node of element e, since the diamatic nonmiat coordinates in the second direction, as given in Figs 2.5.4 and 2.5.5, are only shown for the first sector. The continuation of the diematic normat coordinates beyond the first sector is in accordance with the arrangement shown in Fig 2.5.13. ‘Therefore, the element e in the example of Fig 2.5.12 may be represented by 01,2; 1-1,2] ut (7) U2 6) y Fig 2.5.18 Diamatic normat coordinates in the 2nd ‘direction beyond the borders of the 1st sector 2 Formex Configuration Processing II Further examples of diamatic domes are shown in ‘Fig 2.5.14, The formulations of these domes are left for the reader to carry out as exercise. Fig 2.5.14 Further examples of diamatic domes ‘Any diamatic dome may be transformed into an ellipsoidal form by scaling along coordinate axes, as discussed in Section 1.9 of Ref 1, For example, the diamatic dome whose plan view is shown in Fig 2.5.15b is obtained by scaling of the dome of Fig 255.1Sain the x dicection by the sealofactorO8, Also, one may create an ‘ovate diaimatic dome’ by using different scale factors for the opposite halves, of a diamatic dome in a manner similar to that discussed in Section 1.9 of Ref 1. For example, the ovate diamatic dome of Fig 2.5.15c is obtained by scaling the bottom half of the dome of Fig 2.5.15a in the y direction by the factor 1.3. Also, the ovate diamatic dome of Fig 2.5.15d is obtained by scaling the top and bottom halves of the dome of Fig 2.5.15a in the y direction by factors 1.1 and 0.85, respectively. Tn an ovate dome, due to different scalings of the ‘opposite halves, the pattern in one half is bound tobe more ‘compact’ than that in the other half, as may be seen from Figs 2.5.15c and 2.5.15d. One way of eliminating (or lessening) this effect is to reduce the number of sectors in the half with the more compact pattem. This is illustrated in Fig 2.5.16. The ovate International Journal of Space Structures Vol. 16 No. 1 2001Hoshyar Nooshin and Peter Disney dome shown in this figure is the same as the ovate dome of Fig 2.5.15d except that the aumber of the sectors in the 2” half (bottom half) is reduced from four fo three. Fig 2.5.16 Examples of ellipsoidal and ovate diematic domes Fig 2.5.16 Exemple of an ovate diamatic dome with ‘unequal numbers of sectors in the opposite halves ‘The dome of Fig 2.5.16 may be obtained using the generic scheme shown in the editory display of Fig 2.5.17. This scheme is based on the diamatic pattern of Fig 2.5.11. However, the scheme may be modified to work with any otter diamatic pattern. To do this, thé only part of the scheme that needs modification is the formulation for the formex variable E that represents the configuration of the first sector relative to a diamatic normat. ‘The manner in which scaling is effected in the scheme of Fig 2.5.17 needs some explanation. To International Journal of Space Structures Vol. 16 No. 1 2001 1.8 and 1.9 of Ref 1, the fect retronorms were used for scaling of barrel vault ‘and dome configurations to create elliptic and ovate forms, However, as explained in Section 1,8 of Ref 1, acting in the ‘capacity of a scaling function is not the primary role of a basibifect or basitrifect retronorm, ‘The fundamental role of a retronorm is to transform nomnat coordinates into global x-y-z coordinates, (*) Ovate diamatic domes (*) 0; {*) initial radius of circumsphere (+) (*) number of rhombic openings along a meridian (*) (1) number of sectors in ast half (*) () number of sectors in 2nd half (*) (2) soale factor for 1st half (*) (*) scale factor for 2nd half (*) (¢) sweep angle (*) B=genit(1,m,3,3,0,1)]{(1,0,0; 11,2), 12,4,2; 1,0,9), [1,1,25 1,3,3]}; Fi=bd(R,180/n1,A/(3*m)) Bs F2=bd(R,180/n2,A/(3*m)) |B; G=dil(2,S1) |rosad(0,0,n1,180/n1) [Fit dil(2,$2} |ref(2,0) |rosad(0,0,n2,180/n2) |P2; clear; use &vm{2); draw ni=a; n2=3; SI=14;, <><><> Fig 2.5.17 A generic scheme for ovate diamatic domes ‘The basic ‘scaling function’ in formex algebra is the ‘dilatation function’, where the term ‘dilatation’ implies increase or decrease in size. The particulars of the dilatation function are shown in Fig 2.5.18, also see Section 2.4.3 of the Appendix. ii factor of dilatation {scale factor) direction of dilatation abbreviation for dilatation Fig 2.5.18 Dilatation function Now, referring to the scheme of Fig 2.5.17, the dilatation functions dil(2,S1) and dil(2,S2) 23are used i the formulation of the formex variable G. The dilatation function dil@2,S1) effects the scaling of the first half (top half) of the dome of Fig 2.5.16 by the factor SI = 1.1 and the dilatation function il(2,82) effects the scaling of the second half (bottom half) of the dome by the factor $2 = 0.85. As the last example in this section, consider the double layer diamatic dome of Fig 2.5.19. The plan view of the dome is shown on the left side of the figure with the top layer elements shown by thick lines and the bottom layer elements es well as the web elements shown by thin lines. Plan views of the top layer elements, web elements and bottom layer elements are also shown separately on the right side of Fig 2.5.19, indicated by T, W and B, respectively. SA 7 wy A bs 2 LAN a iN Ni NI vas Me wet RA AY q IY ps y x ID7 xs isa DZ py bs B \ i Sy = SS ey Ny adi SS Vo eI i “7 LI en DISSE x SY DZ Ko ST] NY eS NS aN iF SBS WN De DAIS “ANTS I eV ZN ey, IB ie ui} e 4 xl ‘I oN NSEZ Zz sa 7] Ss (Ms Wes A } Yor x Ds ISA Wes 4 as ez x Ny B L b & é y AALS \ 5h AY x B) Fig 2.5.19 Plan view of a double leyer diamatic dome with the top layer elements (T), web elements (W) and bottom layer elements (B) also shown separately on the right A Formian scheme for the generation of the double layer diamatic dome of Fig 2.5.19 is shown in the editory display of Fig 2.5.20. The scheme is generic involving the following parameters: © Rt denotes the radius of the top circumsphere of the dome. Rb denotes the radius circumsphere of the dome, m denotes the frequency of the top layer of the dome. denotes the number of sectors of the dome. ‘A denotes the sweep angle of the top layer of the dome. of the bottom 24 Formex Configuration Procesing IE (*) Double layer diamatic dome (*) Rt=36; _ (*) radius of top circumsphere (*) Rb=34.5; (*) radius of bottom circumsphere {*)| m=4; — (*) frequency of top layer (*) (*) number of sectors (*) ” A=36; _(*) swoep angle of top layer (*) ‘TOP=genit(i.an,3,3,0,1)] {[RI,0,0; R1,0,3], [RL0,0; Rt.3,3), [R60,3; RL3,3); WEB=genit(1,m,3,3,0,1)|{{Rb,1,2; Rt,0,0], (Rb, 1,2; RtO,3}, [Rb,4,2; Rt3,3]# genit(1,m-1,3,3,0,1)| {(IRb,2,4; Rt,3,6], ~ IRb,2,4; Rt0,3], [Rb,24; RE3,3))3 BOT =geniit{1,m-1.5,3,0.1) |{[Rb,2,4; Rb,1,2], {Rb,2,4; Rb,1,5), [Rb,2,4; Rb,4,5] }# in(,m,3)| [Rb,1,2; Rb,-1,2]; (4, 360/n,A/(9*m) |(TOPAWEB#BOT); ex [rosad(0,0,n,360/n) | F5 clear; use &,vm{2); draw G; <><><> Fig 2.5.20 A generic scheme for the double layer diamatic dome of Fig 2.5.19 ‘Throughout the present section, the circumradii and sweep angles have been used to control the proportions of the domes, However, this control may also be effected by specifying the: span and ttie rise of a dome, see Section 1.9 of Ref 1. The relationships between the circumradius R, sweep angle A, span S and rise H of a dome are given by (see Fig 1.9.2 of Section 1.9 of Ref 1): R=S/(2sin A) A=2 arctan (2H/8) S=2RsinA O H=(S/2) tan (A/2) In the case of the diamatic dome of Fig 2.5.19, the span and the rise are obtained as: S=2,36xsin 36° = 42.321 unit length = (S/2) tan 18° = 6.875 unit length 2.6 INFORMATION EXPORT The objective of this section is to discuss the export of information from Formian into graphics and structural analysis packages. The idea is introduced in terms of the example of the double layer grid whose plan and elevation together with a perspective view are shown in Fig 2.6.1, In this figure, the top layer elements of the grid are shown by thick lines and the bottom layer elements as well as the web elements are shown by thin lines. The grid covers ‘square area of 28 m by 28 m. International Journal of Space Structures Vol. 16 No. 1 2001Hoshyar Noochin and Peter Disney y 28,00 m_ | * The second loading case consists of equal vertical point loads epplied at all the nodes of the left half ofthe top layer of the grid. * "The third loading case consists of a single point load spplied at the central node of the bottom layer of the grid, 28,00 mm SSNZNZRZ SAINT NVANVANVANYAN LERBERNENZNZY\ ISEEERZSZSAIN\ SZINZINANYA NAY “NANNIES AS DYBESERENZRZM LK NP SPR RRA Depp VANVANY IAN VN Perspective view Fig 2.6.1 A double layer grid ‘The grid is supported at 20 nodal points along the perimeter of the bottom layer. The support positions are indicated by little circles on the plan of the grid in Fig 2.6.1. There are two types of supports, The supports indicated by solid circles are constrained in x, ¥ and z directions and the supports indicated by hollow circles have a single constraint in the z direction, ‘The grid is to be analysed using a commercial structural analysis package for the following three loading cases: «The firt loading case consists of equal vertical point loads applied at all the top layer nodes of ‘the grid, International Journal of Space Structures Yol. 16 No. 1 2001 (*) Double layer grid of Fig 2.6.1 (+) d=1.45; (+) depth of the gird (*) inid(7,8,4,4}| (0,0,d; 4,0,d}# tinid(8,7,4,4)|[0,0,4; 0,4,d); BOT=rinid(6,7,4,4) |12.2,0; 6,2,0]# Hinid(7,6,4;4)|[2,2,0; 2,6,0}3 WEBS=rinid(7,7,4,4) |rosad(2,2)] [0,0.4; 2,2,0); GRID=TOP#BOT#WEB; [2,38,0), [18,2,0]); lux{$1) |rosad(14,24)| rin(4,8,4)|[6,2,0]; Lisrinid(6,6,4,4)]10,0.d); La=rinid(4,8,4,4)]{0,0.d}; L3=[14,14,0); use &,vm2),vt(2),vb(14,-8,20¢d, 14,14, 14,14,1}; clear; ‘draw GRID; <><><> Fig 2.6.2 A scheme for the generation of data for ‘the analysis of the grid of Fig 2.6.1 A Formian scheme for the generation of data for the analysis of the grid of Fig 2.6.1 is shown in the editory display of Fig 2.6.2. In this sschoine, the formex variables TOP, BOT and WEB represent the top layer elements, bottom layer elements and web elements of the grid, respectively. Also, the formex variable GRID sepresents all the elements of the grid. Due to the simplicity of the geometry, the formulations are carried out directly in termis of the x-y-z global coordinate system (that is, without the aid of a separate nommat, compare with the example of Fig 1.7.2, Ref I). The support positions are representéd by formex variables $1 and S2, where, SI represents the supports that have constraints in the x, y and z directions and $2 represents the supports that have a single constraint in the z direction, The loed positions are represented by formex variables Li, L2 and L3, where, L1 represents the Joad positions for the first loading case and L2 and L3 represent the load positions for the second and third loading cases, respectively. 25Every one of the formex variables oreated by the scheme of Fig 2.6.2 may be considered to be the ‘name’ of a ‘file’ that contains the information about a formex. The information contained in such a file me be ‘reformatted” to suit 2 graphics or structural analysis package, Reformat Formex File eal Reformat formes fle: Use format: Close [aria Fig 2.6.3 Reformat box For example, let it be required to reformat the “formex file’ GRID for use in the graphics package CorelDRAW. Clicking ‘Transfer’ on the menu bar of the Formian screen will result in the display of a menu whose first item is ‘Reformat Formex File’. Clicking of this menu item will cause the display of the ‘reformat box’ shown in Fig 2.6.3. The reformat box contains a rectangular area on the left in which all the current formex files are listed (The extension for a formex file name is ‘finx’). The formex file GRID that is to be reformatted should be highlighted by clicking its name, ‘The available reformatting styles are listed in an area ‘on the top right comer of the reformat box. ‘The required format for CorelDRAW is HPGL with the associated file name extension ‘plt’, Highlighting of this item will cause the file name GRID pt to appear in an area at the bottom right comer of the reformat box. This name (which can be alfered if desires) will become the name of the reformatted file-for use in CorelDRAW. 26 Formex Configuration Processing I The clicking of the ‘reformat button’ will now place the reformatted file GRID plt in the indicated folder (that is, projects) in the indicated drive (that is, drive ©). If it is required to place the file in a different folder then, before clicking the reformat button, the symbol {...] in the area below the indicated folder should be clicked. This will cause the list of all the current folders to be displayed and the required folder may be chosen by clicking its name, Also, if required, the indicated drive can be changed using the ‘drive box’ shown in Fig 2.6.3. ‘The next step after refoimatting is to ‘import? the reformatted file into CorelDRAW. This can be done through the ‘import” menu item of the ‘file menu” in CorelDRAW. The imported material will resemble the perspective view of the double layer grid shown in Fig 2.6.1 and may be treated as a CorelDRAW ~ graphic object for any further desired treatment in. ) CorelDRAW. A similar procedure may be followed to reformat the formex file GRIDfinx for use in the draughting package AutoCad. In this case, the formatting style to be highlighted in the ‘format box’ of Fig 2.6.3 is “AutoCad(*.dxf), The result of reformatting will bea file in the DXF format called GRID.dxf. Subsequently, the file GRID.dxf may be.opened in AutoCad and be used as though it was an AutoCad file, To insert a formex plot as a ‘picture’ into a word processing package such as “Microsoft Word’, the metafile format, with the associated file name extension ‘wmf, is the most suitable one. It isto be noted that an HPGL file (that is, a fle wit the extension ‘plt’) reflects the current viewing particulars in the Formian environment at the ‘moment of reformatting and the same applies to a metafile but a DXF file is independent of the viewing particulars, ‘All the formex files crested by the scheme of Fig 2.6.2 may be transformed into DXF files for use in the AutoCad environment. Also, commercial structural analysis packages normally accept DXF files (either directly or through AutoCad) for data input. One may thea use 2 DXF file to send the geometric information about each group of entities (clement, support positions, load positions, ...) to a package. Each group of entities will thea be associated with the appropriate attributes (cross- sectional and material properties, constraint particulars, load components, ...) inside the package. International Journal of Space Structures Yol. 16 No. 1 2001APPENDIX ' Basic Formex Functions 2.4.1 Introduction ‘This appendix contains 2 description of the basic formex functions, The formex functions desoribed here are updated and extended versions of the functions discussed in Ref 10, However, the definitions of the formex functions here are given in 2 concise form and only cover the essential particulars of the functions. An important feature of the present definitions of the formex functions is that the functions are allowed to involve noninteger formices. This is in contrast with the definitions in Ref 10 where the functions may only involve integer formices. The reader in assumed to be thoroughly familiar with the basic ideas of formex configuration processing as described in Ref 1, 2.A.2 Transflection Functions “Transflection functions” constitute a major family of formex functions. These functions effect © translation, reflection, vertition (rotation), plissation (shearing), dilatation (Increase or decrease in size) and projection, Glissation, el an Reflection 7 a Projection We Vertition Dilatatic Fig 2.A.1 Effects of transflection funttions International Journal of Space Structures Vol. 16 No. 1 2001 A graphical illustration of transflectional effects is shown in Fig 2.4.1 For a transflection function to have an ‘undistorted’ graphical effect, the coordinate system used for the graphical representation must be of a Cartesian-type with identical uniform graduations along its axes However, reference systems that do not satisfy these requirements are frequently used for the graphical representation of transflectional effects. In these cases, a term such as ‘translation’ should be interpreted in a more general sense than that of 2 imple ‘rigid body’ translational movement, For exemple, consider the two directional curvilinear nomnat shown by thin lines in Fig 2.A.2. Here, the simple L-shaped configuration denoted by C2 is the ‘translation’ by 4 divisions in the Ul direction of the configuration denoted by Cl. In this situation, the sense of ‘translation’ is more general than that of the basic geometric notion of translation, To elaborate, the translation of Cl is guided by the forms and positions of the ‘normat lines’ and the resulting configuration C2 has a shape which is a ‘distorted? form of the shape of Cl. Fig 2.4.2 Transflectional effects in a curvilinear reference system Similarly, in the environment of the normat of Fig 2.A.2, the notion of ‘reflection’ should be interpreted in a more general sense than that of a ‘simple mirror image’, For instance, the configuration denoted by C3 in Fig 2.A.2 is the ‘reflection’ in the U2 direction of C2 with respect to the normat line denoted by a, 27Also, the configuration C4 is the reflection in the UL direction of C3 with respect to the normat line denoted by b. It should bo bome in mind thet, although the names of the transflection functions are suggestive of the basic geometric notions of translation, reflection, projection, ... , the definitions of the ‘rules” of the transflection finctions are ‘algebraic’ rather than ‘geometric’. Therefore, the result of the application of a transflection function is in a ‘numerical form? (in terms of a formex) end the graphical representation Of the result can assume many different geometric shapes depending on the reference system used, ‘There ere three families of transflection fimctions, namely; cardinal functions, toadial functions and provial fmetions. These will bé described below. 2.:A3 Cardinal Functions ‘The family of ‘cardinal functions’ consists of 9 functions that are briefly described in Table 2.4.1. Cardinal functions effect transflections with respect, to the main directions (cardinal directions) of the reference system. The operation rules for the basic cardinal fnctions are described in Table 2.A.2. ‘The general form of the function designator for the first cardinal function, that is, translation function, may be written as ‘ran(4,t) | E where tran(4,t) is the ‘function’, the symbol | is the ‘rallus symbol’, Bis the ‘argument? of the function, tran is the ‘imprint’ of the function and d and t are the ‘canonic. parameters’ of the fanction. ‘The above terminology is used for all the ‘formex functions, However, the number and types of canonic parameters vary from function to function as ‘will be discussed in each case. The examples in Table 2.A.1 contain a number of formex plots. For these plots, the convention is used where a prithe is added to a formex variable to indicate its plot. Thus, the plot of a formex variable Gis denoted by G. Anotier convention that is employed is for the indication of ‘optional parts’. The. compound symbols ~[ and }~ are used for this purpose. These compound symbols are refered to as ‘option brackets’. For example, the presence of the last 28 Formex Configuration Processing It canonic parameter of the vertition fimetion is optional and, therefore, the general form of the vertition function in Table 2.A.1 is given as, ver(al,42,01,¢2~Lr-) This implies that the vertition function is either of the form ver(al,42,¢1,¢2,2) or of the form ver(dt,€2,c1,02) It should be mentioned that in the case of cardinal functions that involve two directions, namely, vertition, rosette and glissation fictions, the directions are specified by dl and.d2 where the value of di may be smaller or greater than that of 2 but the values of dl and d2 may not be equal, a ‘The last column of Table 2.A.1 contains information about the ‘types’ of the canonic parameters of the functions in the context of the programming language Formian. The terms ‘numeric expression” and “integer expression’ in the last column of Table 2.4.1 are ‘Formian grammatical terms” and are briefly described below. A ‘numeric expression’ is a meaningful ‘evaluable’ combination of © numeric constants, © numeric variables, that represent numeric values, function designators that have numeric values, numeric operators, namely, + for addition), ~ (for subtraction and negation), * (Gor multiplication), 1 (for division), * (for exponentiation) and © parentheses. Examples of valid numeric expressions in the programming language Formian are 4Stic[A 5.1203 + sin | (B/12) ~ran | 1.6 8.63°3.2/(75 +C) where © A,Band Care numeric variables, ¢ -5.12e3 is a floatal constant representing the value ~5120.00, ‘© 8.63°3.2 represents 8.63 to the power of 3.2 and tic, sin and ran are numeric fimetions, as described in Section 1.5.3 of Ref 1, is, variables that oO ‘The first of the above examples is an ‘intege ‘expression’ since its value is an integer number and International Journal of Space Structures Vol. 16 No. 1 2001Hoskyar Nooshin and Peter Disney the second and third examples are “floatal expressions” ‘since their values are noninteger, as discussed in Section 1.5.6 of Ref 1. A numeric operator or a rallus symbol may not be followed by a plus or minus sign. Thus m1-25 aid sin|-30 are not acceptable. However, the following forms are acceptable m*(-2.5) and sin | (-30) Forian has a convention that is referred to as the ‘near-integer convention’. The convention provides an interpretation of what may be regarded as an integer value. In order to explain this convention, consider the following examples of rindle and rosette functions in(1, m2, 7.5) and ros(2, 3, 8, 6, 2/2, 36) where it is known that m and n are integer variables whose values are 4 and 20, respectively. ‘Thus, the value of m*2 is 16 and it should be ‘acceptable as the number of replications in the above rindle function. Also, the value of n/2 is 10 and it should be acceptable as the number of replications in the above rosette function. However, there is a problem with both of the above examples. Namely, the exponentiation and division operators * and / in Formian, as in most other programming languages, always give tise to ‘floatal” values iespective of whether their operands have integer or noninteger values. The problem would, therefore, be that numbers for ‘counting’ are given in floating point form. To avoid this problem, the following convention is used. Near-Integer Convention: In any position in a Formian statement where an item with an integer value is required fo appear, an item with a ‘nearly integer’ value will be also acceptable, In this context, a floatal value V is considered to be ‘near enough’ to an integer value provided that abs | (V—rie | V) <1B-5 Here, ‘abs’ is the ‘absolute value function’ and ‘tic? is the ‘rounded integer conversion function’ (Section 1.53 of Ref 1). That is, V is considered to be ‘near enough’ to an integer value provided that the absolute value of the difference between V and its nearest integer is less than 0.00001. ‘Now, referring to the above mentioned examples, aamely, International Journal of Space Structures Vol. 16 No. 1 2001 rin(1, m2, 7.5) and 10s(2,3, 8, 6,n/2,~36), ‘with the near-integer convention, the terms m2 and n/2 will not create any problems’ since they represent floatzl values that are very close to integer values. However, itis to be emphasised that the near-integer convention only applies to situations when an integer Value is ‘required’ to appear. Thus, with the same values for the integer variables m and;n as in the above examples, the Formian statements xamt2, and 8 = [4,3,n52,1n/2}; + will give rise o a floatal variable x (rather than an integer variable) and a floatal formex variable g (rother than an integer formex variable). 2.4.4 Tendial Functions “Tendial functions’ constitute the second major group of transflection functions. These fimctions are divided into five families, namely, tendic functions, tendid functions, tendis functions, tendit functions and tendix fimnctions, “Tendic functions? are extended forms of cardinal functions that allow multiple operations, as described in Table 2.A.3, The imprint of a tendic function is obtained by adding the suffix ‘ic’ to the imprint of the cotresponding cardinal function, There are no tendic functions corresponding to the cardinal functions that involve rotation or shearing, namely, vertition, rosette and glissation functions. On the other hand, the family of tendie functions include a member, namely, ‘gena tendic function’ that has no cardinal counterpart, A gena tendic function always operates along two directions. In contrast, the number of directions along which any other tendie function can operate is unlimited. In Teble 2.A.3, the example for the first tendic finction, that is, tranic function involves the function pen(1,0). This is a ‘pansion function’ that has the effect of adding a zero, as the first uniple, to‘all the signets of its argument, 29Formex Configuration Processing I Table 2.4.1 Cardinal Functions ‘Examples Brief descriptions of functions ‘The argument in all the examples is: E= (01,1551, 0,52,1} “Translation Function dis an integer ‘A ‘translation function’ is of the form Examples: af” expression. Gl =tran(1,2)|E 1s? tran(d,t) and 2. OE yo? tis anumeric where specifies the ‘direction of | G2=tran(2,i)|B 7 expression. translation’ and t specifies the ‘amount of Trea su translation’, Rindle Function Tanda are integer A ‘tindle function’ is of the form Brample: expressions, G=sin(1,3,1.5) [EB rin(énp) pisa numeric ‘where the term ‘rindle” implies ‘translational ee replication’ and where d specifies the 7 ‘direction of replication’, n specifies the - _ ‘number of replications’ and p specifies the 3 ‘pace’, that is, the ‘amount of translation at each step’, Reflection Function dis an integer A ‘reflection function’ is of the form eet) ech fete rofl) and : risa numeric where d spocifies the ‘direction of reflection’ { G2=ref(2,2.5)|E expression, and r specifies the ‘position of the plane of Teflection’. Lambda Function Tis an integer ‘A ‘lambda function’ is of the form ei lead aie te oe lam(4,x) and 4p. G2 ris a numeric where the term ‘lambde’, implies | G2=lam(2,2.5)|E 3 expression, ‘combination of en object with its reflection’ : and where d specifies the ‘direction of €& teflection’ and r specifies the ‘position of the plane of reflection’. ‘Veriition Function Gi and @ are vertti ion i ft Examples: integer eee eee G1 = ver(1,2,1.5,2.5) |E ‘expressions, ver(d1,d2,¢1,¢2 ~Lr}) and where the term ‘vertition’ implies ‘rotation’ | G2 = ver(1,2,3,2,135) | B cl, c2 and rare and where dl and d2 specify the ‘plane of eee rotation’, cl and ¢2 specify the coordinates, Positive See in di and €2 directions, of the ‘centre of : sense of rotation’ and r specifies the ‘rotation’ in x ah \ tation degrees. The sense of rotation is such that i Ss when r is positive then the rotation by r of di-axis about the origin will move the Tose su - positive side of dl-axis towards the positive side of d2-axis. The presence of r is optional and its absence implies that r is equal to 90°. 30 Interriational Journal of Space Structures Vol. 16 No, 1 2001CS Hoshyar Nooshin and Peter Disney Brief descriptions of functions Rosette Function A ‘rosette function’ is of the form ros(41,42,¢1,c2 ~{,n,p]~) where the term ‘rosette’ implies ‘rotational replication’ and where di and 2 specify the ‘plane of rotation’ and e1 and e2 specify the coordinates, in di and d2 directions, of the ‘centre of rotation’ and where n specifies the ‘number of replications’ and p specifies the ‘pace’ (that is, rotation at each step) in degrees, The presence of n and p is optional and their absence implies that n is equal to 4 and p is equal to 90°, The sense of rotation is as defined for the vertition function. Table 2.4.1 Cardinal Functions (Continued) G Examples The argument in all the examples is: Be €(1,1.55 1,1), 1,152,1)} Example: = 105(1,2,3,1,5,-45) | E Testu ‘Types of canonie Parameters in Formian projection’ and p specifies the ‘position of, the plane of projection’. Kecati 7 . dl and d2 are ee oe neers le: integer A ‘glissation function’ is of the form G= f8(1,2,3.5,40) | E expressions, wie 2.08) nand gare where the term ‘glissetion’ implies numeric ‘shearing’ and where dl specifies the oe expressions. ‘direction of glissation’, di and 2 specify siisation cent the ‘plane of glissation’, n specifies the position of the ‘neutral line” (that is, the line 1. ~ positive whose points are not affected by the 4g" \ sense of lissation) and g specifies the ‘angle of Ss glissation Blissation’ in degrees. The neutral line is the Tocus of the “glissation centres? for all the TTP points of the configuration to be glissated. ‘The positive sense of glissation is defined in manner similar to that of rotation for the Yertition function, Dilatation Function U2 dis an integer eee ane Examples: 5) expression, A ‘dilatation function’ is of the form Gl=dilA25)jE 4 [sx 7 7 aa) and 3 held = expression, Where the term ‘dilatation’ implies |? * MIE 27 wy gy! * ‘elongation or contraction’ and where d 7 specifies the ‘direction of dilatation’ and f Tristw specifies the ‘factor of dilatation’, Projection Function is an integer eet : Examples: ua expression. A ‘projection function’ is of the form Gl=proj14y1B 3h got 5 is anumeric proj(.p) and ap oe Gt a j ession, where d specifies the ‘diréction of} O2“PIZ3)IB 1} LET ae International Journal of Space Structures Vol. 16 No. 1 2001 31Formex Configuration Processing IL Table 2.4.2 Operation Rules for Basic Cardinal Functions Operation rules ‘The terms U1, U2, Ud, . Sco eee eee nee eee eee ‘Translation Function Ifa typical signet of E is [U1,U2, ... Ud, ... Uk] thea the conesponding G=tran(d,t)|E signet of G is obtained es [U1,U2, ... JUd+t, ... UK] Reflection Function Ifa typical signet of B is [U1,U2, ... Ud, G=reflds)|E signet of G is obtained as [U1,U2, . If a typical signet of E is [UL,U2, ... Udl, Vertition Function ermabenconpree te G = ver(al,d2,c1,02 ~[1]~) |B responding signet of G is obtained as [U1,U2, V2, ... UK] where ‘V1 = cl + (Udl ~cl) cos r- (Ud2~ 02) sine ‘V2=c2 + (Udl ~cl) sin r+ (Ua2 ~ ¢2) cosr Glissation Function G= glis(dl,42,n,g) | B Ifa typical signet of E is [U1,U2, ... ,Udl, ... ,Ud2, ... Uk) then the cor ) responding signet of G is obtained as [U1,U2, ... ,V, ..- U@2, ... JUK] where V =Udl + (a— Ud2) tan g proj(dp) |B Dilatation Function fa typical signet of B is [U1,U2, ... ,UG, ... ,UK] then the corresponding G=dil(d,p) |B signet of G is obtained as (U1,U2, ... £Ud, ... Uk] Projection Function Brief descriptions of functions ‘The argument in all the examples is: Table 2.4.3 Tendic Functions Examples B= {(1,1.5; 1,1), (1. 2,1} A ‘rinic function’ is of the form tinie(d1,42, ... ,dh, n1,n2, ... P1,p2, ... .ph) rin(@in1-pl) 2 ooh, ‘The effect of the function is equivalent to that of the composite rindle function rin(ah,ah pt) | ..[n(€2n2,p2) | ‘Tranie Function Exampl (franslation Tendic Function) G= tranic(3,2,1.5,2.5) | pan(1,0)* |B are intoger ‘A “tranic funetion’ is of the form us expressions, tranio(d1,€2, ... ,dh, 11,12, ... ,th) : rics 11,2, The effect of the function is equivalent to ie are numeric that of the composite tranistation function expressions, tran(dh,th) |... | tran(d2,t2) | tran(d1,t1) eee res Rinie Function ample: (Rindle Tendic Function) Gemini 3 3,215.1) |pan(2.09° |B Pasa s us See Section 2.A.13 International Journal of Space Structures Vol. 16 No. I 2001Hoshyar Nooshin and Peter Disney ~ Table2.A3 Tet ic Functions (Continued) dl(dhfh) | ... | dil(42,42) | dil(41,£1) 1234 5 ua : Examples ‘Types of canonie Brief descriptions of functions The argument in all the examples is: parameters in (1.1.55 11), [1,15 200) Formian Refic Function Exam | di,@2, ...,db (Reflection Tendic Function) G= refic(2,1,2,3) |B are integer A ‘refic function’ is of the form expressions, refic(d1,€2, ... sdb, 1,22, ... th) 112, sa. gth The effect of the function is equivalent to : are numeric that of the composite reflection function expressions. ef(db,th) |. | xof(42,x2) | ref(d1 11) Peete Tamie Function Example: aa, (Lambda Tendie Function) G= lamio(2,1,2,3) |B are integer ‘A ‘lemic function’ is of the form wig expressions. amio(dt,02, ... db, 11,22, ... ch) a 12... th ‘Tho effect of the function is equivalent to 2 are numeric that of the composite lambda function 1 iol ‘expressions. Jam(4h,rh) |... | lam(42,x2) | Jam(4,r1) “ieee Dilic Function Example: a, ... dh ‘@ilatation Tendic Function) G= dilic(2,3,2.5,2) | pan(1,0)*|E are integer A ‘dilic function’ is of the form us expressions. Ailic(41,d2, ... dh, £1,£2, ... ,£h) 3 1,2, ... fh The effect of the function is equivalent to 2 are numeric ‘that of the composite dilatation function 7 ‘expressions, 0 then the effect of the above genic fanction will be equivalent to that of the rinic function rinio(d1,d2,n1,n2,p1,p2) ‘The parameter b specifies the ‘bias’, that is, the movement in the dl direction for every step in the 42 direction. The parameter t specifies the ‘taper’, that is, the ‘increment’ inn] for every step in the a2 direction. "_ Projic Function “Example: d1,a2, «dh (rejection Tendie Function) G = projie(1,2,4,2.5) [EB are integer A ‘projic function’ is of the form U2 expressions, projic(d1,d2, ... dh, p1,p2, ... ph) 3 g Plp2,... ph The effect of the function is equivalent to are mimeric that of the composite projection function 4 expressions, proj(db,ph) | ... | proj(d2,p2) | proj(41,p1) 12345 U1 Genie Function Example: ai, 2, nl,ad andt (Gena Tendic Function) G= genio(2,3,4,3,1.5,1,0.5,-1) | pan(1,0)*|E | are integer A ‘genic function’ is of the form ‘expressions, genio(d1,d2,n1,n2,p1,p2,b,0) pl, p2 and b ‘The effect of a genic function is to create a are numeric non-tectangular array of objects. If b=0 and expressions, Presse 7 we See Section 2.4.13 International Journal of Space Sructures Vol. 16 No. 1 2001 33Formes Configuration Processing IT Table2.A4 Tendid and Tendix Functions Tranid Function (Translation Tendid Function) {A ‘tranid function’ is of the form tranid(tt,t2) This is equivalent to the composite function ‘ron(2,t2) | tran(14t1) ‘Tranix Function (Translation Tendix Function) are numeric ‘A ‘tranix function’ is of the form expressions, tranix(tl,12,3) : ‘This is equivelent to the composite function tran(3,t3) | tran(2,t2) | tran(1,t1) Rinid Function Rinix Function wai, ad and ad (Rindle Tendid Function) (Rindle Tendix Function) are integer A ‘tinid function” is of the form ‘A ‘rin fometion’ is ofthe form expressions. rinid(n1,n2,p1,p2) tinix(nl,n2,n3,p1,p2,p3) 1, p2 and p3 “This is equivalent io the composite function | This is equivalent to the composite function | Po? Pamesie, rin(2,n2,p2) [rin(1,n1.pl) rin(3,n3,p3) | tin(2,n2,p2) [rin(Lnt.p1)_| expressions, { Refid Function ‘Refix Function Fl, and 3 (Reflection Tendid Fusietion) (Reflection Tendix Function) are numeric A ‘tefid function’ is ofthe form A ‘refx fanotion’ is ofthe form expressions. refid(r,22) refix(t12,33) ‘This is equivalent to the composite function ref(2,22) | ref(1,c1) ‘Thisis equivalent to the composite function efi3,13) | rof(2,r2) | ref(1 x1) Lamid Function ‘Lamix Function Ti, 2 and33 (Lambda Tendid Function) (Lambda Tendix Function) | are numeric A ‘lamid fanction’ is of the form ‘A ‘lamix funetion’ is of the form expressions. lamid(e1,x2) Jamix(el 12,13) ‘This is equivalent to the composite fumetion lam(2,x2) | lam(1r1) This is equivalent to the composite function Jars(3.x3) | Iam2,22) | Iam(71) Dilid Function Dilix Function 7, Rad (Dilatation Tendid Function) (ilatation Tendix Function) are numeric A ‘Gilid funetion’ is of the form A ‘dilix function’ is of the form expressions. (—) auid(01,2) dilix(f1,02,8) ‘This is equivalent to the composite function dil(2,f2) | dil(1,f1) This is equivalent to the composite function 4,8) | dil@2,2) | al(1,£1) Projid Function Projix Function iI, p2 and p3 (Projection Tendid Function) (Projection Tendix Function) ‘are muimeric A ‘projid function’ is of the form A ‘projix fanction’ is of the form expressions, projid(p .p2) rojix(p1.p2,p3) This is equivalent to the composite function proj(2,p2) | proj(1.pt) ‘This is equivalent to the composite function proj(3,p3) | proj(2,p2) | proj(1.p1) Genid Function ni,n2 endt (Gena Tendid Function) are integer A ‘genid function? is of the form expressions. genid(n1 n2,p1,p2,b,) a eee plpdandd ‘This is equivalent to the tendic function are numeric genic(1,2,n1,n2,pl.p2,b.t) expressions, Er — International Journal of Space Structures Vol. 16 No. 1 2001oshyar Nooshin and Peter Disney ‘Thus, for “B= ((ML5; LU, 1452,1)} ‘the formex represented by pan(1,0)|E _ willbe EB [0,1,1.55 0,1,1], [0,1,15 0,2,1]} The pansion ‘function is also used in another 3 examples in Table 2.4.3 with the aim of making the argument E conformable for the intended operations, see Section 2.4.13. “Tendid functions’ are special cases of tendic finetions and are described in Table 2.A.4, A tendid function effects operations in directions 1 and 2 and its imprint is obtained by adding the suffix ‘id’ to the imprint of the corresponding cardinal function. The families of tendis and tendit functions are similar to the tendid fictions. A ‘tendis function’ operates in directions 1 and 3 and its imprint has the suffix ‘is’, A ‘tendit fimction’ operates in directions 2 and 3 and its imprint has the suffix ‘i Except for the directions of the operations, the patticulars of the tendis and tendit fimctions are identical to those of the tendid functions. Therefore, the tendis and tendit functions are not seperately described since their particulars may be deduced from those of the tendid functions in Table 2.4.4, For instance, fiom the description of the tranid function in Table 2.A.4, it may be deduced that the function ‘tranis(2,b) is equivalent to the composite function ‘tran(3,b) | tran(I,a) Also, it may be deduced that the function ‘ranit(a,b) is equivalent to the composite function ‘tran(3,b) | tran(2,a) “Tendix functions’ are special cases of tendic fanctions for operations in directions 1, 2 and 3, ‘These functions are described in Table 2.A.4, Tho imprint of a tendix function is obtained by adding the suffix ‘ix’ to the imprint of the corresponding cardinal function, ‘The gena functions have no counterpart in the tendix family since the effects of the gena functions are confined to two directions. The term ‘gena function’ is used to refer to a gena tendic, gena tendid, gena tendis or gena tendit function. International Journal of Space Seructures Vol. 16 No. 1 2001 us Second Operation uz Second Operation UL Fig 2.A.3 Order of operations for tendid, tendis and tendit functions It should be noted that the order of operitions in tendid, tendis, tendit and tendix functions is always ‘such that the operation along a direction that has a ‘smaller number’ is carried out before the operation along a direction that has a ‘greater number’, This ule does not apply to the tendic functions, where, the order of the operations is dictated by the ‘specified’ order of the directions. The order of operations for the tendid, tendis and tendit functions is illustrated in Fig 2.A.3. 2.4.5 Provial Functions ‘Proviel functions’ are generalisations of cardinal functions that allow operation in any direction, There are four families of provial functions, namely, © ‘proviad functions’, for operations involving directions 1 end 2, ‘© ‘provias fimotions*, for operations involving directions 1 and 3, © ‘proviat functions’, for operations involving directions 2 end3 and © ‘proviax functions’, for operations involving directions 1, 2 and 3, Proviad and proviax functions are described in Table 2.4.5. The imprint of a proviad fimetion is obtained by adding the suffix ‘ad’ to the imprint of the corresponding ‘cardinal function. The imprint of a proviax function is obtained in a similar manner using the suffix ‘ax’, The terms Al, A2, A3, BI, B2 and B3 that appear as” canonic parameters of proviad and proviax functions in Table 2.A.5 represent the coordinates of the end points of @ vector AB, as shown in Figs 2.4.4 and 352.4.5. This vector is referred to as the ‘direction vector". For every proviad function in Table 2..5 there is a corresponding ‘provias fmnction’ ‘thet involves directions 1 and 3 and uses the suffix ‘as’. Also, for every proviad fimetion.in Table 2.A.5 there is a corresponding ‘proviat function” that involves u2 Fig 2.A4: Direction vector for proviad functions Formex Configuration Processing IT directions 2 and 3 and uses the suffix ‘at’, Provies and proviat functions are not separately described since their particulars may be deduced from those of the proviad functions in Table 2.4.5. Ba Fig2.A.8 Direction vector for proviex functions Table 2.A.5 Proviad and Proviax Functions Brief descriptions of factions ‘A “tranad, fimetion’ (franslation proviad function) is of the form trenad(A1,A2, BI,B2~[}4) and a ‘tranax function’ (translation proviax function) is of the form ‘tranax(A1,A2,A3, B1,B2,B3 ~Lt}+) where the direction of translation is given by vector AB and the amount of translation. is given by or the length of AB, in the absence of t ‘A ‘ritad function” (indle proviad function) is of the form rinad(A1,A2, B1,B2, n~f,p}-) and a ‘rinax function’ (rindle proviax function) is of the form ‘inax(A1,A2,A3, B1,B2,B3, n~{,p]~) where the direction of replication is given by veotor AB, the number of replications. is given by n and the amount of translation at each step is given by p or the length of AB, in the absence of p. Examples: and Example: G=sinad( 36 G1 = tranad(1,1,3,2.5)|E G2=tranad(1,1,5.5,2) E “Examples ‘The argument in all the examples is: B= 401.55 141), 0,15 21 Al, A2, AS, BI, B2, B3 andt are numeric: expressions. ieeas aru AT, A2,A3, BI, B2, B3 and p are mumeric expressions. nis an integer expression. 1,125,134) |B d2saee7U International Journal of Space Structures Vol. 16 No. 1 2001Hoshyar Nooshin and Peter Disney Table 2.A.5 Proviad and Proviax Functions (Con Examples ‘The argument in all the examples is: B= {11.53 1,1, 115 2,1)} Brief descriptions of functions ‘A ‘refad fimction’ (feflection proviad function) is of the form refad(A1,A2, B1,B2), a “lamad fanction’ (lambda proviad function) is of the form Jamad(A1,A2, B1,B2), a ‘refax finetion’ (fefleetion proviax finetion) is of the form refax(A1,A2,A3, B1,B2,B3) and a ‘lamax function’ (lambda proviax function) is of the form lamex(A1,A2,A3, BI,B2,B3) Where the direction of reflection is given by vector AB, with the plane of reflection being Examples: GI = refad(1,0,2.75,1) |B and G2= lamad(1,0,2.75,1) | E r2s4 5 ‘Al, A2, A3, BL, B2 and B3 are numeric expressions. where the coordinates of the cente of rotation in plane U1-U2 are given by cl and 02, the amount of rotation is given by r (in dogrees), the sense of rotation is as explained for the cardinal vertition function in Table 2.A.1 and where the absence of r implies that Tis equal to 90°. ani G2 = verad@3,2,135) |B Tieas i Poste sense o Gt] Nee PI ca mY U1 perpendicular to AB at B. A’ ‘verad function’ (vertition proviad | Examples: cl, c2andr function) is of the form G1 = verad(1.5,2.5) | E are numeric verad(cl,c2 ~{,r}~) id expressions. ‘A ‘vorax fimnction” (vertition proviax function) is of the form verax(A1,A2,A3, B1,B2,B3 ~f,1]~) ‘where the axis of rotation is given by vector AB and the amount of rotation is given by r (in degrees), The sense of rotation is such that when the value of r is positive then the rotation causes a right-handed screw to move from A towards B, The absence of r implies that ris equal to 90°. Example: G= verax(2,2,2,0,0,0,50) | pan(,0)* |B See Section 2.A.13 ‘Al, A2, A3, Bl, B2, B3 andr are numeric expressions. ‘A “rosad function’ (rosette proviad function) is of the form rosad(el,c2 ~[n,p}) where the coordinates of the centre of rotation in plane U1-U2 are given by cl and 2, the number of replications is given by n, the amount of rotation at each step (pace) is given by p (in degrees), the sense of rotation is as explained for the cardinal vertition function in Table 2.4.1 and the absence of n and p implies that n is equal to 4 and p is [equal to 90°, Example: G = r0sad(3,1,5,-45) |B International Journal of Space Structures Vol. 16 No. 1 2001 el, c2andp are numeric expressions. nis an integer expression, 37Formex Configuration Processing It Table 2.4.5 Proviad and Proviax Functions (Continued) Brief descriptions of functions Examples ‘The argument in all the examples is: Ex (1,15; 1,1}, (145 2,10} ‘Types of canonic parameters in Formian, “A ‘fosax function’ (rosette proviex function) | Example: ‘Al, A2, AS, is of the form G= rosax(2,2,2,0,0,0,7,25) |panG3,0)*|E | Bl, B2, B3 and p rosex(A1,A2,A3, B1,B2,B3 ~.n,p>) y are numeric where the axis of rotation is given by vector |° xpreesions, AB, the mumber of replications is given by 1, nis an integer the amount of rotation at each step (pace) is expression. given by p (in degrees) and the sense of we rotation is. as described for the verax fanction. The absence of n and p implies that 7B nis equal to 4 and p is equal to 90°, ie ok See Section 2.4.13, ‘A “glisad fianetion” (glissation proviad le Al, A2, BI, function) is of the form G= glised(0,2.5,4,4,45) |E Bland g glisad(A1,A2, B1,B2, g) U2 Direction of glissation fia enrol where the direction of glissation as well as ‘the neutral line is given by vector AB and the: angle of glissation is given by g in egrees. The positive sense of glissation is as described for the glissation function in Table 2A. ‘and neutral line be Spee Teaas ut ‘A ‘glisax function’ (glissation proviax function) is of the form glisax(A1;A2,A3, B1,B2,B3, C1,C2,C3, ¢) where Al, A2, A3, B1, B2, B3, Cl, C2 and C3 specify the coordinates of three noncolinear points A, B and C and where g specifies the ‘glissation angle* in degrees. The points A, B end C define a plane that is referred to as the ‘base plane’. For a point P whose glissation is Q, the “glissation plane’ is the plane ‘that contains point P and is parallel to the base plane, and the ‘glissation Girection’ as well as the ‘neutral line’ is given by the vector DE which is the projection of AB onto the glissation plane. The positive sense of glissation is as indicated by the “broken vector’ ABC. All the canonic parameters of a glisax ‘function are aumeric expressions. ‘A ‘Gilad fiction’ dilatation proviad | Example: ‘Al, A2, AS, function) is of the form G= dilad(0,1.5,1.5,2.5,2)|E BL, B2, B3 and f diled(A1,42, B1,B2, f) and ‘are numeric a ‘dilex function’ (dilatation _proviex expressions. function) is of the form dilex(A1,A2,A3, B1,B2,B3, f) where the direction of dilatation is given vector AB, with point A acting as the ‘origi and where the factor of dilation is given by f. ‘A ‘projad fimction’ (projection proviad | Exampl AL, A2, AB, function) is of the form G= projad(t,0,3.5,1) |B _ | BI, B2 and B3 projad(A1,A2, B1,B2) and are numeric 2 Sprojax function’ (projection “ proviax expressions. function) is of the form projax(A1,A2,A3, B1,B2,B3) ‘whore the direction of projection is given by vector AB, with the plane of projection being dicular to AB at B. 38 International Journal of Space Structures Vol. 16 No. 1 2001Hoshyar Nooshin and Peter Disney For example, consider the formex variable F = [0,0,0; 1,1,15 1,1,2] This is a formex representing a triangular ‘tile’ a plot of which is shown in Fig 2.4.6, denoted by F". The equation G12 = tranad(0,0,1.5,1.5) |F vill transform F into the formex variable G12 whose, plot is shown in Fig 2.46 2s Gl2', ‘The above equation uses the traned function acting in plane 1-2 with the direction vector being indicated by -the dotted vector AB. The equation G13 = tranas(0,0,2.5,1.5)|F will work in a similar way using the tranas fimetion acting in the 1-3 plane, A plot of the resulting formex is shown in Fig 2.A.6, denoted by G13". Similarly, the equation G23 = tranat(0,0,1.5,1.5) | F will give tise to the formex variable G23 whose plot is shown as G23" in Fig 2.A.6, Fig2.A.6 Bifects of tranad, tranas and ‘tranat functions It should be pointed out that the grade of the argument of a proviad function must be greater than or equal to 2 but as far as a provias, proviat or proviax function is concemed, the grade of its argament must be greater than or equal to 3, 2.A6 Introflection Functions ‘Introflection functions’ are formex functions that effect curtailment of formices in various ways. There are a number of introflection functions as follows: © pexum function, © family of rendition functions, International Journal of Space Structures Vel. 16 No. 1 2001 family ofresttion fimetions and * election functions ‘These functions are briefly described below. 2.A.7 Pexum Function ‘The effect of the ‘pexum function’ is to transform a formex E into another formex by deleting every ccantle of E that has the same signets as a preceding cantle of E. The pexum function has no canonic parameter and its imprint is ‘pex’, For example, if E is given by 404,35 1,2), (5,43 2,3), [1,25 4,3], 14,35 5.4), 14,35 1,2)} then the value of the function designator pex | is {14,35 1,2], (5,43 2,3], (4,35 5,47} Here, the third and fifth cantles of B are deleted ‘because their signets are the same as those of the first cantle, Note that a cantle that has the same signets as a preceding cantle is deleted irrespective of the order in which the signets appear in the cantles. Thus, in the above example, the third cantle of B is deleted ‘even though its signets ere not in the same order as those of the frst cantle. The role of the pexum fimnction is to remove the superfluous repeated elements of a configuration. ‘The pexum function has been used in a number of examples throughout the discourse. In particular, see Section 1.4.5 of Ref 1. 2.4.8 Rendition Functions ‘The family of ‘rendition functions’ consists of six functions, as. described in Table 2.A.6. The description of the functions in Table 2.A.6 are given in two different forms. In the left-hand column, the Gescriptions are in terms of numerical procedures and in the right-hand column the descriptions are in terms of graphical effects, A rendition function has a single canonic parameter, In the context of ‘pure’ formex algebra, all that one needs to say about this canonic parameter is thet itis a formex. However, in the context of the programming language Formian, one has to be more specific about the type of the parameter, In Formian, the canonic parameter of a rendition function is required to be a formex expression. A ‘formex expression’ is a meaningful ‘evaluable’ combination of 39Formex Configuration Processing It Table 2.4.6 Rendition Functions ‘A “huxum funetion’ is of the form Ix) where the term ‘luxum’ implies ‘disconnected parts” and where F is a formex expression. If G = lux(F) |B then G is obtained by deleting every cantle of E that includes one or more signets that are in F, ‘The examples are in terms of the plots of B and F shown in 7 The plot of fux(F) | B is obtained by removing {U2 every element of the 6 plot of E that has one or 4 more nodes connected 2 Fn to the nodal points of the plot of F. Tit tbae Plot of lux(F) [E ‘Coluxum Function ‘A ‘coluxum function’ is of the form col) where F is a formex expression. If G = col(F) | E then G is obtained by deleting every cantle of E none ‘of whose signets are in F, The plot of col) TE is puz obtained by removing every clement of the 4. RV plotof that is apart of the plot of ax(F) |B. um ( Ties wain Plot of colfF) [B ‘Nexum Function ‘A ‘nexum fanetion’ is of the form nex(F) where the term ‘nexum’ implies ‘connected parts’ and where F is a formex expression. If G = nex(F) | B then G is obtained’ by deleting every cantle of E that includes one or more signets that are not in F. The plot ofnex() | Eis obtained by. removing _{U? every element’ of the © plot of E thet has one or *7 XXX more nodes not con. # ui nected to the nodal Twins points of the plot of F. Plot of nex(F) |E ‘Conexum Function A‘conexum function’ is of the form con(F) where F is a formex expression. If G = con(F) | B then G is obtained by deleting every cantle of B all whose signets are in F. The plot ofcon(®) [Eis 4uz obtained by removing every element of the 4 plot of E that is apartof 2 the plot of nex(F) | E. uw Tete Plot of con(F) |E Pactum Function A ‘pactim function’ is of the form pac(F) where the term ‘pactum’ implies ‘matching parts’ and where F is a formex expression. If G= pac(F) | E then G is obteined by deleting every cantle of E whose signets are not the same as the signets in a cantle of F. The plot of pact) [Eis obtained by removing U2 every element of the S>>>>. ut plot of E all whose nodes are not coincident “tit tbae Plot of pac(F) [E with all the nodes of an element of the plot of F. Copactam Function A ‘copactum function’ is of the form cop(F) where F is a formex expression. If G = cop(F) | E then G is obtained by deleting every cantle of E all whose signets are the same as the signets in a cantle of F. ‘The plot of cop(F) | Eis U2 obtained by removing . SEES un every element of the plot of B that is a part of tie ebaw Plot of cop(F} |E the plot of pac(F) |B. International Journal of Space Structures Vol. 16 No, 1 2001Hioskyar Noashin and Peter Disney formex constants, formex variables, formex function designators, duplus symbols, parentheses and formex formations, A “formex formation’ is a construct that has the same basic form as a formex constant and in which one or more uniples are given es numeric expressions. For example, if, j and k are numeric variables, then Bits 2) {(1,4,3}, [7,2,ric | (WS), [73.5,-6,11]} are formex formations. and Note that a single formex constant, formex variable, formex function designator or formex formation is counted as a formex expression, Returning to the description of rendition functions, the effect of a rendition funetion is to ‘curtail’ its argument, as dictated by the signets or cantles of the value of its canonic parameter. The first four rendition functions, namely, Juxum, cohixum, nexum and conexum functions use the information provided by their canonic parameters as ‘lists of signets’, irrespective of the manner in which the signets are grouped into. cantles. In contrast, the last two rendition functions, namely, pactum and copactum functions work in terms of the ‘cantles’ of their canonic parameters. U2 v2 246 sie Plot of E un eae eww Plot of F Fig 2.A.7 Plots for the examples of Table 2.4.6 ‘The examples in Table 2.4.6 are in terms of two formex variables E and F. These formex variables are given by EB i6(6,3,2,2) | lamid(2,2) | (1.2; 2,1] and F=rin(,5,2)| {[2,35 2,5}, [2,35 3,4], (3, ‘The plots of B and F are shown in Fig 2.4.7. Rendition functions may be subdivided into three ‘complementary pairs’. To elaborate, the huxum and coluxum functions form a ‘complementary pair’ because their effects are of a complementary nature. In fact, the name ‘colucum’ implies the ‘complement International Journal of Space Structures Vol. 16 No. 12001 of the Iuxum function’. Similarly, the nexum and ‘conexum functions constitute a complementary pair of functions and so'do the pactum and copactum functions. As a consequence of the complementary nature of the effects of the luxum and coluxum functions, the combination of the plots of Inx(F)|E and. col()|E Will givo rise to the plot of B, as may be seen ftom the plots given in Table 2.A.6. This, in fact, is a general rule that applies whatever the values of E and F. This rule is also applicable in the case of the complementary function pairs ‘nexum/eonexum’ and “pactum/copactum'’, 2.A.9 Restition Functions The family of ‘restition functions? consists of six functions", es described in Table 2.4.7. The term ‘restition’ implies ‘cordoning’ (from the Latin word ‘restis’ meaning a cord), A restition fanction effects curtailment of a formex as guided by a ‘region’. An exemple of a region is shown by dotted lines in Fig 2.A.8. Also shown in this figure is a diagonal grid together with a two directional normat, Exterior Border Interior ‘U2 clement element clement Region un 246 8 10 12 14 Fig 2.A.8 Plot of E shown together with a region The formex representing the diagonal grid of Fig 2.A.8 is the same as formex B formulated for the grid of Fig 2.4.7. The border of the region in Fig 2.A.8 may be specified by the formex b=[4,8; 14,8; 14,25; 9.5;2.5] The border is specified by listing the signets that represent the comers of the region. A formex such as _ b above is referred to as a “border specifier’, The elements of a configuration may be divided into three different types, depending on their dispositions with respect to a region, as follows: 4l© ‘interior clement’ whose nodes are inside or on the border of the region, © ‘exterior element’ whose nodes are outside or on the border of the region with at least one node outside the region and © “border element? which has at least one node inside and at least one node outside the region, Examples of these types of elements are shown in Fig 2.A.8, Here, the interior, exterior and border elements are exemplified by “two-noded” clements but the above definitions of interior, exterior and borer clements apply to elements with any number of nodes. IfE is a formex representing a configuration and R is aregion then a cantle of Bis said to be © an ‘interior cantle’ with respect to R, # an ‘exterior cantle' with respect to R or © a fborder cantle’ with respect to R depending on the disposition of the plot of the cantle with respect to the region R. ‘The region shown in Fig 2.A.8 is an example of an ‘ambit region’. The border of an ambit region is Gefined by specifying the coordinates of its comer points. However, there are two other types of regions, namely, ‘rectangular’ and ‘circular’ regions. ‘The border of a rectangular region is defined by specifying the coordinates of two diagonally opposite comers. The border of a circular region is Gefined by specifying © the coordinates of its centre and a point on its, circumference or © the coordinates of its centre and its radius, Rectangular Circular ua Region ua 88 \ \ 2 Fl a ee] pz pa Fig 2.4.9 Rectangular and circular regions ‘Thus, the ‘border specifier’ for the rectangular region shown in Fig 2.4.9 may be given by b= [pl.al; p2.q2] or b=[pl,q2; p2,al] Also, the border specifier for the circular region shown in Fig 2.A.9 may be given by b=[plqlsp2.q2] or b*[plaiz] a2 Formex Configuration Processing A rectangular region may also be specified as an ambit region by giving the coordinates of all four comers but the specification as a ‘rectangular region’ is, obviously, more convenient. To define a region, in addition to its border specifier, it is necessary to provide information about the type and the directions associated with the region. As far as the type of a region is concemed, the ‘region codes’ 1, 2 and 3 are used to specify ambit, rectangular and circular regions, respectively. The reason for the need to specify the directions associated with a region is that regions are not only ‘used in the 1-2 plane, as exemplified so far, but also ‘in planes 1-3 or 2-3 or indeed any other pair of directions. ‘The type and the associated directions of a region are specified by a construct of the form > {¢,d1,d2} where ¢ is the region code and di and d2 are the directions associated with the region. This construct is referred to as the ‘signature’ of the region. For example, the signature of the ambit region of Fig 2.4.8 may be given as (1123 ‘A region may consist of a combination of regions. — - For instance, the region whose boundaries are shown by thick lines in Fig 2.A.10a consists of two seotangular regions and a circular region. This ‘compound region’ may be specified by listing the signatures and border specifiers of its constituent regions one after the other, as follows: {2,23}, [1,15 2.5,2.5], {2,23}, 2,2; 3.5,3.5], {3,2,3}, [3.5,3.5,1] oO When a signature in the specification of a compound region is the same as its preceding one then it may be omitted. Thus, the above specification may, alternatively, be written as 2,2,3}, [1,1 2.5,2.5], [2,25 3.5,3.5], {3,2,3}, (3.5,3.5,1] A region may have a ‘negating’ effect. For instance, the compound region whose boundaries are shown by thick’ lines in Fig 2.A.10b consists of two ‘normal’ rectangular regions and a ‘negating’ circular region. A region with a ‘negating’ effect is referred to as an ‘antiregion’, The specification of an antiregion is identical to that of a normal region except for its region code which is given with a minus sign. For example, the specification of the compound region of Fig 2.A.10b may be given as, {2,1,3}, 15 2.5,2.51, (2,2; 3.5,3.5], £3,1,3}, B.53.5,1) International Journal of Space Structures Vol. 16 No. 1 2001oO Hoshyar Nooshin and Peter Disney Also, the specification of the compound region of Fig 2.4.10c may be written as {21,2}, [1,15 2.5,2.5], {2.1.2}, [2.25 3.5,3.5}, {3,1,2}, [3.5,3.5,1] An antiregion has no effect by itself unless it overlaps a normal region, i Us (a) Fig 2.A.10 Compound regions A region need not necessarily be ‘in one piece’, For example, the compound region of Fig 2.A.10d consists of two separate sections as shown by thick lines. This compound region is obtained as a combination of three simple regions one of which is sm antiregior, The specification of the “disjointed” region of Fig 2.A.10d may'be written as {2,12}, (1,15 2.5,2.5], {-2,1,2},, (2.2; 3.53.51, (3,1,2}, B.5,3.5,1] Incidentally it is interesting to note that a two-noded clement such as ¢ in Fig 2.A.10d will be regarded as an interior element since both of its nodes are inside the region, In general, the specification of a compound region may be written as 81,b1, 52,62, ... hbk (k2 1) where sl, 52, ... , sk are region signatures ond bl, 2, ... , bk are their corresponding border specifiers and where it is understood that a signature that is the ‘same a its preceding one can be omitted, International Journal of Space Structures Vol. 16 No.1 2001 In the specification of a compound region, the ‘order? in which the simple regions are listed has no patticular significance and the shape of the resulting rogion will not be affected by this order. The above general form of the specification of a compound region is used for the canonic parameters of the resttion functions in Table 2.A.7. The family of resttioni fimctions consists of three pairs of complementary functions, namely, ® ducture and coducture functions, ‘© juncture and cojuincture fictions and * vecture and covecture functions. ‘The term ‘ducture’ implies the ‘parts outside the region’, the term ‘juncture’ implies the ‘parts inside the region’ and the term ‘vecture’ implies the ‘parts cut by the region’, The first four restition fictions, namely, ducture, coducture, juncture and cojuncture functions are applicable to configurations with elements having any number of nodes. However, the last two restition fanctions, namely, vecture and covecture fictions, are only relevant in relation to configurations that involve two-noded elements, Gi Interior subelement Uj Region R ‘ig 2.A.11 Interior and exterior subelements Vecture and covecture functions use the concepts of ‘subcantles’ and ‘subelements': These concepts are explained with the aid of the example shown in Fig 2.A.11. In this figure, a region R is shown together with a two-oded border element 1-2, This element is the plot of the cantle [pl,ql; p2,q2] Now, the ‘interior part? of element 1-2 may be represented by the cantle (pLal; 3,43] where p3 and q3 are the coordinates of the ‘border point” of the element. The above cantle is referred to aFarmex Configuration Processing It Table 2.4.7 Restition Functions pane 7 of canons pass Brief descriptions of functions Types ofan neni an Ducture Function I, bl, ...» 8k, BK A ‘ducture? function is of the form are formex uo(61,b1, s2,b2,...,skbk) k21 expressions, ‘The effect of the function is to produce a formex by deleting every cantle of its argument that is not an exterior cantle of the region represented by 24 6 6 m0a274 si,bl, #2,b2, .., sk bk. Flot of ducts) Pet ‘Coducture Function u2 si, bi, 8k, Bk A ‘coducture’ fianction is of the form. 6 are formex cod(s1,b1, s2,b2, :.., sk bk) k21 expressions, ‘The effect of the function is to produce @ formex by} un deleting every cate of argument that i an exterior Stare” cantle of the region represent shbl, 2k. skbke distia teal £ ‘Juncture Function U2 sl, bl, ..., sk, bk A ‘juncture’ function is of the form 6 are formex jun(6l,bl, 82,2, ..., sk,bk) k21 expressions. ‘The effect of the function is to produce a formex by un deleting evry exile of ts epuea Cat isnot an Sta interior cantle of jon represent S1b1, 282, abe id Mt of amis) Cojuncture Function U2 A ‘cojuncture’ function is of the form 6 i coj(si,bl, s2,b2,...,8kbk) k21 « Se, i ‘The effect of the fanction is to produce a formex by 2 UL deleting every cantle of its argument that is an interior Sa cantle of the region represented by Plot ofcofebyle sll, s2,b2, ... sk,bk. ‘Veoture Function™™ | 81, BI, ..., sk, bk A ‘vecture’ function is of the form U2 : are formex, ‘veo(@1,bl, s2,b2,...,akbk) k21 8 expressions. ‘The effect of the function is to produce a formex by fi * i deleting every cantle of its argument that is an exterior fi UL cantle of the region represented by, “Fas ea sl,bl, 52,62, ... , sk,bk. Plot of vee(s,b)]E* and by replacing every cantle of its argument that is a border cantle by its interior subcantle. Covecture Function®* sl, bl, ..., sk, bk A ‘covecture’ function is of the form U2 aie are formex cov(sl,b1, 82,b2,...,sk bk) k21 x %,. expressions. The effect of the fimction is to produce a formex by) > Sages: deleting every cantle of its argument that is an interior a cantle of the region represented by eee rr s1,bl, s2,b2, ... , sk,bk Plot of covis.b)[E* + and by replacing every cantle of its argument that is a border cantle by its exterior subcantle. % E is the formex whose plot is given in Fig 2.A.8 and s and b are, respectively, the signature and border specifier of the region shown in Fig 2.4.8. #k For an element that is not two-noded, vecture and covecture functions act like juncture and cojuncture functions, respectively. “4 Inernational Journal of Space Structures Vol. 16 No. 1 2001Hoshyar Nooshin and Peter Dimney as the ‘interior subcantle’ of element 1-2, Also, the part 1-3, considered as a separate element, is referred. to as the ‘interior subelement’ of clement 1-2. Similarly, the cantle (3,93; p2,q2] is referred to as the ‘exterior subcantle” of element 1- 2 and the part 3-2, considered as a separate element, is referred to as the ‘exterior subelement’ of element 1-2, The concepts of ‘subcantles’ and ‘subelements? only apply to two-noded elements, ‘| Note that the normat directions in Fig 2.A.11 are given as Ui and Uj implying that the concept discussed is spplicable in relation to eny pair of directions. Ibis now necessary to bring out a number of points relating to regions. To begin with, it should be ‘understood that an ambit region is always a ‘convex’ polygon. This convex polygon is obtained by ‘examining the line passing through every pair of points listed in the border specifier of the region. ‘Such a line will be considered to be an edge of the polygon provided that all the listed points are to ‘one side’ of the line (except for two or more of the points that are om the line). Fig 2.4.12 An ambit region For example, consider the ambit region shown in Fig 2.A.12, The signature of this region is {1,Ui,Uj} and its border specifier contains the coordinates of nine points, numbered 1 to 9 in Fig 2.A.12, It can be noticed that not all the specified points are comers of the region. The points that fall within the polygon, namely, points 1, 6 and 9 are then ‘superfluous’ and will be disregarded. Tt should also be noted that the ‘order’ in which the points are listed in the border specifier of an ambit region has no significance and will not affect the shape of the region, ‘The basic simple regions, namely, ambit, rectangular and circular regions are all ‘convex". However, compound regions that consist of combinations of simple regions may be used to create ‘non-convex’ International Journal of Space Structures Vol. 16 No. 1 2001 regional shapes, as may be seen ftom the examples shown in Fig 2.A.10, In general, the border specifier of an ambit region is of the form [pl,al; p2,q25 ... pre] where (P1,41), (2,42), ... (pmax) are the coordinates ofr points and where r = 3. If all the r points are coincident or collinear then the region is a ‘null region’, Also, the following regions are regarded as null regions: © a circular region with a zero radius and © a rectangular region whose border specifier contains the coordinates of two coincident points or two points that give rise to a degenerate rectangle consisting of a line segment. A’ null region is-considered to have no interior, ‘Therefore, with respect to a null region, any element (cantle isan exterior clement (cantle). Now, consider a cantle C and let this cantle be represented by (U11L,U12, ... Uli, ... Uj, U21,U22, ... U2, ... U: Un1,Un2, ... Uni, .-. Uni, ... ,Unm] Let it be required to determine the ‘status’-of this cantle with respect to a region R whose directions are i and j. What is meant by determining the status of the cantle, is to find out if the cantle is an interior, an exterior or a border cantie. ua ua fa) b) UL UW yy UW Fig 2.A.13 Regional prisms Jn determining the status of C, only the i* and uniples of the cantle will be examined. Therefore, the values of the other uniples of the cantle do not have any bearing on the status of the cantle, It is as though the region extends ftom ~ to +m in directions 1 to m excluding the i* and j* directions. Thus, if m=3, a. region such as that shown in Fig 2.4.8 will be effectively like a prism extending from ~c to +00 in the 3" direction. A part of this prism is shown in Fig 2.A.13a, Also, a rectangular region in plane 1-3 will, 45be effectively like an infinitely long prism a part of which is shown in Fig 2,A.13b, ‘Therefore, although as (a part of) the canonic perameter of a resttion fnction, a simple region is always defined. with respect to two disections, the ‘effective directions’ of the region will consist of all the directions associated with the argument of the function. Finally, it should be mentioned that the concept of regions in this section has been described in terms of Cartesian-type normats. However, the idea of a region may be employed in relation to any kind of normat, With nommats that are not of Cartesian-type, the borders of regions will assume shapes that are conformable with the forms of the normat lines and surfaces. For instance, in a spherical normat, the borders of a rectangular region with directions 2 and 3 will be along the parallels and meridians of the normat, 2.A.10 Relection Function ‘A ‘election function’. effects curtailment of ts argument, as dictated by a ‘condition’, For example, consider the configuration shown in Fig 2.A.14a, ‘The configuration consists of an arrangement of two- noded elements. A formex representing the configuration relative to the normat of Fig 2.A.14a is given by PI =rinid(7,5,1,1) | {01,15 2.1), [115 L2]} Now, suppose that it is required to remove the elements sticking out on the right-hand side of the configuration of Fig 2.A.14a. This can be achieved by writing F2= rel(U(2,1)<7.5) | FI where F2 represents the configuration shown in Fig 2.A.14b. In the above equation, the constrict rel(UQs1) < 7.5) is a ‘relection fimetion’ where ‘rel’ is the imprint of the function, standing for ‘relection’, and UG) <75 is the canonic parameter of the function. The term ‘relection’ is a Latin based word meaning ‘re- selection’, ‘The canonic parameter of a relection function is a ‘condition’ relating to the cantles of the argument of the fonction. The effect of the function is to ‘keep’ the cantles that satisfy the condition and ‘delete’ the cantles that do not satisfy the condition. For instance, 46 Formex Configuration Processing II for a cantle of the argument of a relection function, ‘the relation UQN<15 is TRUE if the 1* uniple of the 2™ signet of the cantle is less than 7.5 and is FALSE otherwise, ‘The construct U1) is referred to as a ‘brevant’. The term ‘brevant ‘means a ‘shorthand indicator’ (from Latin ‘brevis’ meaning short). U2 6 &) a Ts Fo=rel(UQ@1)<7.5)|P1 uz 7 @ 1 ser (URI) <7.5 be U(2.2)<8.5)|F2 + FH "0 UL is 8 7 P5erel(U(2,1)<5 |] UG.2}<4) Fs Parel(U(1,2)>2 [| Uli2}>2))F Fig2.A.14 Examples of the application of the relection function ‘The general form of a brevant is UG) representing the j* uniple of the i* signet of a cantle, Jn the context of the programming language Formian, i and j are integer expressions, as explain: in Section 2.4.3. : ‘A relation appearing as the canonic parameter of a relection function is of the general form ERE International Journal of Space Structures Vol. 16 No. 1 2001Hoshyar Nooshin and Peter Disney where E is a numeric expression that may include ‘one or more brevants and where R is a ‘relational operator’. Fomnian has six relational operators, as shown in Teble2.A.8, Table 2.A.8 Relational Operators Operator (Greater than or Greater than or equal to Equal to ‘Not equal to or ‘Less than or equal to a< =| Less than Returning to the examples of Fig 2.A.14, the ‘configuration in Fig 2.A.14c may be represented by F3 = rel(U(2,2) < 5.5) | F2 Here, the effect of the relection function rel(U2.2)< 5.5) is to produce formex F3 from those cantles of F2 that, satisfy the condition ven <55 ‘That is, F3 will be created from all the cantles of F2 in which the 2" uniple of the.2™ signet is less than 5.5. The effect will be to remove the cantles that represent the top fow of the vertical elements in Fig 2.4.14, as shown in Fig 2.A.14c, ‘The configuration of Fig 2.A.14c may also be ‘Tepresented by F3 = rel(U(2,1) < 7.5 && U(2,2)< 5.5) | FL as shown in Fig 2.A.14d. In this case, the above discussed relations are combined ‘using. the compound symbol && which is the “logical AND operator’ in Formian, > = > = = Formian has another logical operator, namely, the ‘OR operator’ which is denoted by the compound symbol ||. The use of the OR operator is exemplified in Figs 2.A.14e and 2.A.14f, In general, the canonic parameter of a relection function is a ‘perdicant’, where a ‘perdicant’ is dofined as any meaningful evaluable combination of ‘© relations that may include brevants, © logical operators and © parentheses, International Journal of Space Structures Yol. 16 No. 1 2001 2.A.11 Elementary Retronorms ‘The term ‘retronormic function’, or ‘retronorm’, is used to refer to a function that transforms the ‘normat coordinates’ of a configuration into ‘global Cartesian coordinates’. In this - process, the ‘configuration” remains the same but the “formex’ ‘that represents the configuration will change. Retronomms play a central role in fommex configuration processing and thore are many etronorms that are frequently used in practice, The objective of this section is to describe a group of 12 basic retronorms that are referred to as the ‘elementary retronorms’. These retronorms are described in Table 2.A. xt) b) 13°57 @ it Fig2.A.15 (a) A web-like configuration with a polarnonmat (b) The web-like configuration and the polar normat shown together with the corresponding rs polar coordinate system and the xy global coordinate system, To explain the idea of a retronorm, consider the web- like configuration shown by thick lines in Fig 2.4.15a. This configuration consists of $8 line elements that are connected together at 35 nodal Points. The web-like configuration in Fig 2.A.15a is shown together with a ‘polar normat”. This normat is to be used as the reference system for the formulation of the compret of the configuration. The web-like configuration and the polar normat are shown together with the comesponding 1-s poler coordinate system and the xy global Cartesian coordinate system in Fig 2.A.15b. a7A formex representing the web-like configuration of Fig 2.A.15 relative to the U1-U2 polar normat may be written as B=rinid(6,5,1,1) | (1.152174 id (7,4,1,1) | [1,15 1,2] If this formex is plotted with respect fo the global x- yy coordinate system, the result will be as shown in Fig2.A.16. y 5p Plot of +t E=rinid(6,5,1,1)| 3 (1,4; 2.0 rinid(7,4,1,1)| at (4,45 2,2] x 703° 5° 7 Fig 2.A.16 Plot of E relative to the x-y global Cartesian coordinate system ‘The configuration in Fig 2.4.16 has the same ‘compret? as that in Fig 2.A.15 but the ‘normic’ properties of these two configurations are quite different, see Section 1.4.4 of Ref 1. To clearly see the relationship between the configurations of Figs 2.4.15 and 2.A.16, the letters A,B, C and D are used to indicate the corresponding comers of the configurations. A formex -whose uniples are the global xy coordinates of the nodal points of the web-like configuration of Fig 2.4.15 may be written as G=t9(1.5,30)|E ‘The effect may be described as follows: If [U1,U2) i signet of E, with Ul and U2 being the normat coordinates of a nodal point of the configuration of Fig2.A.15, then [xy] will be the corresponding signet of G, with x and y being the global coordinates of the same nodal point. ‘The transformation is effected through the function ‘bp(1.5,30) ‘This is a. ‘basipolar’ retronorm which has two ‘canonic parameters. The general form of a basipolar retronorm is bp(b1,2) where, ‘bp’ stands for ‘basipolar” and where bl specifies the ‘scale factor’ in the first direction and “& Formex Configuration Processing IT 2 specifies the ‘scale factor’ in the second direction. ‘The scale factor in the first direction is ‘linear’ and: the scale factor in the second direction is ‘angular’. In the case of the basipolar retronorm for the ‘example of Fig 2.A.15, that is, ‘bp(1.5,30) the first canonic parameter is given as 1.5, indicating that the scale factor for transforming the coordinates, along U1 into those along the radial polar axis r is 1.5, The second canonic parameter of the retronorm is given as 30, This indicates that a division along the second normat direction U2 corresponds to 30° on the circumferential polar axis s. In general, the effect of a basipolar retronoimh is to find the r-s polar coordinates of the nodal points of a configuration from the U1-U2 normat coordinates QO using the scale factors b1 and b2 and then find the equivalent global x-y coordinates using the standard relations x=rcoss y=rsins ‘These relations, in terms of the U1-U2 normat coordinates will be ofthe form IxU1xc0s (62x02) y= blxUlxsin (b2xU2) ‘The prefix ‘basi? in the term ‘basipolar’ implies that the divisions along the U1 and U2 directions are ‘uniform’. Six of the retronorms in Table 2.4.9 have the prefix ‘basi’. Each of these retronorms has a corresponding refronorm with a different prefix, namely, the prefix ‘mtr’ that implies ‘rhythmically” increasing or decreasing. ‘rhythm’ is governed by a ‘geomettic progression’ ¢ YO To elaborate, consider the sequence of the divisions along an axis, as shown in Fig 2.A.17. Here, the first division in the sequence is equal to b and the subsequent divisions are equal to bxm, bm’, bem’, . as shown in the figure. Tf ml then the divisions sly larger as n increases and if mas become progressively smaller as n increases. The terms ‘b’ and ‘m’ are referred to as the ‘basifactor’ end ‘metrifuctor’, respectively, bym_,_ by? o.4 2 3 4 bxnt Fig 2.A.17 Divisions along an axis, as implied by the prefix ‘metr!’ International Journal of Space Structures Vol. 16 No. 1 2001Hoskyar Nooshin and Peter Disney ‘With a ‘thythm’ of the divisions along an axis as shown in Fig 2.4.17, the ‘sum’ of the divisions between O and nis given by bd -m')/(1-m) This represents the sum of the first n terms of the geometric progression b, bxm, bum’, bxrn?,.. Formian has a (aumeric) function thet can be used to find the above som. The function is referred to as the ‘metril’ funtion, In terms of this function, the sum of the first n terms of the above geometric progression may be written as bemet(m) Jn ‘The construct met(m) is a ‘metril fanction’ with ‘met’ being the imprint of the function and with the canonic paremmeter m being the metrifactor. In general, the canonic parameter m and the argument n of @ metril function are numeric expressions, and the value of smnet(n) |[m is determined as follows: * Ifm= 1 orn=0 then met(m) |n is equal to n. * Ifm=#1andn>0 then met) | n is equal to (1=m)/(1-m) © [fms 1 and <0 then met(m) | n is equal to -(-m")/(1-m) One of the retronorms with a ‘metri’ prefix in Table 2.A9 is the ‘metripolar fetronorm’. To exemplify the application of this retronorm, let it be required to create the web-like configuration whose details are given in Fig 2.A.18. In this configuration the divisions in the radial direction have a ‘metri- rhythm’ with a metrifactor of 1.2. Also, the divisions in the circumferential direction have a metri-thythm with « metrifactor of 0.8, Therefore, the divisions in the radial direction become progressively larger and the divisions in the circumferential direction become progressively smaller. To formulate the configuration, it will be convenient to use the UJ-U2 normat that follows the rhythms of the variations of the divisions of the configuration in the radial and circumferential directions, as shown in Fig 2.A.18. In terms of this normat, the configuration is represented by the formex E given in Fig 2.A.16. International Journal of Space Structures Vol, 16 No. 1 2002 Us) 3 xo) 13 587 8 na Fig2.A.18 A configuration with a metri- shythmically varying element sizes The web-like configuration of Fig 2.4.18, relative to the global x-y coordinate system, may then be represented by G=mp(1,30, 1.2,0.8) |B ‘The construct mp(1,30, 1.2,0.8) is a ‘metripolar* retronorm with ‘mp’ being the imprint of the function. The first two canonic parameters of a metripolar retronorm are the ‘basifactors’ specifying the first divisions in the UL and U2 directions, respectively. The remaining two canonic parameters are the ‘metrifactors” specifying the ‘metri-thythms’ along the Ul and U2 directions, Tespectively. Now, tuming the attention to Table 2.A.9, the first six elementary retronorms that are described in the table are for use in relation to Cartesian-type norrnats. This group of retronorms consists of: © ‘basiunifect’ retronorm that effects uniform scaling in the first direction, © ‘metriunifect’ retronorm that effects metri- shythmic scaling in the first direction, * ‘basibifect’ . retronorm that effects uniform scaling in the first and second directions, © ‘metribifect’ retronorm that effects metri- thythmic scaling in the first and second directions, © ‘basitrifect’ retronomm that effects uniform scaling in the first, second and third. directions and © ‘metrtrifect’ retronorm that effects .metri- thythmic scaling in the first, sccond and third _ directions. A mumber of examples involving the use of basibifect and basitrifect retronomms are found in Sections 1.4.4, 1.4.6 and 1.7 of Ref 1, oBrief descriptions of retronorms: Basiunifect Retronorm A ‘basiunifect retronorm’ is of the form bub) ‘The effect of a basiunifect retronorm is to replace the 1* uniple Ul of every signet of its argument by Table 2.4.9 Elementary Retronorms Formex Configuration Processing IT ‘The effect of @ metriunifect retronorm is to replace the 1" uniple U1 of every signet of its argument by ‘bxU1 TTT Ti ‘The canonic parameter b is a numeric expression. Plot of bu(1.5) |E* “Metriunifect Retronorm Yer, § ju A ‘metriunifect retronorm’ is of the form 5 mu(bm) 3 ts 8 7 8 m1 AL bemet{m) [UL 5 ‘The canonic parameters b and m are numeric expressions. Plot of mu(0.8,1.2) | B* / Basibifect Retronorm ‘A *basibifect retronorm' is ofthe form Uh gu ‘bb(b1,b2) uz The effect of a basibifect retronorm is to replace the 1* and 2° uniples Ul and U2 of every signet of its argument by bIxUI and b2xU2 ‘The canonic parameters bi and b2 are numeric expressions. ‘Metribifect Retronorm 4 a ea Plot of bb(1.5,0.9) |E* x pt 3 ut A “netribifect retronorm’ is of the form 5 ua mb(b1,b2, m1,m2) 7 is ‘The effect of a metribifect retronom is to replace the 1* and 2" 1 uniples U1 and U2 of every signet of its argument by + x bixmet(ml) | UL and b2xmet(m2) | U2 Tee a ‘The canonic parameters bl, b2, m1 and m? are numeric expressions. Plot of mb(0.8,1.6, 1.2,0.8) | F ‘Basitrifect Retronorm 2 (vs) A ‘basittifect retronorm’ is of the form ua ‘bi(b1,b2,63) The effect of a basitrifect retronorm is to replace the 1", 2“ and 3"| “mS usiples U1, U2 and U3 of every signet of its argument by bixUl, b2xU2 and b3xU3 ‘The canonio parameters b1, b2 and b3 are numeric expressions. “Metritrifect Retronorm A “nettitrifect retronomm’ is of the form smt(b1,b2,b3, m1,m2,m3) ‘The effect of a metritrifect retronorm is to replace the 1", 2 and 3 uniples U1, U2 and U3 of every signet of its argument by blxmet(mt) |U1, b2xmet(m2)|U2 and b3xmet(m3) | U3 The canonic parameters b1,b2,b3, ml, m2 andm3 are numeric expressions. Plot of b4(1.2,1.5,1.3) | ‘yerat(0,0,35) |:pan(3,0)"* | B* 2 (ua) oy Loo x y Plot of mi(1,1,1, 1.05,1.3,2.5) | (0,0,30) % Eis the formex given in Fig 2.A.16, sx This is a pansion function, as described in Section 2.A.13. 50. International Journal of Space Structures Vol. 16 No. 1 2001Hoshyar Noashin and Peter Disney ‘Table 2.A.9 Elementary Retronorms (Continued Basipolar Retronorm A ‘basipolar retronorm’ is of the form : ‘ bp(bl,b2) 3, The effect of a basipolar retronorm is to replace the 1 and 2" uniples | 3 Ul and U2 of every signet ofits argument by the global x-y coordinates | _ f 1 corresponding to the r-s polar coordinates 7 r=bIxUL and sb2xU2 toss 7 8 it ‘The canonic parameters bl and b2 are numeric expressions, Plot of bp(1.5,8) | EX ‘Metripolar Retronomm YE ue) A ‘metripolar retronorma’ is of the form i amp(b1,b2, ml,m2) ‘The effect of a metripolar retronorm is to replace the 1® and 2" uniples i Ul and U2 of every signet of its argument by the global x-y coordinates OSS > x) corresponding to the r-s polar coordinates rblxmet(ml) |UL and s-boxmet(m2) | U2 rao The canonic parameters bl, b2, ml and m2 are mumeric expressions. Plot of mp(1,20, 0.9,1.2) | E* Basicylindrical Retonom A ‘basioylindrical retronorm’ is of the form bbo(b1,b2,63) ‘The effect of a basicylindrical retronarm is to replace the 1%, 2 and 3¢ uniples Ul, U2 and U3 of every signet of its argument by the global x-y-z coordinates corresponding 1o the r-5-z cylindrical coordinates tb1xUI, s=b2xU2 and z=b3xU3 “The canonic parameters b1, b2 and b3 are numeric expressions. Plot of be(1,15,3) | pan(1,10)"* |B* Metricylindrical Retronorm A ‘motricylindrical retronorm’ is of the form mo(b1,b2,b3, m1,m2.m3) ‘The effect of a metricylindrical retronorm is to teplace the 1, 2 and 3° uniples Ul, U2 and U3 of every signet of its argument by the global x-¥-2 coordinates corresponding to the r-s-z cylindrical coordinates x(t)~_ Jv Fblxmet(ml) | Ul, s-b2xmet{n2)|U2 and zb3xmet(m3)|U3 | prot of m6) The canonie parameters b1, b2, b3, m1, m2 and m3 are numeric expressions, meo(1,25,1, 1,0.8,1.5) | pan(1,10)** | B* Basispherical Retronorm F A *basispherical retronorm’ is of the form bs(bl,b203) ne WS The effect of a basispherical retronorm is to replace the 1%, 2" and 3 uniples Ul, U2 and U3-of every signet of its argument by the global x-yrz coordinates corresponding to the r-st spherical coordinates MY rb1xUI, s-b2xU2 and b3xU3 x(uaz) y ‘The canonic parameters bl, b2 and b3 are numeric expressions. Plot of bs(1,30,5) |pan(1,10)** EX “Mettrispherical Retronomn A ‘metrispherical retronorm’ is of the form A ‘ms(b1,b2,b3, ml,m2,m3) vst) ‘The effect of e metrispherical retronorm is to replace the 1", 2" and 34 uniples U1, U2 and U3. of every signet of its argument by the global U2 (s) x%-y-z coordinates corresponding to the r-s-t spherical coordinates xt) > rbbemet(m1) | Ul, sb2xmet(m2)|U2 and t+b3xmet(m3)|U3 | Plot of ‘The canonic parameters bl, b2, b3, ml, m2 and m3 are mumeric expressions. | ms(1,30,3, 1,0.9,1.2) pan(1,10** | B* % Eis the formex given in Fig 2.A.16, x Thisis a pansion fimotion, as desoribed in Section 2.A.13. z. International Journal of Space Structures Vol. 16 No.1 2001 SI