Matrix Method of Structural Analysis 2/22/2021
Bahir Dar Institute of Technology
Department Of Civil Engineering
(Structural Engineering Stream)
Matrix Methods of Structural Analysis
1.Introduction
Seyfe N.
February 22, 2021
Introduction
1.1. Introduction to Matrix Methods of Structural Analysis
Structural analysis :- determination of response (forces and displacements) of the
structure subjected to loads.
The rapid development of computers and the need for complex and light weight
structures lead to the development of matrix methods of structural analysis.
The analysis procedure can be concisely written using matrix notations and are suitable
for computer programming.
Classical methods (system approach of analysis) ; method of consistent deformation,
slope-deflection methods, etc., that consider the behavior of the entire structure for
developing equations necessary for analysis which are difficult and time-consuming
for larger structures.
Matrix methods of analysis (element approach of analysis) ;the response of the whole
structure is determined using the behavior of the elements or members from which the
structure is made of.
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1.1. Introduction to Matrix Methods of Structural Analysis
The two methods of matrix structural analysis,
Flexibility method (the force method) and Stiffness method ( the displacement method)
The flexibility method of analysis will be discussed in Chapter 2 and stiffness method of
analysis will be discussed in Chapters 3.
In this chapter, the important terms and their definitions used in matrix methods of
analysis will be discussed.
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1.2. Framed Structures
A structure is defined as an assemblage of structural elements that can
withstand load.
Structures can be modeled as ;
Modeled as one-dimensional line elements
• When two dimensions (cross-sectional dimensions) of the elements are very small
compared to the third dimension (length of element),
• Modeled as two-dimensional surface elements
• Elements having one dimension (thickness) very small compared to the other two
dimensions.
Modeled as three-dimensional brick elements
When all the dimensions of the element are of comparable length.
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1.2. Framed Structures
A structure composed of only line elements is known as a framed structure or skeletal
structure.
In a framed structure, joints represent the point of intersection of elements, these joints
can be supports(fixed hinged , roller, or guided-fixed ,elastic (or spring)), or free end of
elements
The framed structures can be grouped into six categories,
(i) beam, (iv) plane frame,
(ii) plane truss, (v) grid, and
(iii) space truss, (vi) space frame
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1.2. Framed Structures
(i) Beam (ii) Plane truss
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1.2. Framed Structures
(i) Beam (ii) Plane truss
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1.2. Framed Structures
(iii) space truss (iv) plane frame
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1.2. Framed Structures
(v) Grid (vi) space frame
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1.2. Framed Structures
Deformations in Framed Structures
• When a structure is acted upon by loads, the members of the structure will undergo
deformations (or small changes in shape), therefore points within the structure will be
displaced to new positions.
• All points of the structure except immovable points of support will undergo such
displacements.
• The calculation of these displacements is an essential part of structural analysis.
• The deformations are called axial, shearing, flexural, and torsional deformations
• Their evaluation is dependent upon the cross-sectional shape of the bar and the
mechanical properties of the material.
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1.3. Coordinate Systems for Forces and Displacements
In order to define a force or displacement, it is necessary to specify
(i) the location on the structure where the force is applied or displacement is measured,
(ii) The magnitude, and
(iii) Direction
Force vector Displacement vector
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1.4. Nodes , Elements and Nodal Degrees of Freedom
The structure ,an assembly of elements, the elements are connected to one other at a
finite number of points known as nodes.
In a framed structure, each element has only two nodes.
plane truss, beam, plane frame
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1.4. Nodes , Elements and Nodal Degrees of Freedom
The degrees of freedom (DOF) of a node are the number of independent displacements
that the node can undergo.
plane truss beam plane frame
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1.4. Nodes , Elements and Nodal Degrees of Freedom
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1.5. Global and Local Coordinate System
For the analysis of structures, two different coordinate systems for forces and displacements
are used.
A single global coordinate system (X, Y, and Z are corresponding global axes) is used to
specify the forces (P) and displacements (D) at the nodes of the structure.
A local coordinate system (x, y, and z are the corresponding local axes) will be used for
each element to indicate the element forces (p) and the corresponding displacements (d).
Global and local coordinates plane truss
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1.5. Global and Local Coordinate System
Global and local coordinates
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1.6. Specification of Geometry of the Structure
The geometry of the structure is developed using the following details:
a) Nodal coordinate data—From this, the coordinates of all the nodes are obtained.
b) Element connectivity data—the details of the elements and the nodes to which elements are
connected.
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1.7.Equivalent Nodal Loads
In the matrix method of analysis, it is assumed that the loads act only at the nodes.
Distributed loads or other types of loads acting on the elements must be replaced by
equivalent loads acting at the nodes.
The loads grouped into two categories, nodal loads P and element loads.
The element loads are replaced by equivalent nodal loads Pe.
Combined nodal loads Pc = P + Pe.
The principle used to find the equivalent nodal load Pe is that the nodal
displacements obtained by analyzing the structure subject to combined nodal
loads Pc will be equal to displacement of the structure caused by actual loads.
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1.7.Equivalent Nodal Loads
The procedure for finding Pe and Pc for two-span continuous beam subjected to
nodal and element loads
a) two-span continuous beam structure
with loads
(b) FBD of the structure subjected to
actual loads
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1.7.Equivalent Nodal Loads
(c) discretized structure,
(d) nodal loads
(e) element loads,
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1.7.Equivalent Nodal Loads
Consider the structure subjected to element loads (Figure e), the structure is restrained
against all nodal displacements (DOF = 0) by introducing nodal restraints (i.e., all the nodes
of the structure are fixed) as shown in Figure f.
The support reactions of the restrained structures are found out. Since all the nodes are
fixed, the support reactions are also known as fixed end actions PF.
(f) restrained structure subjected to
element loads
(g) fixed end actions for elements 1
and 2,
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1.7.Equivalent Nodal Loads
(h) fixed end actions for
restrained structure
(i) Equivalent nodal loads
(j) structure subjected to
combined loads
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1.7.Equivalent Nodal Loads
the nodal displacements due to combined loads will be the same as due to
the real loads.
To observe this, consider the superposition of the nodal displacements of the
structures shown in Figures h and j.
This gives the nodal displacements of the structure subjected to real loads
(Figure a).
However, in Figure h, since all the nodes are fixed, the nodal displacements
in this case are zero.
Hence, the nodal displacements of the structure subjected to combined
loads will be equal to the nodal displacements caused by actual loads.
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1.7.Equivalent Nodal Loads
FBD of structure Fixed end actions for
structure
subjected to
+ restrained structure(h) = subjected to real
combined loads loads
+ =
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Assignment one( For 22-02-2021 )
Determine the nodal loads required for analyzing the continuous beam and plane
frame shown in Figure 1 and 2
Figure 1
Figure 2
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1.8. Kinematic and Static Indeterminacy
Depending on the method of analysis, there are two types of indeterminacy,
namely
1. Degree of static indeterminacy (DSI)
2. Degree of kinematic indeterminacy (DKI) and
When the flexibility method (or force method) is used for analysis, When
actions are the unknowns in the analysis, it becomes necessary to consider
degree of static indeterminacy (DSI).
If the stiffness method (or displacement method) is used, then determination of
degree of kinematic indeterminacy (DKI) is necessary.
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1.8. Kinematic and Static Indeterminacy
1.8.1. Degree of Static Indeterminacy (DSI)
The excess of unknown actions (internal actions and external reactions), as compared to
the number of equations of static equilibrium that are available at the joints.
The number of such equations for each joint depends upon the type of structure.
If these equations are sufficient for finding all actions, both external and internal, then
the structure is statically determinate.
If there are more unknown actions than equations, the structure is statically
indeterminate.
The unknown actions in excess of those that can be found by static equilibrium are
known as static redundants, and the number of such redundants represents the degree of
static indeterminacy of the structure.
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1.8. Kinematic and Static Indeterminacy
1.8.1. Degree of Static Indeterminacy (DSI)
Number of Number of unknown Number of joint
= -
redundants actions equilibrium equations
• Zero - the structure is statically determinate;
• Negative - a mobile structure (Unstable )
• Positive - the structure is statically indeterminate
A structure can be statically indeterminate externally (in this case excess supports will be
present) or statically indeterminate internally (due to excess elements) or both (due to
excess supports and elements).
In flexibility method, forces are treated as unknowns, DSI is very important.
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1.8. Kinematic and Static Indeterminacy
1.8.1. Degree of Static Indeterminacy (DSI)
The beam is externally statically indeterminate to the first degree.
The truss is determinate from the standpoint that we could calculate the reactions given the
loads applied.
However, we would be unable to find the internal forces in the cross members.
The truss is internally indeterminate to the second degree.
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1.8. Kinematic and Static Indeterminacy
1.8.2. Degree of Kinematic Indeterminacy (DKI) Unconstrained DOF
The nodal DOF can be grouped into two,
Unconstrained (active or free) DOF are the unknown
nodal displacements or unknown nodal DOF but the
nodal forces corresponding to these DOF are known.
The nodal forces are the loads acting on the structure which
are known.
Constrained (restrained ) DOF are the known DOF
(they are the restrained DOF at the supports) ,but the
corresponding nodal forces are unknown.
The nodal forces are the support reactions which are not
known.
Constrained DOF
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1.8. Kinematic and Static Indeterminacy
1.8.2. Degree of Kinematic Indeterminacy (DKI) Unconstrained DOF
In the stiffness method of analysis, the nodal
displacements are the unknown quantities therefore the
unconstrained DOF are the basic unknowns.
The number of unconstrained DOF or active DOF of a
structure is the DKI of the structure.
Constrained DOF
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1.8. Kinematic and Static Indeterminacy
1.8.2. Degree of Kinematic Indeterminacy (DKI)
Number of degrees Number of possible Number of restraints
= -
of freedom joint displacements
For the truss
The Number of possible joint displacement = 4 * 2 =8
Number of restraints = 4
Degree of Kinematic Indeterminacy =8 - 4 = 4
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1.8. Kinematic and Static Indeterminacy
Support and Reactions
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1.8. Kinematic and Static Indeterminacy
Example
Determine the DSI of the
structures shown in Figure.
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1.8. Kinematic and Static Indeterminacy
Example
Determine the DSI of the
structures shown in Figure.
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1.8. Kinematic and Static Indeterminacy
Example
Determine the DSI of the
structures shown in Figure.
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1.8. Kinematic and Static Indeterminacy
Example
Determine the DSI and DKI
of the structures shown in
Figure.
Each of the joints D, E, and F
has three degrees of freedom for
joint displacement,therefore, the
truss is kinematically
indeterminate to the ninth
degree.
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1.8. Kinematic and Static Indeterminacy
Example
Determine the DKI of the
structures shown in Figure.
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1.8. Kinematic and Static Indeterminacy
Example
Determine the DKI of the
structures shown in Figure.
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1.8. Kinematic and Static Indeterminacy
Example
Determine the DKI of the
structures shown in Figure.
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1.8. Kinematic and Static Indeterminacy
ASSIGNMENT TWO (22-02-2021)
Determine the DSI and DKI of the
framed structures shown in Figure
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1.9. Structural Mobilities
Both the supports and the members of any structure must be adequate in number and in
geometrical arrangement to insure that the structure is not movable
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1.10. Principle of Superposition
The principle of superposition will be valid whenever
linear relations exist between actions and displacements
of the structure.
This occurs whenever the following three requirements
are satisfied:
1. The material of the structure follows Hooke's law;
2. The displacements of the structure are small; and
3. There is no interaction between axial and flexural
effects in the members.
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Introduction
1.9. Methods of Structural Analysis
Any method in structural analysis uses the three basic equations in mechanics:
1.Equilibrium Equations
If the body is in equilibrium, then as per Newton’s first law of motion, the resultant of the force system
should be equal to zero, i.e., FR = 0 and MR = 0. These equations are known the equations of
equilibrium.
2. Compatibility of Displacements
These conditions refer to the continuity of displacement throughout the structure. The compatibility
conditions are usually considered at the joints in a structure.
3. Force–Displacement Relations
In the case of a linearly elastic structure, the force–displacement relation is linear. Depending on the
method of analysis, the force–displacement relation is expressed either in a flexibility or stiffness
format.
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1.9. Methods of Structural Analysis
Flexibility and Stiffness Method of Analysis
All the methods of analysis of structures fall under two categories, the flexibility method (or
force method) and the stiffness method (or displacement method).
1. Flexibility method
Forces are taken as the basic unknowns
A statically indeterminate structure is made statically determinate (DSI = 0) by removing
the excess or redundant forces (these forces are the basic unknowns and the number of
unknown forces is equal to DSI)
The statically determinate and stable structure thus obtained is known as the primary
structure.
The primary structure is then subjected to external forces (loads) and redundant forces, and
compatibility conditions are used to find the redundant forces.
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1.9. Methods of Structural Analysis
Flexibility and Stiffness Method of Analysis
2. Stiffness method
Displacements are chosen as the basic unknowns.
A kinematically indeterminate structure is made kinematically determinate (DKI = 0) by preventing
all nodal displacements.
These nodal displacements are the basic unknowns and the number of unknown is equal to DKI.
Then the unknown displacements are allowed, and equilibrium equations (using the FBD of nodes)
are used to find the unknown displacements.
Choice of the method
The choice of the method of analysis for a structure depends on the number of unknowns.
If the DSI of the structure is less than DKI, then the flexibility method may be used and vice versa.
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1.9. Methods of Structural Analysis
Flexibility and Stiffness Method of Analysis
Which one is appropriate for computer programming?
Flexibility Method – no unique primary structures , Difficult for computer programming
Statically indeterminate structure.
Possible primary structures.
Stiffness method, the choice of unknown displacements is unique for a structure and the
whole procedure is systematic and is amenable to computer programming.
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