Structural Engineering Lab Manual

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Submitted To: _______________________

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Civil Engineering Department

University of Engineering and Technology,
Taxila.
 Experiment no 02
Comparison of Experimental and Theoretical results of two Degree Statically
Indeterminate beam using Direct Flexibility Method.

1. Introduction

Statically Indeterminate Structures:

• These are the structures that cannot be analyzed using equations of equilibrium only. Or we can say
that if we have reaction more than the equation of equilibrium then the structure

• To solve indeterminate systems, we must combine the concept of equilibrium with compatibility.

For a coplanar structure there is at most three equilibrium equations for each part, so that if there is a total
of n parts and r force and moment reaction components.

r = 3n, statically Determinate structure

r > 3n, statically In-Determinate structure

Examples of 2D indeterminate structure:

What is flexibility?

Flexibility of a member is defined as deformation produced by a unit load.

Advantage of indeterminate structure over determinate structure

For a given loading the maximum stress and deflection of an indeterminate structure are generally smaller
than those of its statically determinate structure
Comparison between determinate and indeterminate structure:

Structural Analysis Methods:

There are two methods to analyze the structural members i.e. Beams, frames etc

1. Flexibility or Force analysis method

2. Displacement or Stiffness method

1. Flexibility or Force analysis method

• In this method redundant constraints are removed and corresponding redundant forces (moment)
are placed.

• An equation of compatibility of deformation is written in terms of these redundants and

corresponding displacements.

• The redundants are determined from these simultaneous equations. Equations of statics are then
used for calculation of designed internal actions. In this method, forces are treated as basic
unknowns.

2. Displacement or Stiffness method

• In this method, the rotation or the nodal displacements are treated as unknown. These are then
related to corresponding forces.

Flexibility or Force analysis method:

This method was originally developed by James Clerk Maxwell in 1864 and later refined by Otto Mohr
and Heinrich Muller .In this method by using known forces we can find unknown reactions. Compatibility
equations are written for displacement and rotations. By solving these equations, redundant forces are
calculated and remaining reactions are evaluated by equations of equilibrium.

Flexibility Method of
Analysis

Indirect Direct

Procedure/Steps of Analysis

1) Division of structure into elements.

2) Formation of element flexibility matrix for all elements.
3) Formation of composite flexibility matrix by combining all elements flexibility matrices.
4) Formation of force transformation matrix.
5) Formation of structure flexibility matrix.
Theoretical calculation procedure:

 The structure is divided into number of elements. The element

consist of the segments with constant cross-sections between the applied loads and points of required
moments. If necessary each segment can also be divided into more than one element.
 The basic variables in the flexibility method are a set of number of independent member forces. A
flexural element of a plane structure actually has six forces acting on it an axial force, shear force & a
moment at each end.

 However axial deformations are usually negligible in a flexural member, therefore only two element
forces remain at each end. But shear forces can be expressed in terms of materials and are not
considered as independent forces.
  Therefore only two out of the remaining four forces are independent. The terms “FORCE” will be
used for moment as well as force. Similarly term deflection or deformation will be used to express
both rotation and translation.

Rotation can be calculated using moment area method, unit load method etc.
Observation and Calculations:
Results:

Remarks:
 EXPERIMENT # 3

Theoretical Investigation of a two Degree Statically Indeterminate beam using

Indirect Flexibility Method.

1. Introduction
Determinacy

The equilibrium equations provide both the necessary and sufficient conditions for equilibrium. When all
the forces in a structure can be determined strictly from these equations, the structure is referred to as
statically determinate. Structure having more unknown‟s forces than available equilibrium equations is
called statically indeterminate structure.

As a general rule, a stable structure is identified as being either statically determinate or indeterminate by
drawing free body diagram of all its members and then comparing its total number of unknown reactive
forces and moment components with the total number of available equilibrium equations.

For a coplanar structure there is at most three equilibrium equations for each part, so that if there is a total
of n parts and r force and moment reaction components.

Examples
Most of the structures designed today are statically indeterminate. For example, reinforced concrete
buildings are almost always statically indeterminate since the columns and beams are poured as continuous
members through the joints and over support.

Advantage of indeterminate structure over determinate structure

For a given loading the maximum stress and deflection of an indeterminate structure are generally smaller
than those of its statically determinate structure
2) Method of Analysis
Requirements to Analyze Indeterminate structures

When analyzing any indeterminate structure, it is necessary to satisfy equilibrium, compatibility and
force displacement requirements for the structure.

1. Equilibrium is satisfied when the reactive forces hold the structure at rest.

2. Compatibility is satisfied when the various segments of structure fit together without intentional
breaks or overlap

3. The force displacement requirements depends upon the way the structural material responds to
loads

Methods of Analysis

In general there are two different ways to satisfy these three requirements.

1. Force or Flexibility Method

2. Displacement or Stiffness Method

Force method

This method was originally developed by James Clerk Maxwell in 1864 and later refined by Otto Mohr
and Heinrich Muller .In this method by using known forces we can find unknown reactions. Compatibility
equations are written for displacement and rotations. By solving these equations, redundant forces are
calculated and remaining reactions are evaluated by equations of equilibrium.
3) Theoretical Calculations

Solve the following frame theoretically by using direct flexibility method

The load P=2N acts at L/3 from left and right support.

The thickness of frame is 2cm and frame is made up of cast iron. Calculate EI value by yourself,
 Experiment No: 4

Comparison of Experimental and Theoretical Results of a two Degree Statically

Indeterminate beam using Indirect Flexibility Method.

Apparatus:

 Beam
 weights
 Moveable Dial gauge
 Measuring Scale

Theory:

For determinate structures, the force method allows us to find internal forces irrespective of the material
information. Material (stress-strain) relationships are needed only to calculate deflections. However, for
indeterminate structures, Statics (equilibrium) alone is not sufficient to conduct structural analysis.
Compatibility and material information are essential.

Indeterminate Structures:

Number of unknown Reactions or Internal forces > Number of equilibrium equations

Note: Most structures in the real world are statically indeterminate.

Advantages

 Smaller deflections for similar members

 Redundancy in load carrying capacity (redistribution)
 Increased stability

Disadvantages

 More material => More Cost

 Complex connections
 Initial / Residual / Settlement Stresses

Procedure for theoretical calculation:

 Determine the number of degrees of indeterminacy.

 Specify the number of redundant forces or moments which must be removed to make the structure
determinate.
 Draw S.I.S and show it to be equal to a sequence of corresponding S.D.S.
 The primary structure supports the same external loads as the S.I.S., and each of other structures
added to the primary structure shows the structure loaded with a separate redundant force or
moment.
 Sketch the elastic curve on each structure and indicate symbolically the displacement or rotation at
the point of each redundant force or moment.
 Write compatibility equation for the displacement or rotation at each point where there is a
redundant force or moment.
 These equations should be expressed in terms of the unknown redundant and their corresponding
flexibility coefficients.
 Determine all the deflections by using moment diagram of primary structure, first and second
redundant, and determine their corresponding flexibility coefficients. For example,
EI . 20 =  M m2 dx
EI . f11 =  m1 m1 dx
EI . f22 =  m2 m2 dx

 Substitute these into the compatibility equations and solve for the unknown redundant.
 If the numerical value for a redundant is negative, it indicates the redundant acts opposite to its
corresponding unit force or unit couple moment.
 Draw a free body diagram of the structure.
 As the redundant forces have been calculated, now calculate the remaining unknown reactions
using equations of equilibrium.
 Now draw the shear and moment diagrams.
Procedure for experimental calculation:

 First of all, place the dial gauges and note down the deflection against the original load.
 Now apply the unit load as a redundant and find out the deflections.
 Put out he values in compatibility equation.
 Compare the experimental calculation with the theoretical calculation.

Calculations:

Sr.# loads (P) Experimental value Theoretical value percentage error

Percentage error = ×100

Precautions:

 Use the digital dial with great care.

 Apply the loads gradually and carefully.
 Don‟t disturb the beam from its position; doing so may alter the results.

Errors:

 Personal error like taking the wrong dimensions.

 Instrumental errors like error in the digital dial or the beam not fully fixed at the free end.

REFRENCES: Structural Analysis by R. C. Hibbeler, 8th edition, chapter 10, article 10.1
 EXPERIMENT # 5

Comparison of Experimental and Theoretical Results of a two Degree Indeterminate

Frame using Direct Flexibility Method.

2. Introduction
Determinacy

The equilibrium equations provide both the necessary and sufficient conditions for equilibrium. When all
the forces in a structure can be determined strictly from these equations, the structure is referred to as
statically determinate. Structure having more unknown‟s forces than available equilibrium equations is
called statically indeterminate structure.

As a general rule, a stable structure is identified as being either statically determinate or indeterminate by
drawing free body diagram of all its members and then comparing its total number of unknown reactive
forces and moment components with the total number of available equilibrium equations.

For a coplanar structure there is at most three equilibrium equations for each part, so that if there is a total
of n parts and r force and moment reaction components.

Most of the structures designed today are statically indeterminate. For example, reinforced concrete
buildings are almost always statically indeterminate since the columns and beams are poured as continuous
members through the joints and over support.
Advantage of indeterminate structure over determinate structure

For a given loading the maximum stress and deflection of an indeterminate structure are generally smaller
than those of its statically determinate structure

2) Method of Analysis
In general there are two different ways to analyze our structure.

3. Force or Flexibility Method

 4. Displacement or Stiffness Method

Force method

This method was originally developed by James Clerk Maxwell in 1864 and later refined by Otto Mohr
and Heinrich Muller .In this method by using known forces we can find unknown reactions. Compatibility
equations are written for displacement and rotations. By solving these equations, redundant forces are
calculated and remaining reactions are evaluated by equations of equilibrium.

5. Experimental procedure
1. As our frame is two degree statically indeterminate with one end with fixed support and other with
a hinge support.
2. First of all we will remove our hinge support and apply a 20N load at the distance of L/3 from left
side which is a fixed support.
3. As we had already attached our gauge with the point D (point of hinge support) so we will note the
deflection produced there due to applied load.
4. After that remove the applied load and attach a 1N load at point D ,first for lateral direction and
then as a gravity load and measure deflection at this point.
5. Now we have both the experimental values from this procedure and theoretical values from
calculations so now we can compare these values.
Table

For 20N load

Experimental (D) Theoretical (D)

For unit load

 Experimental (D) Theoretical (D)

Horizontal

Vertical

Results:

Remarks:
 EXPERIMENT # 6

Comparison of Experimental and Theoretical Results of a two Degree Indeterminate

Frame using Direct Flexibility Method.

3. Introduction
Determinacy

The equilibrium equations provide both the necessary and sufficient conditions for equilibrium. When all
the forces in a structure can be determined strictly from these equations, the structure is referred to as
statically determinate. Structure having more unknown‟s forces than available equilibrium equations is
called statically indeterminate structure.

As a general rule, a stable structure is identified as being either statically determinate or indeterminate by
drawing free body diagram of all its members and then comparing its total number of unknown reactive
forces and moment components with the total number of available equilibrium equations.

For a coplanar structure there is at most three equilibrium equations for each part, so that if there is a total
of n parts and r force and moment reaction components.

Most of the structures designed today are statically indeterminate. For example, reinforced concrete
buildings are almost always statically indeterminate since the columns and beams are poured as continuous
members through the joints and over support.
Advantage of indeterminate structure over determinate structure

For a given loading the maximum stress and deflection of an indeterminate structure are generally smaller
than those of its statically determinate structure

2) Method of Analysis
In general there are two different ways to analyze our structure.

6. Force or Flexibility Method

 7. Displacement or Stiffness Method

Force method

This method was originally developed by James Clerk Maxwell in 1864 and later refined by Otto Mohr
and Heinrich Muller .In this method by using known forces we can find unknown reactions. Compatibility
equations are written for displacement and rotations. By solving these equations, redundant forces are
calculated and remaining reactions are evaluated by equations of equilibrium.

8. Experimental procedure
6. As our frame is two degree statically indeterminate with one end with fixed support and other with
a hinge support.
7. First of all we will remove our hinge support and apply a 20N load at the distance of L/3 from left
side which is a fixed support.
8. As we had already attached our gauge with the point D (point of hinge support) so we will note the
deflection produced there due to applied load.
9. After that remove the applied load and attach a 1N load at point D ,first for lateral direction and
then as a gravity load and measure deflection at this point.
10. Now we have both the experimental values from this procedure and theoretical values from
calculations so now we can compare these values.
Table

For 20N load

Experimental (D) Theoretical (D)

For unit load

 Experimental (D) Theoretical (D)

Horizontal

Vertical

Results:

Remarks:
 EXPERIMENT # 7

THEORETICAL INVESTIGATION OF A TWO DEGREE STATICALLY

INDETERMINATE FRAME USING INDIRECT FLEXIBILITY METHOD
INTRODUCTION:

For determinate structures, the force method allows us to find internal forces (using equilibrium i.e.
based on Statics) irrespective of the material information. Material (stress-strain) relationships are
needed only to calculate deflections.

However, for indeterminate structures, Statics (equilibrium) alone is not enough to conduct structural
analysis. Compatibility and material information are essential.

INDETERMINATE STRUCTURES
Number of unknown Reactions or Internal forces > Number of equilibrium equations
Note: Most structures in the real world are statically indeterminate.

ADVANTAGES

• Smaller deflections for similar members

• Redundancy in load carrying capacity
(redistribution)
• Increased stability

DISADVANTAGES

• Initial / Residual / Settlement Stresses

• Complex connections
• More material => More Cost

METHODS OF ANALYSIS

Structural Analysis requires that the equations governing the following physical relationships be satisfied:

1. Equilibrium of forces and moments

2. Compatibility of deformation among members and at supports
3. Material behavior relating stresses with strains
4. Strain-displacement relations
5. Boundary Conditions
Primarily two types of methods of analysis:

FORCE (FLEXIBILITY) METHOD:

• Convert the indeterminate structure to a determinate one by removing some unknown forces /
support reactions and replacing them with (assumed) known / unit forces.
• Using superposition, calculate the force that would be required to achieve compatibility with
the original structure.
• Unknowns to be solved for are usually redundant forces
• Coefficients of the unknowns in equations to be solved are "flexibility" coefficients.

DISPLACEMENT (STIFFNESS) METHOD:

 Express local (member) force displacement relationships in terms of unknown member

displacements.

 Using equilibrium of assembled members, find unknown displacements.

 Unknowns are usually displacements.

 Coefficients of the unknowns are “Stiffness” coefficients.

MAXWELL'S THEOREM OF RECIPROCAL DISPLACEMENTS; BETTI'S LAW

For structures with multiple degree of indeterminacy

The displacement (rotation) at a point P in a structure due a UNIT load (moment) at point Q is equal
to displacement (rotation) at a point Q in a structure due a UNIT load (moment) at point P.
THEORETICAL CALCULATIONS:

Solve the following frame theoretically by using indirect flexibility method

The load P=20N acts at a distance of L/3 from left support.

Solve, when

a) Support at A is pinned and D is roller.

b) Support at A and D is pinned
c) Support at A is fixed and D is pinned.

The thickness of frame is 2cm and frame is made up of cast iron. Calculate EI value by yourself, if
required.

References

1. Chapter 10. Force Method for Analysis of Indeterminate Structures

2. https://engineering.purdue.edu/~aprakas/CE474/CE474-Ch3-ForceMethod.pdf
 EXPERIMENT NO 8

COMPARISON OF EXPERIMENTAL AND THEORITICAL RESULTS OF A

TWO DEGREE STATICALLY INDETERMIINATE FRAME USING INDIRECT
FLEXIBILITY METHOD

FORCE METHOD:
Developed by James clerk Maxwell in 1864, later refined by Mohr and Muller-B

Method was also referred as compatibility method or method of consistent displacement

Equations are written and solved that satisfy compatibility and force-displacement requirements

It involves the calculation of reaction of the supports and determination of internal action (Normal force,
shear force, and bending moment) within the structure.

FORCE OR FLEXIBILITY METHOD:

In this method redundant constraints are removed and corresponding redundant forces (moment)
are placed. An equation of compatibility of deformation is written in terms of these redundant and
corresponding displacements.

The redundant are determined from these simultaneous equation. Equations of statics are then used for
calculation of designed internal actions. In this method, forces are treated as basic unknowns.

Method of consistent deformation, castigllano‟s 2nd theorem and three moment equations are force
method.

METHODS OF ANALYSIS:
In general there are two different ways to satisfy these three requirements.

9. Force or Flexibility Method

10. Displacement or Stiffness Method

From stress strain relationship.

THEORITICAL PROCEDURE:
1. First of all, we make our indeterminate structure to determinate structure by removing the pin
support.
2. Than by applying the actual load, make moment diagram of the frame.
3. Apply unit load in x and y direction and make the moment diagram.
4. Than by using flexibility chart we find the deflections and forces on the frames.
5. Put the deflection and force values in the matrix equation and find the reactions.
EXPERIMENTAL PROCEDURE:
1. First of all, we make our indeterminate structure to determinate structure by removing the pin
support.
2. Than by applying the actual load, note the value of deflection in x and y direction.
3. Apply unit load in x and y direction and note down the value of deflections.
4. Put the values of the deflections and forces in the computability equations and calculate the values
of the reactions.
5. Compare the experimental results with the theoretical results and note down the percentage of
error if any.
OBSERVATIONS AND CALCULATIONS:

EXPERIMENTAL THEORITICAL

Δ10

Δ20

f11

f12

f21

f22

R1

R2

Percentage error =

CONCLUSIONS:

Results:

Remarks
 Experiment No. 9
Theoretical Investigation of a Truss using Stiffness Method
Objectives:
 Theoretical background of the stiffness method
 Calculation of the truss structure by using of the stiffness method
Theory:
A truss is an assembly of beams or other elements that creates a rigid structure. In engineering, a
truss is a structure that "consists of two-force members only, where the members are organized
so that the assemblage as a whole behaves as a single object". A "two-force member" is a
structural component where force is applied to only two points.
There are only two types of the forces are produced in the truss member, these two forces are
compressive and tensile forces.
 The force which compress the member is called compressive force.
 The force which try to elongate the member is called tensile force.
Analysis:
The purpose of the analysis is to predict the behavior of the structure due to external loads and
temperature etc.
Types of analysis:
There are two types of the analysis, which are described below:
Force analysis:
This method is used to calculate the internal reaction like shear and moment and reaction on the
supports etc.
Deformation analysis:
This method of analysis is basically first calculated the deformation in the structure and then
from the deformation calculation, the reaction on the supports are calculated.
Stiffness Method:
Stiffness method is basically deformation method. In this method, we basically calculate forces
against unit deformation.
Stiffness:
The stiffness of a member is defined as the force which is to be applied at some point to produce
a unit displacement when all other displacement are restrained to be zero.
Mathematically it can be expressed as
K=W/Δ
In other word Stiffness „K‟ is the force required at a certain point to cause a unit displacement at
that point.
The above equation relates the force and displacement at a single point. This can be extended for
the development of a relationship between load and displacement for more than one point on a
structure

Stiffness method for truss:

First of all divide the truss structure into series of the elements, obtain the stiffness matrix for all
the individual element and from these matrix obtain matrix for the whole structure.

For example:

There are two types of the axis system:

Local axis:
the axis which is obtained separately for all the elements and that‟s different for all the element
fig(b) represents local axis system.
Global axis:
The axis which is obtain related to the whole structure and not for the individual element and
fig(c) represents global axis system.
Stiffness matrix for all the individual elements are given below:

Stiffness matrix of element no.1

Stiffness matrix of element no.2

Stiffness matrix of element no.3

Composite stiffness matrix of all elements is given by:

Relationship between forces and displacement from equation
w = kc Δ (1)

The some of the elements in each column of matrix „kc‟ is zero. It is due to the reason that the
stiffness co-efficient in each column represents the force produced by unit deformation of one
end while other is restrained. Since the member is in equilibrium the sum of the forces must be
zero.
However, all co-efficient along the main diagonal must be positive because these terms are
associated with the force acting at the end at which positive deformation is introduced. As
deformation is positive so force produced is also positive
Stiffness matrix of a structure can be generated from stiffness matrices of the elements into
which a structure has been subdivided. The composite stiffness matrix [kc] describes the force
deformation relationship of the individual elements taken one at a time, whereas structure
stiffness matrix [K] describes the load deformation characteristics of the entire structure. In order
to obtain structure stiffness matrix [K] from composite stiffness matrix [kc] a deformation
transformation matrix is used which is described in the subsequent section.
Following is the relationship between element and structure deformation.
δ=TΔ ------------ (2.18)
where
δ = element deformation
Δ = structure deformation
T = deformation transformation matrix
As work done by structure forces = work done by element forces
The final equation obtained by putting all the values in the equation is given below:

is called deformation transformation matrix. Matrix [T] is usually a rectangular matrix. Its
columns are obtained by applying a unit values of structure deformation Δ 1 through Δ n one at a
time and determining the corresponding element deformations δ1 through δm.
In this method, it is difficult to solve the equation (1). So, Dissolve the matrix into simple matrix.
The lines on the matrix basically separation of the matrices.
[Δ u] = [K11]-1 [Wk]

By using this equation,

[Wu] = [K21] [Δu]
The final values of the unknown forces are calculate, which are given below:
The member forces are calculation by using this equation.

The result of this example is given below:

In this way, this method is basically work.

Civil Engineering applications:

Post frame structures
Component connections are critical to the structural integrity of a framing system. In buildings
with large, clear span wood trusses, the most critical connections are those between the truss and
its supports. In addition to gravity-induced forces (a.k.a. bearing loads), these connections must
resist shear forces acting perpendicular to the plane of the truss and uplift forces due to wind.
Depending upon overall building design, the connections may also be required to transfer
bending moment.

Discussion:
………………………………………………………………………………………
………………………………………………………………………………………
………………………………………………………………………………………
………………………………………………………………………………………
………………………………………………………………………………………
………………………………………………………………………………………
………………………………………………………………………………………
………………………………………………………………………………………
………………………………………………………………………………………
………………………………………………………………………………………
…………………………………………………………
 Experiment No. 10
Comparison of Experimental and Theoretical results of a Truss using
Stiffness Method
Objectives:
 Theoretical background of the stiffness method.
 Calculation of truss by using stiffness method.
 Experimentally obtain results of the truss member.
 Comparison between experimental and theoretical results of a truss.
Theory:
A truss is an assembly of beams or other elements that creates a rigid structure. In engineering, a
truss is a structure that "consists of two-force members only, where the members are organized
so that the assemblage as a whole behaves as a single object". A "two-force member" is a
structural component where force is applied to only two points.
There are only two types of the forces are produced in the truss member, these two forces are
compressive and tensile forces.
 The force which compress the member is called compressive force.
 The force which try to elongate the member is called tensile force.
Analysis:
The purpose of the analysis is to predict the behavior of the structure due to external loads and
temperature etc.
Types of analysis:
There are two types of the analysis, which are described below:
Force analysis:
This method is used to calculate the internal reaction like shear and moment and reaction on the
supports etc.
Deformation analysis:
This method of analysis is basically first calculated the deformation in the structure and then
from the deformation calculation, the reaction on the supports are calculated.
Stiffness Method:
Stiffness method is basically deformation method. In this method, we basically calculate forces
against unit deformation.
Stiffness:
The stiffness of a member is defined as the force which is to be applied at some point to produce
a unit displacement when all other displacement are restrained to be zero.
Mathematically it can be expressed as
K=W/Δ
In other word Stiffness „K‟ is the force required at a certain point to cause a unit displacement at
that point.
The above equation relates the force and displacement at a single point. This can be extended for
the develnt of a relationship between load and displacement for more than one point on a
structure

Stiffness method for truss:

First of all divide the truss structure into series of the elements, obtain the stiffness matrix for all
the individual element and from these matrix obtain matrix for the whole structure.
For example:

There are two types of the axis system:

Local axis:
the axis which is obtained separately for all the elements and that‟s different for all the element
fig(b) represents local axis system.
Global axis:
The axis which is obtain related to the whole structure and not for the individual element and
fig(c) represents global axis system.
Stiffness matrix for all the individual elements are given below:
Stiffness matrix of element no.1

Stiffness matrix of element no.2

Stiffness matrix of element no.3

Composite stiffness matrix of all elements is given by:

Relationship between forces and displacement from equation
w = kc Δ (1)

The some of the elements in each column of matrix „kc‟ is zero. It is due to the reason that the
stiffness co-efficient in each column represents the force produced by unit deformation of one
end while other is restrained. Since the member is in equilibrium the sum of the forces must be
zero.
However, all co-efficient along the main diagonal must be positive because these terms are
associated with the force acting at the end at which positive deformation is introduced. As
deformation is positive so force produced is also positive
Stiffness matrix of a structure can be generated from stiffness matrices of the elements into
which a structure has been subdivided. The composite stiffness matrix [kc] describes the force
deformation relationship of the individual elements taken one at a time, whereas structure
stiffness matrix [K] describes the load deformation characteristics of the entire structure. In order
to obtain structure stiffness matrix [K] from composite stiffness matrix [kc] a deformation
transformation matrix is used which is described in the subsequent section.
Following is the relationship between element and structure deformation.
δ=TΔ ------------ (2.18)
where
δ = element deformation
Δ = structure deformation
T = deformation transformation matrix
As work done by structure forces = work done by element forces
The final equation obtained by putting all the values in the equation is given below:
is called deformation transformation matrix. Matrix [T] is usually a rectangular matrix. Its
columns are obtained by applying a unit values of structure deformation Δ 1 through Δ n one at a
time and determining the corresponding element deformations δ1 through δm.
In this method, it is difficult to solve the equation (1). So, Dissolve the matrix into simple matrix.
The lines on the matrix basically separation of the matrices.
[Δ u] = [K11]-1 [Wk]

By using this equation,

[Wu] = [K21] [Δu]
The final values of the unknown forces are calculate, which are given below:
The member forces are calculation by using this equation.

The result of this example is given below:

In this way, this method is basically work.

Apparatus:
 Electric Truss
 Weight hangers
 Weight balance
 weight
  scale
Procedure:
1. First turn on the electronic balance and bring the needle on the electronic meter to 0
until there is no zero error.
2. Find the length of all the members of the truss.
3. Place the weight of 10 N on the hanger and note down the force on the display meter.
4. This force will be considered for the member CD.
5. Now calculate the force member for the CD member by using stiffness method.
6. Find the % difference by using the formula given below:
%age difference= (Experimental value – Theoretical value)/ Theoretical value*100

Diagrams:

Observation and calculation:

Applied load =
Theoretical value of force member =
Experimental value of force member =

Precautions:
 Bring the reading on meter to 0 before taking the readings.
 Do not disturb or put pressure on the apparatus.

Civil Engineering applications:

Post frame structures
Component connections are critical to the structural integrity of a framing system. In buildings
with large, clear span wood trusses, the most critical connections are those between the truss and
its supports. In addition to gravity-induced forces (a.k.a. bearing loads), these connections must
resist shear forces acting perpendicular to the plane of the truss and uplift forces due to wind.
Depending upon overall building design, the connections may also be required to transfer
bending moment.

Discussion:
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References:
 https://en.wikipedia.org/wiki/Truss
 https://www.sciencedirect.com/topics/engineering/stiffness-method
 example is taken from lectures given by sir
 https://en.wikipedia.org/wiki/Truss
 Experiment#11
Study of Models of different types of Bridges and Pre-Stressing

1. What is a bridge
A bridge is a structure which is constructed to avoid obstacles without closing the
way underneath such as a body of water, valley, or road. The purpose of the bridge is to provide
the pass over an obstacle
2. Components of a bridge
Following are the major components of a bridge
 Superstructure
 Substructure
 Foundation
 Span
 Deck
 Beam/Girder
 Bearing
 Pier
 Pier Cap
 Cantilever
 Truss
3. Factors Affecting Structure of the bridge
 Force
 Buckling
 Deformation
 Stability

4. Types of Bridges
The bridges are classified according to:

a) Structure
b) Use/Purpose
c) Materials
a) Classification based on Structures

i) Arch Bridges

Arch bridges have abutments at each end. The

weight of the bridge is thrust into the abutments at
either side. The earliest known arch bridges were
built by the Greeks. These bridges uses arch as a
main structural component (arch is always located below the bridge, never above it). They are
made with one or more hinges, depending of what kind of load and stress forces they must
endure. The arch bridge has great natural strength.

ii) Beam Bridges

These are the oldest and simplest bridge

design consisting of vertical piers and
horizontal beams - e.g. just a simple plank
or stone slab. They are suitable only for
short spans but can used for larger
crossings by adding additional piers

iii) Truss Bridges

These are structures built up by jointing

together lengths of material to form an open
framework - based mainly on triangles
because of their rigidity. They are very
strong and can support heavy loads.

iv) Cantilever Bridges

Cantilever bridges are based on structures that project horizontally into space, supported at
only one end - like a spring board. If two
cantilevers project out from a central pier the
forces are balanced.
Advantages:
more easily constructed at
difficult crossings by virtue of using little or
no falsework.
Disadvantages:
 complex structures and can be difficult to maintain.
v) Suspension Bridges

A suspension bridge (more

precisely, suspended-deck suspension bridge) is a type
of bridge in which the deck (the load-bearing portion)
is hung below suspension cables on vertical
suspenders. The first modern examples of this type of
bridge were built in the early 1800s.[3][4

vi) Cable Stayed Bridges

It may appear to be
similar to suspension bridges, but in fact they are
quite different in principle and in their
construction. There are two major classes of
cable-stayed bridges: Fan type, which are the
most efficient, and Harp or parallel type, which
allow more space for the fixings.

b) Classification based on purpose

Car Traffic bridge–
The most common type of bridge, which is designed for the traffic of
different intensities.
Pedestrian bridge
– Usually made in urban environments, or in terrain where car transport is inaccessible
(rough mountainous terrain, forests, etc.).
Double-decked bridge
Built to provide best possible flow of traffic across bodies of water or rough terrain. Most
often they have large amount of car lanes, and sometimes have dedicated area for train
tracks.
Train bridges
Bridges made specifically to carry one or multiple lane of train tracks.
 Pipelines
Bridges made to carry pipelines across water or inaccessible terrains.
Pipelines can carry water, air, gas and communication cables.
Viaducts
A viaduct is made up of multiple bridges connected into one longer structure.
Ancient structures created to carry water from water rich areas to dry cities.
Commercial bridges
Modern bridges that host commercial buildings such as restaurants

c) Classification by Materials

i. Natural Materials
ii. Wood
iii. Stone
iv. Concrete and Steel
v. Advanced Materials

References
1. Bridges and their types by Prof. A. Balasubramanian Centre for Advanced Studies in
Earth Science, University of Mysore, Mysore.