Force Force and Displacement Matrix Method

Noida Institute of Engineering and Technology,
Greater Noida
Force and Displacement Matrix Method
Aayushi
Assistant Professor
Civil Engg. Department
6/5/2022
1
Unit: 4
Aayushi RCE-502, DOS 1 Unit 4
Subject Name : Design of
Structure I
Course Details : B Tech 5th
Sem

6/5/2022 Aayushi RCE-502, DOS 1 Unit 4 2
Content
 Course Objective
 Course Outcome
 CO-PO & PSO Mapping
 Prerequisite & Recap
 Stiffness Matrix
 Topic outcome
 Objective of topic 1
 Topic outcome & mapping with PO
 Prerequisite & Recap
 Introduction
 Numerical on spring
 Syllabus of unit 4
 Topic objective

Content
 Summary of stiffness matrix method
 The flexibility matrix method
 Topic Objective
 Topic Outcome
 Topic outcome and mapping with PO
 Procedure of flexibility matrix
 Numerical of flexibility matrix
 Matrix in table form
 Summary of flexibility matrix
 Youtube links
 Daily Quiz
 Objective of topic 2
 Introduction

Content
 Old Question Papers
 Expected Questions in University Examination
 Summary
 References
 MCQs
 Weekly Assignment

Objective
1
To impart the principles of elastic structural analysis and behavior
of indeterminate structures.
2
To impart knowledge about various methods involved in the
analysis of indeterminate structures...
3
To apply these methods for analyzing the indeterminate structures
to evaluate the response of structures .
4
To enable the student get a feeling of how real-life structures
behave
5
To make the student familiar with latest computational techniques
and software used for structural analysis. .
6/5/2022
5
Course Objective

Students will be able
CO1 To Identify and analyze the moment distribution in beams and frames by Slope
Deflection Method, Moment Distribution Method and Strain Energy Method.
CO2 To provide adequate learning of indeterminate structures with Muller’s
Principle; Apply & Analyze the concept of influence lines for deciding the
critical forces and sections while designing..
CO3 To learn about suspension bridge, two and three hinged stiffening girders and
their influence line diagram external loading and analyze the same.
CO4 To Identify and analyze forces and displacement matrix for various structural.
CO5 To understand the collapse load in the building and plastic moment formed.
CO6 Apply the concepts of forces and displacements to solve indeterminate structure.
Course Outcome

CO PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12
RCE502.1 3 3 1 2 1 - - - 2 - - 3
RCE502.2 3 2 2 2 3 - - 1 2 - 2 2
RCE502.3 3 - 2 2 1 - - - 2 - 2 2
RCE502.4 3 1 2 2 1 - - - 2 - 2 2
RCE502.5 3 2 2 2 1 - - - 2 - 2 2
RCE502.6 3 2 3 2 1 - - - 2 - 2 3
AVG. 3 2 2 2 1 - - - 2 - 2 2
CO-PO and PSO Mapping
COs PSO1 PSO2 PSO3 PSO4
RCE502.1 2 1 2 3
RCE502.2 2 1 2 2
RCE502.3 2 2 2 2
RCE502.4 2 1 2 3
RCE502.5 2 1 2 3
RCE502.6 2 1 2 3
Avg. 2 1 2 3

Prerequisite and Recap
Basics of strength of material
Basics of engineering mechanics
Basic of Matrix Mathematics

Syllabus of Unit 4
Basic Force and Displacement Matrix method for
analysis of beams, frames and trusses

6/5/2022 10
Objective of Unit 4
Differentiate between the direct stiffness method and the
displacement method. Analysis simple structures by the
direct stiffness matrix and Flexibility matrix.

 Differentiate between the direct stiffness method and the
displacement method.
 Formulate flexibility matrix of member.
 Define stiffness matrix.
 Construct stiffness matrix of a member.
 Analysis simple structures by the direct stiffness matrix.
Topic Objective

 To illustrate the concept of direct stiffness method to obtain the global
stiffness matrix and solve a spring assemblage problem.
6/5/2022 12
Topic Outcomes
Once the student has successfully completed this unit, he/she will be able
to:
 To show how the potential energy approach can be used to both derive
the stiffness matrix for a spring and solve a spring assemblage problem.
 To describe and apply the different kinds of boundary conditions
relevant for spring assemblages.

6/5/2022 13
Objective of Topic
Topic-1 Name
The Stiffness Matrix Method
Objective of Topic-1:
Differentiate between the direct stiffness method and
the displacement method. Analysis simple structures
by the direct stiffness matrix method.

6/5/2022 14
Topic Outcome and mapping with PO
Programme Outcomes (POs)
PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12
TO1 2 1 1 1 1 1
Outcome of Topic-1:
After the successfully competition of this topic students
will be able to Analysis simple structures by the direct
stiffness matrix method.

Degree of static and kinematic
indeterminacy
Basic concepts degree of freedom and
restraints
Generate Matrix
Mathematics Matrix Solving

• This section introduces some of the basic concepts on which the
direct stiffness method is based.
• The linear spring is simple and an instructive tool to illustrate the
basic concepts.
The steps to develop a finite element model for a linear spring
follow our general 8 step procedure.
1. Discretize and Select Element Types - Linear spring elements
2. Select a Displacement Function - Assume a variation of the
displacements over each element.
3. Define the Strain/Displacement and Stress/Strain Relationships -
use elementary concepts of equilibrium and compatibility.
Topic: Introduction to the Stiffness (Displacement) Method

4. Derive the Element Stiffness Matrix and Equations - Define the stiffness
matrix for an element and then consider the derivation of the stiffness matrix
for a linear elastic spring element.
5. Assemble the Element Equations to Obtain the Global or Total Equations
and Introduce Boundary Conditions - We then show how the total stiffness
matrix for the problem can be obtained by superimposing the stiffness matrices
of the individual elements in a direct manner.
The term direct stiffness method evolved in reference to this method.
6. Solve for the Unknown Degrees of Freedom (or Generalized
Displacements) - Solve for the nodal displacements.
7. Solve for the Element Strains and Stresses – The reactions and internal
forces association with the bar element.
8. Interpret the Results
Topic: Procedure of Stiffness Matrix

1. Select Element Type - Consider the linear spring shown below. The
spring is of length L and is subjected to a nodal tensile force, T
directed along the x-axis.
Note: Assumed sign conventions
Continuous…

2. Select a Displacement Function - A displacement function u(x) is
assumed.
u=a1+a2x
In general, the number of coefficients in the displacement function is
equal to the total number of degrees of freedom associated with the
element. We can write the displacement function in matrix forms as:
We can express u as a function of the nodal displacements ui by
evaluating u at each node and solving for a1 and a2.
Continuous…

Continuous….
Solving for a2:
Substituting a1 and a2 into u gives:
In matrix form:
Or in another form:

Where N1 and N2 are defined as:
The functions Ni are called interpolation functions because they
describe how the assumed displacement function varies over the
domain of the element. In this case the interpolation functions are
linear.
Continuous…

Continuous…

3. Define the Strain/Displacement and Stress/Strain Relationships -
Tensile forces produce a total elongation (deformation) delta of the
spring. For linear springs, the force T and the displacement u are
related by Hooke’s law:
where deformation of the spring delta is given as:
Continuous…

4. Step 4 - Derive the Element Stiffness Matrix and Equations - We
can now derive the spring element stiffness matrix as follows:
Rewrite the forces in terms of the nodal displacements:
We can write the last two force-displacement relationships in matrix
form as:
This formulation is valid as long as the spring deforms along the x axis.
The coefficient matrix of the above equation is called the local stiffness
matrix k:
Continuous…

5. Step 4 - Assemble the Element Equations and Introduce
Boundary Conditions
The global stiffness matrix and the global force vector are assembled
using the nodal force equilibrium equations, and force/deformation and
compatibility equations.
where k and f are the element stiffness and force matrices expressed in
the global coordinates.
6. Step 6 - Solve for the Nodal Displacements
Solve the displacements by imposing the boundary conditions and
solving the following set of equations:
Continuous…

7. Step 7 - Solve for the Element Forces
Once the displacements are found, the forces in each element may be
calculated from:
Continuous…

Consider the following two-spring system shown below
where the element axis x coincides with the global axis x.
For element 1:
For element 2:
Both continuity and compatibility require that both elements remain
connected at node 3.
Topic: The Stiffness Method – Spring Example 1

We can write the nodal equilibrium equation at each node as:
Therefore the force-displacement equations for this spring system are:
In matrix form the above equations are:
Continuous…

where F is the global nodal force vector, d is called the global nodal
displacement vector, and K is called the global stiffness matrix.
Assembling the Total Stiffness Matrix by Superposition
Consider the spring system defined in the last example:
The elemental stiffness matrices may be written for each element.
For element 1: For element 2:
Continuous…

Write the stiffness matrix in global format for element 1 as follows:
For element 2:
Apply the force equilibrium equations at each node.
Continuous…

The above equations give:
To avoid the expansion of the each elemental stiffness matrix, we can
use a more direct, shortcut form of the stiffness matrix.
The global stiffness matrix may be constructed by directly adding terms
associated with the degrees of freedom in k(1) and k(2) into their
corresponding locations in the K as follows:
Continuous…

Boundary conditions are of two general types:
1. homogeneous boundary conditions (the most common) occur at
locations that are completely prevented from movement;
2. nonhomogeneous boundary conditions occur where finite non-zero
values of displacement are specified, such as the settlement of a
support.
In order to solve the equations defined by the global stiffness matrix,
we must apply some form of constraints or supports or the structure
will be free to move as a rigid body.
Consider the equations we developed for the two-spring system. We
will consider node 1 to be fixed u1 = 0. The equations describing the
elongation of the spring system become:
Topic :Boundary Conditions

Expanding the matrix equations gives:
The second and third equation may be written in matrix form as:
Once we have solved the above equations for the unknown nodal
displacements, we can use the first equation in the original matrix to
find the support reaction.
Continuous…

For homogeneous boundary conditions, we can delete the row and
column corresponding to the zero-displacement degrees-of-freedom.
Let’s again look at the equations we developed for the two spring
system.
However, this time we will consider a nonhomogeneous boundary
condition at node 1: u1=delta. The equations describing the elongation
of the spring system become:
Expanding the matrix equations gives:
Continuous…

By considering the second and third equations because they have
known nodal forces we get:
In matrix form the above equations are:
For nonhomogeneous boundary conditions, we must transfer the terms
from the stiffness matrix to the right-hand-side force vector before
solving for the unknown displacements.
Once we have solved the above equations for the unknown nodal
displacements, we can use the first equation in the original matrix to
find the support reaction.
Continuous…

 The stiffness coefficients are
defined in this section.
 Construction of stiffness
matrix for a simple member
is explained.
 A few simple problems are
solved by the direct stiffness
method.
 The difference between the
slope-deflection method and
the direct stiffness method is
clearly brought out. .
Summary of Stiffness Matrix Method

 To analysis statically indeterminate structure of degree one.
 To solve the problem by either treating reaction or moment as
redundant.
 To draw shear force and bending moment diagram for
statically indeterminate beams.
 To state advantages and limitations of force method of
analysis.
Topic Objective

 Concept of force method for analysis of statically indeterminate
structure.
6/5/2022 38
Topic Outcomes
Once the student has successfully completed this unit, he/she will be able
to:
 Selection of the basic determinate structure
 Illustration of force method by numerical examples.
 Analysis simple structures by the direct stiffness matrix

6/5/2022 39
Objective of Topic
Topic-2 Name
The Flexibility Matrix Method
Objective of Topic-2:
To analysis statically indeterminate structure by
Flexibility matrix Method.

6/5/2022 40
Topic Outcome and mapping with PO
Programme Outcomes (POs)
PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12
TO1 1 1 1 1 1 1 1
Outcome of Topic-2:
After the successfully competition of this topic students
will be able to analysis of Different beam and Frames
by Flexibility matrix method

Concept of force method for analysis of
statically indeterminate structure.
Selection of the basic determinate
structure.
Illustration of force method by
numerical examples.

• In 1864 James Clerk Maxwell published the first consistent
treatment of the flexibility method for indeterminate structures .
• With the flexibility method equations of compatibility involving
displacements at each of the redundant forces in the structure are
introduced to provide the additional equations needed for solution.
• This method is somewhat useful in analyzing beams, frames and
trusses that are statically indeterminate to the first or second degree.
• For structures with a high degree of static indeterminacy such as
multi-story buildings and large complex trusses stiffness methods
are more appropriate.
• Nevertheless flexibility methods provide an understanding of the
behavior of statically indeterminate structures.
Topic: The Flexibility Matrix

The fundamental concepts that underpin the flexibility method
will be illustrated by the study of a two span beam. The
procedure is as follows
1.Pick a sufficient number of redundants corresponding to the
degree of indeterminacy.
2.Remove the redundants.
3.Determine displacements at the redundants on released
structure due to external or imposed actions.
4.Determine displacements due to unit loads at the redundants on
the released structure.
5.Employ equation of compatibility, e.g., if a pin reaction is
removed as a redundant the compatibility equation could be the
summation of vertical displacements in the released structure
must add to zero.
Topic: The procedure of Flexibility Method

Original
Beam
FBD
Released
Beam
The beam to the left is statically
indeterminate to the first degree.
The reaction at the middle support
RB is chosen as the redundant.
The released beam is also shown.
Under the external loads the
released beam deflects an amount
DB.
A second beam is considered
where the released redundant is
treated as an external load and the
corresponding deflection at the
redundant is set equal to DB.
Continuous…

A more general approach consists in finding the displacement at B
caused by a unit load in the direction of RB. Then this displacement can
be multiplied by RB to determine the total displacement
Also in a more general approach a consistent sign convention for
actions and displacements must be adopted. The displacements in the
released structure at B are positive when they are in the direction of the
action released, i.e., upwards is positive here.
The displacement at B caused by the unit action is
The displacement at B caused by RB is δB RB. The displacement caused
by the uniform load w acting on the released structure is
Continuous…

Thus by the compatibility equation
Continuous…

.
• If a structure is statically
indeterminate to more than one
degree, the approach used in the
preceding example must be
further organized and more
generalized notation is introduced.
• Consider the beam to the left. The
beam is statically indeterminate to
the second degree. A statically
determinate structure can be
obtained by releasing two
redundant reactions. Four possible
released structures are shown.
Topic: Numerical

The redundants chosen are at Band C.
The redundant reactions are designated
Q1and Q2.
The released structure is shown at the left
with all external and internal redundants
shown.
DQL1is the displacement corresponding
to Q1and caused by only external actions
on the released structure
to Q2caused by only external actions on
Both displacements are shown in their
assumed positive direction.
6/5/2022
Aayushi RCE-502, DOS 1
Unit 4
48
Continuous…
The redundants chosen are at Band C.
The redundant reactions are designated
Q1and Q
The released structure is shown at the left
with all external and internal redundants
shown.
to Q1and caused by only external actions
on the released structure
to Q2caused by only external actions on
Both displacements are shown in their
assumed positive direction.

• We can now write the compatibility equations for this structure. The
displacements corresponding to Q1 and Q2 will be zero. These are
labeled DQ1 and DQ2 respectively
• In some cases DQ1 and DQ2 would be nonzero then we would write
Continuous…

The equations from the previous page can be written in matrix format
as
where:
{DQ } - vector of actual displacements corresponding to the redundant
{DQL } - vector of displacements in the released structure
corresponding to the redundant action [Q] and due to the loads
[F] - flexibility matrix for the released structure corresponding to the
redundant actions [Q]
{Q} - vector of redundants
Continuous…

The vector {Q} of redundants can be found by solving for them from
the matrix equation on the previous overhead.
To see how this works consider the previous beam with a constant
flexural rigidity EI. If we identify actions on the beam as
Since there are no displacements imposed on the structure
corresponding to Q1 and Q2, then
Continuous…

The vector [DQL] represents the displacements in the released structure
corresponding to the redundant loads. These displacements are
The positive signs indicate that both displacements are upward. In a
matrix format
The flexibility matrix [F ] is obtained by subjecting the beam to unit
load corresponding to Q1 and computing the following displacements
Continuous…

Similarly subjecting the beam to unit load corresponding to Q2 and
computing the following displacements
The flexibility matrix is
The inverse of the flexibility matrix is
As a final step the redundants [Q] can be found as follows
Continuous…

All matrices used in the flexibility method are summarized in the
following tables
Topic: Matrix in Table form

Continuous…

 Analysis of released structure for unit
values of redundant
 Determination of redundants through
the superposition equations.
 Determine the other displacements
and actions. The following are the
four flexibility matrix equations for
calculating redundants member end
actions, reactions and joint
displacements where for the released
structure
 All matrices used in the flexibility
method.
Summary of Flexibility Matrix Method

Youtube/other Video Links
Youtube Video Links
Topic Links Process
Stiffness matrix
method
https://www.youtube.com/watch?v
=AbvrPVT9OSM
Click on
the link
Flexibility
matrix method
https://www.youtube.com/watch?v
=AbvrPVT9OSM
Click on
the link
Matrix method youtube.com/watch?v=Mlmp9s0tl
gc
Click on
the link

 Flexibility coefficients depend upon loading of the primary
structure.
State whether the above statement is true or false
a) True
b) False
 Define Stiffness coefficient and flexibility coefficient?
 Define Degree of freedom?
 What do you understand by term Redundant?
Daily Quiz

 Develop the flexibility matrix for the cantilever with coordinate as
shown in Fig. , take uniform flexural rigidity.
 Analyze the following continuous beam using the flexibility or
stiffness method of matrix analysis.
 Using the force method, determine the reactions and moments over
supported of a continuous beam shown in Fig.
Daily Quiz

 Analyze the continuous beam shown in Fig. using flexibility method
and draw BMD.
 Using flexibility matrix method , find reaction at supports in
following beam of Fig . Take EI as constant.
Daily Quiz

 Compare flexibility method and stiffness method.
 What are the type of structures that can be solved using stiffness
matrix method?
 List the properties of the stiffness matrix
 Why is the stiffness matrix method also called equilibrium method
or displacement method?
 Calculate the support reactions in the continuous beam ABC due to
loading as shown in Fig.1.1 Assume EI to be constant throughout.
Weekly Assignment

 Analyse the continuous beam as shown in Fig . If the downward
settlement of supports B and C are 10 mm and 5 mm respectively .
Take EI = 184 x 10 " N - mm ' . Use flexibility matrix method
 Analyze the continuous beam shown in fig by stiffness method.
Draw bending moment diagram and elastic curve.
Weekly Assignment

How many compatibility equations should be written if we have n no. of
redundant reactions?
a) n – 1
b) n
c) n + 1
d) n + 2
Flexibility matrix is always:-
a) symmetric
b) non-symmetric
c) anti-symmetric
d) depends upon loads applied
MCQ s

Which of the following primary structure is best for computational
purposes?
a) symmetric
b) non-symmetric
c) anti-symmetric
d) depends upon loads applied
For computational purposes, deflected primary structure ans
actual structure should be ___________
a) as different as possible
b) as similar as possible
c) it doesn’t matter
d) in between
MCQ s

Numerical accuracy of solution increases if flexibility coefficients
with larger values are located:-
a) near main diagonal
b) near edges
c) in between
d) near side middles
 For stable structures, one of the important properties of
flexibility and stiffness matrices is that the elements on the main
diagonal
(i) Of a stiffness matrix must be positive
(ii) Of a stiffness matrix must be negative
(iii) Of a flexibility matrix must be positive
(iv) Of a flexibility matrix must be negative
MCQ s

 The Castigliano's second theorem can be used to compute
deflections
(A) In statically determinate structures only
(B) For any type of structure
(C) At the point under the load only
(D) For beams and frames only
 When a uniformly distributed load, longer than the span of the
girder, moves from left to right, then the maximum bending
moment at mid section of span occurs when the uniformly
distributed load occupies
(A) Less than the left half span
(B) Whole of left half span
(C) More than the left half span
(D) Whole span
MCQ s

 If in a pin-jointed plane frame (m + r) > 2j, then the frame is
(Where ‘m’ is number of members, ‘r’ is reaction components
and ‘j’ is number of joints)
(A) Stable and statically determinate
(B) Stable and statically indeterminate
(C) Unstable
(D) None of the above
 Principle of superposition is applicable when
(A) Deflections are linear functions of applied forces
(B) Material obeys Hooke's law
(C) The action of applied forces will be affected by small deformations
of the structure
(D) None of the above
MCQ s

Old Question Papers

• A Fixed beam AB of constant flexural rigidity is shown . The beam
is subjected to a uniform distributed loadof w moment M=wL2 kNm.
Draw Shear force and bending moment diagrams by force method.
• Define coefficient of stiffness and flexibility matrix?
Expected Questions for University Exam

 Analyze the beam shown in fig by stiffness matrix method. Take EI
as constant.
 Analyse the continuous beam as shown in figvby stiffness matrix
method if the support B sink by 10 mm. Take EI = 6000 KN cubic
meter
Expected Questions for University Exam

The analysis of a structure by the matrix method may be described by
the following steps:
1. Problem statement
2. Selection of released structure
3. Analysis of released structure under loads
4. Analysis of released structure for other causes
5. Analysis of released structure for unit values of redundant
6. Determination of redundants through the superposition equations.
7. Determine the other displacements and actions. The following are
the four flexibility matrix equations for calculating redundants member
end actions, reactions and joint displacements
where for the released structure
8.All matrices used in the matrix method are summarized in the
following tables
Summary

 Jain, A. K., “Advanced Structural Analysis “, Nem Chand
& Bros., Roorkee.
 Hibbeler, R.C., “Structural Analysis”, Pearson Prentice
Hall, Sector - 62, Noida-201309
 C. S. Reddy “Structural Analysis”, Tata Mc Graw Hill
Publishing Company Limited,New Delhi.
 Timoshenko, S. P. and D. Young, “ Theory of Structures”
, Tata Mc-Graw Hill BookPublishing Company Ltd., New
Delhi.
 Dayaratnam, P. “ Analysis of Statically Indeterminate
Structures”, Affiliated East-WestPress.
 Wang, C. K. “ Intermediate Structural Analysis”, Mc
Graw-Hill Book PublishingCompany Ltd.
References

References

Force Force and Displacement Matrix Method

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