This document outlines a project to implement the direct stiffness matrix method for analyzing beam and grid structures. It discusses the objectives of understanding basic matrix analysis principles and applying them to beam and grid structures. For beams, it describes generating the stiffness matrix manually and through coding in MATLAB. For grids, it similarly involves analytical and coded approaches to matrix analysis. The document presents the methodology for each approach and shows sample results of the analyses.

1. 5/15/2023 Implementation Direct Stiffness Matrix in Beam and Grids 1
#PROJECT
Implementation of Direct Stiffness Matrix Method in a Beam & Grids
Structures using Analytical and Codal Method
ADVANCED STRUCTURE ANALYSIS
(MSTR 501)
KATHMANDU UNIVERSITY
SCHOOL OF ENGINEERING
DEPARTMENT OF CIVIL ENGINEERING
M.E STRUCTURAL ENGINEERING
DHULIKHEL

2. Presentation Outline :
1. Introduction
 Objectives
 Problem Statement
2. Literature Review
 Direct Stiffness Method
 Beam Element
 Grids Element
3. Methodology
 Analytical and Codal Method for Beam
 Analytical and Codal Method for Grids
4. Results
5. Conclusion
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3. 1. INTRODUCTION
Objectives :-
• To understand the basic principles of matrix analysis.
 To understand the implementation of direct stiffness matrix method in beam and grid structures
(2D and 3D)
 To properly understand working phenomenon of commercially available software and check with
analytical results.
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4. 1. INTRODUCTION
Problem Statement :
 Beam Element
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Figure 1 : Problem Statement of Beam Element.(Main Girder Beam of T-Beam Bridge)

5. 1. INTRODUCTION
Problem Statement :
 Grid Element
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Figure 2: Problem Statement of Grid Element

6. 2. LITERATURE REVIEW
Structural analysis, is the process of predicting the performance of a given structure under a
prescribed loading condition.
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Methods of Structural Analysis
Oldest
Classical
Method
After
Classical
Matrix
Method
Newest
Finite
Element
Method
Matrix Analysis of Structures:
Two Formulations are possible:-
1. Force Method (Flexibility Method)
2. Displacement Method (Stiffness Method)
 It is much easier to formulate the necessary matrices
for the computer using the stiffness method.

7. 2. LITERATURE REVIEW
Direct Stiffness Method :
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 Express local (member) force-displacement relationships in terms of unknown member
displacements.
 Suitabilitywhen DSI is more than DOF for a same structureor for highly redundantstructure.
▪ Most of Framed Structure in real practices are highly redundant. So, Force Method will consume
lots of time than stiffness method.
 Very formalized and computer friendly.
 Versatile , Suitable for large problems.

8. 2. LITERATURE REVIEW
BEAM
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 Beam is structurein which loads are subjected normal to the longitudinalcentroidalaxis.
 Generally, StrongerAxis of beam is oriented in a direction so that load acts perpendicularto this axis.
 Equilibrium Equation in case of Beam.
Analytical Model :
 Its an ideal representation of the actual structure for analytical purposes.
 Modeled such as unknown external reaction force acts only on the joint..
 Nodes are placed on the support and points where the cross sectional area changes.
 The start and end nodes are identified by arrows pointing to the end of the connection.
Figure 3: Analytical Model for Beam

9. 2. LITERATURE REVIEW
BEAM
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Degree of Freedom (DoF)
 The degrees of freedom (or free coordinates) of a beam are simply its unknown joint
displacements (translations and rotations).
 The translation of the joint in the global X direction is 0 in beam.
 The beam joint can have up to two degrees of freedom, a translation in the global Y direction and
one rotation around the global Z axis.

10. 2. LITERATURE REVIEW
BEAM
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Member Stiffness Relation :
 It is the equation expressing the forces as a function of displacement.
In matrix Form ,
Here , member stiffness matrix ,

11. 2. LITERATURE REVIEW
BEAM
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Fixed End Forces :
 The forces at the member ends, due to external loads, if both ends of the member were fixed against
translations and rotations .
Co-ordinate Transformation:
 The global and local co-ordinate concise in case of beam so no need for transformation of forces and
displacement from local to global and vice versa.
{F} = {Q} and {v} = {u}
Equilibrium Checks:
Figure 4: Fix End Forces for Beam

12. 2. LITERATURE REVIEW
GRIDS
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 Two-dimensional frames of straight elements that are interconnected by rigid bodies and / or flexible joints
and that receive loads and reaction forces perpendicular to the plane of the structure.
Figure 5: Example of Grid Element

13. 2. LITERATURE REVIEW
GRIDS
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Analytical Model :
• Divided into members and joints so that members are straight prismatic, and all the external reactions act
only at the joints.
Figure 6: Grid Analytical Model

14. 2. LITERATURE REVIEW
GRIDS
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Degree of Freedom :
• A free joint of a grid has three degrees of freedom , the translation in the Y direction and the rotations about
the X and Z axes.
• For any joint The translation in the Y direction is numbered first, followed by the rotation about the X axis,
and then the rotation about the Z axis.
Member Stiffness Relation in the Local Co-ordinate System:
 Shear force, Torsional moment and bending moment are developed at both beginning and end node.
Expression ,
Q=ku+Qf
Figure 7: Twist on Grid

15. 2. LITERATURE REVIEW
GRIDS
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Member Stiffness Relation in the Local Co-ordinate System:
i. For member rigidly connected to both end (MT=0) ii. For a member with a hinge at the beginning (MT = 1):
iii. For a member with a hinge at the end (MT = 2): iii. For a member with hinges at both ends (MT = 3):
k=0

16. 2. LITERATURE REVIEW
GRIDS
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Co-ordinate Transformations :
‘T’ matrix can be used to transform member end
forces and displacements from the local
coordinate system to global coordinate system.
Figure 8: Inclination of Grid Element with Global Axis

17. 3. METHODOLOGY
BEAM
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Beam Analytical Method:
8. Equilibrium Checks
7. Determination of Rection Vector 'R'
6. Computation of member end forces Q=k.u+Qf
5. Computation of member end displacement u
4. Determination of joint displacement d from P-Pf=S.d
3. Determination of Joint load P
2. Evaluation of Structure stiffness matrix S and fixed-joint vector Pf
1. Preparation of Analytical Model
So , on the basis of mentioned steps of
analysis of beam I have taken main girder
beam of T-beam bridge where loads on
beam and its cross-section is already
calculated on bridge engineering and after
that to fulfill the requirement of this project
all other steps of direct stiffness method is
done manually.
Beam Manual Analytical.pdf

18. 3. METHODOLOGY
BEAM
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Beam Codal Method:
First We divide the code into two Module i.e
i. Input Module
ii. Analysis module
And developed code on MATLAB 2018a Software.
Input Module
Joint data Support Data Material
Property Data
Cross-
sectional
Property Data
Member Data Joint Load Data
Member Load
Data

19. 3. METHODOLOGY
BEAM
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Analysis
Module
Assignment of
Structure Co-ordinate
Number
Generation of the
structure stiffness matrix
and the equivalent joint
load vector
Storage of joint
Loads into the
Structure Load
Vector
Solution for
Joint
Displacement
Calculation of
Member forces
and Support
Reactions
Beam Codal Methodology.pdf

20. 3. METHODOLOGY
BEAM
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Summary of Beam Codal Method:
Beam MATLAB Code.pdf

21. 3. METHODOLOGY
GRIDS
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Grids Analytical Method:
To fulfill the requirement of this project
all the steps of direct stiffness method is
done for selected problem of grid using
Analytical approach by Manually.
1)11.Equilibrium Checks
10. Determination of support reaction vector ‘R’.
9.Computation of member end forces Q=ku+Qf , F=TT.Q
8. Determination of member end displacement ‘u’.
7. Substitute P, Pf and S into the structure stiffness relation, p-pf=S.d
6. Determine the joint displacement ‘d’.
5. Computation of external joint loads P.
4. Evaluation of structure stiffness matrix S and fixed- joint vector Pf=TT.Qf
3. Computation of member global Stiffness Matrix K=TT.k.T
2. Computation member stiffness matrix ‘k’and fixed joint force Qf
1. Preparation of Analytical Model
Grid Manual Analytical.pdf

22. 3. METHODOLOGY
GRIDS
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Grids Codal Method:
Identify degrees of Freedom d and restrained co-
ordinates of the beam.
For each member :
Evaluate k and Qf , T
Store K=TT.k.T
Store K in S & Pf=TT.Qf
Form Joint Load Vector P
Solve P-Pf=S.d for d
For each member obtain v form d
u=T.v
Calculate Q=k.u +Qf
& F=TT.Qf
Store F in R
To fulfill the requirement of this project all the
steps of direct stiffness method is done for
selected problem of grid using Codal approach
on MATLAB 2018a Software (Attached on
ANNEX-IV of Report)
Grids MATLAB Code.pdf

23. 4. RESULTS
BEAM
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24. 4. RESULTS
BEAM
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25. 4. RESULTS
GRIDS
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26. 4. RESULTS
GRIDS
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27. 5. CONCLUSION
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 By combining matrix analysis of structure with FEM , structure engineer can develop a relevant
structural analysis program.
 After doing programs in MATLAB software , it allows structural analyst like us to know how the
structural analysis programs analyze any kind of structure.
 The results from Codal approach in MATLAB 2018a is accurate than analytical approach which is
done manually.
 As real life structure are highly indeterminant , analytical method becomes highly time consuming
and has great chances of error .
 So that , scripting language like MATLAB is best for analysis.

28. 5. CONCLUSION
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 Matrix Analysis of Structures (2010) , Aslam Kassimali (2E)
 Matrix analysis of Structures (1990), William Weaver , JR.James M.Gere (2E)
 Menon D (2005) Advanced Structural Analysis
 Lecture notes (Kameshwar Sahani , Lecturer , Kathmandu University)
 https://www.cengageasia.com/
 https://nptel.ac.in/courses/105/106/105106050/

29. THANK YOU!!!