Unit - 5
Stiffness method
1. Fundamental Concepts of Stiffness Method of Analysis
Let 
and 
have been positive (anticlockwise).
3. Applying imaginary restraints at the joints to prevent all joint displacements.
4. The restrained structure is assumed to be acted upon by all loads except those that correspond to the unknown displacements.
5. The actions (moments) ADL, and ADL are corresponding to
and 
respectively and caused by the loads acting on the structure.
It is a matrix method that makes use of the members' stiffness relations for computing member forces and displacements in structures. The direct stiffness method is the most common implementation of the finite element method (FEM).
It is also called a displacement method.
[S]+[D]+[AM]=[AJ]
S = Stiffness Matrix
[D]=unknown joint displacement
i.e. 
AM=Fixed end moments at particular points
AJ=External moment
2. Important Points
When 
is present in your frame
3. Steps for analysis at stiffness matrix
1. Find known rotation i.e., 
2. Find fixed end moment
3. Find AM value
4. Find S matrix
5. Put in compatibility equation [S]+[D]+[AM]=[AJ]
6. Apply S.D. equation
This method is advanced tool for analyzing determinate structures. It is also referred as a displacement method. One of its advantages over the flexibility method is that it is conductive to computer programming. In the flexibility method the unknown quantities are redundant reactions that must be arbitrarily chosen, but in the stiffness method the unknowns are the joint displacements in the structure, which are automatically specified.
In this method, the structure is restrained from undergoing displacements at the joints. The restraining forces are determined at the joints. The number of artificial restraints added to make the structure restrained however, structure does not satisfy the equilibrium of forces at the joints. Displacements are then allowed to take place at the joints until artificial restraining forces vanish. The displacements are evaluated by satisfying the equilibrium conditions of the joints with the joint displacements known, the forces on the structure are determined by the superposition of the forces in the restrained structure and the forces due to displacements at the joints.
Essential features of stiffness method
Key takeaway:
Problems:
Step 1) Find fixed end moment
For Span AB
Span BC
Step 2) Find AM value and Find S matrix
= -5+13.33
= 8.33
Step 3) Use compatibility equation
[S]+[D]+[AM]=[AJ]
Apply SD equation for span AB
2. Analyze the given figure by Stiffness Method
Step 1) Find Dki 
Step 2) Fixed end moments
Step 3) S.D. equation
Step 4) Stiffness Matrix
=1.33
= -3.34
=-13.33
Step 5) Compatibility Equation
[S+[D]+[AM]= [AJ]
/EI
/EI
3. Analyses the given figure by Stiffness Method
Step 1) 
Step 2) Find fixed end moments
Step 3) S.D. equation
-------(4)
Step 4) Stiffness Matrix
=1.11 
= -75
= -60+80
= 20
Step 5) Use compatibility equation
[S]{D} +{AM}={AJ}
4. Analyze the given beam shown in the figure by stiffness method Take EI=
Step 1) Dki =?
Step 2) Find the fixed end
Step 3) Find AM value
=125
= -135+60
=-75
Step 4) S.D. equation
Step 5) Stiffness Matrix
Step 6) Use compatibility equation
[S]{D} +{AM}={AJ}
KNM
KNM
KNM
5. Analyze the given beam shown in the figure by stiffness method
Step 1) Dki = 2
Step 2) Fixed End Moments
Step 3) S.D. equation
Step 4) Find AM value
=-12.5+13.33
=0.83
Step 5) Apply compatibility equation
[S]{D} +{AM}={AJ}
/EI
6. Analyze the given fig. By stiffness Matrix method take EI IS Constant
Step 1) Find Dki
Step 2) Fixed end moment
Step 3) SD equation
Step 4) Find value AM
Step 5) Stiffness matrix
Stiffness Matrix
Step 6) Apply compatibility equation
[S]{D} +{AM}={AJ}
7. Analyze the given beam by stiffness method B is sink by 10mmEI=
=126.33KNM
Apply S.D. equation
8. Analyze the beam by stiffness matrix method draw SFD & BMD
Apply S.D. equation
Draw SFD &BMD
Key takeaway:
Step 1) Find Dki
Step 2) Fixed end moment
Step 3) SD equation
Step 4) Find value AM
Step 5) Stiffness matrix
Step 6) Apply compatibility equation
Step 7) Draw BMD and SFD
Problem 1:
1. Analyses portal frame by stiffness matrix. Draw BMD Take EI= constant
Solution:
Step 1:
=?
Step 2: Find fixed end Moments
For 
for
for 
No loading
Step 3: Find ADL
Step 4: Find stiffness matrix
Step 5: Compatibility Equation
Step 6: Find final moments by slope deflection equation
For
δ
For 
Step 7 Draw BMD
2. Analyze the portal frame by stiffness matrix method take EI=cost.
Step 1:
Step 2: Find fixed end moments
Step 3: Find 
Step 4: Stiffness Matrix
Step 5: Find final moments by S.D Equation for AB
3. Analyze the portal frame by stiffness matrix method take EI=cost. Draw SFD and BMD.
Step 1: Find Dki
= ? 
= 3 
Step 2: Find fixed end moments.
Step: 3 Find ADL
Step 4: Stiffness of Matrix
Step 5: Compatibility Equation
Step 6: Find final moments for AB
Key takeaway:
Steps:
References:
1. Structural Analysis: Deodas Menon---Narosa Publishing House.
2. Structural Analysis: Thandavamoorthy---Oxford University Press.
3. Structural Analysis: A Matrix Approach by Pundit and Gupta, McGraw Hills.
4. Structural Analysis by Hibbler, Pearson Education.
5. Structural Analysis: M. M. Das, B. M. Das---PHI Learning Pvt Ltd. Delhi.
6. Fundamentals of Structural Analysis: 2nded---West---Wiley.
7. Theory of Structures: Vol. I & II by B. C. Punmia, Laxmi Publication.
8. Theory of Structures: Vol. I & II by Perumull& Vaidyanathan, Laxmi Publication.
9. Fundamentals of Structural Analysis: K. M. Leet, Vang, Gilbert—McGraw Hills
10. Matrix Methods for structural engineering.by Gere, Weaver.
11. Introduction to the Finite element method, Dr. P.N. Godbole, New Age Publication, Delhi.
12. Finite element Analysis, S.S. Bhavikatti, New Age Publication, Delhi.
13. Basic Structural Analysis: Wilbur and Norris.
