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Unit - 5

Stiffness method

Q1) Explain concept of fundamental concepts of flexibility and stiffness relation between them?

A1) 1. Fundamental Concepts of Stiffness Method of Analysis

The degree of kinematic indeterminacy Dki is two.

The unknown displacements

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

Q2) What is stiffness method – Structure approach only?

A2) 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

This method is a matrix version of classical generalized slope deflection method.

Joint displacements are treated as primary unknowns in this method.

Number of unknowns is equal to the degree of kinematic indeterminacy of the structure.

The unknown joint displacements for a particular structure are uniquely defined.

Conditions of joint equilibrium are used to form equations in unknown displacement.

Member approach of the stiffness method is suitable for developing general computer programme. Hence, the stiffness method has acquired immense popularity.

Q3) Analyze the beam shown in the figure by stiffness method Take EI= constant

A3) 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

Q4) Analyze the given figure by Stiffness Method

A4) 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

Q5) Analyse the given figure by Stiffness Method

A5) 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}

Q6) Analyze the given beam shown in the figure by stiffness method Take EI=

A6) 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

Q7) 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

Q8) Analyze the given fig. By stiffness Matrix method take EI IS Constant

A8) 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}

Q9) Analyze the given beam by stiffness method B is sink by 10mmEI=

A9)

=126.33KNM

Apply S.D. equation

Q10) Analyze the beam by stiffness matrix method draw SFD & BMD

A10)

Apply S.D. equation

Draw SFD &BMD

Q11) Analyse portal frame by stiffness matrix. Draw BMD Take EI= constant

A11)

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

Step 7 Draw BMD

Q12) Analyze the portal frame by stiffness matrix method take EI=cost.

A12) 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

Q13) Analyze the portal frame by stiffness matrix method take EI=cost. Draw SFD and BMD

A13) 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

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