Method of Consistent Deformation, Summaries of Theory of Structures

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Method of Consistent

Deformation

Theory of Structures-II

M Shahid Mehmood

Department of Civil Engineering

Swedish College of Engineering and Technology, Wah Cantt

Structural Analysis

By

R. C. Hibbeler

FRAMES

• Method of consistent deformation is very useful for

solving problems involving statically indeterminate

frames for single story and unusual geometry.

• Problems involving multistory frames, or with high

indeterminacy are best solved using the slope deflection

or moment distribution or the stiffness methods.

2

Solution

Principle of Superposition

• By inspection the frame is indeterminate to the first

degree.

4

5 m

4 m

A

B

8 kN/m

Actual Frame

Solution

Principle of Superposition

• We will choose the horizontal reaction at support B as

the redundant.

• The pin at B is replaced by the roller, since a roller will

not constraint B in the horizontal direction.

5

A

B

8 kN/m

Primary Structure

Δ B

Solution

Compatibility Equation

7

A

B

8 kN/m

Primary Structure

Δ B

5 m

4 m

A

B

8 kN/m

Actual Frame

=

A

B

redundant B x

applied

B x

f BB

B x

B x BB

 Δ  B f 



Solution

Compatibility Equation

• The terms Δ B

and f BB

will be computed using the method

of virtual work.

• The frame ’ s x coordinates and internal moments are

shown in figure.

• It is important that in each case the selected coordinate

x 1

or x 2

be the same for both the real and virtual loadings.

• Also the positive directions for M and m must be same.

8

B x BB

0  Δ  B f

Solution

10

A

B

8 kN/m

x 1

20 kN

20 kN

x 2

2

1 1 1

20 4 x

M  x 

0 2

M 

A

B

x 1

0.8 kN

0.8 kN

x 2

1 1

m  0. 8 x

2 2

m  1 x

1 kN

1 kN

EI EI

dx

EI

x dx

EI

x x x dx

dx

EI

Mm

L

B

5

0

4

0

1 1 2 2

2

1 1

0

Solution

Compatibility Equation

• For f BB

we require application of real unit load acting at B

and a virtual unit load acting at B

11

A

B

x 1

0.8 kN

0.8 kN

x 2

1 1

m  0. 8 x

2 2

m  1 x

1 kN

1 kN

Solution

Compatibility Equation

• Substituting the data in Eq. (1)

13

B x BB

 Δ  B f

EI

B

EI

x

  3. 47 kN ANS

x

B

Solution

Equilibrium Condition

• Showing B x

on the free body diagram of the frame in the

correct direction, and applying the equations of

equilibrium, we have

14

2.5 m

4 m

A

B

40 kN

2.5 m

B y

3.47 kN

A y

A x

Example 7

Determine the moment at fixed support A for the frame

shown. EI is constant.

16

8 ft

A

B

Actual Frame

C

4 ft

3 ft

5 ft

100 lb/ft

Solution

Principle of Superposition

• By inspection the frame is indeterminate to the first

degree. 17

8 ft

A

B

Actual Frame

C

4 ft

3 ft

5 ft

100 lb/ft

Solution

Compatibility Equation

Reference to point A

19

=

A

B

actual frame

C

100 lb/ft

A

B

primary structure

C

100 lb/ft

θ A

A

B

Redundant M A

applied

C

M A

α AA

M A

  0 (1)

A A AA

    M 

Solution

Compatibility Equation

• The terms θ A

and α AA

will be computed using the method

of virtual work.

• The frame ’ s x coordinates and internal moments are

shown in figure.

20

  0 (1)

A A AA

    M 