Muller-Breslau Principle StevenVukazich

SanJoseStateUniversity

Cy 1.5

9 m 6 m 3 m

D

CB

A

0

x = Cy

0 05 m 0

9 – m 012 m 0.515 m 1.018 m 1.5

+

Influence Lines for Our Model Problem

Ay

– 0.5

9 m 6 m 3 m

D

CB

A

0

x = Ay

0 15 m 1

9 – m 112 m 0.515 m 018 m – 0.5

+

1.0

Influence Lines for Our Model Problem

MA

4.5 m

9 m 6 m 3 m

D

CB

A

0

x = MA

0 05 m – 5 m

9 – m – 9 m12 m – 4.5 m15 m 018 m 4.5 m

+

– 9.0 m

Influence Lines for Our Model Problem

Influence Lines for Our Model Problem

9 m 6 m 3 m

D

CB

A5 m

O

VO

– 0.5

0

Influence Lines for Our Model Problem

9 m 6 m 3 m

D

CB

A5 m

O

MO

2.0 m

0

– 4.0 m

1. Remove the ability for the structure to resist the response quantity (e.g. reaction, internal shear, internal bending moment at a particular point). For a determinate structure this will result in an unstable structure.

2. Apply the response quantity to the modified unstable structure from Step 1.

3. The rigid body motion of the modified unstable structure is the shape of the influence line for the response quantity.

General procedure for the applying the Muller-Breslau principle to find the shape of influence lines

Shape of Cy Influence Line

9 m 6 m

3 m

D

CB

A

x

Use the Muller-Breslau Principle to find the shape of the influence line for the support reaction at C for our model problem:

1. Remove the ability for the structure to resist the response quantity (e.g. reaction, internal shear, internal bending moment at a particular point). For a determinate structure this will result in an unstable structure.

Remove the roller support at C

9 m 6 m

3 m

DB

A

x

Use the Muller-Breslau Principle to find the shape of the influence line for the support reaction at C for our model problem:

2. Apply the response quantity to the modified unstable structure from Step 1.

Apply Cy consistentwith positive sign converntion

Cy

Shape of Cy Influence Line

9 m 6 m

3 m

DB

A

x

Use the Muller-Breslau Principle to find the shape of the influence line for the support reaction at C for our model problem:

3. The rigid body motion of the modified unstable structure is the shape of the influence line for the response quantity.

Cy Shape of Cy Influence Line!

Shape of Cy Influence Line

Cy

A B DC

Shape of Ay Influence Line

9 m 6 m

3 m

D

CB

A

x

Use the Muller-Breslau Principle to find the shape of the influence line for the vertical support reaction at A for our model problem:

Remove the ability of the Support to resist vertical motion.Rotation is still restrained

Shape of Ay Influence Line

9 m 6 m

3 m

D

CB

A

x

Apply Ay consistentwith positive sign convention

Ay

Shape of Ay Influence Line

9 m 6 m

3 m

D

CB

A

x

Ay Shape of Ay Influence Line!

Ay

A B DC

Shape of MA Influence Line

Use the Muller-Breslau Principle to find the shape of the influence line for the moment reaction at A for our model problem:

Remove the ability of the Support to resist rotation.Vertical movement is restrained

9 m 6 m 3 m

D

CB

A

Shape of MA Influence Line

9 m 6 m 3 m

D

CB

A

Apply MA consistentwith positive sign convention

MA

Shape of MA Influence Line

9 m 6 m 3 m

D

CBA

Shape of MA Influence Line!

MA

MA

A B DC

9 m 6 m 3 m

D

C

BA

5 mO

Use the Muller-Breslau Principle to find the shape of the influence line for the internal shear force at O for our model problem:

Shape of VO Influence Line

Remove the ability of point O toresist shear force.Rotation is still restrained

9 m 6 m 3 m

D

C

BA

5 m

O

Shape of VO Influence Line

Apply VO consistentwith positive sign convention

VO

VO

9 m 6 m 3 m

D

C

BA

5 m

O

Shape of VO Influence Line

Shape of VO influence line!

VO

VO

A B D

VO

CO

9 m 6 m 3 m

D

C

BA

5 m

O

Use the Muller-Breslau Principle to find the shape of the influence line for the internal shear force at O for our model problem:

Shape of MO Influence Line

Remove the ability of point O toresist bending moment.Shear is still restrained.

9 m 6 m 3 m

D

C

BA

5 m

O

Shape of MO Influence Line

MOMO

Apply MO consistentwith positive sign convention

9 m 6 m 3 m

D

C

BA

5 m

O

Shape of MO Influence Line

MOMO

Shape of MO influence line!

A B D

MO

CO