160.12.3 influence lines mb principle...use the muller-breslau principle to find the shape of the...
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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