Muller-Breslau's Principle and Influence Lines for Structural Analysis, Study notes of Structural Analysis

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INFLUENCE LINE

Reference:

Structural Analysis

Third Edition (2005)

By

Aslam Kassimali

DEFINITION

An influence line is a graph of a response function of astructure as a function of the position of a downward unitload moving across the structure

,^

0

1

,

y

B

y

x

C

x

a

L

S

x

A

a

x^

L

L

 −

= −

≤

<

 = 



=

−

<

≤



(^

)^

(^

0

( )

1

,

y

B

y

x

C

L

a

L

a

x^

a

L

M

x

A

a

a

a

x^

L

L

^

−

=

−

≤

≤

 = 

^



^

=

−

≤

≤

^



^

^





Example 8.

kN

Draw the influence lines for the verticalreactions at supports A and C, and the shearand bending moment at point B, of the simplysupported beam shown in Fig. 8.3(a). Influence line for A

y

(^

)^

(^

(^

0 : 20

1 20

0

1 20

1

20

20

c y y M A

x

x^

x

A

=

−

−

=

−

=

=

−

∑

kN/kN

Fig. 8.3(c)

Influence line for C

y

(^

)^

(^

(^

1

20

0

1 20

20

A

y

y M x

C x^

x

C

=

−

=

=

=

∑

Fig. 8.3(d)

Influence line for M

B

Place the unit load to the left of point B, determine the bending moment at Bby using the free body of the portion BC:

B^

y

M

C

x

ft

Place the unit load to the right of point B, determine the bending moment atB by using the free body of the portion AB:

B

y

M

A

ft

x

ft

gives

,^

y

B

y

x

C

x

ft

M

x

A

ft

x

ft

Fig. 8.3(f)

Example 8.

Draw the influence lines for the vertical reactions atsupports A, C, and E, the shear just to the right ofsupport C, and the bending moment at point B ofthe beam shown in Fig. 8.5(a).

Influence line for C

y

A 20

y^

y

y^

y

M x

C

E

x

C

E

∑^ By substituting the expressions for E

y

, we obtain

y

x

x

x

ft

C

x

x

x

ft

x

ft

Fig. 8.5(d)

Influence line for A

y

y^1 y^

y^

y

y^

y^

y

F

A

C

E

A

C

E

∑

By substituting the expressions for C

y^

and E

y

, then

y

x

x

x

ft

A

x

x

x

ft

x

ft

Fig. 8.5(e)

Influence line for M

B

(^

y

B

y A

x

x

ft

M

A

ft

x

ft

By substituting the expressions for A

, we obtainy

(^

B

x

x

x

x

ft

x

x

M

ft

x

ft

x

x

ft

x

ft

Fig. 8.5(g)

MULLER-BRESLAU’S PRINCIPLE AND

QUALITATIVE INFLUENCE LINES

•Developed by Heinrich Muller-Breslau in 1886. •Muler-Breslau’s principle:

The influence line for a force (or

moment) response function is given by the deflected shape of thereleased structure obtained by removing the restraintcorresponding to the response function from the originalstructure and by giving the released structure a unitdisplacement (or rotation) at the location and in the direction ofthe response function, so that only the response function and theunit load perform external work. •Valid only for influence lines for response functions involvingforces and moments, e.g. reactions, shears, bending moments orforces in truss members, not valid for deflections.

Qualitative Influence Lines

In many practical applications, it is necessary to determine only thegeneral shape of the influence lines but not the numerical values ofthe ordinates.

A diagram showing the general shape of an influence

line without the numerical values of its ordinates is called aqualitative influence line

. In contrast, an influence line with the

numerical values of its ordinates known is referred to as a quantitative influence line

Example 8.6^ Draw the influencelines for thevertical reactions atsupports B and Dand the shear andbending moment atpoint C of the beamshown in the Figure8.9(a).

Example 8.7^ Draw the influence linesfor the vertical reactionsat supports A and E, thereaction moment atsupport A, the shear atpoint B, and the bendingmoment at point D of thebeam shown in Fig.8.10(a).