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Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931

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Section
BuyFindarrow_forward

Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931
Chapter 9, Problem 19P
Textbook Problem
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Determine the absolute maximum shear in a 60 ft long simply supported beam due to the series of four moving concentrated loads shown in Fig. P9.14

FIG. P9.14, P9.16, P9.19, P9.23

Chapter 9, Problem 19P, Determine the absolute maximum shear in a 60 ft long simply supported beam due to the series of four

To determine

Find the absolute maximum shear in a 60 ft long simply supported beam.

Explanation of Solution

Calculation:

Sketch the simply supported beam as shown in Figure 1.

The simply supported beam the maximum shear occurs at support. Hence, find the influence line diagram for support reactions.

Case 1: Consider the Absolute Maximum Shear occurs at Support A.

Influence line diagram for support reaction at A.

Apply a 1 k unit moving load at a distance of x from left end A.

Sketch the free body diagram of beam as shown in Figure 2.

Refer Figure 2.

Find the equation of support reaction (Ay) at A using equilibrium equation:

Take moment about point B.

Consider moment equilibrium at point B.

Consider clockwise moment as positive and anticlockwise moment as negative.

Sum of moment at point B is zero.

ΣMB=0Ay(60)1(60x)=0Ay(60)60+x=060Ay=60x

Ay=1x60        (1)

Find the equation of support reaction (By) at B using equilibrium equation:

Apply vertical equilibrium equation of forces.

Consider upward force as positive (+) and downward force as negative ().

Ay+By=1

Substitute 1x60 for Ay.

1x60+By=1By=11+x60By=x60        (2)

Consider Equation (1).

Find the value of influence line ordinate of reaction Ay at support A.

Substitute 0 for x in Equation (1).

Ay=1060=1k

Similarly calculate the influence line ordinate of reaction Ay for different value of x and summarize the result in Table 1.

x (ft)Ay(k/k)
01
600

Draw the influence line diagram for the vertical reactions at support A using Table 1 as shown in Figure 3.

Refer Figure 3.

Find the slope (θAB) of influence line diagram of Ay.

θAB=1L

Here, L is the length of the beam.

Substitute 60 ft for L.

θAB=160

Sketch the loading position as shown in Figure 4.

Find the shear force when load 1 at A.

Sketch the loading position on the beam when the load 1 placed at just right of A as shown in Figure 5.

Refer Figure 5.

(SA)1=[10(L)+20(L15)+20(L25)+5(L35)](θAB)

Substitute 60 ft for L and 160 for θAB.

(SA)1=[10(60)+20(6015)+20(6025)+5(6035)](160)=10+20(4560)+20(3560)+5(2560)=10+15+11.67+2.08=38.75k

Sketch the loading position on the beam when the load 2 placed at just right of A as shown in Figure 6.

Refer Figure 6.

Find the shear force when load 2 at A.

(SA)2=[10(0)+20(L)+20(L10)+5(L20)](θAB)

Substitute 60 ft for L and 160 for θAB.

(SA)2=[10(0)+20(60)+20(6010)+5(6020)](160)=0+20+20(5060)+5(4060)=20+16

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