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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 18P
Textbook Problem
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Determine the absolute maximum shear in a 15 m long simply supported beam due to the series of three moving concentrated loads shown in Fig. P9.13.

Chapter 9, Problem 18P, Determine the absolute maximum shear in a 15 m long simply supported beam due to the series of three

FIG. P9.13, P9.17, P9.18, P9.22

To determine

Find the absolute maximum shear in a 15 m 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 1.

Refer Figure 1.

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(15)1(15x)=0Ay(15)15+x=015Ay=15x

Ay=1x15        (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 1x30 for Ay.

1x15+By=1By=11+x15By=x15        (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=1015=1kN

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

x (m)Ay(kN/kN)
01
150

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 15 m for L.

θAB=115

Sketch the loading position as shown in Figure 4.

Find the maximum absolute maximum shear for case 1.

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.

Find the shear force of case 1.

SA=[125(L)+100(L2)+50(L5)](θAB)

Substitute 15 m for L and 115 for θAB

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