   # Draw the influence lines for vertical reactions at supports B and D and the shear and bending moment at point C of the beams shown in Figs. P8.7 and P8.8. FIG. P8.7

#### Solutions

Chapter
Section
Chapter 8, Problem 7P
Textbook Problem
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## Draw the influence lines for vertical reactions at supports B and D and the shear and bending moment at point C of the beams shown in Figs. P8.7 and P8.8. FIG. P8.7

To determine

Draw the influence lines for the vertical reactions at supports B and D.

Draw the influence lines for the shear and bending moment at point C.

### Explanation of Solution

Calculation:

Apply a 1 k unit moving load at a distance of x from right end D.

Sketch the free body diagram of beam as shown in Figure 1.

Refer Figure 1.

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

Consider moment equilibrium at point D.

Consider clockwise moment as positive and anticlockwise moment as negative.

Sum of moment at point D is zero.

ΣMD=0By(20)1(x)=020By=xBy=x20 (1)

Find the equation of support reaction (Dy) at D using equilibrium equation:

Apply vertical equilibrium equation of forces.

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

By+Dy=1

Substitute x20 for By.

x20+Dy=1Dy=1x20 (2)

Consider Equation (1).

Find the value of influence line ordinate of reaction By at support B.

Substitute 0 for x in Equation (1).

By=020=0

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

 x (ft) from right end By (k/k) 0 0 10 12 20 1 25 54

Draw the influence line diagram for the vertical reactions at support B using Table 1 as shown in Figure 2.

Consider Equation (2).

Find the influence line ordinate of reaction Dy at support D.

Substitute 0 for x in Equation (1).

Dy=1020=1

Similarly calculate the influence line ordinate of reaction Dy for different value of x and summarize the result in Table 2.

 x (ft) from right end Dy (k/k) 0 0 10 12 20 0 25 −14

Draw the influence line diagram for the vertical reactions at support D using Table 2 as shown in Figure 3.

Find the equation of shear force at C of portion DC (0x<10ft).

Sketch the free body diagram of the section DC as shown in Figure 4.

Refer Figure 4.

Apply equilibrium equation of forces.

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

ΣFy=0

SC+Dy1=0SC=Dy+1

Substitute 1x20 for Dy.

SC=(1x20)+1=1+x20+1=x20

Find the equation of shear force at C of portion CA (10ft<x25ft).

Sketch the free body diagram of the section CA as shown in Figure 5.

Refer Figure 5.

Apply equilibrium equation of forces.

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

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