   # 8.1 through 8.3 Draw the influence lines for vertical reactions at supports A and C and the shear and bending moment at point B of the beams shown in Figs. P8.1 through P8.3. FIG. P8.3

#### Solutions

Chapter
Section
Chapter 8, Problem 3P
Textbook Problem
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## 8.1 through 8.3 Draw the influence lines for vertical reactions at supports A and C and the shear and bending moment at point B of the beams shown in Figs. P8.1 through P8.3. FIG. P8.3

To determine

Draw the influence lines for the vertical reactions at supports A and C.

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

### Explanation of Solution

Calculation:

Apply a 1 kN 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:

Consider moment equilibrium at point C.

Consider clockwise moment as positive and anticlockwise moment as negative

Sum of moment at point C is zero.

ΣMC=0Ay(15)1(15x)=0Ay(15)15+x=015Ay=15xAy=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+Cy=1

Substitute 1x15 for Ay.

1x15+Cy=1Cy=11+x15Cy=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/kN

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) 0 1 10 13 15 0

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

Consider Equation (2).

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

Substitute 15 for x in Equation (2).

Cy=1515=1kN/kN

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

 x (m) Cy (kN/kN) 0 0 10 23 15 1

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

Find the equation of shear force at B of portion AB (0x<10m).

Sketch the free body diagram of the section AB 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

AySB1=0SB=Ay1

Substitute 1x15 for Ay.

SB=1x151=x15

Find the equation of shear force at B of portion BC (10m<x15m).

Sketch the free body diagram of the section BC 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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