   # Draw the influence lines for the shear and bending moment at point E of the beam shown in Fig. P8.20. FIG. P8.20, P8.21, P8.22

#### Solutions

Chapter
Section
Chapter 8, Problem 22P
Textbook Problem
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## Draw the influence lines for the shear and bending moment at point E of the beam shown in Fig. P8.20. FIG. P8.20, P8.21, P8.22

To determine

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

### Explanation of Solution

Calculation:

Apply a 1 kN unit moving load at a distance of x from left end B.

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

Refer Figure 1.

Consider the unit load at a variable position x to the left hinge C. (placed portion AC of the beam 0x<12m).

Find the vertical support reaction (Ay) at A for portion AC.

Take Moment at hinge C from left end A.

Consider clockwise moment as positive and anticlockwise moment as negative.

ΣMCAC=0Ay(12)1(12x)=012Ay=12xAy=1x12

Consider the unit load at a variable position x to the right hinge C. (Place portion CG of the beam 12m<x30m).

Find the vertical support reaction (Ay) at A for portion CG.

Take Moment at hinge C from left end A.

Consider clockwise moment as positive and anticlockwise moment as negative.

ΣMCAC=0Ay(12)=0Ay=0

Thus, the equations of the influence line ordinate for Ay are,

Ay=1x12, 0x<12m (1)

Ay=0, 12m<x30m (2)

Refer Figure 1.

Find the equation of influence line ordinate for the vertical reaction (Dy) using equilibrium equation.

Apply moment equilibrium at F.

Consider clockwise moment as positive and anticlockwise moment as negative.

ΣMF=0Ay(24)+Dy(6)1(24x)=06Dy=24x24AyDy=4x64Ay (3)

Find the influence line ordinate of vertical reaction (Dy) for portion AC (0x<12m).

Substitute 1x12 for Ay in Equation (3).

Dy=4x64(1x12)=4x64+x3=x+2x6=x6

Find the influence line ordinate of vertical reaction (Dy) for portion CG (12m<x30m).

Substitute 0 for Ay in Equation (3).

Dy=4x64(0)=4x6

Thus, the equations of the influence line ordinate for Dy are,

Dy=x6, 0x<12m (4)

Dy=4x6, 12m<x30m (5)

Refer Figure 1.

Find the equation of influence line ordinate for the vertical reaction (Fy) using force equilibrium equation.

Consider the vertical forces equilibrium condition, take the upward force as positive (+) and downward force as negative ().

ΣFy=0Ay+Dy+Fy=1Fy=1AyDy (6)

Find the influence line ordinate of vertical reaction (Fy) for portion AC (0x<12m).

Substitute 1x12 for Ay and x6 for Dy in Equation (6).

Fy=1(1x12)(x6)=11+x12x6=x12

Find the influence line ordinate of vertical reaction (Fy) for portion AC (12m<x30m).

Substitute 0 for Ay and 4x6 for Dy in Equation (6).

Fy=1(0)(4x6)=14+x6=x63

Thus, the equations of the influence line ordinate for Fy are,

Fy=x12, 0x<12m (7)

Fy=x63, 12m<x30m (8)

Find the equation of shear at E of portion AE (0x<21m).

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

Refer Figure 2.

Apply equilibrium equation of forces.

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

ΣFy=0

Ay1+DySE=0SE=Ay+Dy1 (9)

Find the influence line ordinate of shear at E (SE) for portion AC (0x<12m).

Substitute 1x12 for Ay and x6 for Dy in Equation (9).

SE=(1x12)+(x6)1=1x12+2x121=x12

Find the influence line ordinate of shear at E (SE) for portion CE (12m<x<21m)

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