   # Draw the influence lines for the shears and bending moments at points B and E of the beam shown in Fig. P8.31. FIG. P8.31, P8.32

#### Solutions

Chapter
Section
Chapter 8, Problem 32P
Textbook Problem
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## Draw the influence lines for the shears and bending moments at points B and E of the beam shown in Fig. P8.31. FIG. P8.31, P8.32

To determine

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

### Explanation of Solution

Calculation:

Use Muller-Breslau's principle.

Influence line ordinate for shear at B.

Cut the section at B to obtain the released structure of the beam. Next apply a small relative displacement in positive direction of SB by moving end B of the portion BC upward by Δ and end B of the portion AB can neither translate nor rotate due to fixed support of A to obtain the deflected shape.

Sketch the deflected shape of the released beam when the beam cut at B as shown in Figure 1.

The general shape of the influence line for shear (SB) at b is the similar of deflected shape.

Sketch the general shape of the influence line for shear (SB) at B as shown in Figure 2.

Find the numerical value of influence line ordinate of SB at B.

Place the 1 kN load first just to the left of B and then just to the right of B.

Sketch the free body diagram of beam as shown in Figure 3.

Refer Figure 3.

Find the shear at B(SB,L) when unit load 1 kN placed just left of B using the equation:

The portion AB can neither translate nor rotate. Hence, the shear at B just left of B is 0.

SB,L=0

Thus, the value of influence line ordinate just left of B is 0.

Find the shear at B(SB,R) when unit load 1 kN placed just right of B using the equation:

The influence line ordinate of portion BC is 1 kN/kN and the shear at B is 1 kN when 1 kN load placed at just right of B.

Thus, the value of influence line ordinate just right of B is 1kN/kN.

The numerical values of influence line ordinate in different points on the beam for shear SB are determined using geometry of the qualitative (similar triangle) of the influence line diagram and summarize the values as in Table 1.

 x (m) Points Influence line ordinate of SB(kN/kN) 0 A 0 5 B− 0 5 B+ 1 10 C 1 16 D 0 19 E 0 22 F 0

Sketch the influence line diagram for shear at point B using Table 1 as shown in Figure 4.

Influence line ordinate for moment at B.

Insert a hinge at B to obtain the released structure of the beam. Next apply a small rotation θ in positive direction of MB at B in the released structure. The rotation θ rotate the portion BC in counterclockwise and the portion AB can neither translate nor rotate due to fixed support of A to obtain the deflected shape.

Sketch the deflected shape of the released beam when insert a hinge at B as shown in Figure 5.

The general shape of the influence line for bending moment (MB) at B is the similar of deflected shape.

Sketch the general shape of the influence line for bending moment (MB) at B as shown in Figure 6.

Find the numerical value of influence line ordinate for MB at B.

Refer Figure 6.

Find the moment at C from B.

Apply 1 kN load at B.

Consider clockwise moment as positive and anticlockwise moment as negative.

MC=1(5)=5kN-m

Thus, the value of influence line ordinate of MB at C is 5kN-m/kN.

The remaining values of influence line ordinate are determined by geometry (similar triangle) of the influence line and summarize the values as in Table 2.

 x (m) Points Influence line ordinate of MB(kN-m/kN) 0 A 0 5 B 0 10 C ‑5 16 D 0 19 E 0 22 F 0

Sketch the influence line diagram for moment at point B as shown in Figure 7.

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