   # Draw the influence lines for the shears and bending moments at points D and F of the beam shown in Fig. P8.33. FIG. P8.33, P8.34

#### Solutions

Chapter
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Chapter 8, Problem 34P
Textbook Problem
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## Draw the influence lines for the shears and bending moments at points D and F of the beam shown in Fig. P8.33.FIG. P8.33, P8.34 To determine

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

### Explanation of Solution

Calculation:

Use Muller-Breslau's principle.

Influence line ordinate for shear at D.

Cut the beam at D to obtain the released structure of the beam. Next apply a small relative displacement in positive direction of SD by moving end D of the portion CD downward by Δ1 and end D of the portion DE upward by Δ2 to obtain the deflected shape.

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

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

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

Find the numerical value of influence line ordinate of SD at D.

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

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

Refer Figure 3.

The reaction at the hinge E and C is 12kN because the point D is center of the hinge CD.

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

SD,L=R=12kN

Thus, the value of influence line ordinate just left of D is 12kN/kN.

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

SD,R=R=12kN

Thus, the value of influence line ordinate just right of D is 12kN/kN.

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

 x (m) Points Influence line ordinate of SB(kN/kN) 0 A 0 8 B 0 14 C 1 18 (D′)− −12 22 (D′)+ 120 25 E 0 28 F 0 36 G 0

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

Influence line ordinate for moment at D.

Insert a hinge at D to obtain the released structure of the beam. Next apply a small rotation θ in positive direction of MD at D in the released structure. The rotation θ rotate the portion CD in counterclockwise and the portion DE in counterclockwise to obtain the deflected shape.

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

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

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

Find the numerical value of influence line ordinate for MD at D.

Refer Figure 3.

Find the moment at C from B.

Apply 1 kN load at D.

Consider clockwise moment as positive and anticlockwise moment as negative.

MD=R(4)=12(4)=2kN-m

Thus, the value of influence line ordinate of MD at D is 2kN-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 8 B 0 14 C 0 22 D 2 25 E 0 28 F 0 36 G 0

Sketch the influence line diagram for moment at point D using Table 2 as shown in Figure 7

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