   # 7.20 and 7.21 Use the virtual work method to determine the slope and deflection at point B of the beam shown. FIG. P7.21, P7.23, P7.58

#### Solutions

Chapter
Section
Chapter 7, Problem 21P
Textbook Problem
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## 7.20 and 7.21 Use the virtual work method to determine the slope and deflection at point B of the beam shown. FIG. P7.21, P7.23, P7.58

To determine

Find the slope and deflection at point B of the beam using virtual work method.

### Explanation of Solution

Given information:

The beam is given in the Figure.

The value of E is 70 GPa and I is 164(106)mm4.

Calculation:

Consider the real system.

Sketch the real system of the beam as shown in Figure 1.

Refer Figure 1.

Consider the section at x distance from the end B.

Calculate the moment M as follows:

M=50

Consider the virtual system.

Remove all the real loads and apply a unit couple at the point on the beam where the slope is desired.

Sketch the virtual system of the beam with unit couple at point B as shown in Figure 2.

Let an equation expressing the variation of bending moment due to virtual couple be Mv1.

Refer Figure 2.

Mv1=1

Find the slope at B using the virtual work expression:

1(θB)=0LMv1MEIdx (1)

Here, L is the length of the beam, E is the Young’s modulus, and I is the moment of inertia.

Substitute 1 for Mv1, 4 for L, and 50 for M in Equation (1).

θB=1EI04(1)(50)dx=1EI04(50)dx=1EI[50x]04=200kNm2EI

Substitute 70 GPa for E and 164(106)mm4 for I.

Therefore, the slope at point B of the beam is 0

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