   # 7.24 through 7.27 Use the virtual work method to determine the deflection at point C of the beam shown. FIG. P7.27, P7.62

#### Solutions

Chapter
Section
Chapter 7, Problem 27P
Textbook Problem
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## 7.24 through 7.27 Use the virtual work method to determine the deflection at point C of the beam shown. FIG. P7.27, P7.62

To determine

Find the deflection at point C of the beam using virtual work method.

### Explanation of Solution

Given information:

The beam is given in the Figure.

The value of E is 29,000 ksi and I is 350in.4.

Apply the sign conventions for calculating reactions, forces and moments using the three equations of equilibrium as shown below.

• For summation of forces along x-direction is equal to zero (Fx=0), consider the forces acting towards right side as positive (+) and the forces acting towards left side as negative ().
• For summation of forces along y-direction is equal to zero (Fy=0), consider the upward force as positive (+) and the downward force as negative ().
• For summation of moment about a point is equal to zero (Matapoint=0), consider the clockwise moment as negative and the counter clockwise moment as positive.

Calculation:

Consider the real system.

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

Find the reactions at the supports A and B:

Summation of moments about A is equal to 0.

MA=0By(15)2(21)(212)=0By=29.4k

Summation of forces along y-direction is equal to 0.

+Fy=0Ay+By2(21)=0Ay+29.42(21)=0Ay=12.6k

Consider the virtual system.

Remove all the real loads and apply unit load where the point to find the deflection.

Let the bending moment due to virtual load be Mv.

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

Find the reactions at the supports A and B:

Summation of moments about A is equal to 0.

MA=0By(15)1(21)=0By=1.4k

Summation of forces along y-direction is equal to 0.

+Fy=0Ay+By1=0Ay+1.41=0Ay=0.4k

Find the equations for M and Mv for the 2 segments of the beam as shown in Table 1.

 Segment x-coordinate M (k-ft) Mv (k-ft) Origin Limits (ft) AB A 0−15 12.6x−x2 −0.4x CB C 0−6 −x2 −x

Find the deflection at C using the virtual work expression:

1(ΔC)=0LMvMEIdx (1)

Rearrange Equation (1) for the limits 015 and 06 as follows

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