# Determine the rotation of joint D of the frame shown in Fig. P7.43 by the virtual work method. Use the graphical procedure (Table 7.6) to evaluate the virtual work integrals.

### Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931

Chapter
Section

### Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931
Chapter 7, Problem 44P
Textbook Problem
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## Determine the rotation of joint D of the frame shown in Fig. P7.43 by the virtual work method. Use the graphical procedure (Table 7.6) to evaluate the virtual work integrals.

To determine

Find the rotation at joint D of the frame using virtual work method.

### Explanation of Solution

Given information:

The frame is given in the Figure.

The value of E is 70 GPa and I is 1,290×106mm4

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 joint is equal to zero (Matapoint=0), consider the clockwise moment as negative and the counter clockwise moment as positive.

Calculation:

Consider the real system.

Draw a diagram showing all the given real loads acting on it.

Let the bending moment due to real load be M.

Sketch the real system of the frame and the member end forces as shown in Figure 1.

Find the reactions at the supports:

Summation of moments about A is equal to 0.

MA=0MA200(2)=0MA=400kNm

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

+Fx=0Ax+200=0Ax=200kN

Sketch the moment diagram of the real system as shown in Figure 2.

Consider the virtual system.

Draw a diagram of frame without the given real loads. For rotation apply unit couple at the joint in the desired direction.

Let the bending moment due to virtual couple be Mv.

Sketch the virtual system of the frame with unit couple at joint D and the member end forces as shown in Figure 3.

Find the reactions at the supports:

Summation of moments about A is equal to 0.

MA=0MA1=0MA=1kNm

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

+Fx=0Ax=0

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

+Fy=0Ay=0

Sketch the moment diagram of the virtual system as shown in Figure 4.

Find the slope at D using the virtual work expression:

θD=0LMvMEIdx (1)

Moment of inertia of span AB is 2I, moment of inertia of span BC is I, and moment of inertia of span DC is I.

Rearrange Equation (1) for the limits 05, 04, and 03 as follows.

θD=1EI[1205(MvM)dx+04(MvM)dx+03(MvM)dx] (2)

Refer Table 7.6, “Integrals 0LMvMdx for moment diagrams of simple geometric shapes” in the textbook.

For the moment diagram in Figure 2 and Figure 4, the value of 05MvMdx is 12Mv1(M1+M2)L, the value of 04MvMdx is Mv1M1L, and the value of 03MvMdx is 12Mv1M1L

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