7.34 and 7.35 Use the virtual work method to determine the slope and deflection at point D of the beam shown.
FIG. P7.35, P7.64
Find the slope and deflection at point D of the beam using virtual work method.
Given information:
The beam is given in the Figure.
Value of E is 30,000 ksi, I is
Calculation:
Consider the real system.
Draw a diagram showing all the given real loads acting on it.
Let an equation expressing the variation of bending moment due to real loading be M.
Sketch the real system of the beam as shown in Figure 1.
Find the reactions and moment at the supports:
Consider portion BCD, Summation of moments about B is equal to 0.
Summation of forces along ydirection is equal to 0.
Summation of moments about A is equal to 0.
Consider the virtual system.
Draw a diagram of beam without the given real loads. For deflection apply unit load at the point and in the desired direction.
For slope calculation, 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 D as shown in Figure 2.
Let an equation expressing the variation of bending moment due to virtual couple be
Consider portion BCD, Summation of moments about B is equal to 0.
Summation of forces along ydirection is equal to 0.
Summation of moments about A is equal to 0.
Sketch the virtual system of the beam with unit load at point D as shown in Figure 3.
Let an equation expressing the variation of bending moment due to virtual load be
Consider portion BCD, Summation of moments about B is equal to 0.
Summation of forces along ydirection is equal to 0.
Summation of moments about A is equal to 0.
Find the equations for M,
Segment  xcoordinate 
M 

 
Origin  Limits (m)  
DC  D  
CB  D  
AB  A 
Find the slope at D using the virtual work expression:
Here, L is the length of the beam, E is the young’s modulus, and I is the moment of inertia.
Rearrange Equation (1) for the limits
Substitute
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