Find the strain energy of the prismatic beam AB.
Answer to Problem 40P
The strain energy of the prismatic beam AB is
Explanation of Solution
Given information:
Taking into account the effect of both normal and shearing stresses.
Calculation:
Calculate the moment of inertia (I) for the rectangular cross section as shown below.
Here, b is the width of the cross section and d is the depth of the cross section.
Calculate the area of the cross section (A) as shown below.
Calculate the centroid (c) as shown below.
Calculate the reactions as shown below.
Take moment about B is Equal to zero.
Summation of forces along vertical direction is Equal to zero.
Calculate the shear force as shown below.
Shear force at A is
Shear force at B is
Calculate the bending moment as shown below.
Bending moment at A is
Bending moment at B is
Sketch the shear force and bending moment diagram as shown in Figure 1.
Refer to Figure 1.
Maximum shear force
Maximum bending moment
Bending moment at a distance
Calculate the strain energy due to bending
Substitute
Substitute
Calculate the shear stress
Calculate the strain energy density
Here, G is the modulus of rigidity.
Substitute
Substitute
The value of
Differentiate both sides of the Equation as shown below.
Calculate the strain energy due to shear as shown below.
Substitute
Substitute
Calculate the total strain energy as shown below.
Substitute
Hence, the strain energy of the prismatic beam AB is
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Chapter 11 Solutions
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