Concept explainers
A = 2300 mm2, I = 9.5(106) mm4.
R14–1
Answer to Problem 14.1RP
The total axial and bending strain energy in the A992 steel beam is
Explanation of Solution
Given information:
The cross-sectional area of the beam is
Moment of inertia of the beam is
Assumption:
The modulus of elasticity or Young’s modulus of theA992 steelis
Explanation:
Determine the reactions:
Entire beam:
Show the free body diagram of the entire beam as in Figure 1.
Moment about the point A:
Determine the vertical reaction at point B by taking moment about point A.
Along the vertical direction:
Determine the vertical reaction at point B by resolving the vertical component of forces.
Along the horizontal direction:
Determine the horizontal reaction at point A by resolving the horizontal component of force.
Show the calculation of reaction as follows:
Solve Equation (1).
Substitute 7.5kN for
Solve Equation (3).
Region
Show the free-body diagram of the section as in Figure 2.
Moment about the section:
Determine the moment at section by taking moment about the section.
Along the horizontal direction:
Determine the normal axial force at the section by resolving the horizontal component of forces.
Show the calculation of values as follows:
Substitute 7.5kN for
Substitute 15 kN for
Strain energy due to axial load:
Determine the strain energy of a bar of constant cross-sectional area A and constant internal axial load N using the equation.
Here, N is the axial load, L is the length of beam, E is Young’s modulus or modulus of elasticity, and A is cross-sectional area of the beam.
Substitute 15 kN for N, 10 m for L,
Strain energy due to Bending:
Determine the strain energy in the beam due to bending using the equation.
Here, M is the moment in the beam and I is the moment of inertia of the beam.
Substitute 10 m for L,
Total strain energy:
Determine the total strain energy by adding the strain energy due to axial load and the strain energy due to bending.
Substitute 2.4456 J for
Thus, the total axial and bending strain energy in the A992 steel beam is
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