Answer:
The formula for the strain energy due to bending for a beam is:
U = (1/2) * M * EI
where:
U = strain energy due to bending
M = maximum bending moment
E = modulus of elasticity of the beam
I = moment of inertia of the beam
We are given that the modulus of elasticity of the timber beam is E = 1.5 x 10^6 psi and the moment of inertia of the beam is I = 113 in^4. To find the maximum bending moment, we need to determine the shear force and bending moment diagrams for the beam.
Assuming that the load is applied at the midpoint of the beam, we can determine that the shear force at any point x along the beam is:
V(x) = 6 - 2x
The maximum shear force occurs at x = 0, where V(0) = 6 kips.
To determine the bending moment, we integrate the shear force function:
M(x) = ∫ V(x) dx = ∫ (6 - 2x) dx = 6x - x^2 + C
where C is the constant of integration. To solve for C, we use the boundary condition that the bending moment is zero at x = 12:
M(12) = 0
6(12) - 12^2 + C = 0
C = 72 - 144
C = -72
Therefore, the bending moment at any point x along the beam is:
M(x) = 6x - x^2 - 72
The maximum bending moment occurs at the midpoint of the beam, where x = 6:
Mmax = 6(6) - 6^2 - 72
Mmax = -36 kip-ft
Now we can calculate the strain energy due to bending:
U = (1/2) * Mmax * EI
U = (1/2) * (-36) * (1.5 x 10^6) * 113
U = 1.0854 x 10^9 in-lbs
Converting to foot-pounds, we get:
U = 1.0854 x 10^9 / 12
U = 90,450,000 ft-lbs
Therefore, the strain energy due to bending for the timber beam and loading shown is approximately 90,450,000 foot-pounds.
Step-by-step explanation: