Strength of materials: Determine the total strain energy stored in a prismatic beam of length L when it is subjected to a constant bending moment along its entire span (assume linear elasticity with modulus E and second moment of area I).

Introduction / Context:
When a beam bends under moment, elastic strain energy is stored throughout its length. For constant bending moment, this energy can be computed directly from strength of materials relations. This question checks fluency with the strain energy formula and how it reduces when the bending moment is uniform along the member.

Given Data / Assumptions:

• Prismatic beam of length L with constant E and I.
• Constant bending moment M along the entire span.
• Linear elastic behavior and small deflections.

Concept / Approach:
Elastic strain energy in bending is U = ∫[M(x)^2 / (2 * E * I)] dx over the member. If M is constant, the integral simplifies significantly. The derivation relies on the standard energy expression for beams and does not require curvature integration beyond the uniform case.

Step-by-Step Solution:
1) Use U = ∫ from 0 to L of [M^2 / (2 * E * I)] dx.2) Since M is constant, take M^2 / (2 * E * I) outside the integral: U = [M^2 / (2 * E * I)] * ∫0→L dx.3) Evaluate the integral: ∫0→L dx = L.4) Therefore, U = (M^2 * L) / (2 * E * I).

Verification / Alternative check:
Dimensional check: M has dimensions of force*length; M^2 * L gives force^2 * length^3. Dividing by E * I (force/area * length^4) yields force * length = energy, which is dimensionally consistent.

Why Other Options Are Wrong:
Options with L^2 or 1/L arise from incorrect integration or misuse of the energy formula.Options involving M * L terms miss the required square dependence on moment.

Common Pitfalls:
Confusing strain energy in bending with that in axial members; forgetting the square of bending moment in the integrand; or inserting incorrect limits while integrating.

Final Answer:
U = (M^2 * L) / (2 * E * I)