For a prismatic beam of length L and second moment of area I subjected to a constant bending moment M along its span, determine the total strain energy stored due to bending (express your answer in terms of M, L, E, and I).

Introduction / Context:
Strain energy methods are powerful for deflection calculations (Castigliano's theorem) and for understanding how members store elastic energy under load. For a constant bending moment along a span (e.g., pure bending region), the energy expression simplifies neatly.

Given Data / Assumptions:

• Constant bending moment M over the full length L.
• Beam has constant E and I; linear elastic behavior.
• No shear deformation energy is considered (pure bending).

Concept / Approach:
The bending strain energy density per unit length is (M^2) / (2 E I). For variable moment, one would integrate this along x. With M constant, the integral reduces to multiplication by L.

Step-by-Step Solution:
1) General formula: U = ∫ (M(x)^2 / (2 E I)) dx.2) For constant M: U = (M^2 / (2 E I)) ∫ dx from 0 to L.3) Evaluate the integral: U = (M^2 / (2 E I)) * L = M^2 L / (2 E I).

Verification / Alternative check:
Dimensional check: M^2 has units of (force*length)^2; dividing by E I (force/area * length^4) and multiplying by length yields force*length (work), consistent with energy.

Why Other Options Are Wrong:

• U = M^2 L / (E I): Missing the factor 1/2; overestimates energy by 2×.
• U = M^2 L / (4 E I): Underestimates by 2×.
• U = 2 M^2 L / (E I): Incorrect scaling.

Common Pitfalls:

• Forgetting the 1/2 factor that arises from integrating stress–strain energy density.
• Applying this constant-moment formula to non-uniform bending without integration.

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