energy-methods

name	energy-methods
description	Energy methods in structural analysis — strain energy, Castigliano's theorems, virtual work, unit load method, complementary energy, deflection of beams/frames/trusses.
metadata	{"priority":7,"promptSignals":{"phrases":["Castigliano","virtual work","strain energy","energy method","unit load method","complementary energy"],"minScore":3}}

Energy Methods — Complete Skill

Strain Energy

Strain Energy by Load Type

Axial: U = ∫ N² dx / (2AE) Bending: U = ∫ M² dx / (2EI) Shear: U = ∫ f_s V² dx / (2GA) (f_s = form factor: 1.2 rectangular, 10/9 circular) Torsion: U = ∫ T² dx / (2GJ)

Castigliano's First Theorem (for linear elastic systems): Force = ∂U/∂δ (force is derivative of strain energy w.r.t. corresponding displacement)

Castigliano's Second Theorem: δ_i = ∂U/∂P_i (displacement in direction of load P_i = partial derivative of U) θ_i = ∂U/∂M_i (rotation at point of applied moment M_i)

Castigliano's Method for Deflection

Procedure

1. Apply all real loads PLUS a dummy load Q at point of interest (in direction of desired deflection)
2. Write M(x), N(x), T(x) in terms of Q
3. Compute U = ∫ M²dx/(2EI) + ...
4. δ = ∂U/∂Q, then set Q = 0

Shortcut (differentiate before integrating): δ = ∫ (∂M/∂Q) × M / (EI) dx + ∫ (∂N/∂Q) × N / (AE) dx + ... Evaluate at Q = 0 after differentiating

Example — Cantilever Beam, End Deflection

M(x) = -P(L-x), ∂M/∂P = -(L-x) δ = ∫₀ᴸ M (∂M/∂P) dx / EI = ∫₀ᴸ [-P(L-x)][-(L-x)] dx / EI = PL³/(3EI) ✓

Example — Truss Deflection

δ = Σ N_i (∂N_i/∂P) L_i / (A_i E_i) N_i = bar forces under real load ∂N_i/∂P = bar forces under unit dummy load (unit load method)

Virtual Work Principle

Statement

For any virtual displacement δu compatible with constraints: δW_external = δW_internal Σ F_i × δu_i = ∫ σ_ij × δε_ij dV

Unit Load Method (Mohr's integral): Deflection at point A in direction of unit load: δ_A = ∫ m(x) × M(x) / (EI) dx m(x) = moment due to unit virtual load at A M(x) = moment due to real loads

Rotation at point B: θ_B = ∫ m_θ(x) × M(x) / (EI) dx m_θ = moment from unit virtual moment at B

Mohr's Product Integral Table

For linear M and linear m: ∫₀ᴸ m × M dx = L/6 × (2m_1 M_1 + m_1 M_2 + m_2 M_1 + 2m_2 M_2) Standard integrals for common shapes (triangular × triangular, triangular × rectangular, etc.):

• Rectangle × Rectangle: L × a × b
• Triangle (0→a) × Rectangle (b): L × a × b / 2
• Triangle × Triangle (same direction): L × a × b / 3
• Triangle × Triangle (opposite): L × a × b / 6
• Parabola (max c at center) × Rectangle (b): 2L × c × b / 3

Minimum Complementary Energy

For statically indeterminate structures: ∂U/∂X_i = Δ_i (X_i = redundant reaction, Δ_i = displacement at redundant)

For fixed supports: Δ_i = 0: ∂U/∂X_i = 0 → compatibility equations (force method)

Force Method (Compatibility)

1. Remove redundant reactions → statically determinate base structure
2. Compute primary structure deflection at redundant location δ₀ (from external loads)
3. Compute flexibility coefficient f_ii (deflection at i from unit redundant at i)
4. Compatibility: δ₀ + f_ii × X_i = Δ_prescribed (0 for fixed support)
5. Solve X_i

For n redundants: [f] {X} = -{δ₀} (flexibility matrix equation)

Energy Theorems

Maxwell-Betti Reciprocal Theorem

f_ij = f_ji Deflection at i due to unit load at j = deflection at j due to unit load at i

Principle of Minimum Potential Energy

Of all admissible displacement fields, the true one minimizes total potential energy: Π = U - W_external → δΠ = 0

Variational Principle (Ritz Method)

Assume: u(x) = Σ a_i φ_i(x) (admissible functions satisfying BCs) Minimize Π with respect to a_i: ∂Π/∂a_i = 0 → algebraic equations for a_i

Output

Provide: deflection δ [mm] and rotation θ [rad] at specified points using Castigliano or unit load method, compatibility equations for indeterminate structure, redundant reactions [N or N·m], deflection components by load type (bending/axial/shear/torsion).