beam-solver

name	beam-solver
description	Complete beam analysis — any loading, any support condition. Reactions, shear/moment diagrams, deflection, slope, stress. Handles simply supported, cantilever, fixed-fixed, propped, overhanging, continuous beams.
metadata	{"priority":9,"promptSignals":{"phrases":["beam","simply supported","cantilever","shear diagram","moment diagram","deflection","beam loading","reactions"],"minScore":4}}

Beam Solver — Complete Skill

Step 1: Classify the Beam

Statically determinate:

• Simply supported (pin + roller): 2 unknowns, 2 equations
• Cantilever (fixed + free): 2 unknowns (R, M), 2 equations
• Overhanging: pin + roller with extension

Statically indeterminate:

• Propped cantilever: 3 unknowns → use compatibility
• Fixed-fixed: 4 unknowns → use superposition or slope-deflection
• Continuous: use 3-moment equation (Clapeyron) or moment distribution

Step 2: Reactions

Use ΣFy = 0 and ΣM = 0. For indeterminate, add compatibility: δ_released_support = 0 (deflection at removed support must be zero)

Step 3: Shear Force Diagram (SFD)

• Start from left: V(x) = ΣF_left(x)
• Point load P at x=a: V jumps by -P (downward load)
• UDL w: V changes linearly, slope = -w
• Concentrated moment M₀: no jump in V
• Sign convention: positive V = left face up, right face down

Step 4: Bending Moment Diagram (BMD)

• M(x) = ΣM_left(x) (sum of moments of all forces left of x)
• Slope of M = V(x) → dM/dx = V
• M is zero at pins, rollers, free ends
• M = fixed value at fixed supports (the reaction moment)
• Maximum M occurs where V = 0

Step 5: Key Formulas by Load Case

Simply Supported — Point Load P at center (a = L/2): R_A = R_B = P/2 M_max = PL/4 (at center) δ_max = PL³/48EI (at center)

Simply Supported — UDL w: R_A = R_B = wL/2 M_max = wL²/8 (at center) δ_max = 5wL⁴/384EI (at center)

Simply Supported — Point Load P at distance a from left: R_A = Pb/L, R_B = Pa/L (b = L-a) M_max = Pab/L (at load point) δ_at_load = Pa²b²/3EIL

Cantilever — Point Load P at tip: R_A = P, M_A = PL M_max = PL (at fixed end) δ_tip = PL³/3EI θ_tip = PL²/2EI

Cantilever — UDL w: R_A = wL, M_A = wL²/2 M_max = wL²/2 (at fixed end) δ_tip = wL⁴/8EI

Fixed-Fixed — UDL w: R_A = R_B = wL/2 M_A = M_B = -wL²/12 (fixed end moments, hogging) M_mid = +wL²/24 (sagging) δ_max = wL⁴/384EI

Propped Cantilever — UDL w: R_B (prop) = 3wL/8 R_A = 5wL/8, M_A = wL²/8 M_max = 9wL²/128 (at x = 5L/8 from fixed)

Step 6: Bending Stress

σ = My/I → σ_max = Mc/I = M/S Section modulus S = I/c For standard sections: use section database (W, S, C, HSS, pipe)

Step 7: Deflection by Superposition

Combine standard cases. Use:

• Table of standard beam deflections (Roark's Formulas for Stress and Strain, Table 3)
• Superposition: δ_total = Σδ_individual cases
• Castigliano's theorem for complex loading: δ_i = ∂U/∂P_i

Step 8: Check Deflection Limit

L/240 (floor beams, live load) L/360 (floor beams, total load, plastered ceiling) L/480 (sensitive equipment L/600 (precision machinery)

Output

Always draw ASCII SFD + BMD, state max M location, compute σ_max, check deflection limit.