frame-analysis

name	frame-analysis
description	Rigid frame analysis — slope-deflection method, moment distribution, portal method, cantilever method, sidesway, fixed-end moments, stiffness matrix method.
metadata	{"priority":6,"promptSignals":{"phrases":["frame analysis","rigid frame","slope deflection","moment distribution","portal method","cantilever method","sidesway","fixed end moment"],"minScore":4}}

Frame Analysis — Complete Skill

Fixed-End Moments (FEM) — Standard Cases

Sign convention: +M = clockwise at far end; -M = counterclockwise

Loading	FEM_AB	FEM_BA
UDL w over full span L	-wL²/12	+wL²/12
Point load P at midspan	-PL/8	+PL/8
Point load P at distance a from A	-Pab²/L²	+Pa²b/L²
Triangular load (max w at B)	-wL²/20	+wL²/30
Moment M_A at A	-M/2	—
Settlement Δ at B (w/o rotation)	+6EIΔ/L²	+6EIΔ/L²

Slope-Deflection Method

For prismatic member AB: M_AB = 2EI/L × (2θ_A + θ_B - 3ψ) + FEM_AB M_BA = 2EI/L × (2θ_B + θ_A - 3ψ) + FEM_BA

Where:

• θ_A, θ_B = joint rotations (+ = clockwise)
• ψ = chord rotation = Δ/L (relative settlement or sidesway)
• EI/L = stiffness of member

Procedure:

1. Write slope-deflection equations for all members
2. Apply equilibrium at each joint: ΣM = 0
3. Apply shear equation for sidesway: ΣF_x = 0
4. Solve simultaneous equations for θ, ψ
5. Back-substitute to find moments

Moment Distribution Method (Hardy Cross)

Concepts: Distribution Factor: DF = K/(ΣK) where K = EI/L (or 3EI/4L for pin end) Carry-Over Factor: CO = 0.5 (fixed-far end), CO = 0 (pin/roller)

Procedure:

1. Lock all joints → compute FEMs
2. Release one joint → unbalanced moment distributed to members → CO to far ends
3. Repeat for all joints → iterate until moments converge (<1%)
4. Sum all distributed moments + FEMs for final M at each end

Relative stiffness adjustments: Near end fixed, far end pin: K_eff = ¾K (modified stiffness, CO = 0) Far end is free: K = 0 (cantilever gets no distribution) Symmetric span, symmetric load: only half-frame needed

Stiffness Method (Matrix)

Global stiffness matrix: [K]{d} = {F} For 2D beam element (4 DOF: v_1, θ_1, v_2, θ_2): [k] = EI/L³ × | 12 6L -12 6L | | 6L 4L² -6L 2L² | |-12 -6L 12 -6L | | 6L 2L² -6L 4L² |

Assemble global [K], apply BCs, solve for {d}, recover member forces.

Portal Method (Approximate — Lateral Loads)

Assumptions:

1. Inflection points at midspan of all beams and midheight of all columns
2. Interior columns carry 2× shear of exterior columns

Procedure:

1. Distribute total story shear: V_ext = V/(2(n+1)), V_int = 2V_ext (n = bays)
2. Find column moments from shear × half-height
3. Beam moments from joint equilibrium

Accuracy: ±15% for low-rise buildings (< 5 stories), rectangular frames

Cantilever Method (Approximate — Tall Buildings)

Assumption: axial force in columns proportional to distance from centroid of column areas

Procedure:

1. Locate centroid of column cross-section areas
2. Moment from lateral force = H × y (overturning)
3. Axial force in column i: P_i = A_i × x_i / Σ(A_j × x_j²) × M_story
4. Column shears from equilibrium, beam moments from shear

More accurate than portal method for tall frames (>5 stories)

Sidesway Analysis

Frame sways when loading or geometry is unsymmetric. Unknown: Δ (lateral displacement), causing chord rotation ψ = Δ/L

In slope-deflection: add shear equilibrium equation: ΣF_x = 0: (M_AC + M_CA)/L_AC + (M_BD + M_DB)/L_BD + H = 0

In moment distribution: apply unit sway Δ=1 → find moments → sway correction factor.

Influence Lines for Frames

Müller-Breslau principle: influence line shape = deflected shape from unit deformation at response quantity For frames: place unit load at various positions, find response — Müller applies for statically indeterminate frames too

Elastic vs. Plastic Analysis

Elastic: linear, unique solution Plastic (mechanism method): find collapse load by equating internal work (Σ M_p×θ) to external work (P×Δ) Number of plastic hinges at collapse = degree of indeterminacy + 1

Output

Provide: FEMs, slope-deflection equations or DF table, final end moments [kN·m], reactions [kN], sway magnitude Δ [mm].