| 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:
- Write slope-deflection equations for all members
- Apply equilibrium at each joint: ΣM = 0
- Apply shear equation for sidesway: ΣF_x = 0
- Solve simultaneous equations for θ, ψ
- 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:
- Lock all joints → compute FEMs
- Release one joint → unbalanced moment distributed to members → CO to far ends
- Repeat for all joints → iterate until moments converge (<1%)
- 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:
- Inflection points at midspan of all beams and midheight of all columns
- Interior columns carry 2× shear of exterior columns
Procedure:
- Distribute total story shear: V_ext = V/(2(n+1)), V_int = 2V_ext (n = bays)
- Find column moments from shear × half-height
- 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:
- Locate centroid of column cross-section areas
- Moment from lateral force = H × y (overturning)
- Axial force in column i: P_i = A_i × x_i / Σ(A_j × x_j²) × M_story
- 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].
