truss-analysis

name	truss-analysis
description	Truss analysis — method of joints, method of sections, zero-force members, statically determinate/indeterminate trusses, space trusses, deflection by virtual work/unit load method.
metadata	{"priority":7,"promptSignals":{"phrases":["truss","truss analysis","method of joints","method of sections","pin-jointed","zero force member","space truss"],"minScore":4}}

Truss Analysis — Complete Skill

Determinacy Check

Plane truss: m + r = 2j → determinate m = members, r = reactions (support conditions), j = joints m + r < 2j → mechanism (unstable) m + r > 2j → statically indeterminate (degree = m+r-2j)

Space truss: m + r = 3j → determinate

Common supports:

• Pin: 2 reactions (H, V)
• Roller: 1 reaction (perpendicular to surface)
• Fixed: 3 reactions (H, V, M) — not common in trusses

Zero-Force Members (Identify First — Simplifies Analysis)

1. If only 2 non-collinear members at unloaded joint → both are zero-force
2. If 2 members are collinear at a joint with a 3rd non-collinear and no load → 3rd is zero-force
3. If joint has 3 members, 2 collinear, and load along the collinear direction → 3rd is zero-force

Method of Joints

Start at joint with ≤ 2 unknown member forces. Sign convention: Assume tension (+) for all members. If result is negative → compression.

At each joint: ΣFx = 0, ΣFy = 0

Procedure:

1. Find support reactions (whole body equilibrium)
2. Identify zero-force members
3. Start at joint with 2 unknowns
4. Solve 2 equilibrium equations
5. Move to next joint, repeat until all members solved

Method of Sections

Use when: need to find specific member force without solving entire truss Procedure:

1. Cut through truss cutting ≤ 3 members (including target)
2. Isolate one portion
3. Apply ΣFx=0, ΣFy=0, ΣM=0 to find cut member forces
4. Choose moment center to eliminate 2 unknowns at once

For Pratt/Warren truss diagonals: Take moment about intersection of the other two cut members → directly find diagonal force

Typical Truss Types

Pratt truss: vertical members in compression, diagonals in tension under downward load Warren truss: alternating diagonals, no verticals (efficient for uniform loading) Howe truss: opposite of Pratt (diagonals in compression — wood construction) K-truss: for very long spans (diagonals only go half-depth) Vierendeel truss: no diagonals — moment-resisting joints (not pin-jointed, more complex)

Member Force Efficiency

Tension members: use rods, angles, cables (high Sut/ρ) Compression members: use tubes, wide flanges (buckling governs → need r_min) Critical compression member: P_cr = π²EI_min/(KL)² [Euler, fixed K based on truss connection detail]

Truss Deflection — Unit Load Method (Virtual Work)

δ_j = Σ(n_i·F_i·L_i)/(A_i·E_i)

Where:

• δ_j = deflection at joint j in direction of interest
• n_i = force in member i due to unit virtual load at j
• F_i = actual member force (from real loads)
• L_i, A_i, E_i = length, area, modulus of member i

Procedure:

1. Solve truss under real loads → F_i
2. Remove real loads, apply unit load at joint of interest → n_i
3. Calculate Σ(n_i·F_i·L_i/A_iE_i) → δ

Redundant Truss (Indeterminate)

Degree of indeterminacy = i = m+r-2j (plane) Remove i members as releases (redundant forces X₁, X₂...) Compatibility: deflection at released cut = 0 for each redundant δ_0j + Σ(X_k·δ_kj) = 0 (unit load theorem system of equations) Solve for X_k → back-substitute for all member forces

Space Trusses

Joints: ΣFx=0, ΣFy=0, ΣFz=0 (3 equations per joint) Determinacy: m+r=3j Method of joints applies in 3D — work from joints with ≤ 3 unknowns Member direction cosines: l, m, n → force components = F·(l, m, n)

Output

Provide: all member forces (tension/compression, zero), critical compression member (lowest FOS vs. buckling), max deflection, design recommendations for overstressed members.