Truss Analysis — Method of Joints, Sections & Computer Methods

Steel truss analysis techniques: method of joints, method of sections, zero-force member identification, matrix stiffness method, and practical modeling guidance.

Analysis methods overview

Truss analysis determines the axial force in each member for a given set of applied loads. Three classical methods exist, plus modern computer-based approaches:

• Method of joints — isolates each joint as a free body diagram and solves equilibrium (sum Fx = 0, sum Fy = 0). Works outward from the supports. Best for finding all member forces in a simple truss.
• Method of sections — cuts through the truss and takes a free body of one side. Three equilibrium equations (sum Fx = 0, sum Fy = 0, sum M = 0) solve for up to three unknown member forces at the cut. Best for finding the force in a specific member without solving the entire truss.
• Zero-force member identification — geometric inspection of unloaded joints identifies members that carry zero force. This simplifies the analysis and serves as a quick check on computer results.
• Matrix stiffness method — the standard computer approach. Assembles global stiffness matrix, applies loads, solves for displacements, then back-calculates member forces. Handles any geometry, loading, and support conditions.

Zero-force member rules

Two rules identify zero-force members by inspection:

Rule 1: At an unloaded joint where only two members meet and they are not collinear, both members are zero-force members.

Rule 2: At an unloaded joint where three members meet and two are collinear, the third member (the non-collinear one) is a zero-force member.

These rules must be applied iteratively — removing a zero-force member may create a new two-member joint, revealing another zero-force member.

Zero-force members are not useless. They provide stability under other load cases (wind uplift, asymmetric snow) and reduce the unbraced length of compression chords. Removing them from the fabrication is an error.

Worked example — method of sections

Pratt truss: span 18 m, depth 3 m, 6 panels at 3 m each. Total factored UDL on top chord = 10 kN/m. Determine the force in the bottom chord at panel 3 (mid-span) and the diagonal at panel 2.

Reactions: R = wL/2 = 10 x 18 / 2 = 90 kN at each support.

Bottom chord at mid-span (member BC between panels 3 and 4): Cut through panel 3, take the left portion. Take moments about the top chord joint directly above the cut to eliminate the top chord and diagonal forces from the equation.

Sum M about top joint at x = 9 m: R x 9 - w x 9 x 4.5 - F_bottom x 3 = 0. 90 x 9 - 10 x 9 x 4.5 - F_bottom x 3 = 0. 810 - 405 - 3 x F_bottom = 0. F_bottom = 135 kN (tension).

Cross-check: M_max = wL^2/8 = 10 x 18^2 / 8 = 405 kN-m. Chord force = M / depth = 405 / 3 = 135 kN. Matches.

Diagonal at panel 2: Cut between panels 2 and 3. Take the left portion. Sum vertical forces: R - w x 6 - F_diag x sin(theta) = 0, where theta = arctan(3/3) = 45 degrees.

90 - 60 - F_diag x 0.707 = 0. F_diag = 30 / 0.707 = 42.4 kN (tension, as expected for a Pratt diagonal).

Modeling guidance for computer analysis

When using finite element software for truss analysis:

• Pin-jointed vs rigid-jointed model. A pin-jointed model gives only axial forces — appropriate for preliminary design. A rigid-jointed model captures secondary bending moments from gusset plate fixity. For final design, rigid-jointed analysis with member self-weight applied as distributed load (not panel-point concentrated loads) gives more accurate results.
• Member releases. If modeling as beam elements (not truss elements), apply moment releases at both ends of web members to simulate pin connections. Do not release chord members — they carry bending between panel points.
• Support conditions. Model one support as pinned (fixed in x and y) and the other as roller (fixed in y only) to allow thermal expansion. If both supports are pinned, temperature changes induce large horizontal forces.

Span-to-depth ratios by application

Code provisions for truss analysis

AISC Direct Analysis Method eliminates the need for K-factor calculation in most cases by applying notional loads and using reduced member stiffness (0.8EI for all members, additional tau_b for compression members). This simplifies truss analysis significantly.

Common pitfalls

• Applying loads between panel points without accounting for local bending. If purlins or equipment loads are applied between the truss nodes, the loaded chord member experiences combined axial force and local bending. The interaction must be checked per AISC H1 (or equivalent).
• Using K = 1.0 for all compression members. In-plane K for continuous top chords is less than 1.0 (approximately 0.85-0.90) because adjacent panels provide rotational restraint. Out-of-plane K depends on bracing layout and may be 1.0 (braced at each panel) or up to 2.0 (cantilever).
• Ignoring deflection at mid-span. Truss deflection under live load is often L/500 to L/800 for roof trusses but must be verified. Cambering the truss for dead load deflection is standard practice for spans over 15 m.
• Not checking member reversal under pattern loading. A top chord member that is in compression under full gravity load may go into tension under wind uplift. A web member designed for tension only (rod or flat bar) will buckle under compression reversal.

Run this calculation

• Section Properties Database
• Beam Calculator

Related references

• Compact Section Limits
• Bolt Hole Sizes
• Connection Types
• Plate Girder Design
• Steel Space Frame
• How to Verify Calculations
• steel beam capacity calculator

Disclaimer

This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against the applicable standard and project specification before use. The site operator disclaims liability for any loss arising from the use of this information.