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 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. Simplifies analysis and checks computer results.
- Matrix stiffness method -- the standard computer approach. Assembles global stiffness matrix, applies loads, solves for displacements, then back-calculates member forces.
Method selection guide
| Task | Best Method | Speed | Accuracy |
|---|---|---|---|
| Find all member forces | Method of joints | Slow | Exact (for ideal truss) |
| Find one specific member force | Method of sections | Fast | Exact |
| Identify zero-force members | Visual inspection | Very fast | Exact |
| Complex geometry/loading | Computer analysis | Fast | Depends on model |
| Verify computer results | Method of sections | Moderate | Exact |
| Final design with 2nd-order | Computer + DA method | Fast | Highest |
Zero-force member rules
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 member examples
| Configuration | Members | Zero-Force? | Why |
|---|---|---|---|
| Two members, L-joint | Both | Yes | Rule 1: non-collinear |
| Three members, T-joint | Non-collinear | Yes | Rule 2: collinear pair balanced |
| Two members, straight | Neither | No | Collinear, force transmitted |
| Loaded joint, 2 members | Depends | No | Load present, rules don't apply |
Zero-force members provide stability under other load cases (wind uplift, asymmetric snow) and reduce unbraced length of compression chords. Do not remove them from fabrication.
Chord force formulas -- parallel chord truss
For a simply supported, parallel chord truss with uniform load w:
Top chord compression = M / d
Bottom chord tension = M / d
Diagonal force = V / sin(theta)
Vertical force = V (directly)
Member forces by truss type (6-panel Pratt, uniform load)
| Member | Force Formula | End Panel | Mid Panel |
|---|---|---|---|
| Top chord | M/d (compression) | Low | Maximum |
| Bottom chord | M/d (tension) | Low | Maximum |
| Diagonal (Pratt) | V/sin(theta) (tension) | Maximum | Near zero |
| Vertical (Pratt) | Local load (compression) | Moderate | Moderate |
| Diagonal (Warren) | Alternates T/C | Maximum | Moderate |
Maximum chord force by span (w = 1.0 klf, d = span/12)
| Span (ft) | Depth (ft) | M_max (kip-ft) | Chord Force (kips) |
|---|---|---|---|
| 30 | 2.5 | 112.5 | 45.0 |
| 40 | 3.3 | 200.0 | 60.0 |
| 50 | 4.2 | 312.5 | 75.0 |
| 60 | 5.0 | 450.0 | 90.0 |
| 80 | 6.7 | 800.0 | 120.0 |
| 100 | 8.3 | 1250.0 | 150.0 |
Chord force = wL^2/(8d). Deeper trusses reduce chord force proportionally.
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.
Reactions: R = wL/2 = 10 x 18 / 2 = 90 kN at each support.
Bottom chord at mid-span: Cut through panel 3, take the left portion. Take moments about the top chord joint directly above the cut:
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 / d = 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 = 42.4 kN (tension, as expected for Pratt).
Worked example -- method of joints
Same Pratt truss. Determine forces at the left support joint (bottom chord + first vertical + first diagonal).
Known: Reaction R = 90 kN (upward). Bottom chord horizontal. First diagonal at 45 degrees.
Sum Fy = 0: R - F_vert - F_diag x sin(45) = 0. Sum Fx = 0: F_bottom - F_diag x cos(45) = 0.
At the support (no applied load directly at the support joint), F_vert = 0 (no purlin at support). So: 90 - 0 - F_diag x 0.707 = 0. F_diag = 127.3 kN. Wait -- this is the first diagonal from the support which has no intermediate panel load, so F_diag = R / sin(45) = 127.3 kN. The previous method of sections result of 42.4 was at panel 2, not the end diagonal. The end diagonal carries the full reaction.
F_bottom = 127.3 x cos(45) = 90.0 kN (tension).
Modeling guidance for computer analysis
When using finite element software for truss analysis:
- Pin-jointed vs rigid-jointed model. Pin-jointed gives only axial forces (preliminary). Rigid-jointed captures secondary bending from gusset fixity (final design).
- Member releases. If using beam elements (not truss elements), apply moment releases at both ends of web members to simulate pins. Do not release chord members -- they carry bending between panel points.
- Support conditions. One pinned (fixed x and y), one roller (fixed y only) to allow thermal expansion. Both pinned induces thermal forces.
- Notional loads. Apply 0.002Yi per AISC Direct Analysis Method for stability-sensitive trusses.
Recommended modeling approach by analysis stage
| Design Stage | Model Type | Loading | Purpose |
|---|---|---|---|
| Preliminary | Pin-jointed | Panel point loads | Member sizing |
| Design development | Rigid-jointed | Distributed + point | Secondary bending |
| Final design | Rigid + DA method | Distributed + notional | Code compliance |
| Connection design | Rigid-jointed | Envelope forces | Gusset plate sizing |
Span-to-depth ratios by application
| Application | Typical span/depth | Reason |
|---|---|---|
| Roof truss (light load) | 10-15 | Economical; deflection rarely governs |
| Floor truss (heavy load) | 12-18 | Deflection governs; vibration limit |
| Transfer truss | 3-5 | Very heavy loads; uses full story height |
| Pedestrian bridge | 10-15 | Aesthetic depth limits; deflection limit L/500 |
| Long-span roof (> 40 m) | 15-20 | Weight economy; deeper = lighter chords |
Method of Joints — Step-by-Step Procedure
The method of joints isolates each joint as a free-body diagram and solves two equilibrium equations (sum Fx = 0, sum Fy = 0) for the unknown member forces. It proceeds systematically from the supports outward.
Detailed Procedure
Step 1: Calculate the support reactions using global equilibrium (sum Fy = 0, sum M = 0).
Step 2: Start at a support joint that has no more than two unknown member forces. Typically this is a support joint with one horizontal member and one diagonal.
Step 3: Draw the free-body diagram of the joint. Assume all unknown member forces are tension (pointing away from the joint). A negative result means the member is in compression.
Step 4: Apply equilibrium:
- Sum Fx = 0 (horizontal forces)
- Sum Fy = 0 (vertical forces)
Step 5: Solve the two equations for the two unknowns.
Step 6: Move to an adjacent joint that now has no more than two unknowns (solved members from previous joints count as known forces). Repeat Steps 3-5.
Step 7: Continue until all member forces are determined.
Step 8: Verify by checking equilibrium at the last joint (all forces should balance; this is a built-in check).
Method of Joints Worked Example
Consider a simple Warren truss: span = 24 ft, depth = 4 ft, 4 panels at 6 ft each. Bottom chord nodes at 0, 6, 12, 18, 24 ft. Total factored load = 48 kips applied as 12 kips at each upper node.
Reactions: RA = RB = 24 kips (symmetric).
Joint A (bottom-left support): Bottom chord horizontal, diagonal rises at angle theta = arctan(4/6) = 33.69 degrees.
Sum Fy = 0: 24 - F_diag x sin(33.69) = 0. F_diag = 24/0.5547 = 43.3 kips (tension). Sum Fx = 0: F_bottom - F_diag x cos(33.69) = 0. F_bottom = 43.3 x 0.832 = 36.0 kips (tension).
Joint B (upper-left): Loads from two diagonals and one top chord segment. Applied load = 12 kips downward.
Sum Fy = 0: F_diag1 x sin(33.69) - 12 - F_diag2 x sin(33.69) = 0. Sum Fx = 0: F_diag1 x cos(33.69) + F_top - F_diag2 x cos(33.69) = 0.
Solving simultaneously gives the remaining member forces. The method of joints becomes increasingly tedious for large trusses, which is why the method of sections or computer analysis is preferred for trusses with more than 8-10 panels.
Method of Sections — Detailed Worked Example
The method of sections is the most efficient way to find the force in a specific member without solving the entire truss. It cuts through the truss and uses three equilibrium equations on one half.
Example: Pratt Truss with Concentrated Load
Given: Pratt truss, span = 48 ft, depth = 6 ft, 8 panels at 6 ft each. A single concentrated factored load of 60 kips at the mid-span top chord node. Fy = 50 ksi.
Reactions: RA = RB = 30 kips.
Find: Force in the bottom chord at mid-span (member between nodes at x = 18 ft and x = 30 ft).
Step 1 — Cut the truss through the mid-span panel. Cut members: top chord (x = 24 ft), bottom chord (x = 18 to 30 ft segment), and two diagonals in the panel.
Step 2 — Take the left half as a free body.
Step 3 — Apply sum of moments about the top chord node at x = 24 ft:
Sum M = RA x 24 - F_bottom x 6 = 0
30 x 24 - F_bottom x 6 = 0
F_bottom = 720 / 6 = 120 kips (tension)
Verification: M_max at mid-span = RA x 24 - P/2 x 0 = 720 kip-ft. Chord force = M/d = 720/6 = 120 kips. Confirms the section result.
Step 4 — Find diagonal force by cutting through the leftmost diagonal and applying sum Fy:
Sum Fy = RA - F_diag x sin(theta) = 0
30 - F_diag x sin(45) = 0
F_diag = 42.4 kips (tension in Pratt diagonal)
Advantages of Method of Sections
- Finds any member force in one calculation (no need to solve the whole truss)
- Three equilibrium equations (sum Fx, sum Fy, sum M) solve for up to three unknowns at each cut
- Taking moments about the intersection of two cut members isolates the force in the third member directly
- Excellent for verifying computer analysis results
Pratt vs Warren vs Howe Truss Comparison
| Feature | Pratt Truss | Warren Truss | Howe Truss |
|---|---|---|---|
| Diagonal direction | Angles toward center | Alternating V-pattern | Angles toward ends |
| Diagonal force type | Tension (under gravity) | Alternating T and C | Compression |
| Vertical force type | Compression | Zero-force ( unloaded panels) | Tension |
| Best use | Long spans, heavy loads | Medium spans, uniform loads | Short spans, timber |
| Typical span range | 60-200 ft | 30-120 ft | 20-80 ft |
| Material efficiency | Excellent (tension diag) | Very good (balanced) | Good (timber diag) |
| Connection type | gusset plates, welded | Direct chord connections | Timber joints, steel |
| Weight efficiency | Very good | Excellent (fewest members) | Good |
| Common applications | Bridges, transfer truss | Roof trusses, pedestrian bridges | Floor joists, small roofs |
Force Distribution Comparison (Uniform Load)
For a 60-ft span, 5-ft depth truss with uniform load of 2 klf:
| Member Type | Pratt Force Pattern | Warren Force Pattern | Howe Force Pattern |
|---|---|---|---|
| Top chord | Max compression at mid-span | Max compression at mid-span | Max compression at mid-span |
| Bottom chord | Max tension at mid-span | Max tension at mid-span | Max tension at mid-span |
| End diagonals | Maximum tension | Maximum T and C alternating | Maximum compression |
| Mid-span diagonals | Near zero | Moderate alternating T/C | Near zero |
| End verticals | Moderate compression | Near zero | Maximum tension |
All three truss types produce the same chord forces (M/d) at mid-span. The difference lies in how the shear is distributed to the web members. Pratt trusses are preferred for steel construction because the diagonals are in tension (no buckling concern), while Warren trusses are lighter because they use fewer total members.
Typical Chord and Web Member Sizes Table
| Truss Span (ft) | Truss Depth (ft) | Top Chord (compression) | Bottom Chord (tension) | End Diagonal | Interior Diagonal |
|---|---|---|---|---|---|
| 20-30 | 2-3 | HSS 4x4x3/8 | HSS 4x4x3/8 | HSS 4x4x1/4 | HSS 3x3x1/4 |
| 30-50 | 3-4 | W8x24 | W8x24 | HSS 5x5x3/8 | HSS 4x4x3/8 |
| 50-80 | 4-7 | W10x33 | W10x30 | W8x24 | HSS 6x6x3/8 |
| 80-120 | 7-10 | W12x40 | W12x35 | W8x31 | W8x24 |
| 120-160 | 10-13 | W14x48 | W14x38 | W10x39 | W8x31 |
| 160-200 | 13-17 | W14x68 | W14x54 | W12x40 | W10x33 |
These sizes are approximate for gravity-only loading (floor trusses) with Fy = 50 ksi and typical office live loads. Seismic or heavy industrial loads will require larger members.
Connection Design for Truss Joints
Truss connections must transfer member forces between chords and web members. The connection type depends on the member type and force magnitude:
| Connection Type | Application | Typical Force Range | Key Design Check |
|---|---|---|---|
| Direct welded (HSS) | HSS-to-HSS truss joints | 20-150 kips | Branch wall local yielding, chord plastification |
| Gusset plate (bolted) | W-shape-to-W-shape joints | 50-500+ kips | Bolt shear, bearing, block shear, Whitmore section |
| Gusset plate (welded) | HSS-to-W-shape or W-to-W joints | 50-500+ kips | Weld capacity, base metal shear, Whitmore section |
| Knife plate | HSS web to W-shape chord | 20-100 kips | HSS wall local yielding, weld capacity |
| Split-tee | Heavy chord-to-column joints | 200-1000+ kips | Flange bolt tension, prying action |
Gusset Plate Design Per AISC
Gusset plates are designed using the Uniform Force Method (AISC Manual Part 9). Key limit states for each gusset plate connection:
- Bolt shear: phiRn = 0.75 x Fnv x Ab x n (per bolt, single or double shear)
- Bolt bearing on gusset: phiRn = 0.75 x 2.4 x d x t x Fu (deformation considered)
- Block shear: phiRn = 0.75 x (0.60 x Fu x Anv + Ubs x Fu x Ant) per AISC J4.3
- Whitmore section: Effective width for checking tension/compression at the end of the gusset = spread angle of 30 degrees from each exterior bolt
- Gusset-to-chord weld: Designed for the horizontal component of the member force
- Gusset buckling: For compression diagonals, check the gusset as a compression element using the Whitmore section width and the unsupported length to the nearest line of bolts
Deflection Calculation for Trusses
Truss deflection can be estimated using the virtual work method (unit load method):
delta = sum(F_i x f_i x L_i) / (E x A_i)
Where F_i = force in member i due to actual loads, f_i = force in member i due to a unit virtual load at the deflection point, L_i = length of member i, A_i = area of member i.
Quick Deflection Estimate for Parallel Chord Trusses
For a simply supported, parallel chord truss with uniform load w, the mid-span deflection is approximately:
delta = 5 x w x L^4 / (384 x E x I_eff)
Where I_eff = A_chord x d^2 / 2 (approximate effective moment of inertia, d = truss depth).
| Span (ft) | Depth (ft) | Chord Area (in^2) | I_eff (in^4) | Delta (LL=1 klf) | L/delta | Deflection Check |
|---|---|---|---|---|---|---|
| 40 | 3.3 | 3.0 | 2,614 | 0.48 in | 1000 | OK (< L/360) |
| 60 | 5.0 | 4.0 | 7,200 | 0.72 in | 1000 | OK (< L/360) |
| 80 | 6.7 | 5.0 | 16,015 | 0.92 in | 1043 | OK (< L/360) |
| 100 | 8.3 | 6.0 | 29,575 | 1.15 in | 1043 | OK (< L/360) |
| 120 | 10.0 | 7.0 | 50,400 | 1.37 in | 1052 | OK (< L/360) |
For floor trusses, the typical live load deflection limit is L/360. For roof trusses, L/240 is common. The span-to-depth ratios in this table (12:1) generally produce trusses that easily meet deflection limits.
Truss Design per AISC 360
AISC 360-22 governs truss member design through the following provisions:
| Limit State | AISC Section | Application to Truss Members |
|---|---|---|
| Tension yielding | D2 | Bottom chord, tension diagonals |
| Tension rupture | D3 | Bottom chord and tension diagonals at connections |
| Compression | E3 | Top chord, compression diagonals |
| Flexure (chords) | F2 | Chord bending between panel points |
| Combined axial + bending | H1 | Chords with inter-nodal loads |
| Shear | G1 | Rare for truss members; check at concentrated loads |
| Connection design | J | Bolted and welded joints at panel points |
Important Design Considerations
Effective length of compression chords: In-plane K is approximately 0.9 (continuous chord with adjacent panel restraint). Out-of-plane K depends on purlin or brace spacing.
Secondary bending: Loads applied between panel points create bending in the chord. This must be checked using AISC Chapter H (combined axial + flexure). The interaction equation is:
Pu/(phiPn) + 8/9 x (Mu/(phiMn)) <= 1.0 [when Pu/(phiPn) >= 0.2]
Member reversals: Under wind uplift, gravity-loaded compression members may become tension members (and vice versa). Warren truss diagonals alternate between tension and compression under different load patterns.
Minimum member sizes: AISC recommends a minimum member size for handling and erection stability, typically at least an HSS 3x3x3/16 or W6x8.5 for web members.
Slenderness limits: AISC recommends KL/r <= 200 for compression members and L/r <= 300 for tension members (the latter is a serviceability recommendation, not a strength limit).
Multi-code comparison
| Aspect | AISC 360 | AS 4100 | EN 1993 | CSA S16 |
|---|---|---|---|---|
| Analysis method | Ch. C (Direct Analysis) | Cl. 4.4 (second-order) | Cl. 5.2.2 | Cl. 8 (stability) |
| Notional loads | 0.002Yi per level | Cl. 4.4.2 (0.003Vi) | EN 1993-1-1 Cl. 5.3.2 | Cl. 8.4.1 (0.005W) |
| Compression member | Ch. E (KL/r <= 200) | Cl. 6.3 (KL/r <= 180) | Cl. 6.3.1 | Cl. 13.3 (KL/r <= 200) |
| Effective length | K per alignment chart or DA | K per Cl. 4.6.3 | Non-dimensional lambda_bar | K per Cl. 10.4 |
| Tension member | Ch. D | Cl. 7 | Cl. 6.2 | Cl. 13.2 |
| Deflection limits | IBC Table 1604.3 | AS 1170.0 Table B1 | EN 1990 Annex A | NBCC 4.1.8.13 |
Common mistakes
Applying loads between panel points without local bending check. If purlins or equipment loads are between nodes, the chord experiences combined axial + bending. Check per AISC H1.
Using K = 1.0 for all compression members. In-plane K for continuous top chords is approximately 0.85-0.90 (adjacent panels provide restraint). Out-of-plane K depends on bracing.
Ignoring deflection at mid-span. Truss deflection under live load is often L/500 to L/800. Cambering for dead load is standard for spans over 15 m.
Not checking member reversal under pattern loading. A top chord in compression under gravity may go into tension under wind uplift. Warren diagonals alternate T/C.
Using only panel-point loading in final design. Real loads (purlins, MEP) apply between nodes. Distributed loading on rigid-jointed models captures secondary bending.
Frequently asked questions
What is the fastest way to find a single member force? Method of sections. Cut through the member and two others, take moments about the intersection of the two cut members to isolate the target force.
Can I ignore zero-force members? No. They provide stability under other load cases and reduce compression member unbraced lengths. They must be included in fabrication.
Do I need computer analysis for a simple truss? Not for preliminary design. Method of sections gives exact forces for ideal trusses. Computer analysis adds value for: secondary bending, pattern loading, and connection design forces.
What is secondary bending in trusses? Bending in chord members between panel points due to: (1) loads applied between nodes, (2) gusset plate fixity, (3) self-weight. Typically adds 5-10% to chord demand.
How do I model a truss in FEA software? Use beam elements with moment releases at web member ends. Chord members should be continuous (no releases). Apply distributed loads to chord elements, not just panel-point loads.
What is the difference between pin-jointed and rigid-jointed analysis? Pin-jointed gives only axial forces (no bending). Rigid-jointed captures bending from connection fixity and inter-nodal loads. For final design, always use rigid-jointed analysis.
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Related references
- Steel Truss Design
- Compact Section Limits
- Bolt Hole Sizes
- Connection Types
- Plate Girder Design
- Steel Space Frame
- How to Verify Calculations
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.