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 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:

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:

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

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:

  1. Bolt shear: phiRn = 0.75 x Fnv x Ab x n (per bolt, single or double shear)
  2. Bolt bearing on gusset: phiRn = 0.75 x 2.4 x d x t x Fu (deformation considered)
  3. Block shear: phiRn = 0.75 x (0.60 x Fu x Anv + Ubs x Fu x Ant) per AISC J4.3
  4. Whitmore section: Effective width for checking tension/compression at the end of the gusset = spread angle of 30 degrees from each exterior bolt
  5. Gusset-to-chord weld: Designed for the horizontal component of the member force
  6. 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

  1. 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.

  2. 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]
  1. 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.

  2. 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.

  3. 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

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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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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.