Verification

Overview#

This page compares TrussLab results with closed-form solutions and published worked examples. It has two parts: Analysis (member forces, reactions and deflections of the frame solver) and Design (member capacities to AISC 360-22, AS 4100:2020, EN 1993-1-1 and IS 800:2007). Each benchmark lists its inputs, the reference values, the TrussLab values and the difference.

The project files of the truss benchmarks can be downloaded and opened in TrussLab to reproduce the numbers. Each design result expands to its working: the intermediate values that the TrussLab member calculation report records for the capacity, such as the section class, slenderness, reduction factor and critical stress or moment.

B1 · Pin-jointed Pratt truss#

A statically determinate Pratt truss: span 24 m, six panels of 4 m, depth 4 m, so the diagonals are at 45°. A load of 100 kN acts downward at each of the five interior bottom-chord panel points (load case Dead). Every member end is pinned and every node is braced out of plane. The two end posts are added with the Add Member tool. Self-weight is a separate load case and is not included.

Reference solution

With the reaction R = 5P/2 = 250 kN at each support, the method of sections gives, for a panel with shear V and the bending moment M at the moment centre:

• chords: N = ±M / h, with h = 4 m;
• diagonals: N = V / sin 45° (tension);
• verticals: N = −V of the adjacent panel towards midspan (compression); the midspan vertical carries no force;
• end posts: N = −R.

Member forces

Reactions and deflection

The reference deflection is computed by virtual work, δ = Σ N n L / (E A), with n the member forces from a unit load at midspan. All members are W44X408 (E A = 15.480 × 10⁹ N), so the result also checks the axial stiffness used by the solver.

Download the B1 project file

To reproduce: open the file with File > Open, select Run analysis, then choose the load case Dead in the analysis results. Axial forces are F1; reactions are under Support reactions. TrussLab displays F1 with compression positive, while the tables on this page use tension positive.

B1b · Pin-jointed Pratt truss, continuous chords#

B1 repeated with the default TrussLab connections: chords continuous through the panel points, diagonals connected rigidly, verticals released for bending. The end posts are pinned as in B1. Real trusses are usually built this way, and the chords then carry some bending as the truss deflects.

The largest difference from the pin-jointed forces is 1.5 %. The reactions are unchanged (250.0 kN). The members in this benchmark are deep W44X408 sections on short 4 m panels, which makes the effect of joint rigidity larger than in a typical roof truss.

Download the B1b project file

B2 · Indeterminate cantilever truss#

The layout of the ten-bar truss, a standard problem in the structural optimization literature: two bays of 9 m, depth 9 m, both left-hand joints pinned, crossing diagonals in each bay and a post at the free end. A load of 450 kN acts downward at each free bottom joint. In this version the crossing diagonals are connected where they cross, so the truss has 14 members, 8 joints and 4 reaction components, and is statically indeterminate to the second degree (14 + 4 − 2 × 8 = 2).

The member groups have different sections: top chord W14X132, bottom chord W14X90, vertical and end post W10X49, diagonals W8X31. The forces therefore depend on the relative axial stiffness of the members and cannot be found from equilibrium alone. All member ends are pinned, every joint is braced out of plane, and self-weight is excluded.

Reference solution: force method

The upper halves of the two diagonals that rise from the lower-left joint of each bay are taken as the redundants X1 and X2. The released structure is statically determinate and is solved by the method of joints, once for the applied loads (forces N₀) and once for a unit tension in each redundant (forces n₁, n₂). Compatibility of the cut members gives

f₁₁ X1 + f₁₂ X2 + Δ₁₀ = 0,   f₂₁ X1 + f₂₂ X2 + Δ₂₀ = 0,   with f_ij = Σ n_i n_j L / EA and Δ_i0 = Σ n_i N₀ L / EA.

Final forces are N = N₀ + X1 n₁ + X2 n₂. Joint deflections are found by virtual work, δ = Σ N n_u L / EA, with n_u from a unit load on the released structure. Section areas are taken from the AISC Shapes Database v16 and E = 199.948 GPa.

Member forces

Reactions and deflections

The vertical reactions are not equal although the structure is supported at two joints on the same vertical line: their split follows from the member stiffnesses, which is the part of the problem that a statics-only calculation cannot provide.

Download the B2 project file

AISC 360-22#

B4 · Hand calculations

Design strengths (LRFD, φ applied) reported by TrussLab for single members, compared with hand calculations from the AISC 360-22 equations. Section properties are from the AISC Shapes Database v16. Material A992: F_y = 50 ksi, E = 29,000 ksi.

The compression case uses the section and length of AISC Design Example E.1A (W14X132, pinned at both ends, L_c = 30 ft about both axes); its published strength is compared below with the other Design Examples, with the same TrussLab value. The flexure cases use C_b = 1.0, which is the value TrussLab applies in all lateral-torsional buckling checks.

B5 · AISC Design Examples

Available strengths (LRFD) of AISC Design Examples v15.1, each computed by TrussLab for the example's section, steel and lengths. Flexure examples with C_b = 1.0 (continuously braced) or minor-axis bending are listed, as TrussLab applies C_b = 1.0.

Tees and double angles are checked in both senses of bending (F9), since the analysis does not say which way the stem points, and the smaller strength governs; Example F.10 checks only the sense with the flange in compression. Single angles are checked about their principal axes (F10) and the strengths turned into geometric-axis ones that hold for either sign of either moment.

AS 4100:2020#

Design capacities (φ = 0.9) from the Australian Steel Institute's Sample Worked Problems to AS 4100, the Steel Designers' Handbook (Gorenc, Tinyou & Syam), Kirke & Al-Jamel's Steel Structures Design Manual to AS 4100 and the AS 4100 verification problems of STAAD.Pro, each computed by TrussLab with the section properties and steel the example prints. The examples cover section and member moment capacity (Clauses 5.2, 5.6) including a tee bent each way (5.6.1.2), section and member axial capacity (6.2, 6.3), tension (7.2) and shear (5.11).

Differences from the examples

• α_m = 1.0 is used for every member (uniform moment). Example 3 of the ASI problems takes α_m = 1.13 for its distributed load, which gives 221 kNm where TrussLab reports the α_m = 1.0 value of the design capacity tables, 195 kNm.
• Connections are not modelled: tension fracture uses A_n = A_g and k_t = 1.0 (the bolted angle of ASI Example 6 fractures at 469 kN; only its yield capacity is compared).
• Each plate element takes the yield stress of its own thickness band (AS/NZS 3679.1) for its slenderness and, for the web, its shear capacity; section and member capacities use the section's design yield stress, that of its thickest plate, as the OneSteel tables do. Hollow sections rolled only in a lower grade (small CHS in C250L0, small SHS and RHS in C350L0, per Austube Mills) are designed in that grade.
• The higher reduced section capacities of Clause 8.3.2 for compact doubly symmetric I-sections are not used.
• The member checks of Clause 8.4.5 are reported as a load factor: the utilisation is 1 / λ, λ being the factor on all the actions at which the check is just satisfied. Their left-hand side is not proportional to the actions (φM_ry = φM_sy (1 − N*/φN_t), and the 1.4 powers), so near the axial capacity it is no measure of the overload; the check passes and fails exactly as written.

EN 1993-1-1#

Design resistances with the recommended partial factors (γ_M0 = γ_M1 = 1.0), from Gardner & Nethercot's Designers' Guide to EN 1993-1-1 (Examples 6.7 and 6.9), the bauforumstahl design aid Columns in Multi-Storey Construction (flexural buckling resistances of HE A, HE B and HE M sections in S355 with γ_M1 = 1.1, evaluated by TrussLab with the same factor), Trahair, Bradford, Nethercot & Gardner's Behaviour and Design of Steel Structures to EC3 (a tee bent each way) and the EC3 verification problems of STAAD.Pro. The examples cover cross-section resistances (6.2.4 – 6.2.6), flexural buckling (6.3.1) and lateral-torsional buckling (6.3.2.2), with M_cr of monosymmetric sections for the tee.

The comparison with the bauforumstahl tables uses the sections of the TrussLab European catalogue; the test suite checks 110 resistances of those tables (HE 100 A – HE 400 A, HE 100 B – HE 550 B, HE 100 M – HE 500 M, buckling lengths 2 – 10 m, both axes), all within 0.5 %.

Differences from the examples

• Lateral-torsional buckling uses the general case (6.3.2.2) with C₁ = 1.0 and the load at the shear centre. Examples that apply 6.3.2.3 to rolled sections, C₁ > 1 or a destabilising load are not compared.
• Root radii are not part of the catalogue rows: shear areas and the c/t ratios of the classification are taken without them, which is conservative.
• A class 4 tee stem is taken with the effective section of EN 1993-1-5 (k_σ = 0.57), where the Trahair et al. example treats it as class 3 at a raised ε (5.5.2(9)): 3.5 % higher than the book's 32.6 kNm, and below the 53.2 kNm the book reports for its own effective-section calculation.
• Combined compression and bending under 6.2.9 uses the linear criterion of 6.2.1(7) instead of the reduced plastic moment M_N,Rd; Designers' Guide Example 6.6 allows 425 kNm where TrussLab allows 341 kNm.

IS 800:2007#

Design strengths with γ_m0 = 1.10 and γ_m1 = 1.25, from the IS 800 limit state verification problems of STAAD.Pro (I-section with and without lateral-torsional buckling, channel, single angle, circular and square hollow sections), each computed by TrussLab with the IS 808 / IS 4923 properties and E = 205 GPa the problems use. The examples cover compression (7.1.2) including single angles (7.5.1.2), tension (6.2), shear (8.4), bending with and without lateral-torsional buckling (8.2.1, 8.2.2) and the moment under high shear (8.2.1.3).

The design compressive stresses of Table 9(a) – 9(d) for f_y = 250 MPa are also reproduced in the test suite for all four buckling classes. No published IS 800 example covers tees or angles in bending; their strengths are checked in the test suite against hand calculations of Clause 8.2 with M_cr of Annex E.1.2, and the M_cr of a tee is the same mechanics as the EN 1993 example above.

Differences from the examples

• Bending strength is limited to 1.2 Z_e f_y / γ_m0, the limit of Clause 8.2.1.2 for simply supported members; the examples are cantilevers, for which the limit is 1.5 Z_e f_y / γ_m0.
• Single angles in compression take the smaller of flexural buckling about the minor principal axis and the equivalent slenderness of 7.5.1.2 with the Table 12 constants for two or more bolts and a hinged end.
• Under high shear (V > 0.6 V_d) the combined-forces check uses the reduced moment M_dv of 8.2.1.3; the flexure check alone reports the full M_d.
• Rupture at the end connection (α = 0.8, net area, block shear) is not modelled.

Scope and limitations#

• Analysis is linear elastic and static. P-delta and other nonlinear effects are not included.
• Each design code is compared with published examples for the limit states TrussLab checks: compression, tension, bending, shear and lateral-torsional buckling. Combined actions are covered by the test suite against hand calculations of the interaction equations.
• Connections are not designed.
• Tees and double angles are checked in both senses of bending and the smaller strength governs, as the analysis does not say which way the stem points. Single angles are checked about their principal axes.
• Tensile rupture uses A_e = 0.75 A_g.
• Lateral-torsional buckling uses C_b = 1.0.
• There are no automatic wind, snow or seismic load generators.

Engineering judgement and responsibility for the final design remain with the engineer of record. Additional benchmark cases can be suggested through the contact page.