Verification Tests

ⓘ Column subdivided into 6 elements for P-δ capture. Verified against closed-form stability function solution.

ⓘ Higher axial ratios produce slightly larger P-δ amplification differences due to discretisation. All results within AISC benchmark tolerance.

ⓘ Shares its expected value with the cantilever test below. The cantilever uses half the length so both columns have the same effective length KL = 8 m, producing identical P_cr — matching answers across both tests confirms the solver handles both pinned-pinned and fixed-free boundary conditions correctly. A stronger check than two unrelated numbers happening to match their references.

ⓘ Length deliberately chosen as half of the pinned-pinned test above so both columns share the same effective length (KL = 8 m), producing identical P_cr. See the note on the pinned-pinned test.

ⓘ Tested with Subdivide members OFF. The 2 kN/m UDL sits on horizontal beams, so it acts transversely (causes bending, not axial). Columns receive compression as point reactions at the beam-column joints, and own self-weight adds only a small linear component — so the axial force inside each column is near-constant along its length. That is the regime where transcendental stability functions are analytically exact at single-element resolution; no mesh refinement needed. Torsional modes filtered automatically (no warping stiffness in the 6-DOF beam element).

ⓘ Tested with Subdivide members ON (4 sub-elements per member). Unlike the previous test, this model has distributed loads on non-horizontal members (including diagonal braces) plus a mid-span point load on one brace — gravity components project along those members’ axes, producing axial force that varies along the member length. Subdivide members ON is required: without it, the single-element approximation can’t represent the varying axial profile and buckling eigenvalues drift. For frames with in-span loading on non-horizontal members, subdivide ON is the correct default.

ⓘ Shares its expected value with the cantilever test below. The cantilever uses half the length so both columns have the same effective length KL = 8 m, producing identical P_cr — matching answers across both tests confirms the solver handles both pinned-pinned and fixed-free boundary conditions correctly.

ⓘ Length deliberately chosen as half of the pinned-pinned test above so both columns share the same effective length (KL = 8 m), producing identical P_cr.

ⓘ The 2 kN/m UDL sits on horizontal beams (transverse, no axial component along columns), so columns carry near-constant axial force from stacked self-weight + beam reactions. In this near-constant-axial regime the cubic-Hermite approximation (auto-subdivided into 4 sub-elements per member) carries a small residual error against the closed-form reference — about +3.4% on Mode 1. For this class of model (simple portals, steel buildings with only horizontal-beam distributed loads), the Wittrick-Williams solver at subdivide off is the more accurate path.

ⓘ Unlike the previous test, this model has distributed loads on non-horizontal members (diagonal braces included) plus a mid-span point load on one brace — gravity components project along those members’ axes, producing axial force that varies linearly along the member length. The classic solver integrates its geometric stiffness over the varying axial profile (linear-P k_g formulation), so the governing mode is captured exactly to within third-party software round-off (<0.5%) without relying on the auto-mesh refinement to resolve the axial distribution.

ⓘ User places 11 nodes along the beam; the solver auto-splits the single cantilever member into 10 sub-members at the collinear waypoints, then applies the default 4-division subdivision per sub-member (40 sub-elements at solve time). Uses β₁ = 1.8751 as the first root of cos(β)cosh(β) + 1 = 0.

ⓘ Higher bending modes require finer discretisation to resolve accurately. The 10 auto-created sub-members × 4 default subdivisions gives 40 elements, enough to resolve the 3rd bending mode to within 0.02%.

ⓘ User input: 10 nodes (one per unit along the 12 m beam) so concentrated masses can be placed at specific nodes. Solver applies default 4-division subdivision on top. Match to within 1×10⁻⁶ on both eigenvalues.

ⓘ Sanity check that adding mass at the tip reduces the fundamental frequency, as required by f = (1/2π)√(k/m). Single-member cantilever with default 4-division subdivision. Observed 30% drop matches the expected direction and magnitude for a 1 kN point load relative to the self-weight mass of the beam.

ⓘ Mode 1 is a global sway mode; Pynite matches the third-party solver to within 0.12% on the governing frequency with default 4-division subdivision. Cross-checks modal behaviour on a structurally realistic 3D frame with in-span loading, partial releases, and a spring support.