FEA Engine Verification

The solver's member and frame linear-static and P-Delta capabilities originated from Pynite, an open-source 3D structural analysis library, and have since been adapted and modified for this application. This page verifies the current implementation across reactions, internal forces, deflections, plate and shell behavior, second-order effects, Classic and Wittrick-Williams buckling, and modal natural frequencies. Results computed directly from each listed test case are compared with published benchmarks, closed-form solutions, independent numerical calculations, and independently checked third-party results.

Test Cases

Verified benchmarks

Avg Difference

0.20%

Across all tests

Verification Model

Larger 3-storey frame and shell-slab case study comparing AutoCalcs against a separate OpenSees reference model for linear, P-Delta, modal, member, node, and raw plate results.

Beams

Verification of reactions, bending moments, shear forces, and deflections for simply supported, continuous, and overhanging beams under distributed and point loads.

Test	Structure	Expected	Result	Status
Continuous beam - maximum moment (3 supports, UDL)CISC Handbook of Steel Construction, 11th Ed., Diagram 24	2-span continuous beam, 10 ft + 10 ft, w = 10 kip/ft	M = 18,000 kip·in	18,000 kip·in	Exact
Overhanging beam - max moment at supportCISC Handbook of Steel Construction, 11th Ed.	2-support beam, L = 10 ft, a = 3 ft overhang, w = 10 /ft	M = 45.0	45.0	Exact
Overhanging beam - min interior momentCISC Handbook of Steel Construction, 11th Ed.	2-support beam, L = 10 ft, a = 3 ft overhang, w = 10 /ft	M = −103.5	−103.5	Exact
Simply supported beam - shear at support (biaxial)Analytical: V = wL/2	SS beam, 10 ft, w_y = 0.5 kip/ft, w_z = 0.75 kip/ft	V = 2.50 kip	2.50 kip	Exact
Simply supported beam - midspan moment (biaxial)Analytical: M = wL²/8	SS beam, 10 ft, w_y = 0.5 kip/ft, w_z = 0.75 kip/ft	M = 6.25 kip·ft	6.25 kip·ft	Exact
Simply supported beam - midspan deflectionAnalytical: δ = 5wL⁴/384EI	SS beam, L = 120 in, UDL, E = 29,000 ksi, I = 150 in⁴	δ = 0.0259 in	0.0259 in	Exact
End release (pinned) - midspan momentAnalytical: M = wL²/8	Fixed-fixed beam with Rz releases, L = 120 in, w = 0.5 kip/ft	M = 900 kip·in	900 kip·in	Exact

2D & 3D Frames

Verification of frame analysis across XY, YZ, and XZ planes. Includes multi-member portal frames and 3D frames with internal hinges. Reference solutions from established FEM textbooks.

Test	Structure	Expected	Result	Status
2D frame XY - horizontal reaction at baseLogan, A First Course in FEM, 4th Ed., Problem 5.30	5-member portal frame, 2 × 30 kip gravity loads	R_x = 11.69 kip	11.69 kip	Exact
2D frame XY - vertical reaction at baseLogan, A First Course in FEM, 4th Ed., Problem 5.30	5-member portal frame, 2 × 30 kip gravity loads	R_y = 30.00 kip	30.00 kip	Exact
2D frame XY - base momentLogan, A First Course in FEM, 4th Ed., Problem 5.30	5-member portal frame, 2 × 30 kip gravity loads	M = 1,810 kip·in	1,810 kip·in	Exact
2D frame YZ - reactions (rotated plane verification)Logan, A First Course in FEM, 4th Ed., Problem 5.30	Same frame as FR-01 in YZ plane	R_z = 11.69 kip	11.69 kip	Exact
2D frame YZ - midspan vertical displacementLogan, A First Course in FEM, 4th Ed., Problem 5.30	Same frame as FR-01 in YZ plane	δ = −6.667 in	−6.667 in	Exact
XZ simple beam - end reactionsAnalytical: R = P/2	SS beam, 14 ft, P = 5 kip at midspan	R = −2.50 kip	−2.50 kip	Exact
3D frame with internal hinge - base horizontal reactionKassimali, Structural Analysis, Example 3.35	4-member 3D frame, point + distributed loads	R_x = 15.46 kip	15.46 kip	0.03%

P-Delta Analysis (Second-Order)

Verification of the linearised two-pass P-Delta formulation against AISC benchmark problems. These tests confirm second-order amplification of moments and deflections for the benchmark frames under combined axial and lateral loading; they do not validate large-rotation or material-nonlinear response.

Test	Structure	Expected	Result	Status
Cantilever column - second-order base momentAISC Benchmark Problem (2nd-order elastic)	20 ft cantilever, P = 100 kip axial + H = 5 kip lateral	M = 435.6 kip·ft	435.6 kip·ft	Exact
ⓘ Six nodes create five sub-elements for P-δ capture. Verified against the closed-form stability-function solution.
Cantilever column - second-order tip deflectionAISC Benchmark Problem (2nd-order elastic)	20 ft cantilever, P = 100 kip axial + H = 5 kip lateral	δ = 40.27 in	40.27 in	Exact
Pin-base column - midheight moment (P = 150 kip)AISC Benchmark Case 1, Combo 2	28 ft pin-base, W-section, w = 0.2 kip/ft + axial	M = 269 kip·in	268.9 kip·in	0.06%
Pin-base column - midheight moment (P = 300 kip)AISC Benchmark Case 1, Combo 3	28 ft pin-base, W-section, w = 0.2 kip/ft + axial	M = 313 kip·in	313.3 kip·in	0.10%
Pin-base column - midheight moment (P = 450 kip)AISC Benchmark Case 1, Combo 4	28 ft pin-base, W-section, w = 0.2 kip/ft + axial	M = 375 kip·in	374.8 kip·in	0.06%
Fixed-base column - base moment (P = 100 kip)AISC Benchmark Case 2, Combo 2	28 ft fixed-base, W-section, H = 1 kip lateral + axial	M = 469 kip·in	469.1 kip·in	0.01%
Fixed-base column - base moment (P = 150 kip)AISC Benchmark Case 2, Combo 3	28 ft fixed-base, W-section, H = 1 kip lateral + axial	M = 598 kip·in	598.6 kip·in	0.11%
Fixed-base column - base moment (P = 200 kip)AISC Benchmark Case 2, Combo 4	28 ft fixed-base, W-section, H = 1 kip lateral + axial	M = 848 kip·in	848.9 kip·in	0.11%
ⓘ Higher axial ratios produce slightly larger P-δ amplification differences due to discretisation. All results within AISC benchmark tolerance.

Eigenvalue Buckling Analysis - Wittrick-Williams (exact member)

Verification against analytical Euler solutions, textbook frame benchmarks, an independently assembled 3D truss tangent matrix, and an independent differential-equation solution for changing axial force. The member solution is exact when axial force is constant; changing-force cases are reported only after the refined results converge.

Test	Structure	Expected	Result	Status
Pinned-pinned column - exact Euler (K = 1.0)Euler: π²EI/L² = 1,128.8 kN	1 element, E = 200 GPa, I = 366 cm⁴, L = 8 m, P = 1 kN	λ = 1,128.8	1,128.8	Exact
ⓘ 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.
Fixed-free cantilever - exact Euler (K = 2.0)Euler: π²EI/(2L)² = 1,128.8 kN	1 element, E = 200 GPa, I = 366 cm⁴, L = 4 m, P = 1 kN	λ = 1,128.8	1,128.8	Exact
ⓘ 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.
Fixed-fixed column - exact Euler (K = 0.5)Euler: 4π²EI/L² = 4,515.3 kN	2 elements, E = 200 GPa, I = 366 cm⁴, L = 8 m, P = 1 kN	λ = 4,515.3	4,515.3	Exact
Two-story sway frame - critical loadMcGuire, Gallagher & Ziemian, MSA 2nd Ed., Ex 9.5	2-story, 2-bay portal, E = 200 GPa, P = 6,000 kN	P_cr = 6,630 kN	6,609.4 kN	0.31%
Portal frame (imperial) - load factorMcGuire, Gallagher & Ziemian, MSA 2nd Ed., Ex 9.6	W10x45 columns, W27x84 beam, lateral + gravity loads	λ = 2.200	2.198	0.10%
Portal frame - Timoshenko analyticalTimoshenko & Gere, Theory of Elastic Stability, Art. 2.4	Fixed-base portal, L = 100 in, E = 1,000 ksi	P_cr = 737.9 lbf	737.5 lbf	0.06%
Braced column - elastic intermediate restraintMcGuire, Gallagher & Ziemian, MSA 2nd Ed., Ex 9.8	Column braced at midheight, E = 29,000 ksi, P = 200 kips	P_cr = 215.7 kips	214.1 kips	0.75%
3D tetrahedral truss with pin-ended bars and tension prestressIndependent eigenvalue solution from an assembled 3D tangent-stiffness matrix	Four nodes, six released bars, and a combined 3D reference load	λ = 402,943	402,943	0.00%
ⓘ The reference load puts some bars in compression and others in tension, checking that both destabilising and stabilising geometric-stiffness contributions are included.
Greenhill cantilever - continuously changing axial forceIndependent beam-column differential-equation shooting solution	L = 4 m cantilever under a uniform axial body load, N(x) = w(L - x)	λ = 734.7513	734.7498	0.00%
ⓘ Axial force changes continuously along the cantilever, so the solver compares successively finer member divisions before accepting the result.
Cantilever with an interior axial point loadEuler closed form for the loaded lower segment: π²EI/(4a²)	L = 4 m cantilever, axial point load at a = 1.3 m	λ = 8,760.003906	8,760.003927	0.00%
ⓘ The point load creates a step in axial force, testing whether the solver splits the member at the load position before buckling analysis.

Eigenvalue Buckling Analysis - Classic (cubic-Hermite LBA)

Verification of the Classic solver (default) against companion analytical, textbook, and third-party benchmarks. Classic uses automatic member subdivision (4 sub-elements per member) to capture changing axial-force profiles. The listed differences show its agreement with each analytical, textbook, or independently checked reference.

Test	Structure	Expected	Result	Status
Pinned-pinned column - exact Euler (K = 1.0)Euler: π²EI/L² = 1,128.8 kN	1 element, E = 200 GPa, I = 366 cm⁴, L = 8 m, P = 1 kN	λ = 1,128.8	1,129.4	0.05%
ⓘ 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.
Fixed-free cantilever - exact Euler (K = 2.0)Euler: π²EI/(2L)² = 1,128.8 kN	1 element, E = 200 GPa, I = 366 cm⁴, L = 4 m, P = 1 kN	λ = 1,128.8	1,128.9	0.00%
ⓘ 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.
Fixed-fixed column - exact Euler (K = 0.5)Euler: 4π²EI/L² = 4,515.3 kN	2 elements, E = 200 GPa, I = 366 cm⁴, L = 8 m, P = 1 kN	λ = 4,515.3	4,517.7	0.05%
Two-story sway frame - critical loadMcGuire, Gallagher & Ziemian, MSA 2nd Ed., Ex 9.5	2-story, 2-bay portal, E = 200 GPa, P = 6,000 kN	P_cr = 6,630 kN	6,609.5 kN	0.31%
Portal frame (imperial) - load factorMcGuire, Gallagher & Ziemian, MSA 2nd Ed., Ex 9.6	W10x45 columns, W27x84 beam, lateral + gravity loads	λ = 2.200	2.198	0.10%
Portal frame - Timoshenko analyticalTimoshenko & Gere, Theory of Elastic Stability, Art. 2.4	Fixed-base portal, L = 100 in, E = 1,000 ksi	P_cr = 737.9 lbf	737.6 lbf	0.03%
Braced column - elastic intermediate restraintMcGuire, Gallagher & Ziemian, MSA 2nd Ed., Ex 9.8	Column braced at midheight, E = 29,000 ksi, P = 200 kips	P_cr = 215.7 kips	214.2 kips	0.71%
3D tetrahedral truss with pin-ended bars and tension prestressIndependent eigenvalue solution from an assembled 3D tangent-stiffness matrix	Four nodes, six released bars, and a combined 3D reference load	λ = 402,943	402,943	0.00%
ⓘ The reference load puts some bars in compression and others in tension, checking that Classic includes both destabilising and stabilising geometric-stiffness contributions.
3D frame with non-uniform axial - third-party software verification (Mode 1)Third-party structural-analysis software, 3D asymmetric spring-supported frame	12 nodes, 17 members, distributed loads + mid-span point load + fixity release	λ = 0.391	0.393	0.51%
ⓘ Distributed loads on inclined members and a point load on one brace produce changing axial force. Classic matches the independently checked third-party governing mode within 0.51%.
Greenhill cantilever - continuously changing axial forceIndependent beam-column differential-equation shooting solution	L = 4 m cantilever under a uniform axial body load, N(x) = w(L - x)	λ = 734.7513	734.9000	0.02%
ⓘ Classic uses automatic member subdivision to represent the continuously changing axial-force profile.
Cantilever with an interior axial point loadEuler closed form for the loaded lower segment: π²EI/(4a²)	L = 4 m cantilever, axial point load at a = 1.3 m	λ = 8,760.003906	8,844.272020	0.96%
ⓘ This checks how automatic member subdivision in Classic represents the step in axial force at the load position.

Verification of undamped free-vibration modal analysis against analytical solutions and published numerical benchmarks. Cantilever tests compare against Euler-Bernoulli beam theory (Blevins); the Petersen beam eigenvalues trace back to Petersen's Dynamik der Baukonstruktionen (2000).

Physical-mass verification also checks finite eigenvalues, massless-coordinate recovery, repeated eigenspaces, positive-mass rigid modes, and rejection of corrupted eigenpairs. The mass policy is identified in each analysis result; legacy stabilized results retain an artificial-inertia audit. These checks do not certify that a sparse candidate search found the complete spectrum, or establish prestressed modal capability.

Test	Structure	Expected	Result	Status
Cantilever beam - fundamental frequency (Mode 1)Analytical Euler-Bernoulli: fₙ = (βₙ² / 2πL²)·√(EI/ρA)	11-node cantilever (auto-split into 10 sub-members), L = 20 m, E = 200 GPa, I = 8.33×10⁻⁶ m⁴, ρ = 7,800 kg/m³	f₁ = 0.6404 Hz	0.6404 Hz	Exact
ⓘ 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). This is standard production usage - users commonly place waypoint nodes for supports, connections, or visualisation. Uses β₁ = 1.8751 as the first root of cos(β)cosh(β) + 1 = 0.
Cantilever beam - Mode 2Analytical Euler-Bernoulli, β₂ = 4.6941	11-node cantilever (auto-split into 10 sub-members), L = 20 m, E = 200 GPa, I = 8.33×10⁻⁶ m⁴, ρ = 7,800 kg/m³	f₂ = 4.0132 Hz	4.0133 Hz	0.00%
Cantilever beam - Mode 3Analytical Euler-Bernoulli, β₃ = 7.8548	11-node cantilever (auto-split into 10 sub-members), L = 20 m, E = 200 GPa, I = 8.33×10⁻⁶ m⁴, ρ = 7,800 kg/m³	f₃ = 11.237 Hz	11.239 Hz	0.02%
ⓘ 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%. Further refinement (more user-placed nodes or a higher modal_mesh_divisions) tightens the residual further.
Petersen beam - first eigenvalue ω²Petersen, Dynamik der Baukonstruktionen (2000), p. 252	10-element vertical beam (L = 12 m) with distributed + concentrated mass	ω² = 115.188	115.188	Exact
ⓘ 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.
Petersen beam - second eigenvalue ω²Petersen, Dynamik der Baukonstruktionen (2000), p. 252	10-element vertical beam (L = 12 m) with distributed + concentrated mass	ω² = 3,056.953	3,056.953	Exact
Mass-increase frequency drop - physics sanity checkBasic dynamics: f ∝ √(k/m), so adding mass at the tip lowers f	Cantilever, self-weight only vs self-weight + 1 kN tip point load	f drops	3.271 → 2.289 Hz	0.00%
ⓘ 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.
3D frame with non-uniform axial - third-party software verification (Mode 1)Third-party software, 3D asymmetric spring-variant frame (mass source: LC1 self-weight + superimposed DL)	12 nodes, 17 members, asymmetric spring-variant frame with distributed loads + mid-span point load + fixity release	f₁ = 0.342 Hz	0.3424 Hz	0.12%
ⓘ 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.

Plate and Shell Elements

Plate and shell meshes are checked against published plate theory and named benchmarks. These tests cover very thin plate bending, transverse deflection, principal bending moments, recovered surface stress, Mindlin shear deformation, a singly curved cylindrical roof, and systematic mesh refinement.

Test	Structure	Expected	Result	Status
Simply supported square thin plate - centre transverse deflectionTimoshenko & Woinowsky-Krieger, Theory of Plates and Shells, Navier series	1 m × 1 m, 10 mm thick, 1 kPa pressure; 4×4 → 8×8 → 16×16 meshes	0.221676 mm	0.221965 mm	0.13%
ⓘ Independent Navier-series centre-deflection target. The 16×16 result changes by 0.46% from the preceding 8×8 mesh.
Very thin simply supported square plate - centre transverse deflectionTimoshenko & Woinowsky-Krieger, Theory of Plates and Shells, Navier series	1 m × 1 m, 1 mm thick, 1 Pa pressure; 4×4 → 8×8 → 16×16 meshes	0.221676 mm	0.221528 mm	0.07%
ⓘ Span/thickness = 1,000 and deflection/thickness = 0.222. The pressure is scaled to keep this linear thin-plate benchmark in the small-deflection range.
MacNeal-Harder thin square plate - simply supported centre transverse deflectionMacNeal & Harder (1985), Figure 7 and Table 4	2 × 2 plate, thickness 0.0001 in the same unspecified length unit (span/thickness = 20,000), uniform pressure q = 10^-4 in consistent force/area units; 4×4 → 8×8 → 16×16 meshes	4.062 length units	4.057254 length units	0.12%
ⓘ Span/thickness = 20,000. The paper is unit-consistent rather than inch-specific; this reproduction uses N and mm.
MacNeal-Harder thin rectangular plate - simply supported centre transverse deflectionMacNeal & Harder (1985), Figure 7 and Table 4	10 × 2 plate, thickness 0.0001 in the same unspecified length unit (short-span/thickness = 20,000), uniform pressure q = 10^-4 in consistent force/area units; 30×6 → 60×12 → 120×24 meshes	12.97 length units	12.935510 length units	0.27%
MacNeal-Harder thin square plate - clamped centre transverse deflectionMacNeal & Harder (1985), Figure 7 and Table 4	2 × 2 plate, thickness 0.0001 in the same unspecified length unit (span/thickness = 20,000), uniform pressure q = 10^-4 in consistent force/area units; 4×4 → 8×8 → 16×16 meshes	1.26 length units	1.261647 length units	0.13%
MacNeal-Harder thin rectangular plate - clamped centre transverse deflectionMacNeal & Harder (1985), Figure 7 and Table 4	10 × 2 plate, thickness 0.0001 in the same unspecified length unit (short-span/thickness = 20,000), uniform pressure q = 10^-4 in consistent force/area units; 30×6 → 60×12 → 120×24 meshes	2.56 length units	2.586123 length units	1.0%
MacNeal-Harder thin rectangular plate - mixed triangular/quadrilateral mesh centre transverse deflectionMacNeal & Harder (1985), Figure 7 and Table 4	10 × 2 plate, thickness 0.0001 in the same unspecified length unit (short-span/thickness = 20,000), simply supported, uniform pressure q = 10^-4 in consistent force/area units; conforming 120×24 mixed mesh	12.97 length units	12.935272 length units	0.27%
Scordelis-Lo cylindrical roof shell benchmark - free-edge vertical deflectionMacNeal & Harder (1985), Figure 8, Table 5(a) and Table 13	Radius 25, length 50, thickness 0.25, vertical load 90 per unit area; 4×4 → 8×8 → 16×16 quarter-shell meshes	0.3024 downward	0.301656 downward	0.25%
ⓘ The paper quotes 0.3086 as a theoretical value but uses 0.3024 to normalize its shell-element results. The final mesh changes by 0.73%.
Morley 60-degree thin skew plate - centre transverse deflectionMorley, Skew Plates and Structures (1963); published commercial-solver benchmark reproduction	1 m rhombus, 10 mm thick, simply supported, 1 Pa pressure; 4×4 → 8×8 → 14×14 meshes	-0.932 mm	-0.923076 mm	0.96%
ⓘ The source defines this as a thin-plate skew-sensitivity benchmark; span/thickness = 100.
Morley 60-degree thin skew plate - maximum principal bending momentMorley, Skew Plates and Structures (1963); published commercial-solver benchmark reproduction	Same 1 m rhombus; smoothed centre result from the 14×14 mesh	0.0425 N·m/m	0.0421879 N·m/m	0.73%
ⓘ Principal bending moment per unit plate width. The result shown is the smoothed value reported to users.
Morley 60-degree thin skew plate - minimum principal bending momentMorley, Skew Plates and Structures (1963); published commercial-solver benchmark reproduction	Same 1 m rhombus; smoothed centre result from the 14×14 mesh	0.0333 N·m/m	0.0329341 N·m/m	1.1%
ⓘ Principal bending moment per unit plate width. The result shown is the smoothed value reported to users.
NAFEMS P18.LE6 skew plate - bottom-centre maximum principal stressNAFEMS P18.LE6, The Standard NAFEMS Benchmarks (TNSB Rev. 3)	1 m × 1 m, 30-degree skew, 10 mm thick, simply supported, -0.7 kPa pressure; 17×17 → 33×33 → 65×65 meshes	0.802 MPa	0.808621 MPa	0.83%
ⓘ Genuine named NAFEMS plate benchmark, checked against a published commercial-solver reproduction and source input deck. The 65×65 result changes by 1.75% from the 33×33 mesh.
Annular thick plate - transverse deflection with shear deformationRoark & Young (1975), p. 376, via CSI SAP2000 Verification Problem 2-012	Inner radius 1.4 in, outer radius 2 in, thickness 0.5 in; inner edge simply supported and 800 lb/in circular line load at radius 1.8 in	-0.00534 in	-0.0053432 in	0.06%
Rectangular plate bending - pinned edges, 8×4 mesh, von Mises stressThird-party commercial FEA software comparison (2020), rectangular plate - pinned edges	2 m × 1 m steel plate, 10 mm thick, 1 kPa pressure	4.63425 MPa	4.71616 MPa	1.8%
Rectangular plate bending - pinned edges, 16×8 mesh, von Mises stressThird-party commercial FEA software comparison (2020), rectangular plate - pinned edges	2 m × 1 m steel plate, 10 mm thick, 1 kPa pressure	5.11488 MPa	5.13836 MPa	0.46%
Rectangular plate bending - fixed edges, 8×4 mesh, von Mises stressThird-party commercial FEA software comparison (2020), rectangular plate - fixed edges	2 m × 1 m steel plate, 10 mm thick, 1 kPa pressure	1.66868 MPa	1.71097 MPa	2.5%
Rectangular plate bending - fixed edges, 16×8 mesh, von Mises stressThird-party commercial FEA software comparison (2020), rectangular plate - fixed edges	2 m × 1 m steel plate, 10 mm thick, 1 kPa pressure	2.89678 MPa	2.91354 MPa	0.58%

Special Cases

Verification of rotated members, spring elements, self-weight, torsion, support settlement, tension-only members, and inclined geometry. These tests confirm correct coordinate transformation, load resolution, and advanced analysis features.

Test	Structure	Expected	Result	Status
Rotated member (45°) - strong-axis momentAnalytical: M·cos(45°)	SS beam, 10 ft, 45° roll, w = 2 kip/ft global Y	M_z = −17.68 kip·ft	−17.68 kip·ft	Exact
Rotated member (45°) - weak-axis momentAnalytical: M·sin(45°)	SS beam, 10 ft, 45° roll, w = 2 kip/ft global Y	M_y = 17.68 kip·ft	17.68 kip·ft	Exact
Rotated column (90°) - strong-axis moment at midspanAnalytical: M = PL/4	10 ft column, 90° roll, P = 1 kip at midspan	M_z = −2.50 kip·ft	−2.50 kip·ft	Exact
Spring support - vertical reactionAnalytical: R = P (equilibrium)	Single node, k_y = 5 kip/in, P = 5 kip	R_y = 5.00 kip	5.00 kip	Exact
Sloped beam - gravity load distributionAnalytical: statics equilibrium	45° inclined member, 2 × 5 kip gravity loads, factor 1.4	R_y = 7.00 kip	7.00 kip	Exact
Spring element - displacement (series springs)Logan, A First Course in FEM, 4th Ed., Example 2.1	3 springs in series, k = 1000/2000/3000 lb/in, P = 5000 lb	δ_3 = 0.909 in	0.909 in	0.00%
Spring element - displacement at loaded nodeLogan, A First Course in FEM, 4th Ed., Example 2.1	3 springs in series, k = 1000/2000/3000 lb/in, P = 5000 lb	δ_4 = 1.364 in	1.364 in	0.00%
Self-weight - W14×34 computed unit weightAISC Steel Construction Manual	Continuous beam, W14×34, ρ = 490 pcf	w = 0.034 kip/ft	0.034 kip/ft	Exact
Axial distributed load - reactionsAnalytical: R = wL/2	5 m cantilever, w_axial = 10 N/m, fixed-fixed	R = −25.0 N	−25.0 N	Exact
Torsion - left support reactionAnalytical: statics (torsion equilibrium)	14 ft beam, T = 5 kip·ft at 3 ft + T = 10 kip·ft at 11 ft	M_x = −6.07 kip·ft	−6.07 kip·ft	Exact
Torsion - right support reactionAnalytical: statics (torsion equilibrium)	14 ft beam, T = 5 kip·ft at 3 ft + T = 10 kip·ft at 11 ft	M_x = −8.93 kip·ft	−8.93 kip·ft	Exact
Support settlement - reaction at interior support BKassimali, Structural Analysis, 3rd Ed., Example 13.14	3-span beam, 60 ft, prescribed settlements at B, C, D	R_B = 122.4 kip	122.5 kip	0.14%
Support settlement - reaction at interior support CKassimali, Structural Analysis, 3rd Ed., Example 13.14	3-span beam, 60 ft, prescribed settlements at B, C, D	R_C = −61.5 kip	−61.5 kip	0.15%
Tension-only member - active under tensionAnalytical: statics (force resolution)	Truss with tension-only braces, P = 10 kip	F = −100.5 kip	−100.5 kip	Exact
Tension-only member - deactivated under compressionAnalytical: member deactivated (no compression)	Truss with tension-only braces, P = 10 kip	F = 0 kip	0 kip	Exact

References

Logan, D.L. - A First Course in the Finite Element Method, 4th Edition
Kassimali, A. - Structural Analysis, 3rd Edition
CISC - Handbook of Steel Construction, 11th Edition
AISC - Second-Order Elastic Analysis Benchmark Problems
AISC - Steel Construction Manual, 15th Edition
McGuire, W., Gallagher, R.H. & Ziemian, R.D. - Matrix Structural Analysis, 2nd Edition
Timoshenko, S.P. & Gere, J.M. - Theory of Elastic Stability
Petersen, C. - Dynamik der Baukonstruktionen, Vieweg (2000)
Blevins, R.D. - Formulas for Natural Frequency and Mode Shape
Timoshenko, S.P. & Woinowsky-Krieger, S. - Theory of Plates and Shells
MacNeal, R.H. & Harder, R.L. - A Proposed Standard Set of Problems to Test Finite Element Accuracy, Finite Elements in Analysis and Design 1 (1985): rectangular plates and Scordelis-Lo roof, Figure 7, Figure 8, Table 4, Table 5(a), and Table 13
Morley, L.S.D. - Skew Plates and Structures (1963), analytical basis for a published commercial-solver thin skew-plate reproduction
NAFEMS - P18.LE6, The Standard NAFEMS Benchmarks, TNSB Rev. 3; published commercial-solver reproduction
Roark & Young - annular thick plate reference via SAP2000 Verification Problem 2-012
Euler-Bernoulli beam theory - Analytical closed-form solutions

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