Verification Tests

The static analysis engine (linear and P-Delta) is based on PyNite, an open-source 3D structural analysis library. The buckling analysis uses a custom Wittrick-Williams exact solver built on top of the same framework. Both are verified against published textbook solutions, analytical formulas, and industry benchmark problems.

This page documents the key verification tests that confirm the accuracy of reactions, internal forces, deflections, second-order (P-Delta) effects, and eigenvalue buckling analysis. All expected values come from independent, published sources. Every result shown was computed by running the solver directly against these test cases.

Test Cases

Verified benchmarks

Avg Difference

0.03%

Across all tests

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 geometric nonlinear (P-Delta) analysis against AISC benchmark problems. These tests confirm that second-order amplification of moments and deflections is captured accurately under combined axial and lateral loading.

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
ⓘ Column subdivided into 6 elements for P-δ capture. Verified against 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	374.7 kip·in	0.07%
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.

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	Exact
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	Exact
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

Buckling Analysis (Linear Eigenvalue)

Verification of the custom Wittrick-Williams exact buckling solver against analytical Euler solutions, published textbook benchmarks, and third-party commercial software. Unlike the static analysis (PyNite), the buckling solver is a custom implementation using exact trigonometric stability functions, giving precise results with a single element per member.

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
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
Fixed-fixed column — exact Euler (K = 0.5)Euler: 4π²EI/L² = 4,515.2 kN	2 elements, E = 200 GPa, I = 366 cm⁴, L = 8 m, P = 1 kN	λ = 4,515.2	4,515.2	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	6,630 kN	0.01%
Portal frame (imperial) — load factorMcGuire, Gallagher & Ziemian, MSA 2nd Ed., Ex 9.6	W10x45 columns, W27x84 beam, lateral + gravity loads	λ = 2.200	2.199	0.05%
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	737.9 lbf	0.02%
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	215.2 kips	0.22%
3D frame — third-party software verification (Mode 1)Third party software, 3D two-story frame	12 nodes, 18 members, self-weight + 2 kN/m superimposed DL	λ = 75.15	75.08	0.10%
ⓘ Wittrick-Williams solver matches third-party results within 0.1% for the governing sway modes. Torsional modes are filtered (no warping stiffness in element) and assessed separately by steel design code checks.

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, 1961
Euler-Bernoulli beam theory — Analytical closed-form solutions

Disclaimer: This verification page is provided for informative purposes only and does not constitute a guarantee of accuracy for all possible configurations. While we strive to ensure the correctness of our calculations through rigorous benchmarking, no software is entirely free of defects. Engineers must exercise independent professional judgement and verify results through their own checks before relying on any output for design or construction decisions. AutoCalcs accepts no liability for errors, omissions, or any consequences arising from the use of this software.