Simulation data with error bars
We solved 2,952 points (1,450 of them stress) automatically while changing the shape step by step, and checked the error bar of each. All of it is available as CSV. Everything was computed with the free NovaSolver Studio.
Updated 4 November 2026
22 kinds of problem, with all the data
Each point went through these checks.
- Three meshes
- CalculiX as a second solver
- Similarity
- Symmetry
- Equilibrium
- Neighbours
- Direction
- Exact solutions
| Problem | Points | Error bar | Compared with | Result | All points |
|---|---|---|---|---|---|
| Stress | |||||
| L-bracket (load spread over the top) | 300 | CalculiX, a second solver | difference within 0.7 % | CSV | |
| Plate with a hole (tension) | 300 | CalculiX, a second solver | difference within 0.01 % | CSV | |
| Stepped shaft (tension) | 200 | CalculiX, a second solver | difference within 0.8 % | CSV | |
| Round bar with a U-groove (tension) | 150 | CalculiX, a second solver | difference within 0.4 % | CSV | |
| Flat bar with shoulders (tension) | 200 | CalculiX, a second solver | difference within 0.7 % | CSV | |
| Thick cylinder (internal pressure, plane strain) | 100 | CalculiX, a second solver | difference within 0.15 % | CSV | |
| Bar with a spherical cavity (tension) | 200 | CalculiX, a second solver | difference within 0.08 % | CSV | |
| Nonlinear | |||||
| L-bracket collapse load (elastic-plastic) | 29 | plastic-hinge hand calculation | 0.913 – 1.036 | CSV | |
| Cantilever, large deflection | 120 | elastica (exact) | 0.999 – 1.002 | CSV | |
| Column after buckling | 30 | elastica (differs by the small imperfection) | 1.000 – 1.017 | CSV | |
| Cylinder pressed on a block | 18 | Hertz | 0.992 – 1.057 | CSV | |
| Vibration and buckling | |||||
| Cantilever, first natural frequency | 100 | Euler-Bernoulli beam | 0.990 – 1.025 | CSV | |
| L-bracket, first natural frequency | 80 | cantilever formula | 0.896 – 1.088 | CSV | |
| Column, linear buckling load | 100 | Engesser | 1.002 – 1.044 | CSV | |
| Thermal stress and rotation | |||||
| Thick cylinder, thermal stress | 60 | Timoshenko (exact) | 0.988 – 1.003 | CSV | |
| Spinning thick cylinder | 60 | spinning-cylinder solution (exact) | 0.992 – 1.003 | CSV | |
| Heat conduction | |||||
| Fin efficiency | 240 | 1D fin formula | 0.983 – 1.000 | CSV | |
| Round pipe in a rectangular block | 150 | strip shape factor | 1.000 – 1.385 | CSV | |
| Electric | |||||
| Eccentric coax, capacitance and peak field | 200 | bipolar-coordinate solution (exact) | 1.000 – 1.001 | CSV | |
| Conductor with a hole, resistance | 200 | Maxwell dilute limit | 1.000 – 1.542 | CSV | |
| Flow | |||||
| Rectangular duct, pressure drop | 80 | Shah-London series (exact) | 0.928 – 0.993 | CSV | |
| Spheroid drag | 35 | sphere drag formula | 0.970 – 1.179 | CSV | |
An error bar is the width 95 % of the points stay within. Each shape had the checks that apply to it. On the flat bar, the 10 hardest shapes, including the one point the neighbour check flagged, were solved again on a finer mesh. Nine moved by 0.6 % or less; the flagged point moved 3.7 %, and its error bar was widened to match.
The error bar is the width that holds 95 % of the points, for the problem's widest answer. The ratio to theory is the range over all points. Only in the exact rows is the distance from 1 itself the error; the other theories have a limited range, so their ratio also holds the effect of the shape. The collapse-load error bar includes the load step.
The CSV files are published under CC BY 4.0. Credit "NovaSolver" with a link to this page and you can use them freely, commercial use included. No warranty.
How to read the CSV, and the input ranges
- Input columns: all ratios are dimensionless. For example
t_over_Lis thickness / length andnuis Poisson's ratio. <answer>_band: the error bar of that answer (relative; 0.03 means 3 %).<answer>_over_…: ratio to a hand formula or theory.mirror_err,equilibrium_err: the symmetry and force-balance residuals.…_loo_err,…_neighbour_flag: the difference when the point is left out and predicted from the others, and a mark on points that disagree with them.…_recheck_diff,…_flag_resolved: the difference after a finer re-solve, and its outcome.has_gci,has_similarity,has_calculix: whether the point had the three-mesh, similarity and CalculiX checks.nodes: node count,solve_s: solve time in seconds,studio: version used.
| Problem | Input ranges | Answer columns | Description |
|---|---|---|---|
| L-bracket (load spread over the top) | h_over_L 0.4–1.0, t_over_L 0.04–0.15, w_over_L 0.25–0.75, r_over_t 0.25–1.5, nu 0.25–0.35 | sigma_star, delta_star | Read |
| Plate with a hole (tension) | d_over_W 0.1–0.6, t_over_W 0.05–0.5, nu 0.25–0.35 | kt_net, compliance_star | Read |
| Stepped shaft (tension) | D_over_d 1.2–2.0, r_over_h 0.2–0.9, nu 0.25–0.35 | kt | Read |
| Round bar with a U-groove (tension) | D_over_d 1.2–2.0, r_over_h 0.25–0.85, nu 0.25–0.35 | kt | Read |
| Flat bar with shoulders (tension) | D_over_d 1.2–2.0, r_over_h 0.2–0.9, t_over_d 0.1–1.0, nu 0.25–0.35 | kt | Read |
| Thick cylinder (internal pressure, plane strain) | b_over_a 1.1–3.0, nu 0.2–0.45 | hoop_star, ur_star | Read |
| Bar with a spherical cavity (tension) | a_over_R 0.1–0.6, nu 0.0–0.45 | kt | Read |
| L-bracket collapse load (elastic-plastic) | h_over_L 0.4–1.0, t_over_L 0.04–0.15, w_over_L 0.25–0.75, r_over_t 0.25–1.5, nu 0.25–0.35 | lam_limit, amp_15 | Read |
| Cantilever, large deflection | t_over_L 0.01–0.03, w_over_t 1.0–4.0, nu 0.25–0.35 | delta_1, short_1, delta_3, short_3, delta_6, short_6 | Read |
| Column after buckling | t_over_L 0.01–0.03, w_over_t 1.2–4.0, nu 0.25–0.35 | delta_106, delta_120, delta_150 | Read |
| Cylinder pressed on a block | a_over_R 0.005–0.03, h_over_R 0.1–0.4, nu 0.2–0.35 | p_over_hertz, a_over_hertz | Read |
| Cantilever, first natural frequency | t_over_L 0.02–0.15, w_over_t 1.2–4.0, nu 0.2–0.4 | f_star | Read |
| L-bracket, first natural frequency | h_over_L 0.4–1.0, t_over_L 0.04–0.15, w_over_L 0.25–0.75, r_over_t 0.25–1.5, nu 0.25–0.35 | f_star | Read |
| Column, linear buckling load | t_over_L 0.02–0.15, w_over_t 1.2–4.0, nu 0.2–0.4 | p_star | Read |
| Thick cylinder, thermal stress | b_over_a 1.1–3.0, nu 0.2–0.45 | hoop_star, axial_star | Read |
| Spinning thick cylinder | b_over_a 1.1–3.0, nu 0.2–0.45 | hoop_star, ur_star | Read |
| Fin efficiency | mL 0.2–3.0, t_over_L 0.05–0.25, w_over_t 1.0–5.0 | q_star, eta | Read |
| Round pipe in a rectangular block | d_over_W 0.1–0.8, H_over_W 1.0–3.0 | s_star | Read |
| Eccentric coax, capacitance and peak field | a_over_b 0.1–0.6, e_over_gap 0.0–0.7 | c_star, emax_star | Read |
| Conductor with a hole, resistance | d_over_W 0.05–0.6, e_over_W 0.0–0.15 | dl_over_W, kj | Read |
| Rectangular duct, pressure drop | aspect 1.0–5.0, Re 1.0–20.0 | fRe | Read |
| Spheroid drag | aspect 0.5–2.0, Re 2.0–150.0 | cd | Read |
What it is for
Against exact solutions and textbook values
Thick cylinder vs Lamé (exact)
Within 0.22 % at all 100 points.
Spherical cavity vs Goodier (exact)
Within 0.25 % at Poisson's ratios 0, 0.3 and 0.45.
Plate with a hole vs Howland
Plates thinner than a fifth of the hole diameter agree within 2 %; thick plates run a few percent higher, a 3D effect.
Stepped shaft vs Peterson's curve fit
10 to 20 % above the fit, the same trend as published reports that the chart runs low. The coefficients come from a third-party site and were not checked against the book, so this is a trend check only.
The error bar measures how accurately the equations were solved, not how a real part matches (material scatter, tolerances).