Verification

Last run Sep 17, 2026.

Method

Each tool is compared against a separate calculation of the same quantities, mostly in Python with sympy, numpy or CoolProp. The test inputs are run through the tool's own code in a browser. The checks are re-run after any change to a tool's math.

Beam formula calculator

Reference: Singularity-function beam solver (sympy), solving reactions and integration constants from the boundary conditions. Closed forms from Roark's Formulas for Stress and Strain, 8th ed.
Result: 95 formula checks over all 17 cases, non-round inputs. 106 values from the tool. 4 formulas corrected before release.
Limits: Constant EI, one span, small deflection, no shear deformation.

Shear and moment diagram calculator

Reference: Reactions from equilibrium, V(x) and M(x) by exact integration over one side of the cut (sympy).
Result: 13 beams, all support and load types, 3 points per load interval. 190 values, within 1 part in 10⁹.
Limits: One span, determinate supports. No axial load or deflection.

Mohr's circle calculator

Reference: Eigenvalues and eigenvectors of the stress or strain tensor, and direct rotation R·S·R^T (numpy).
Result: 27 stress states, 10 strain states. 245 values, within 1 part in 10⁹.
Limits: 3D mode takes σz as principal. No out-of-plane shear.

Moment of inertia and centroid calculator

Reference: Section cut into non-overlapping grid cells on every rectangle edge, exact properties summed per cell. Method checked against closed forms for a hollow rectangle and an I-section.
Result: 15 sections: I, T, channel, angle, Z, hollow rectangle, plate with voids, 8 random stacks. 150 values, within 1 part in 10⁹.
Limits: Rectangles and rectangular voids. Overlapping shapes are flagged.

Vector resultant calculator

Reference: Component sums (numpy).
Result: 26 2D systems including the page's worked example, 15 3D systems. 209 values, within 1 part in 10⁹.
Limits: Concurrent forces. No moments about a point.

Thermodynamics cycle solver

Reference: CoolProp (IAPWS-95 water, R-134a, ideal-gas air c_p), cold air-standard closed forms, Cengel and Boles Examples 9-3, 9-5, 9-6.
Result: Steam: 22 states, 8 kPa to 20 MPa, within 0.02% (the IF97 vs IAPWS-95 difference). R-134a: 11 states, within 1 part in 10⁹. Otto: 4 cycles, within 0.003%. Brayton: 102 checks. Diesel and Atkinson: 250 checks. Forward and backward solving.
Limits: Rankine without reheat or regeneration. IF97 Regions 1, 2, 4: no near-critical region, no steam above 800 C. R-134a only.

Equation of motion and transfer function solver

Reference: Python version of the engine (sympy, numpy); the JS mirrors it function for function.
Result: 139 checks: determinants and cofactors to 4×4, Cramer's rule transfer functions, roots, natural frequencies and mode shapes. Tool matches the reference to 12 significant figures on a 2-mass and 3-mass chain.
Limits: 1 to 4 masses, linear, zero initial conditions, up to second order.

Free body diagram maker

Reference: None. Drawing tool, no calculation.