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.

Each tool has worked cases below: the inputs, what the tool returns, and the reference value. Enter the same inputs in the tool to reproduce them.

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: 151 formula checks over all 24 cases, non-round inputs. 154 values from the tool. 5 formulas corrected before release.
Limits: Constant EI, one span, small deflection, no shear deformation.

Worked cases (3)

Simply supported, off-centre point load

P = 40 kN at a = 2 m, L = 6 m, E = 200 GPa, I = 8000 cm⁴

	Tool	Reference
R_A	26.6667 kN	26.6667 kN
R_B	13.3333 kN	13.3333 kN
M_max	53.3333 kN·m	53.3333 kN·m
δ_max	9.677 mm	9.677 mm
at x	2.73401 m	2.73401 m

Reference: Macaulay integration of EI y″ = M(x), with the reactions and both integration constants solved from the support conditions (sympy). δ_max is found where the slope is zero.

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Propped cantilever, full UDL

Fixed at A, roller at B. w = 10 kN/m, L = 5 m, E = 200 GPa, I = 8000 cm⁴

	Tool	Reference
R_A	31.25 kN	31.25 kN
R_B	18.75 kN	18.75 kN
M_A (hogging)	31.25 kN·m	31.25 kN·m
M₊	17.5781 kN·m	17.5781 kN·m
at x	3.125 m	3.125 m
δ_max	2.11567 mm	2.11567 mm
at x	2.89232 m	2.89232 m

Reference: Macaulay integration of EI y″ = M(x), with the reactions and both integration constants solved from the support conditions (sympy). δ_max is found where the slope is zero.

Open this case in the tool

Fixed both ends, off-centre point load

P = 30 kN at a = 1.5 m, L = 5 m, E = 200 GPa, I = 8000 cm⁴

	Tool	Reference
R_A	23.52 kN	23.52 kN
R_B	6.48 kN	6.48 kN
M_A (hogging)	22.05 kN·m	22.05 kN·m
M_B (hogging)	9.45 kN·m	9.45 kN·m
M under the load	13.23 kN·m	13.23 kN·m
δ_max	0.837402 mm	0.837402 mm
at x	2.08333 m	2.08333 m

Reference: Macaulay integration of EI y″ = M(x), with the reactions and both integration constants solved from the support conditions (sympy). δ_max is found where the slope is zero.

Open this case in the tool

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: 18 beams, all support and load types, five with overhangs, 3 points per load interval. 284 values, within 1 part in 10⁹.
Limits: One span, determinate supports. No axial load or deflection.

Worked cases (2)

Simply supported, point load and partial UDL

L = 8 m. 10 kN down at x = 2 m. 6 kN/m down from x = 3 m to 8 m.

	Tool	Reference
R_A	16.875 kN	16.875 kN
R_B	23.125 kN	23.125 kN
M at x = 2 m	33.75 kN·m	33.75 kN·m
M at x = 3 m	40.625 kN·m	40.625 kN·m
V = 0 at x	4.14583 m	4.14583 m
M_max	44.5638 kN·m	44.5638 kN·m

Reference: R_B from moments about A, R_A from vertical equilibrium, M(x) by integrating the left free body (sympy). By hand: R_B = (10·2 + 30·5.5) / 8.

Cantilever with UDL and a couple

Fixed at x = 0, L = 4 m. 3 kN/m down over the full length. 8 kN·m counterclockwise at x = 2 m.

	Tool	Reference
Wall reaction	12 kN	12 kN
M at the wall	−16 kN·m	−16 kN·m
M just left of x = 2 m	2 kN·m	2 kN·m
M just right of x = 2 m	−6 kN·m	−6 kN·m
M at x = 3 m	−1.5 kN·m	−1.5 kN·m

Reference: M(x) from the free body to the right of each cut, which needs no reactions (sympy). Sagging positive. The couple makes the 8 kN·m jump at x = 2 m.

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.

Worked cases (3)

Plane stress, rotated element

σ_x = 80 MPa, σ_y = −40 MPa, τ_xy = 25 MPa, element rotated 30° counterclockwise.

	Tool	Reference
σ₁	85 MPa	85 MPa
σ₂	−45 MPa	−45 MPa
τ_max (in plane)	65 MPa	65 MPa
θ_p	11.3099°	11.3099°
σ_x′	71.6506 MPa	71.6506 MPa
σ_y′	−31.6506 MPa	−31.6506 MPa
τ_x′y′	−39.4615 MPa	−39.4615 MPa

Reference: eigenvalues and eigenvectors of the stress tensor, and the rotated element from R·S·R^T (numpy). By hand: centre 20, radius √(60² + 25²) = 65.

3D, σ_z principal

σ_x = 50 MPa, σ_y = 30 MPa, τ_xy = 20 MPa, σ_z = −40 MPa.

	Tool	Reference
σ₁	62.3607 MPa	62.3607 MPa
σ₂	17.6393 MPa	17.6393 MPa
σ₃	−40 MPa	−40 MPa
τ_{abs max}	51.1803 MPa	51.1803 MPa

Reference: eigenvalues of the 3×3 stress tensor (numpy).

Plane strain

ε_x = 500 µε, ε_y = −200 µε, γ_xy = 300 µε.

	Tool	Reference
ε₁	530.789 µε	530.789 µε
ε₂	−230.789 µε	−230.789 µε
γ_max (in plane)	761.577 µε	761.577 µε

Reference: eigenvalues of the strain tensor, which holds γ_xy/2 off the diagonal (numpy).

Moment of inertia and centroid calculator

Reference: Rectangles: section cut into non-overlapping grid cells on every rectangle edge, exact properties summed per cell, checked against closed forms for a hollow rectangle and an I-section. Circles, arcs and polygons: exact integration in sympy for every part in every orientation, and composites summed from exact moments about a fixed origin, with no parallel axis step per part.
Result: 15 rectangle sections (I, T, channel, angle, Z, hollow rectangle, plate with voids, 8 random stacks): 150 values within 1 part in 10⁹. Circles, arcs and polygons: 245 checks, including Beer and Johnston Sample Problem 5.1, a pipe against π(D⁴ − d⁴)/64 and the half and quarter circle closed forms; the tool matches the 8 composites in 80 values within 1 part in 10⁹.
Limits: Straight edges and circular arcs in whole quarter turns; no ellipses or fillets. Overlapping shapes are flagged.

Worked cases (3)

T-section

Flange 120 × 20 mm on top of a web 20 × 100 mm, centred.

	Tool	Reference
A	4400 mm²	4400 mm²
ȳ	82.7273 mm	82.7273 mm
I_x	5.67394 × 10⁶ mm⁴	5.67394 × 10⁶ mm⁴
I_y	2.94667 × 10⁶ mm⁴	2.94667 × 10⁶ mm⁴
J	8.62061 × 10⁶ mm⁴	8.62061 × 10⁶ mm⁴

Reference: parallel axis theorem by hand, Σ(bh³/12 + Ad²) over the rectangles, voids subtracted. Axes through the centroid; x̄ from the left edge, ȳ from the bottom.

Hollow rectangle

100 mm wide, 150 mm tall, 10 mm wall. Drawn as a solid rectangle with an 80 × 130 mm void.

	Tool	Reference
A	4600 mm²	4600 mm²
I_x	1.34783 × 10⁷ mm⁴	1.34783 × 10⁷ mm⁴
I_y	6.95333 × 10⁶ mm⁴	6.95333 × 10⁶ mm⁴
S_x	1.79711 × 10⁵ mm³	1.79711 × 10⁵ mm³

Reference: (BH³ − bh³) / 12, and S_x = I_x / 75 mm.

Z-section, product of inertia

Bottom flange 60 × 10 mm pointing left, web 10 × 80 mm, top flange 60 × 10 mm pointing right, 100 mm deep overall.

	Tool	Reference
A	2000 mm²	2000 mm²
x̄, ȳ	55, 50 mm	55, 50 mm
I_x	2.86667 × 10⁶ mm⁴	2.86667 × 10⁶ mm⁴
I_y	1.11667 × 10⁶ mm⁴	1.11667 × 10⁶ mm⁴
I_xy	1.35 × 10⁶ mm⁴	1.35 × 10⁶ mm⁴

Reference: parallel axis theorem by hand, Σ(bh³/12 + Ad²) over the rectangles, voids subtracted. Axes through the centroid; x̄ from the left edge, ȳ from the bottom. I_xy = ΣA d_xd_y, with y up.

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.

Worked cases (3)

Two forces at right angles

200 N at 30°, 150 N at 120°. The worked example on the tool page.

	Tool	Reference
R_x	98.2051 N	98.2051 N
R_y	229.904 N	229.904 N
\|R\|	250 N	250 N
θ	66.8699°	66.8699°

Reference: component sums (numpy). By hand: the forces are perpendicular, so |R| = √(200² + 150²) = 250 N at 30° + tan⁻¹(150/200).

Three forces

400 N at 20°, 250 N at 135°, 300 N at 250°.

	Tool	Reference
R_x	96.4943 N	96.4943 N
R_y	31.677 N	31.677 N
\|R\|	101.561 N	101.561 N
θ	18.1739°	18.1739°

Reference: component sums (numpy).

3D

F₁ = (120, −50, 80) N, F₂ = (−30, 90, 60) N, F₃ = (10, 40, −100) N.

	Tool	Reference
R	(100, 80, 40) N	(100, 80, 40) N
\|R\|	134.164 N	134.164 N
θ_x, θ_y, θ_z	41.81, 53.4, 72.65°	41.81, 53.4, 72.65°

Reference: component sums and direction cosines (numpy).

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, 9-7, 10-4.
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. Dual, regenerative Brayton and reheat Rankine: 33 reference checks and 393 values from the tool. Forward and backward solving.
Limits: Rankine with at most one reheat and no regeneration. IF97 Regions 1, 2, 4: no near-critical region, no steam above 800 C. R-134a only.

Worked cases (3)

Steam: superheated state, isentropic expansion

State 1: 3 MPa, 350 °C. State 2: 10 kPa, s₂ = s₁.

	Tool	Reference
h₁	3116.06 kJ/kg	3116.06 kJ/kg
s₁	6.74492 kJ/kg·K	6.74493 kJ/kg·K
v₁	0.090555 m³/kg	0.0905564 m³/kg
T₂	45.8075 °C	45.8063 °C
x₂	0.812796	0.812806
h₂	2136.08 kJ/kg	2136.08 kJ/kg

Reference: CoolProp, IAPWS-95. The tool uses IAPWS-IF97, the industrial fit to IAPWS-95, so the two differ in the fifth or sixth figure.

Brayton, Cengel and Boles Example 9-5

Ideal cycle, pressure ratio 8, T₁ = 300 K, P₁ = 100 kPa, T₃ = 1300 K, air tables.

	Tool	Textbook
T₂	539.752 K	540 K
h₂	544.173 kJ/kg	544.35 kJ/kg
T₄	770.397 K	770 K
h₄	789.604 kJ/kg	789.37 kJ/kg
Back work ratio	0.4024	0.403
η_th	0.4254	0.426

Reference: the worked answers in Cengel and Boles, which read the air table by hand interpolation, hence the last-digit differences.

Diesel, Cengel and Boles Example 9-3

r = 18, cutoff ratio 2, P₁ = 14.7 psia (101.35 kPa), T₁ = 80 °F (300 K), cold air, k = 1.4.

	Tool	Closed form	Textbook
T₂	953.301 K	953.301 K	1716 °R (953.3 K)
P₂	5797.19 kPa	5797.19 kPa	841 psia (5798 kPa)
T₃	1906.6 K	1906.6 K	3432 °R (1906.7 K)
T₄	791.705 K	791.705 K	1425 °R (791.7 K)
P₄	267.472 kPa	267.472 kPa	38.8 psia (267.5 kPa)
η_th	0.6316	0.6316	0.632

Reference: cold air closed forms, T₂ = T₁r^k−1, T₃ = r_cT₂, T₄ = T₃(r_c/r)^k−1, η = 1 − (r_c^k − 1) / (k r^k−1(r_c − 1)), and the textbook answers.

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.

Worked cases (2)

Two equal masses in a chain

Wall, spring k, mass m, spring k, mass m. m = 2 kg, k = 800 N/m. Body 1: 2x₁″ = −800x₁ + 800(x₂ − x₁) + F. Body 2: 2x₂″ = −800(x₂ − x₁).

	Tool	Reference
ω₁	12.3607 rad/s	12.3607 rad/s
Mode 1 shape	0.618034, 1	0.618034, 1
ω₂	32.3607 rad/s	32.3607 rad/s
Mode 2 shape	1, −0.618034	1, −0.618034

Reference: eigenvalues of M⁻¹K (numpy). By hand: ω² = (3 ∓ √5)k / 2m.

Two-mass chain with damping

m₁ = 10 kg, m₂ = 5 kg, c₁ = 20, c₂ = 10 N·s/m, k₁ = 1200, k₂ = 800 N/m, force F on mass 1. Body 1: 10x₁″ = −1200x₁ − 20x₁′ + 800(x₂ − x₁) + 10(x₂′ − x₁′) + F. Body 2: 5x₂″ = −800(x₂ − x₁) − 10(x₂′ − x₁′).

	Tool	Reference
X₁/F	(s² + 2s + 160) / (10s⁴ + 50s³ + 3640s² + 5600s + 192000)	(5s² + 10s + 800) / (50s⁴ + 250s³ + 18200s² + 28000s + 960000)
Poles, pair 1	−0.510194 ± 8.05392j	−0.510194 ± 8.05392j
ζ, pair 1	0.063221	0.063221
Poles, pair 2	−1.98981 ± 17.0544j	−1.98981 ± 17.0544j
ζ, pair 2	0.11589	0.11589
Undamped ω_n	8.069, 17.1724 rad/s	8.069, 17.1724 rad/s

Reference: det Z(s) and the adjugate of Z(s) = Ms² + Cs + K in sympy, roots in numpy. The tool divides the transfer function through by 5.

Free body diagram maker

Reference: None. Drawing tool, no calculation.

Verification

Method

Simply supported, off-centre point load

Propped cantilever, full UDL

Fixed both ends, off-centre point load

Simply supported, point load and partial UDL

Cantilever with UDL and a couple

Plane stress, rotated element

3D, σz principal

Plane strain

T-section

Hollow rectangle

Z-section, product of inertia

Two forces at right angles

Three forces

3D

Steam: superheated state, isentropic expansion

Brayton, Cengel and Boles Example 9-5

Diesel, Cengel and Boles Example 9-3

Two equal masses in a chain

Two-mass chain with damping

3D, σ_z principal