Generated solver evidence

Structural Solver Verification Centre

Trace the current analytical benchmarks from the reference equations to the exact production-solver result. Every value below comes from the generated evidence package used by the release tests.

Review benchmark cases Open the calculator
Current evidence scope Analytical beam, bar, release and coordinate-transformation cases. Truss, frame, mixed-model and browser evidence will be added as those suites are completed.

How the evidence is produced

Reference first. Production solver second.

Expected answers are established independently using closed-form structural mechanics. The production solver then analyzes the same model, and every declared output is evaluated against a tolerance chosen before the test runs.

  1. 01

    Define the reference

    Geometry, properties, loads, equations and expected answers are recorded without using a production-solver result as the source.

  2. 02

    Run the real solver

    The benchmark runner imports the same solver file delivered to calculator users. There is no separate demonstration calculator.

  3. 03

    Apply declared tolerances

    Each numerical check records its expected value, displayed engineering unit, absolute tolerance and relative tolerance.

  4. 04

    Publish generated evidence

    This page reads the resulting evidence artifact. Counts and calculated values are not maintained as a second handwritten copy.

Numerical acceptance rule

|calculated − expected| ≤ absolute tolerance + relative tolerance × |expected|

For these well-conditioned closed-form cases, the default relative tolerance is 1e-8 and the relative equilibrium residual must not exceed 1e-10. Case-specific near-zero tolerances are shown in each result table.

Current benchmark record

Every current case, equation and checked result

11 reproducible models · 75 declared checks · generated July 25, 2026 · 9:52 AM EDT

Axial response

1 case
OB-V-BAR-001 Horizontal axial bar with end force 5/5 checks passed · closed-form · residual 0

Reference basis

Uniform prismatic bar fixed at one end and loaded axially at the other.

Nodes
2
Members
1
Loads
1
E
200 GPa
A
5000 mm²
I
2.0e+8 mm⁴
R_x = -P delta = P L / (E A) N = P
Checked result Reference Calculated Difference Acceptance Status
Node N1 · Horizontal reaction -100 kN -100 kN 0 kN ±1.010e-6 kN Pass
Node N2 · Horizontal displacement 0.3 mm 0.3 mm 5.551e-17 mm ±1.300e-8 mm Pass
Member M1 · Peak absolute axial force 100 kN 100 kN 0 kN ±1.010e-6 kN Pass
Member M1 · Peak absolute bending moment 0 kN m 0 kN m 0 kN m ±1.000e-9 kN m Pass
Solver · Relative equilibrium residual 1.000e-10 0 1.000e-10 ≤ 1.000e-10 Pass
  • axial stiffness
  • nodal force
  • reaction
  • axial displacement

Cantilever response

3 cases
OB-V-BEAM-001 Cantilever with transverse tip force 5/5 checks passed · closed-form · residual 4.547e-16

Reference basis

Prismatic cantilever with a vertical point force at the free end.

Nodes
2
Members
1
Loads
1
E
200 GPa
A
5000 mm²
I
2.0e+8 mm⁴
R_y = P M_root = P L delta_tip = P L^3 / (3 E I)
Checked result Reference Calculated Difference Acceptance Status
Node N1 · Vertical reaction 10 kN 10 kN 7.105e-15 kN ±1.100e-7 kN Pass
Node N1 · Support moment 40 kN m 40 kN m 3.553e-14 kN m ±4.100e-7 kN m Pass
Node N2 · Vertical displacement -5.33333333 mm -5.33333333 mm 2.665e-15 mm ±6.333e-8 mm Pass
Member M1 · Peak absolute bending moment 40 kN m 40 kN m 3.553e-14 kN m ±4.100e-7 kN m Pass
Solver · Relative equilibrium residual 1.000e-10 4.547e-16 1.000e-10 ≤ 1.000e-10 Pass
  • flexural stiffness
  • nodal force
  • reaction
  • tip displacement
  • peak moment
OB-V-BEAM-002 Cantilever with applied tip moment 6/6 checks passed · closed-form · residual 5.457e-16

Reference basis

Prismatic cantilever with a positive moment applied at the free end.

Nodes
2
Members
1
Loads
1
E
200 GPa
A
5000 mm²
I
2.0e+8 mm⁴
M_root = -M theta_tip = M L / (E I) delta_tip = M L^2 / (2 E I)
Checked result Reference Calculated Difference Acceptance Status
Node N1 · Support moment -10 kN m -10 kN m 1.066e-14 kN m ±1.100e-7 kN m Pass
Node N2 · Vertical displacement 2 mm 2 mm 1.332e-15 mm ±3.000e-8 mm Pass
Node N2 · Rotation 0.001 rad 0.001 rad 6.505e-19 rad ±1.001e-8 rad Pass
Member M1 · Peak absolute bending moment 10 kN m 10 kN m 0 kN m ±1.100e-7 kN m Pass
Member M1 · Peak absolute shear force 0 kN 0 kN 0 kN ±1.000e-9 kN Pass
Solver · Relative equilibrium residual 1.000e-10 5.457e-16 1.000e-10 ≤ 1.000e-10 Pass
  • nodal moment
  • rotation
  • curvature
  • constant moment
OB-V-BEAM-009 Cantilever with full uniform load 9/9 checks passed · closed-form · residual 2.339e-16

Reference basis

Four metre prismatic cantilever with a ten kilonewton-per-metre uniform load.

Nodes
2
Members
1
Loads
1
E
200 GPa
A
5000 mm²
I
2.0e+8 mm⁴
R_y = w L = 40 kN M_root = w L^2 / 2 = 80 kN m theta_tip = -w L^3 / (6 E I) = -0.002666666667 rad delta_tip = -w L^4 / (8 E I) = -8 mm
Checked result Reference Calculated Difference Acceptance Status
Node N1 · Vertical reaction 40 kN 40 kN 7.105e-15 kN ±4.100e-7 kN Pass
Node N1 · Support moment 80 kN m 80 kN m 4.263e-14 kN m ±8.100e-7 kN m Pass
Node N2 · Vertical displacement -8 mm -8 mm 5.329e-15 mm ±9.000e-8 mm Pass
Node N2 · Rotation -0.00266667 rad -0.00266667 rad 2.168e-18 rad ±1.003e-8 rad Pass
Member M1 · Maximum displacement 8 mm 8 mm 7.105e-15 mm ±9.000e-8 mm Pass
Member M1 · Maximum displacement location 4 m 4 m 0 m ±5.000e-8 m Pass
Member M1 · Peak absolute shear force 40 kN 40 kN 7.105e-15 kN ±4.100e-7 kN Pass
Member M1 · Peak absolute bending moment 80 kN m 80 kN m 4.263e-14 kN m ±8.100e-7 kN m Pass
Solver · Relative equilibrium residual 1.000e-10 2.339e-16 1.000e-10 ≤ 1.000e-10 Pass
  • distributed load
  • cantilever reaction
  • tip rotation
  • tip displacement

Simply supported response

4 cases
OB-V-BEAM-003 Simply supported beam with full uniform load 8/8 checks passed · closed-form · residual 3.638e-16

Reference basis

Six metre simply supported prismatic beam with a ten kilonewton-per-metre uniform load.

Nodes
2
Members
1
Loads
1
E
200 GPa
A
5000 mm²
I
2.0e+8 mm⁴
R_A = R_B = w L / 2 V_max = w L / 2 M_max = w L^2 / 8 delta_max = 5 w L^4 / (384 E I)
Checked result Reference Calculated Difference Acceptance Status
Node N1 · Vertical reaction 30 kN 30 kN 0 kN ±3.100e-7 kN Pass
Node N2 · Vertical reaction 30 kN 30 kN 3.553e-15 kN ±3.100e-7 kN Pass
Member M1 · Peak absolute shear force 30 kN 30 kN 0 kN ±3.100e-7 kN Pass
Member M1 · Peak absolute bending moment 45 kN m 45 kN m 1.421e-14 kN m ±4.600e-7 kN m Pass
Model · Maximum displacement 4.21875 mm 4.21875 mm 0 mm ±5.219e-8 mm Pass
Member M1 · Start-end moment 0 kN m 1.091e-14 kN m 1.091e-14 kN m ±1.000e-8 kN m Pass
Member M1 · Finish-end moment 0 kN m 1.091e-14 kN m 1.091e-14 kN m ±1.000e-8 kN m Pass
Solver · Relative equilibrium residual 1.000e-10 3.638e-16 1.000e-10 ≤ 1.000e-10 Pass
  • distributed load
  • released rotations
  • reactions
  • shear
  • moment
  • member displacement
OB-V-BEAM-005 Simply supported beam with triangular distributed load 6/6 checks passed · closed-form · residual 2.165e-16

Reference basis

Six metre simply supported beam with a triangular load increasing from zero to twelve kilonewtons per metre.

Nodes
2
Members
1
Loads
1
E
200 GPa
A
5000 mm²
I
2.0e+8 mm⁴
R_A = w_max L / 6 = 12 kN R_B = w_max L / 3 = 24 kN V(x) = 12 - x^2 x_Mmax = sqrt(12) = 3.464101615 m M_max = 16 sqrt(3) = 27.712812921 kN m
Checked result Reference Calculated Difference Acceptance Status
Node N1 · Vertical reaction 12 kN 12 kN 7.105e-15 kN ±1.300e-7 kN Pass
Node N2 · Vertical reaction 24 kN 24 kN 7.105e-15 kN ±2.500e-7 kN Pass
Member M1 · Peak absolute shear force 24 kN 24 kN 7.105e-15 kN ±2.500e-7 kN Pass
Member M1 · Peak absolute bending moment 27.71281292 kN m 27.71281292 kN m 1.421e-14 kN m ±2.871e-7 kN m Pass
Member M1 · Peak moment location 3.46410162 m 3.46410162 m 2.147e-12 m ±4.464e-8 m Pass
Solver · Relative equilibrium residual 1.000e-10 2.165e-16 1.000e-10 ≤ 1.000e-10 Pass
  • linearly varying load
  • asymmetric reactions
  • off-grid critical moment
  • critical-station recovery
OB-V-BEAM-006 Simply supported beam with off-centre point load 7/7 checks passed · closed-form · residual 2.046e-16

Reference basis

Six metre simply supported beam with a ten kilonewton point load two metres from the left support.

Nodes
2
Members
1
Loads
1
E
200 GPa
A
5000 mm²
I
2.0e+8 mm⁴
R_A = P b / L = 6.666666667 kN R_B = P a / L = 3.333333333 kN M_max = P a b / L = 13.333333333 kN m x_delta,max = L - sqrt((L^2 - a^2) / 3) = 2.734013676 m delta_max = 0.967699651 mm
Checked result Reference Calculated Difference Acceptance Status
Node N1 · Vertical reaction 6.66666667 kN 6.66666667 kN 0 kN ±7.667e-8 kN Pass
Node N2 · Vertical reaction 3.33333333 kN 3.33333333 kN 0 kN ±4.333e-8 kN Pass
Member M1 · Peak absolute bending moment 13.33333333 kN m 13.33333333 kN m 1.776e-15 kN m ±1.433e-7 kN m Pass
Member M1 · Peak moment location 2 m 2 m 0 m ±3.000e-8 m Pass
Member M1 · Maximum displacement 0.96769965 mm 0.96769965 mm 4.441e-16 mm ±1.968e-8 mm Pass
Member M1 · Maximum displacement location 2.73401368 m 2.73401368 m 8.002e-13 m ±3.734e-8 m Pass
Solver · Relative equilibrium residual 1.000e-10 2.046e-16 1.000e-10 ≤ 1.000e-10 Pass
  • member point load
  • asymmetric reactions
  • point-load moment
  • off-grid displacement maximum
OB-V-BEAM-007 Simply supported beam with partial uniform load 6/6 checks passed · closed-form · residual 1.001e-16

Reference basis

Six metre simply supported beam with an eight kilonewton-per-metre load from x = 1 m to x = 4 m.

Nodes
2
Members
1
Loads
1
E
200 GPa
A
5000 mm²
I
2.0e+8 mm⁴
W = 24 kN at x = 2.5 m R_A = 14 kN R_B = 10 kN V(x) = 14 - 8(x - 1) within the loaded interval x_Mmax = 2.75 m M_max = 26.25 kN m
Checked result Reference Calculated Difference Acceptance Status
Node N1 · Vertical reaction 14 kN 14 kN 1.776e-15 kN ±1.500e-7 kN Pass
Node N2 · Vertical reaction 10 kN 10 kN 0 kN ±1.100e-7 kN Pass
Member M1 · Peak absolute shear force 14 kN 14 kN 1.776e-15 kN ±1.500e-7 kN Pass
Member M1 · Peak absolute bending moment 26.25 kN m 26.25 kN m 0 kN m ±2.725e-7 kN m Pass
Member M1 · Peak moment location 2.75 m 2.75 m 4.441e-14 m ±3.750e-8 m Pass
Solver · Relative equilibrium residual 1.000e-10 1.001e-16 1.000e-10 ≤ 1.000e-10 Pass
  • partial distributed load
  • load boundaries
  • off-grid critical moment

Restraints and releases

2 cases
OB-V-BEAM-004 Fixed-pinned beam with full uniform load 5/5 checks passed · closed-form · residual 0

Reference basis

Six metre prismatic member fixed at the start and rotationally released at the restrained finish.

Nodes
2
Members
1
Loads
1
E
200 GPa
A
5000 mm²
I
2.0e+8 mm⁴
R_A = 5 w L / 8 R_B = 3 w L / 8 |M_A| = w L^2 / 8 M_B = 0
Checked result Reference Calculated Difference Acceptance Status
Node N1 · Vertical reaction 37.5 kN 37.5 kN 7.105e-15 kN ±3.850e-7 kN Pass
Node N2 · Vertical reaction 22.5 kN 22.5 kN 0 kN ±2.350e-7 kN Pass
Member M1 · Start-end moment 45 kN m 45 kN m 0 kN m ±4.600e-7 kN m Pass
Member M1 · Finish-end moment 0 kN m 0 kN m 0 kN m ±1.000e-8 kN m Pass
Solver · Relative equilibrium residual 1.000e-10 0 1.000e-10 ≤ 1.000e-10 Pass
  • one-end release
  • release condensation
  • reaction distribution
  • end moment
OB-V-BEAM-008 Fixed-fixed beam with full uniform load 9/9 checks passed · closed-form · residual 0

Reference basis

Six metre fixed-fixed prismatic beam with a ten kilonewton-per-metre uniform load.

Nodes
2
Members
1
Loads
1
E
200 GPa
A
5000 mm²
I
2.0e+8 mm⁴
R_A = R_B = w L / 2 = 30 kN |M_A| = |M_B| = w L^2 / 12 = 30 kN m delta_max = w L^4 / (384 E I) = 0.84375 mm
Checked result Reference Calculated Difference Acceptance Status
Node N1 · Vertical reaction 30 kN 30 kN 3.553e-15 kN ±3.100e-7 kN Pass
Node N2 · Vertical reaction 30 kN 30 kN 0 kN ±3.100e-7 kN Pass
Node N1 · Support moment 30 kN m 30 kN m 3.553e-15 kN m ±3.100e-7 kN m Pass
Node N2 · Support moment -30 kN m -30 kN m 3.553e-15 kN m ±3.100e-7 kN m Pass
Member M1 · Maximum displacement 0.84375 mm 0.84375 mm 0 mm ±1.844e-8 mm Pass
Member M1 · Maximum displacement location 3 m 3 m 0 m ±4.000e-8 m Pass
Member M1 · Peak absolute bending moment 30 kN m 30 kN m 2.842e-14 kN m ±3.100e-7 kN m Pass
Solver · Stability classification fully-restrained fully-restrained — Exact match Pass
Solver · Relative equilibrium residual 1.000e-10 0 1.000e-10 ≤ 1.000e-10 Pass
  • fully restrained system
  • fixed-end forces
  • member displacement recovery

Coordinate transformations

1 case
OB-V-XFORM-001 Inclined simply supported member with global point load 9/9 checks passed · closed-form · residual 1.819e-16

Reference basis

Three-four-five inclined member with a ten kilonewton global vertical point load at midspan.

Nodes
2
Members
1
Loads
1
E
200 GPa
A
5000 mm²
I
2.0e+8 mm⁴
R_Ay = R_By = P / 2 P_local,x = -P sin(theta) = -8 kN P_local,y = -P cos(theta) = -6 kN |N|max = 4 kN |V|max = 3 kN M_max = P_local,y L / 4 = 7.5 kN m
Checked result Reference Calculated Difference Acceptance Status
Node N1 · Horizontal reaction 0 kN 0 kN 0 kN ±1.000e-8 kN Pass
Node N1 · Vertical reaction 5 kN 5 kN 0 kN ±6.000e-8 kN Pass
Node N2 · Vertical reaction 5 kN 5 kN 0 kN ±6.000e-8 kN Pass
Member M1 · Peak absolute axial force 4 kN 4 kN 0 kN ±5.000e-8 kN Pass
Member M1 · Peak absolute shear force 3 kN 3 kN 0 kN ±4.000e-8 kN Pass
Member M1 · Peak absolute bending moment 7.5 kN m 7.5 kN m 8.882e-16 kN m ±8.500e-8 kN m Pass
Member M1 · Start-end moment 0 kN m -9.095e-16 kN m 9.095e-16 kN m ±1.000e-8 kN m Pass
Member M1 · Finish-end moment 0 kN m 9.095e-16 kN m 9.095e-16 kN m ±1.000e-8 kN m Pass
Solver · Relative equilibrium residual 1.000e-10 1.819e-16 1.000e-10 ≤ 1.000e-10 Pass
  • inclined member
  • global member load
  • local force transformation
  • reaction transformation

Interpret the evidence correctly

Verified scope and explicit limitations

This evidence supports the documented mathematical domain. It is not a statement that every possible structure, modelling decision or real-world condition has been validated.

Covered by the solver formulation

  • First-order, linear-elastic 2D static response
  • Axial and Euler-Bernoulli flexural stiffness
  • Nodal, point and linearly varying member loads
  • Fixed, pinned, roller and custom restraints
  • Continuous, one-end-released and pin-ended members
  • Local and global member-load directions
  • Load cases, linear combinations and self-weight

Not provided by this solver

  • P-Delta or other second-order effects
  • Geometric or material nonlinearity
  • Tension-only or compression-only behavior
  • Shear deformation or support settlement
  • Thermal, buckling, modal or dynamic analysis
  • Member resistance or connection design
  • Certification of a particular engineering design
Engineering-use statement

This release passed the documented benchmark suite within the stated tolerances for first-order, linear-elastic 2D analysis. Verification does not constitute design certification or replace independent professional review of geometry, restraints, loading, assumptions and governing requirements.

Traceable release evidence

Download the evidence behind this page

The JSON package contains the exact benchmark models, reference notes, declared tolerances, calculated values, differences, diagnostics, runtime information and production-solver fingerprint.

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Suite
Optimal Beam priority analytical verification
Generated
July 25, 2026 · 9:52 AM EDT
Solver SHA-256
7cb2472d5ee0864d1b5c5b0db4fbbc8413c8fa13185c784403fb8591c3ebb89c
Result
Passed

Methodology references

Credibility guidance and comparison resources

ASME V&V 10 Verification and validation guidance for computational solid mechanics. NASA-STD-7009B Credibility practices for models and simulations. OpenSees Examples Manual Independent structural-analysis examples for the planned comparison suite.

These references inform the verification approach. Their inclusion does not represent certification or a claim of formal compliance.