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Reference calculations & regression benchmarks

14 Worked Structural Analysis Examples

Check bending moments, reactions, deflections and axial forces against classical mechanics. Every example includes assumptions, analytical equations, expected results, a downloadable JSON model and a one-click link to open the structure in FramePad.

14Worked models
55Numerical checks
3Structural categories
Validation scope: These are hand-calculation benchmark comparisons for selected linear-elastic cases, not third-party verification, certification or a guarantee for arbitrary models. Assumptions include small displacement, linearly elastic material behavior, prismatic members and idealized supports. Forces are in kN, lengths in m, stiffnesses in kN and kN·m²; reported global Y is positive upward. The benchmarks are not member-strength, buckling, connection or code-design checks.

Jump to Beam examples · Hinges · Truss and axial examples. Each model link opens the free editor; your currently unsaved editor state may be replaced.

Beams

11 reference examples · FramePad uses the direct-stiffness method for analysis.

BENCHMARK 01 · BEAMS

Simply supported beam · UDL

6 m pin–roller beam under 10 kN/m, with a midspan bending moment and deflection check.

4 checks passed
ABq = 10 kN/m
Reference resultMidspan moment: +45 kN·m
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Support reactions
    R_A = R_B = qL/2 = 10 × 6/2 = 30 kN
  2. Maximum sagging moment
    M(L/2) = qL²/8 = 45 kN·m
  3. Maximum deflection
    v(L/2) = −5qL⁴/(384EI) = −10.547 mm

Numerical checks

CheckAnalytical reference
Left reaction+30 kN
Right reaction+30 kN
Midspan moment+45 kN·m
Midspan deflection-10.547 mm

Modeling note: Prismatic Euler–Bernoulli beam. Deflection is downward; FramePad uses positive global Y upward.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

BENCHMARK 02 · BEAMS

Simply supported beam · central point load

6 m pin–roller beam carrying a 24 kN point load at the center.

4 checks passed
ABP = 24 kN
Reference resultMoment at load: +36 kN·m
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Reactions
    R_A = R_B = P/2 = 12 kN
  2. Moment beneath load
    M_max = PL/4 = 24 × 6/4 = 36 kN·m
  3. Midspan deflection
    v(L/2) = −PL³/(48EI) = −6.750 mm

Numerical checks

CheckAnalytical reference
Left reaction+12 kN
Right reaction+12 kN
Moment at load+36 kN·m
Center deflection-6.750 mm

Modeling note: Point load acts vertically downward at x = 3 m.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

BENCHMARK 03 · BEAMS

Simply supported beam · eccentric point load

6 m beam carrying 30 kN at 2 m from the left support; reactions are unequal.

3 checks passed
ABP = 30 kN
Reference resultPeak moment at load: +40 kN·m
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Statics
    a = 2 m; b = 4 m; L = 6 m
  2. Reactions
    R_A = Pb/L = 20 kN; R_B = Pa/L = 10 kN
  3. Moment at load
    M(a) = Pab/L = 40 kN·m

Numerical checks

CheckAnalytical reference
Left reaction+20 kN
Right reaction+10 kN
Peak moment at load+40 kN·m

Modeling note: Maximum moment occurs at the applied load; deflection maximum occurs elsewhere.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

BENCHMARK 04 · BEAMS

Cantilever · end point load

4 m cantilever with a 20 kN downward force at its free end.

3 checks passed
ABP = 20 kN
Reference resultFixed vertical reaction: +20 kN
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Vertical reaction
    R_A = P = 20 kN
  2. Fixed-end moment
    |M_A| = PL = 80 kN·m
  3. Free-end displacement
    v(L) = −PL³/(3EI) = −26.667 mm

Numerical checks

CheckAnalytical reference
Fixed vertical reaction+20 kN
Fixed reaction moment magnitude+80 kN·m
Tip deflection-26.667 mm

Modeling note: Fixed-end reaction moment sign depends on the chosen global axis convention.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

BENCHMARK 05 · BEAMS

Cantilever · uniform load

4 m cantilever carrying 8 kN/m over its full length.

3 checks passed
ABq = 8 kN/m
Reference resultFixed vertical reaction: +32 kN
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Reaction force
    R_A = qL = 32 kN
  2. Fixed reaction moment
    |M_A| = qL²/2 = 64 kN·m
  3. Tip deflection
    v(L) = −qL⁴/(8EI) = −16.000 mm

Numerical checks

CheckAnalytical reference
Fixed vertical reaction+32 kN
Fixed reaction moment magnitude+64 kN·m
Tip deflection-16.000 mm

Modeling note: Constant flexural rigidity over the span.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

BENCHMARK 06 · BEAMS

Cantilever · interior point load

5 m cantilever with a 25 kN downward load 3 m from the fixed end.

3 checks passed
ABP = 25 kN
Reference resultFixed vertical reaction: +25 kN
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Reactions
    R_A = 25 kN; |M_A| = Pa = 75 kN·m
  2. Tip deflection
    v(L) = −Pa²(3L−a)/(6EI) = −28.125 mm

Numerical checks

CheckAnalytical reference
Fixed vertical reaction+25 kN
Fixed reaction moment magnitude+75 kN·m
Tip deflection-28.125 mm

Modeling note: Unloaded segment beyond x = 3 m has zero curvature but rotates and translates.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

BENCHMARK 07 · BEAMS

Fixed-fixed beam · uniform load

6 m built-in beam with 10 kN/m. Tests the restraint moments and smaller midspan deflection.

5 checks passed
ABq = 10 kN/m
Reference resultLeft end moment magnitude: +30 kN·m
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Support reactions
    R_A = R_B = qL/2 = 30 kN
  2. End hogging moments
    |M_A| = |M_B| = qL²/12 = 30 kN·m
  3. Center sagging moment
    M(L/2) = qL²/24 = 15 kN·m
  4. Center deflection
    v(L/2) = −qL⁴/(384EI) = −2.109 mm

Numerical checks

CheckAnalytical reference
Left vertical reaction+30 kN
Right vertical reaction+30 kN
Left end moment magnitude+30 kN·m
Midspan positive moment+15 kN·m
Midspan deflection-2.109 mm

Modeling note: End rotations are prevented by two fixed supports.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

BENCHMARK 08 · BEAMS

Fixed-fixed beam · center load

6 m built-in beam with a centered 24 kN load, checking equal moments and reactions.

5 checks passed
ABP = 24 kN
Reference resultLeft end moment magnitude: +18 kN·m
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Reactions
    R_A = R_B = P/2 = 12 kN
  2. End and midspan moments
    |M_A| = |M_B| = M_mid = PL/8 = 18 kN·m
  3. Deflection
    v(L/2) = −PL³/(192EI) = −1.688 mm

Numerical checks

CheckAnalytical reference
Left vertical reaction+12 kN
Right vertical reaction+12 kN
Left end moment magnitude+18 kN·m
Midspan positive moment+18 kN·m
Midspan deflection-1.688 mm

Modeling note: Continuity and symmetry produce equal hogging and sagging moment magnitudes.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

BENCHMARK 09 · BEAMS

Propped cantilever · uniform load

6 m beam, fixed left and vertically supported right, under 10 kN/m.

4 checks passed
ABq = 10 kN/m
Reference resultLeft end moment magnitude: +45 kN·m
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Compatibility
    Enforce zero vertical displacement at the propped end
  2. Reactions
    R_A = 5qL/8 = 37.5 kN; R_B = 3qL/8 = 22.5 kN
  3. Left end moment
    |M_A| = qL²/8 = 45 kN·m

Numerical checks

CheckAnalytical reference
Left vertical reaction+37.5 kN
Right vertical reaction+22.5 kN
Left end moment magnitude+45 kN·m
Right end moment0 kN·m

Modeling note: Statically indeterminate by one vertical reaction; reference values assume constant EI.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

BENCHMARK 10 · BEAMS

Two-span continuous beam · uniform loads

Two equal 3 m spans with full-length 10 kN/m loads; continuity causes hogging at the center.

4 checks passed
ABCq = 10 kN/mq = 10 kN/m
Reference resultCenter hogging moment: -11.25 kN·m
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Three-moment / symmetry solution
    For equal spans of 3 m and q = 10 kN/m
  2. Support moments
    M_B = −qL²/8 = −11.25 kN·m
  3. Vertical reactions
    R_A = R_C = 3qL/8 = 11.25 kN; R_B = 5qL/4 = 37.5 kN

Numerical checks

CheckAnalytical reference
Left support reaction+11.25 kN
Middle support reaction+37.5 kN
Right support reaction+11.25 kN
Center hogging moment-11.25 kN·m

Modeling note: Central roller support does not release beam-to-beam bending continuity; it restrains only vertical translation.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

BENCHMARK 14 · BEAMS

Simply supported beam · uplift

4 m simply supported beam with an upward UDL of 6 kN/m, testing negative reaction and moment signs.

3 checks passed
ABq = -6 kN/m
Reference resultCenter moment: -12 kN·m
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Resultant applied uplift
    Total upward load = |q|L = 24 kN
  2. Reactions
    R_Ay = R_By = −|q|L/2 = −12 kN
  3. Center moment
    M(L/2) = −|q|L²/8 = −12 kN·m

Numerical checks

CheckAnalytical reference
Left reaction-12 kN
Right reaction-12 kN
Center moment-12 kN·m

Modeling note: Supports capable of resisting uplift are assumed. Standard pin and roller symbols alone do not guarantee this physically.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

Hinges

1 reference examples · FramePad uses the direct-stiffness method for analysis.

BENCHMARK 11 · HINGES

Two spans with central hinge · uniform loads

The same 3 m + 3 m beam as above, but a hinge at the middle support removes continuity.

6 checks passed
ABCq = 10 kN/mq = 10 kN/m
Reference resultHinge moment on left span: 0 kN·m
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Independent simple spans
    Each span behaves as simply supported when the central rotation is released
  2. Reactions
    R_A = R_C = qL/2 = 15 kN; R_B = qL = 30 kN
  3. Moments
    M at central hinge = 0; M_midspan = qL²/8 = 11.25 kN·m

Numerical checks

CheckAnalytical reference
Left support reaction+15 kN
Middle support reaction+30 kN
Right support reaction+15 kN
Hinge moment on left span0 kN·m
Hinge moment on right span0 kN·m
First span midmoment+11.25 kN·m

Modeling note: Compare with the continuous beam case above to isolate the effect of rotational releases.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

Trusses

2 reference examples · FramePad uses the direct-stiffness method for analysis.

BENCHMARK 12 · TRUSSES

Symmetric triangular truss · joint equilibrium

Three axial-only bars, a 40 kN downward apex load, and an exact method-of-joints solution.

5 checks passed
ABCP = 40 kN
Reference resultAB axial force: -31.24 kN
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Geometry
    L_diag = √(3² + 2.5²) = 3.905 m; sinθ = 2.5/L_diag
  2. Reactions
    R_Ay = R_Cy = 20 kN
  3. Sloping members
    N_AB = N_BC = −20/sinθ ≈ −31.241 kN (compression)
  4. Bottom chord
    N_AC = 20(3/2.5) = +24 kN (tension)

Numerical checks

CheckAnalytical reference
Left vertical reaction+20 kN
Right vertical reaction+20 kN
AB axial force-31.24 kN
BC axial force-31.24 kN
AC axial force+24 kN

Modeling note: Pin-jointed idealization. Negative axial force denotes compression in this example.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

BENCHMARK 13 · TRUSSES

Axially loaded bar · extension

A 5 m axial-only bar is held at the left and pulled at the right with 30 kN.

3 checks passed
ABN = 30 kN
Reference resultAxial tension: +30 kN
Open model in FramePadDownload model JSON
View analytical derivation, expected values and assumptions

Reference equations

  1. Force balance
    R_Ax = −P = −30 kN
  2. Axial force
    N = +P = 30 kN (tension)
  3. Axial extension
    Δ = PL/EA = (30 × 5)/(21,000,000) = 0.007143 mm

Numerical checks

CheckAnalytical reference
Left horizontal reaction-30 kN
Axial tension+30 kN
Right axial displacement+0.007 mm

Modeling note: The roller suppresses vertical motion of the right node; no transverse load is applied.

All numerical checks compare the FramePad solver output with the stated reference values; actual numerical tolerances are included in the downloadable test fixture.

How these examples were checked

The reference values were calculated separately from the FEM solver using static force and moment equilibrium, classical Euler–Bernoulli beam solutions, compatibility for indeterminate beams and the method of joints for the truss. The automated checks load each published JSON model through FramePad's own model importer and compare reactions, member forces, moments and selected displacements against fixed numerical references.

The numerical regression tests pass on the included FramePad v3.1 reference source. Because the deployed application can change, run the included test script against your current local application before publishing and again after solver updates. Comparisons use absolute numerical tolerances appropriate for the modeled units.

What these checks do not cover

General member buckling, geometric or material nonlinearity, imposed moments, load combinations, construction stages, support settlement, arbitrary inclined-member load conventions, plate or shell behavior, seismic analysis and all national design-code resistance checks require separate validation. The node hinge tool releases all attached frame-member ends rather than a selected end only.

Responsible use: FramePad is an educational and preliminary analysis tool. Do not rely on these examples as evidence of verified professional design software. Independently confirm models, loads, stability and calculations for safety-critical decisions.