Combined Loading

Combined Loading (combined-loading) — beta

Purpose

Evaluate combined axial, bending, torsion, and shear stresses in a rectangular cross-section and compute von Mises equivalent stress and safety factor. Used for quick screening of machine elements and structural members under multiaxial loading without full 3D FEA.

Physics & theory

Real components rarely experience a single stress mode. Axial force $F$ produces uniform normal stress ${\sigma }_a=F/A$. Bending moment $M$ creates linear normal stress ${\sigma }_b=Mc/I$. Torque $T$ generates torsional shear ${\tau }_t=Tc/J$ for a rectangular section using the thin-wall approximation $J\approx w{h}^3/3$. Direct shear from transverse force adds ${\tau }_v=V/A$.

Normal stresses from axial and bending load superpose: $\sigma ={\sigma }_a+{\sigma }_b$. For ductile materials under combined normal and shear stress, the von Mises (distortion energy) criterion gives equivalent stress ${\sigma }_{\mathrm{vm}}=\sqrt{{\sigma }^2+3{\tau }^2}$. Safety factor is $SF={\sigma }_y/{\sigma }_{\mathrm{vm}}$.

Stress components are evaluated at the section centroid for a prismatic rectangular cross-section. The module assumes elastic behavior and does not model local buckling, stress concentrations, or warping restraint — use dedicated beam or shell analysis when those effects govern.

Inputs must specify positive width, height, and material yield strength; zero-area sections are rejected at validation.

Governing equations

Numerical method

Closed-form evaluation: section properties $A$, $I_{xx}$, and $J$ are computed from rectangular width and height. Individual stress components are calculated algebraically; von Mises stress and safety factor follow directly. Design status flags safe, warning, or critical based on threshold ratios (SF ≥ 2 safe, ≥ 1.25 warning).

Inputs

Outputs

• Section properties $A$, $I_{xx}$, $J$
• stress components
• von Mises stress
• safety factor
• design status.

Design codes & checks

• Indicative: Von Mises combined stress
• US: AISC 360-22 Chapter H (combined forces)
• EU: EN 1993-1-1 Clause 6.2.1 equivalent stress
• ISO: ISO 10828 equivalent stress methods

Assumptions & limitations

• Solid rectangular section; not I-beams, tubes, or arbitrary profiles.
• Elastic linear superposition; no buckling or local instability.
• Torsion uses rectangular approximation; thin-wall or circular sections need dedicated checks.
• Shear stress from transverse force is averaged over area (not parabolic distribution).

Verification

• CI: combined-loading-indicative-01.json
• Engineer sign-off: validation-master-checklist.md

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed. McGraw-Hill.
2. Gere, J. M., & Goodno, B. J. Mechanics of Materials, 9th ed.
3. AISC. Specification for Structural Steel Buildings (ANSI/AISC 360-22), Chapter H.
4. EN 1993-1-1:2005. Eurocode 3 — Clause 6.2.
5. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.