Plate Bending

Plate Bending (plates)

Purpose

Analyze bending of thin rectangular plates under uniform pressure or point loads with various edge boundary conditions. Computes maximum deflection, bending moments, and membrane/bending stresses for flat plate components in machinery housings, panels, and structural decks.

Physics & theory

Kirchhoff–Love plate theory extends beam bending to two dimensions. For a thin plate of thickness $t$, flexural rigidity is $D=E{t}^3/[12(1-{\nu }^2)]$. The biharmonic equation $D{\nabla }^{4}w=p$ governs out-of-plane deflection $w(x,y)$ under transverse pressure $p$.

Bending moments relate to curvature: $M_x=-D\left(\frac{{\partial }^{2}w}{\partial x^2}+\nu \frac{{\partial }^{2}w}{\partial y^2}\right)$. Maximum stress at the surface is $\sigma =6M/t^2$ for pure bending. Edge conditions — simply supported (SS), clamped (C), or free — strongly influence peak deflection and stress concentration at corners.

For rectangular plates, Navier or Levy series solutions exist for simply supported edges; the solver uses a finite-element discretization on a rectangular mesh for general boundary mixes.

Each edge can be simply supported (SS), clamped (C), or free, independently on all four sides. Mixed edge conditions change moment distribution and peak deflection significantly compared with fully simply supported plates.

Transverse pressure and point loads superpose linearly in elastic analysis. The solver validates positive plate dimensions and flexural rigidity before meshing.

Governing equations

Numerical method

2D plate FEM on a structured rectangular mesh (femSolver). Mindlin–Reissner or Kirchhoff plate elements assemble stiffness from $D$ and mesh geometry. Transverse loads are applied as consistent nodal forces. The linear system yields nodal deflections; moments and stresses are recovered by differentiation of shape functions.

Inputs

Outputs

• Deflection field $w(x,y)$, maximum deflection, bending moments $M_x
• M_y$, maximum bending stress, utilization vs allowable stress and deflection limits.

Design codes & checks

• Indicative: Plate bending stress and deflection screening
• US: ASME BPVC Section VIII, Div. 1 flat plate context (screening)
• EU: EN 13445 flat ends and plates (screening)

Assumptions & limitations

• Thin plate theory ($t/L<1/20$ typically); thick-plate shear deformation not included.
• Linear elastic, small deflection.
• Flat plate only; no stiffeners or large membrane stretching.
• Fewer centralized validation benchmarks than beam/column modules.

References

1. Timoshenko, S., & Woinowsky-Krieger, S. Theory of Plates and Shells, 2nd ed. McGraw-Hill.
2. Roark, R. J., Young, W. C., & Budynas, R. G. Formulas for Stress and Strain, 8th ed.
3. Ugural, A. C. Stresses in Plates and Shells, 4th ed. CRC Press.
4. ASME BPVC Section VIII, Division 1 (flat plate design rules).
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.