Circular Plates (`circular-plates`)

Check	IND	US	EU	ISO
Plate deflection	implemented	—	—	—
Plate bending stress	implemented	—	—	—

Purpose

Compute deflection and bending stress in solid circular plates under uniform transverse pressure with simply supported or clamped outer edges. Combines Roark closed-form benchmarks with an axisymmetric finite-difference solver for mesh-controlled accuracy.

Physics & theory

Axisymmetric circular plates under uniform pressure $p$ exhibit radially symmetric deflection $w (r)$ . Flexural rigidity is $D = E t^{3} / [12 (1 - ν^{2})]$ . For a clamped edge ( $w = 0$ , $d w / d r = 0$ at $r = a$ ), peak deflection at center scales as $w_{m a x} \propto p a^{4} / D$ ; for simply supported edges, boundary moments vanish and deflection is larger.

Roark's tabulated coefficients provide quick screening: clamped plate $w_{m a x} = α p a^{4} / D$ with $α \approx 0.0138$ ; simply supported $α \approx 0.171$ . Maximum bending stress at the surface follows $σ \approx β p a^{2} / t^{2}$ with $β \approx 0.75$ for uniform pressure.

The axisymmetric FDM solver discretizes the biharmonic operator on a radial grid, iterating until convergence between applied pressure and plate curvature.

Outer-edge support is modeled as either clamped ( $w = 0$ , $d w / d r = 0$ at $r = a$ ) or simply supported ( $w = 0$ , $M_{r} = 0$ at $r = a$ ). Switching between these edge conditions changes center deflection by an order of magnitude and shifts the location of peak bending stress from center to edge.

The solver requires positive radius, thickness, and flexural rigidity; non-axisymmetric loading and annular plates are outside scope.

Governing equations

D (\frac{d ^{4} w}{d r ^{4}} + \frac{2}{r} \frac{d ^{3} w}{d r ^{3}} - \frac{1}{r ^{2}} \frac{d ^{2} w}{d r ^{2}} + \frac{1}{r ^{3}} \frac{d w}{d r}) = p

D = \frac{E t ^{3}}{12 ( 1 - ν ^{2} )}

σ_{m a x} = \frac{β p a ^{2}}{t ^{2}}

Numerical method

Dual approach: (1) Roark closed-form coefficients for benchmark comparison; (2) axisymmetric Kirchhoff FDM on a radial line with meshSegments (4–64). Jacobi-style iteration (~800 steps) enforces boundary conditions. FEM deflection error vs Roark is reported as femDeflectionErrorPercent.

Inputs

Parameter	Description
`radius`	Outer plate radius $a$
`thickness`	Plate thickness $t$
`modulus`, `poisson`	$E$ , $ν$
`pressure`	Uniform transverse pressure $p$
`boundary`	`clamped` or `simply_supported`
`meshSegments`	Radial FDM segments (default 12)

Outputs

Maximum deflection and stress, flexural rigidity $D$
Roark benchmark values
FEM–Roark error percentage, mesh segment count.

Design codes & checks

Indicative: Plate deflection and bending stress screening
US: ASME BPVC UG-34 flat head context (screening)
EU: EN 13445 flat ends (screening)

Assumptions & limitations

Solid circular plate; annular plates use simplified extensions only.
Thin Kirchhoff plate theory; no transverse shear deformation.
Uniform pressure; no point loads or thermal gradients.
Linear elastic, small deflection.

Verification

CI: circular-plates-indicative-01.json
Engineer sign-off: validation-master-checklist.md

References

Roark, R. J., Young, W. C., & Budynas, R. G. Formulas for Stress and Strain, 8th ed., Case 11.
Timoshenko, S., & Woinowsky-Krieger, S. Theory of Plates and Shells, 2nd ed.
Ugural, A. C. Stresses in Plates and Shells, 4th ed.
ASME BPVC Section VIII, Division 1, UG-34.
Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Circular Plates

Assumptions

Limitations

Engineering checks