Circular Plate — Deflection, Slope & Stress

Uniform pressure or central point load on a solid circular plate — simply supported or fixed edge

Circular Plate — Deflection, Slope & Stress Distribution

This calculator finds the complete response of a solid circular plate of uniform thickness under axisymmetric loading using classical Kirchhoff thin-plate theory. The governing differential equation for the lateral deflection w(r) is

\nabla^4 w = \frac{q}{D}, \quad D = \frac{Et^3}{12(1-\nu^2)}

where D is the plate rigidity, E the elastic modulus, t the thickness and \nu Poisson's ratio. The axisymmetric operator \nabla^2 = d^2/dr^2 + (1/r)\,d/dr is applied twice. The closed-form solutions are exact for the assumed linear-elastic, thin-plate model.

Four load cases are available:

• Uniform pressure, simply supported (SS): uniform lateral pressure q over the whole plate, edge free to rotate. Maximum deflection at the centre.
• Uniform pressure, fixed edge: uniform pressure q, edge slope is zero. Maximum deflection at centre; maximum moment at the fixed edge.
• Central point load, simply supported (SS): total load W applied over a small circular area of radius r_o at the plate centre. A Westergaard equivalent radius r_o' = \sqrt{1.6r_o^2 + t^2} - 0.675t removes the logarithmic singularity for r_o < 0.5t. Maximum deflection at the centre.
• Central point load, fixed edge: same central load W with a built-in edge. Edge moment M_{ra} = -W/(4\pi) is independent of plate geometry; it is negative (hogging).

The calculator reports the plate rigidity D, key scalar results (y_c, y_{max}, \theta_a, M_c, M_{ra}), the maximum moment M_{max} and its location, the maximum bending stress \sigma_{max} = 6|M_{max}|/t^2, and four radial distribution curves (y(r), \theta(r), M_r(r), M_t(r)) for plotting.

Sign convention: deflection y is positive downward (in the direction of the load). Moments are positive when the lower face of the plate is in tension (sagging).

Model limits: isotropic linear-elastic thin plate, axisymmetric loading, small deflections (y \ll t), uniform thickness, solid plate (no holes). The results are structural response quantities; there is no pass/fail safety check.

Covered cases of this table:

• Solid circular plate of uniform thickness
• Uniform pressure or central point load, with simply supported or fixed edge (4 cases)
• Not included: annular plates, other loads and edge conditions of Table 11.2

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