Circular Plate under Uniform Pressure (Clamped vs Simply Supported)

stress

A flat circular plate closing off a pressure — a tank head, a submarine porthole, a valve cover, a manhole lid — is one of the most common structural jobs there is, and its answer depends almost entirely on one detail at the rim. Weld or bolt the edge down (clamped) and the plate is stiff and cracks first at the edge. Set the same plate on a ring gasket (simply supported) and it sags about four times as far and cracks first at the center. This page puts both classical cases side by side for a uniformly loaded plate of radius $a$ and thickness $t$ under pressure $q$.

A plate bends in two directions at once, so its stiffness is not a beam’s $EI$ but the flexural rigidity

$D = \frac{E\,t^3}{12\,(1 - \nu^2)}, \qquad \nabla^4 w = \frac{q}{D}$

— the $1/(1-\nu^2)$ is the plate paying for curving both ways at once: the material cannot contract sideways freely, so it comes out stiffer than a strip of the same thickness. The small transverse deflection $w(r)$ then obeys the axisymmetric plate equation $\nabla^4 w = q/D$, and the only thing that separates the two cases is the boundary condition at $r = a$.

Two rims, two very different plates

$\delta_c = \frac{q\,a^4}{64\,D}, \qquad \sigma_c = \frac{3\,q\,a^2}{4\,t^2}\quad\text{(clamped: max at the EDGE)}$

$\delta_{ss} = \frac{5+\nu}{1+\nu}\,\frac{q\,a^4}{64\,D}, \qquad \sigma_{ss} = \frac{3\,(3+\nu)\,q\,a^2}{8\,t^2}\quad\text{(simply supported: max at the CENTER)}$

Clamping the edge does two things: it holds the plate down and it holds it flat ($w(a)=0$ and $w'(a)=0$), so a bending moment builds up at the clamp — that edge moment is the clamped plate’s hot spot. A simple support holds the plate down but lets it tilt, so the moment at the rim is zero and the plate is hottest at its center instead. Divide the two center deflections and everything but the rim condition cancels:

$\frac{\delta_{ss}}{\delta_c} = \frac{5+\nu}{1+\nu} \approx 4.08, \qquad \frac{\sigma_{ss}}{\sigma_c} = \frac{3+\nu}{2} \approx 1.65$

Simply supported is worse on both counts — four times the sag and about two-thirds again the peak stress. The clamp was quietly carrying that much. The widget draws the two dished profiles together: watch the clamped plate meet its wall with a flat tangent while the simply-supported plate tilts into its ring support — the boundary condition is visible in the shape of the curve.

The page where Poisson’s ratio moves a stress

On most stress pages, swapping material moves only stiffness-driven quantities (deflection) and the strength margin; the stresses themselves are pure statics. Not here. Look again at the two peak stresses: $\sigma_c$ contains no material property at all — not $E$, not $\nu$ — so the clamped-edge stress is identical for steel, aluminium, or gray iron. But $\sigma_{ss}$ carries $\nu$ directly, because the simply-supported plate is defined by a moment condition at the rim, and the plate bending moment is where Poisson’s ratio lives:

$M_r = -D\left(w'' + \nu\,\frac{w'}{r}\right), \qquad M_r(a) = 0 \ \text{(simply supported)}$

So of the four plate outputs, swapping material moves three ($\delta_c$, $\delta_{ss}$, $\sigma_{ss}$) and leaves one ($\sigma_c$) exactly where it was. Switch the widget from A36 steel ($\nu = 0.30$) to gray iron ($\nu = 0.26$): $\sigma_{ss}$ shifts, $\sigma_c$ does not budge, and the deflections jump (gray iron’s $E$ is well under half of steel’s, so the plate sags more than twice as far). This is the “Ti deflects more than steel” material moment, sharpened — here a material change reaches all the way into a stress.

A stiffness-and-Poisson material axis (no yield column)

The material enters through exactly two properties, $E$ and $\nu$, and both are genuinely load-bearing — $E$ and $\nu$ set $D$ and hence both deflections, and $\nu$ additionally sets $\sigma_{ss}$. There is deliberately no bound yield strength on this page, because the demonstration pair spans ductile steel (which yields) and brittle gray cast iron (which has no yield point — it fractures). No single “yield strength” column honestly describes both, so the strength check takes an allowable stress you set as a design input, and reports a margin at each plate’s hot spot, $\mathrm{SF}_c = \sigma_{allow}/\sigma_c$ and $\mathrm{SF}_{ss} = \sigma_{allow}/\sigma_{ss}$. The failure note says more about why a brittle flat head is a different design problem from a ductile one.

Where it stops being true

Two global warnings mark the edges of the model, and both are honest about which way they are wrong. Past $t/a \approx 0.1$ the plate is thick enough that transverse shear — which this thin-plate (Kirchhoff) theory neglects — matters, and the real plate deflects a bit more than the widget says. Past a center deflection of about half the thickness, membrane (in-plane stretching) action kicks in, the real plate stiffens, and the linear numbers under-predict the strength — wrong on the safe side, but still wrong. This flat-plate bending calculation is exactly the one the pressure-vessel page warns about for flat heads; every number’s provenance is recorded on the /verification/ page.

Try it

Governing relations

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

How it fails

The widget answers a narrow elastic question — given this circular plate, this pressure, and this rim condition, how far does it deflect and where is it hottest? Flat plates get into trouble in ways that mostly live at the edges of that question.

• A flat head is the expensive way to close a pressure vessel. This page is precisely the calculation the pressure-vessel page warns about: bending stress grows as $q\,a^2/t^2$, so a flat plate large enough to matter needs to be thick. That is why real vessels are capped with dished, torispherical, or hemispherical heads — a curved head carries pressure mostly in membrane tension, at a small fraction of the stress and weight of the flat plate the same pressure would demand. Reach for a flat head only for small diameters, low pressures, or where a flat face is required. The widget shows you the penalty you are paying.

• The real edge is neither perfectly clamped nor perfectly simply supported — the two cases bracket it. A welded-in plate is close to clamped; a plate resting on a gasket with a few bolts is close to simply supported; a bolted flange with a real gasket is somewhere in between, and its true deflection and stress lie between the two curves the widget draws. That is the practical value of showing both: the honest answer for a partially restrained edge is bracketed, not pinpointed, and reading only one case can be unconservative by up to the factor $(3+\nu)/2$ in stress or $(5+\nu)/(1+\nu)$ in deflection.

• A brittle flat head fractures at the tensile hot spot with no warning. The demonstration includes gray cast iron on purpose. Gray iron has no yield point — it is roughly three times stronger in compression than in tension and fails by brittle cleavage from the tensile surface. On a pressure-loaded plate that tensile surface is the loaded-side face at the clamped edge, or the far face at a simply-supported center, and a graphite flake or casting pore there is a ready crack starter. A ductile steel plate yields and redistributes first; a cast-iron one simply cracks. Same geometry, completely different failure mode — which is exactly why this page binds stiffness and Poisson’s ratio but not a single “yield strength”.

• Stress concentrations sit on top of the nominal number. The clamped-edge stress is the nominal bending stress at a clean built-in rim. A real bolted head has bolt holes, a fillet or weld toe at the shell junction, and a gasket seating groove — each a stress raiser that multiplies the local peak on the already-hot edge. The central point-load, edge-moment, and annular-plate cases (each its own family in the references) are out of scope here; this page is the uniform-pressure solid plate.

• Static, small-deflection, axisymmetric, elastic only. No pressure-cycling fatigue (a head that survives one proof pressure can still crack after thousands of cycles at the edge), no large-deflection membrane stiffening past $\delta \approx t/2$ (the widget warns rather than lie), no transverse-shear correction past $t/a \approx 0.1$, no thermal gradients or bolt-preload stresses. Read the page as how the rim condition and the material set the stiffness and the stress, not as a finished head design.

Sources

• Timoshenko, S. P., & Woinowsky-Krieger, S., Theory of Plates and Shells, 2nd ed., McGraw-Hill, 1959 — Ch. 3 (Symmetrical Bending of Circular Plates), §15-16: the symmetrical-bending ODE (§15) and the uniformly loaded circular plate (§16), flexural rigidity D = E t³/(12(1-ν²)), the clamped case (max deflection q a⁴/(64 D) at the center, max stress 3 q a²/(4 t²) at the edge) and the simply-supported case (center deflection (5+ν)/(1+ν)·q a⁴/(64 D), center stress 3(3+ν) q a²/(8 t²)).
• Young, W. C., Budynas, R. G., & Sadegh, A. M., Roark's Formulas for Stress and Strain, 8th ed., McGraw-Hill, 2012 — Table 11.2 (Flat circular plates of constant thickness, uniform load over the entire plate): case 10a (simply supported edge) and case 10b (fixed/clamped edge), used here as an independent oracle for the four closed forms.