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As shown in the figure provided, a Uniform load acts upon a rectangular plate with edges at Y = 0 and Y = b which are simply supported.

Rectangular Plate Simply supported on edges with Uniform Acting Load

Using this form of the Plate Equation:

$\frac{\partial^4 \omega}{\partial x^4} + 2\frac{\partial^4 \omega}{\partial x^2\partial y^2}+\frac{\partial^4\omega }{\partial y^4} = \frac{P}{D}$

Where P is the Uniform Load and,

$ D = \frac{Et^3}{12(1-v^2)}$

How do you determine an equation that describes the Deflection of the plate?

Also, how is load 'P' put into the equation?

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    $\begingroup$ If the edges are supported as shown the deflections only change with $y$, not with $x$, so this is the same as a beam. $\endgroup$ – alephzero Apr 16 at 12:39
  • $\begingroup$ Do you need to derive the equation? If not there are many formulas available Roark's Formulas of Stress and Strain have lots of equations for plates. $\endgroup$ – ShadowMan Apr 16 at 17:20
  • $\begingroup$ See researchgate.net/publication/… for various solutions of plate problems. Will give you an idea how these equations are solved. $\endgroup$ – Biswajit Banerjee Apr 17 at 0:42
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Analyses of one way slabs deflections are complicated. Boundary conditions and even the radius of curvature affect the behavior of the slab and contribution of twisting of hypothetical strip sections to carrying moment.

Many codes adopt using effective width for design of strength and deflection of short, wide slabs. for example for a ratio of support b to span a of 1.6 $(effective\ width)/a\quad is\ 1.519\ and\ E= E/(1-v^2)$, (in pp:170 Roark’s Formulas for Stress and Strain by YOUNG and BUDYNAS 7th ed).

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