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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
    Commented Apr 16, 2019 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
    Commented Apr 16, 2019 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
    Commented Apr 17, 2019 at 0:42
  • $\begingroup$ Here is a PowerPoint slide that will help you in this regard. Note, if the length and width are much larger than the thickness (a/h>8), this is a "plate", though sometimes can be simplified as a one-way beam. However, as the a/h ratio getting larger, it becomes a thin plate or membrane, which behaves very differently than the beam. slideshare.net/vaignan/… $\endgroup$
    – r13
    Commented Oct 4, 2021 at 15:10
  • $\begingroup$ If you are getting so technical that the simple beam theory is no good, then I'd point out that as drawn you will get membrane stiffening effects due to the axial constraint. $\endgroup$
    – Greg Locock
    Commented Jun 26, 2023 at 23:15

3 Answers 3

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This Plate is simply supported. You may find the deflection of the plate by using simple beam theory in this particular case. If this doesn't work for you, you may also check Timoshenko theory of plates and shells.

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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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Roak's Formulas suggest using the beam equation with some modifications. It states:

11.4 Bimetallic Plates

A very wide beam of rectangular cross section can be treated as a beam if E is replaced by $\dfrac {E}{1-\nu^2}$ and $I$ by $t^3/12$ (see Sec. 8.11). It can also be treated as a plate with two opposite edges free as shown in Figs. 8.16 and 11.2. For details see Ref. 88. To use the beam equations in Tables 8.1, 8.5, 8.6, 8.8, and 8.9 for plates like that shown in Fig. 11.2 with two opposite edges free, the loadings must be uniformly distributed across the plate parallel to side b as shown. At every position in such a plate, except close to the free edges a, there will be bending moments $Mz = \nu Mx$. If the plate is isotropic and homogeneous, and in the absence of any in-plane loading, there will be no change in length of any line parallel to side b.

Table 8.1, Case 2e: enter image description here

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