# 3.7: Stress Formula for Plates

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- 21735

\)In the section on beams, it was shown that the profile of axial stress can be determined from the known bending moment \(M\) and axial force \(N\), see Equation (3.4.8). A similar procedure can be developed for plates by comparing Equations (3.6.10-3.6.29) with Equation (3.6.1). The stress-strain curve for the plane stress can be expressed in terms of the middle surface strain tensor \(\epsilon_{\alpha \beta}^{\circ}\) and curvature tensor \(\kappa_{\alpha \beta}\) by combining Equations (3.6.1) and (3.6.5).

\[\sigma_{\alpha \beta} = \frac{E}{1 − \nu^2} [(1 − \nu)\epsilon_{\alpha \beta}^{\circ} + \nu\epsilon_{\gamma \gamma}^{\circ}\delta_{\alpha \beta}] \\ + \frac{E}{1 − \nu^2} [(1 − \nu)\kappa_{\alpha \beta} + \nu\kappa_{\gamma \gamma}\delta_{\alpha \beta}] z \]

From the moment-curvature relation, Equation (3.6.10):

\[(1 − \nu)\kappa_{\alpha \beta} + \nu\kappa_{\gamma \gamma}\delta_{\alpha \beta} = \frac{M_{\alpha \beta}}{D}\]

Similarly, from Equation (3.6.24)

\[(1 − \nu)\epsilon_{\alpha \beta}^{\circ} + \nu\epsilon_{\gamma \gamma}^{\circ}\delta_{\alpha \beta} = \frac{N_{\alpha \beta}}{C}\]

where \(D = \frac{Eh^3}{12(1 − \nu^2)}\) is the bending rigidity, and \(C = \frac{Eh}{1 − \nu^2}\) is the axial rigidity of the plate.

From the above system, one gets

\[\sigma_{\alpha \beta} = \frac{Ez}{1 − \nu^2} \frac{M_{\alpha \beta}}{D} + \frac{E}{1 − \nu^2} \frac{N_{\alpha \beta}}{C}\]

or using the definitions of \(D\) and \(C\)

\[\sigma_{\alpha \beta} = \frac{N_{\alpha \beta}}{h} + \frac{zM_{\alpha \beta}}{h^3/12}\]

The above equation is dimensionally correct, because both \(N_{\alpha \beta}\) and \(M_{\alpha \beta}\) are respective quantities per unit length. In particular stress in the case of cylindrical bending is

\[\sigma_{xx} = \frac{N_{xx}}{h} + \frac{zM_{xx}}{h^3/12}\]

Multiplying both the numerators and denominators of the two terms above by \(b\) yields

\[\sigma_{xx} = \frac{N_{xx}b}{hb} + \frac{zM_{xx}b}{bh^3/12}\]

Now, observing that \(N_{xx}b = N\) is the beam axial force, \(bM_{xx} = M\) is the beam bending moment, \(hb = A\) is the cross-section of the rectangular section beam, and \(\frac{bh^3}{12}\) is the moment of inertia, the familiar beam stress formula is obtained

\[\sigma = \frac{N}{A} + \frac{Mz}{I}\]