3.7: Stress Formula for Plates

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\)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

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