# 3.7: Stress Formula for Plates

\)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

$\sigma = \frac{N}{A} + \frac{Mz}{I}$