Plates
After completing this chapter, you should be able to:
- Extend Euler-Bernoulli beam theory to plates by defining resultant forces (\(Q_x\), \(Q_y\)) and moments (\(M_{xx}\), \(M_{yy}\), \(M_{xy}\))
- Derive the moment-curvature relations \(M_{xx} = -K(w_{,xx} + \nu w_{,yy})\) with flexural rigidity \(K = Eh^3/[12(1-\nu^2)]\)
- Apply Kirchhoff’s plate equation \(\nabla^4 w = p/K\) to analyze plate deflection problems
Plates
Stress
Kirchhoff plate theory is the straightforward generalization of Euler-Bernoulli beam theory to plates. We abandon the plane situation in which all derivatives in \(y\)-direction vanish. The weak boundary conditions then become \[\begin{align} \label {eq:beamweakforcex} Q_x(x,y) &= \int_h \mathrm{d}z\, \tau_{xz}(x, y, z) \\ \label {eq:beamweakforcey} Q_y(x,y) &= \int_h \mathrm{d}z\, \tau_{yz}(x, y, z) \\ \label {eq:beamweakmomentxx} M_{xx}(x, y) &= \int_h \mathrm{d}z\, z \sigma_{xx}(x, y, z) \\ \label {eq:beamweakmomentyy} M_{yy}(x, y) &= \int_h \mathrm{d}z\, z \sigma_{yy}(x, y, z) \\ \label {eq:beamweakmomentxy} M_{xy}(x, y) &= \int_h \mathrm{d}z\, z \tau_{xy}(x, y, z), \end{align}\] where the integral is over the height \(h\) of the plate. \(Q_x\) and \(Q_y\) are called shear forces, \(M_{xx}\) and \(M_{yy}\) are bending moments and \(M_{xy}\) is the torsional moment.
Note that employing static equilibrium \(\sigma_{ij,j}=0\) we obtain \[\begin{equation} Q_{x,x} + Q_{y,y} = \int_{-h/2}^{h/2} \mathrm{d}z\, \left ( \tau_{zx,x} + \tau_{zy,y} \right ) = -\int_{-h/2}^{h/2} \mathrm{d}z\, \tau_{zz,z} \end{equation}\]but \[\begin{equation} \int_{-h/2}^{h/2} \mathrm{d}z\, \tau_{zz,z} = \tau_{zz}(x, y, h/2) - \tau_{zz}(x, y, -h/2) \equiv p(x,y) \end{equation}\]where \(p(x,y)\) is the pressure on the plate (cf. also the corresponding equation Eq. \(\eqref{eq:beambczz}\) for the beam). Similarly \[\begin{equation} M_{xx,x} + M_{xy,y} = \int_{-h/2}^{h/2} \mathrm{d}z\, z\left ( \tau_{xx,x} + \tau_{xy,y} \right ) = -\int_{-h/2}^{h/2} \mathrm{d}z\, z\tau_{xz,z} \end{equation}\]and \[\begin{equation} \int_{-h/2}^{h/2} \mathrm{d}z\, z\tau_{xz,z} = \left [z\tau_{xz}\right ]_{-h/2}^{h/2} - \int_{-h/2}^{h/2} \mathrm{d}z\, \tau_{xz} = -Q_x. \end{equation}\]where the last equality holds because \(\tau_{xz}(x,y,h/2)=-\tau_{xz}(x,y,-h/2)\). The condition for static equilibrium \(\sigma_{ij,j}=0\) therefore becomes \[\begin{align} \label {eq:plateeq1} Q_{x,x} + Q_{y,y} &=-p(x,y) \\ \label {eq:plateeq2} M_{xx,x} + M_{xy,y}&=Q_x(x,y) \\ \label {eq:plateeq3} M_{xy,x} + M_{yy,y}&=Q_y(x,y) \end{align}\] in the weak form. Note that this can be written in the compact form \(Q_{i,i}=-p\) and \(M_{ij,j}=Q_i\).
As in the Euler-Bernoulli case, we assume that the components \(\sigma_{xx}\), \(\sigma_{yy}\) and \(\tau_{xy}\) vary linearly with \(z\). We can write \[\begin{align} \sigma_{xx}(x, y, z) &= \frac {M_{xx}(x, y)}{I} z \\ \sigma_{yy}(x, y, z) &= \frac {M_{yy}(x, y)}{I} z \\ \tau_{xy}(x, y, z) &= \frac {M_{xy}(x, y)}{I} z \end{align}\] with \(I=\int dz\, z^2=h^3/12\). The remaining components of the stress tensor are obtained from static equilibrium. Static equilibrium yields \[\begin{equation} \tau_{xz,z} = -\frac {z}{I} Q_x \quad \text {and}\quad \tau_{yz,z} = -\frac {z}{I} Q_y \end{equation}\]which can be integrated under the condition \(\tau_{xz}(x, y, h/2)=\tau_{xz}(x, y, -h/2)=0\) to \[\begin{equation} \tau_{xz}(x, y, z) = \frac {Q_x}{2I} \left (\frac {h^2}{4}-z^2\right ) \quad \text {and}\quad \tau_{yz}(x, y, z) = \frac {Q_y}{2I} \left (\frac {h^2}{4}-z^2\right ). \end{equation}\]This is analogous to Eq. \(\eqref{eq:beamstressxz}\) for the beam.
We are finally left with finding an expression for \(\sigma_{zz}\). Again we use static equilibrium to obtain \[\begin{equation} \sigma_{zz,z} = -\tau_{xz,x}-\tau_{yz,y} = \frac {p(x,y)}{2I} \left (\frac {h^2}{4}-z^2\right ). \end{equation}\]Integration under the condition that the loads on top and bottom surface of the plate balance, \(\sigma_{zz}(x,h/2)=-\sigma_{zz}(x,-h/2)\), gives \[\begin{equation} \label {eq:platestresszz} \sigma_{zz}(x, y, z) = \frac {p(x, y)}{2I} \left (\frac {h^2}{4} - \frac {z^2}{3}\right ) z. \end{equation}\]At the top and bottom of the plate we find \(\sigma_{zz}(x,h/2)=-\sigma_{zz}(x,-h/2)=p(x,y)/2\).
Displacements
Now that we know the stress inside the plate, we can again compute the displacements from Hooke’s law. In the full three-dimensional case, Hooke’s law,
\[\begin{align} \label {eq:platehooke1} \varepsilon_{xx} \equiv u_{x,x} &= (\sigma_{xx} - \nu \sigma_{yy} - \nu \sigma_{zz})/E \\ \label {eq:platehooke2} \varepsilon_{yy} \equiv u_{y,y} &= (\sigma_{yy} - \nu \sigma_{xx} - \nu \sigma_{zz})/E \\ \label {eq:platehooke3} 2\varepsilon_{xz} \equiv u_{x,z} + u_{z,x} &= 2(1+\nu )\tau_{xz}/E \\ \label {eq:platehooke4} 2\varepsilon_{yz} \equiv u_{y,z} + u_{z,y} &= 2(1+\nu )\tau_{yz}/E \\ \label {eq:platehooke5} 2\varepsilon_{xy} \equiv u_{x,y} + u_{y,x} &= 2(1+\nu )\tau_{xy}/E, \end{align}\] and taking the derivative of Eq. \(\eqref{eq:platehooke3}\) with respect to \(x\) and of Eq. \(\eqref{eq:platehooke4}\) with respect to \(y\), we obtain \[\begin{align} \label {eq:ux1} u_{x,xz} + u_{z,xx} &= 2(1+\nu )\tau_{xz,x}/E \\ \label {eq:uy1} u_{y,yz} + u_{z,yy} &= 2(1+\nu )\tau_{yz,y}/E \end{align}\] and \[\begin{align} \label {eq:ux2} u_{x,xz} + u_{z,xx}=\partial_z( u_{x,x}) + u_{z,xx}&=u_{z,xx} + (\sigma_{xx,z} - \nu \sigma_{yy,z} - \nu \sigma_{zz,z})/E \\ \label {eq:uy2} u_{y,yz} + u_{z,yy}=\partial_z( u_{y,y}) + u_{z,yy}&=u_{z,yy} + (\sigma_{yy,z} - \nu \sigma_{xx,z} - \nu \sigma_{zz,z})/E. \end{align}\] Combining Eqs. \(\eqref{eq:ux1}\), \(\eqref{eq:ux2}\) and Eqs. \(\eqref{eq:uy1}\), \(\eqref{eq:uy2}\) and noting that \(\sigma_{zz,z}=-\tau_{xz,x}-\tau_{xz,y}\) yields \[\begin{align} u_{z,xx} &= \left [(2+\nu )\tau_{xz,x} - \nu \tau_{yz,y} - \sigma_{xx,z} + \nu \sigma_{yy,z} \right ]/E \\ u_{z,yy} &= \left [(2+\nu )\tau_{yz,y} - \nu \tau_{xz,x} - \sigma_{yy,z} + \nu \sigma_{xx,z}\right ]/E. \end{align}\] We now create linear combination of these expressions such that \(\sigma_{xx,z}=M_{xx}/I\) or \(\sigma_{yy,z}=M_{yy}/I\) drop out,
\[\begin{align} u_{z,xx} + \nu u_{z,yy} &= \left [(2+\nu -\nu ^2)\tau_{xz,x} - \nu (1+\nu )\tau_{yz,y} - (1-\nu ^2)M_{xx}/I \right ]/E \\ u_{z,yy} + \nu u_{z,xx} &= \left [(2+\nu -\nu ^2)\tau_{yz,y} - \nu (1+\nu )\tau_{xz,x} - (1-\nu ^2)M_{yy}/I \right ]/E. \end{align}\] We now only consider the displacement at the surface, \(w(x,y)\equiv u_z(x,y,h/2)\). Since the surfaces are traction free, all terms involving \(\tau_{xz}\) and \(\tau_{yz}\) vanish. Hence \[\begin{align} \label {eq:plateMxx} M_{xx} &= -K (w_{,xx} + \nu w_{,yy}) \\ \label {eq:plateMyy} M_{yy} &= -K (w_{,yy} + \nu w_{,xx}) \end{align}\] with the flexural rigidity \(K=EI/(1-\nu ^2)=Eh^3/[12(1-\nu ^2)]\).
Finally, we are looking for an expression for \(M_{xy}=I\tau_{xy,z}\). We have from Eqs. \(\eqref{eq:platehooke3}\)-\(\eqref{eq:platehooke5}\) \[\begin{equation} \frac {2(1+\nu )}{EI} M_{xy} = u_{x,yz} + u_{y,xz} = \frac {2(1+\nu )}{E} (\tau_{xz,y} + \tau_{yz,x}) - 2u_{z,xy}, \end{equation}\]which yields \[\begin{equation} \label {eq:plateMxy} M_{xy} = -K(1-\nu ) w_{,xy}, \end{equation}\]the desired expression.
We now plug Eqs. \(\eqref{eq:plateMxx}\), \(\eqref{eq:plateMyy}\) and \(\eqref{eq:plateMxy}\) into the equilibrium conditions Eqs. \(\eqref{eq:plateeq2}\) and \(\eqref{eq:plateeq3}\). This yields \[\begin{align} -K(w_{,xxx} + w_{,xyy}) &= Q_x(x, y) \\ -K(w_{,yyy} + w_{,xxy}) &= Q_y(x, y) \\ -K(w_{,xxxx} + 2w_{,xxyy} + w_{,yyyy}) &= -p(x, y). \end{align}\] The last expression is Kirchhoff’s equation, \[\begin{equation} w_{,xxxx} + 2w_{,xxyy} + w_{,yyyy} = \nabla ^2 \nabla ^2 w = \nabla ^4 w = \frac {p}{K}, \end{equation}\]that governs the deformation of plates.
This chapter extended beam theory to 2D plate problems:
- Kirchhoff plate theory: Generalizes Euler-Bernoulli beam theory to two dimensions
- Resultant quantities: Shear forces \(Q_x\), \(Q_y\) and moments \(M_{xx}\), \(M_{yy}\), \(M_{xy}\) are integrals through thickness
- Equilibrium: \(Q_{i,i} = -p\) and \(M_{ij,j} = Q_i\) in weak form
- Stress distributions: \(\sigma_{xx}\), \(\sigma_{yy}\), \(\tau_{xy}\) linear in \(z\); \(\tau_{xz}\), \(\tau_{yz}\) parabolic in \(z\)
- Moment-curvature: \(M_{xx} = -K(w_{,xx} + \nu w_{,yy})\) with flexural rigidity \(K = Eh^3/[12(1-\nu^2)]\)
- Kirchhoff equation: \(\nabla^4 w = p/K\) is the biharmonic governing equation for plate deflection
- Biharmonic operator: \(\nabla^4 = \nabla^2\nabla^2\) appears in both plate bending and fracture mechanics
Plate theory is essential for analyzing thin structural elements like floors, panels, and MEMS devices.