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6.1: Beam Deflection Equation

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  • The three group of equations for the plate bending problem, formulated in Chapter 1, 2 and 3 are:

    \[\text{ Geometry } \quad \kappa_{\alpha \beta} = −w_{,\alpha \beta} \label{7.1}\]

    \[\text{ Equilibrium } \quad M_{\alpha \beta,\alpha \beta} + p = 0 \label{7.2}\]

    \[\text{ Elasticity law } \quad M_{\alpha \beta} = D[(1 − \nu)\kappa_{\alpha \beta} + \nu\kappa_{\gamma\gamma}\delta_{\alpha \beta}] \label{7.3}\]

    Eliminating \(\kappa_{\alpha \beta}\) between Equations \ref{7.1} and \ref{7.2}

    \[M_{\alpha \beta} = D[(1 − \nu)w_{,\alpha \beta} + \nu w_{,\gamma\gamma}\delta_{\alpha \beta}] \label{7.4}\]

    and substituting the result into Equation \ref{7.3} gives

    \[D[(1 − \nu)w_{,\alpha \beta} + \nu w_{,\gamma\gamma}\delta_{\alpha \beta}]_{, \alpha \beta} + p = 0 \]

    The second term in the brackets is non-zero only when \(\alpha = \beta\). Therefore Equation \ref{7.4} transforms to

    \[Dw_{,\alpha\alpha \beta\beta}[−1 + \nu − \nu] + p = 0 \label{7.6}\]

    or finally

    \[Dw_{,\alpha\alpha \beta\beta} = p\]

    Introducing the definition of Laplacian \(\nabla^2\) and bi-Laplacian \(\nabla^4\) in the rectangular coordinate system,

    \[\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}, \nabla^4 = \nabla^2\nabla^2\]

    an alternative form of Equation \ref{7.6} is

    \[D\nabla^4w = p \label{7.9}\]

    This is a linear inhomogeneous differential equation of the fourth order. The boundary conditions in the local coordinate system were given by Equations (2.8.4 - 2.8.7).

    A separate set of equations must be stetted for the in-plane response of the plate

    \[\text{ Geometry } \quad \epsilon_{\alpha \beta}^{\circ} = \frac{1}{2}(u_{\alpha , \beta} + u_{\beta , \alpha})\]

    \[\text{ Equilibrium } \quad N_{\alpha \beta , \beta} = 0\]

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

    Eliminating the strains \(\epsilon_{\alpha \beta}^{\circ}\) and membrane force \(N_{\alpha \beta}\) between the above system, one gets two coupled partial differential equations of the second order for \(u_{\alpha} (u_1, u_2)\).

    \[(1 − \nu)u_{\alpha , \beta \beta} + (1 + \nu)u_{\beta , \alpha \beta} = 0 \]

    Such system is seldom solved, because in practical application constant membrane forces are considered.

    In either case the in-plane and out-of-plane response of plates is uncoupled in the classical, infinitesimal bending theory of plates. These two system are coupled through the finite rotation term \(N_{\alpha \beta}w_{,\alpha \beta}\). The extended governing equation in the theory of moderately large deflection is

    \[D\nabla^4w + N_{\alpha \beta}w_{,\alpha \beta} = 0 \]

    The above equation will be re-derived and solved for few typical loading cases in Chapter 9. The analysis of the differential equation \ref{7.9} in the classical bending theory of plates along with exemplary solutions can be found in the lecture notes of the course 2.081 plates and shells. In this section we will look into the bending problem of circular plates, which is governed by the linear ordinary differential equation.