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Engineering LibreTexts

1.6: Strain-Displacement Relation of Thin Plates

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  • The present course 2.080 is a prerequisite for a more advanced course 2.081 on Plates and Shells. A complete set of lecture notes for 2.081 is available on OpenCourseWare. The interested reader will find there a complete presentation of the theory of moderately large deflection of plates, derived from first principles. Here only a short summary is given.

    1.6.1: Notation

    In the lectures on plates and shells two notations will be used. The formulation and some of the derivation will be easier (and more elegant) by invoking the tensorial notation. Here students should flip briefly to Recitation 1 where the above mathematical manipulations are explained. For the purpose of the solving plate problems, the expanded notation will be used.

    Points on the middle surface of the plate are described by the vector \(\{x_1, x_2\}\) or \(x_{\alpha}\), \(\alpha = 1, 2\) in tensor notation or \(\{x, y\}\) in expanded notation.

    Likewise, the in-plane components of the displacement vector are denoted by \(\{u, v\}\). The vertical component of the displacement vector in the \(z\)-direction is denoted by \(w\).

    1.6.2: Plate versus Beam Theory

    The plate theory requires fewer assumptions and is more self-consistent than the beam theory. For one, there are no complications arising from the concept of the centroidal axis for arbitrarily shaped prismatic beams. The \(z\)-coordinate is measured from the middle plane which is self explanatory. Finally, the flexural/torsional response of non-symmetric and/or thin-walled cross-section beams is not present in plates. The complexity of the plate formulation comes from the two-dimensionality of the problem. The ordinary differential equations in beams are now becoming partial differential equations.