SOME NOTES ON THE MATHEMATICAL THEORY OF A LOADED ELASTIC PLATE RESTING ON AN ELASTIC FOUNDATION

In this paper the approximations necessary for a mathematical treatment of the general problem are first discussed. The main approximation involved is in the relationship between foundation reaction and plate deflexion, and a brief survey is made of the relationships which have been suggested by previous authors. The case in which the loading is static and the foundation reaction is directly proportional to the local plate deflexion is discussed in detail for both infinite and semi-infinite rectangular plates. Fourier integrals are used to solve problems in which all plate-edges are simply supported. The difficulties which arise when other edge conditions exist are discussed, and a method indicated for dealing with clamped edges. The problem of a semi-infinite rectangular plate under a given distribution of shear and bending moment along its free boundary is solved. The problem of a uniformly travelling load on an infinite plate is discussed in detail. It is shown that there exists a certain critical velocity, beyond which stresses and deflexions become infinite. The ratios of the maximum deflexion and bending moment to their static values are expressed as functions of λ, the ratio of the actual velocity to its critical value. It is found that while these deflexions and stresses are greater than their static values, the increase is small unless λ approaches unity.