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The Kirchhoff assumption in thin elastic plates results in a biharmonic equation for the lateral deflection and a C 1 deflection field is therefore required in the finite element method for their approximate solution. By considering the thin plate as a three dimensional solid and by discarding the Kirchhoff assumption, the continuity requirement for the displacements is reduced to C 0 . The stiffness matrix produced in this way becomes, however, violently ill-conditioned as the thickness t of the structure is reduced. It is shown here that the factor 1/t 2 causing this ill-conditioning can be removed from the stiffness matrix and consequently from its condition number by relating the thickness t to the diameter of the element h, without losing the rate of convergence provided by the degree of the shape functions inside the element. This is used here to construct a well-conditioned 9-degrees-of-freedom plate bending element which is only C 0 but which converges quadratically to the C 1 solution (Kirchhoff solution) of thin plates. Addition of shear to C 1 elements is also considered.

The master stiffness matrix is derived from the biharmonic plate equation. The application of Galerkin's method to the biharmonic plate equation results in a system of algebraic equations. These equations contain area integrals which must be evaluated over the plate domain. It is shown that by considering the domain of these integrals as a collection of polygonal sub-domains and approximating the displacement function with polynomial spline functions over each sub-domain, the original set of simultaneous algebraic equations can be approximated by a new set, the coefficient matrix of which is identical to the master stiffness matrix of the Finite Element Method. Criteria are postulated for best plate elements and development of an 18 degree-of-freedom plate element, conforming with these criteria, is outlined.

Two methods are described to obtain the solution of the transient thermoelastic stress distribution in a homogeneous spherical fuel element. The first is the classical approach involving the simultaneous solution of the displacement potential equation ( ∇ 2 χ = (1 + μ 1 − μ)αΔT ) for the particular integral and the biharmonic stress function equation (∇4 ψ = 0) for the complementary solution. In this method the solution for the particular integral produces a stress distribution which does not satisfy the real boundary conditions. However a correction is obtained using the complementary solution so that the sum of the two solutions satisfy the boundary conditions. It is shown that the classical solution leads to great computational difficulties even for the case where the temperature is a function only of the radius. As an alternative method it is shown that the complete solution satisfies the displacement equation ∇ 2 χ = (1 + μ 1 − μ)αΔT + B (where B is a constant). The problem simplifies to the solution of Poisson's equation. This second solution, which has been called the direct displacement method, is much easier to program and takes up much less computer storage. In order to check the program accuracy and limitations, the temperature distribution is initially assumed to be a function of radius only; closed form solutions are available for this case for steady state conditions. Further results are obtained for the transient case of a heated sphere where the temperature is a function of R and θ, axial symmetry being assumed about a vertical axis. Typical results indicate that maximum stresses occur when steady state temperatures are reached and occur near the center of the sphere rather than at the surface. A third technique which has been investigated is the finite element method assuming a linear displacement variation. However the effort involved is much greater than that associated with the solution of Poisson's equation for the regular system used. Work is progressing on a transient 3-dimensional solution.

Publisher Summary This chapter discusses finite elements with harmonic interpolation functions. The basis of all finite element solutions of partial differential equations must be the satisfaction of three types of condition for best fit, namely, in the element interior on the interfaces, and on the external boundary of the entire region considered. It is pointed out that these conditions can be satisfied exactly, or in the mean, and that they are independent. Convergence of a solution may be obtained either by using the same type of element and increasing the number of elements toward infinity, while the size of each element tends to zero, or by keeping the division into elements unchanged and letting the number of degrees of freedom in each individual element increase toward infinity. Many engineering problems involve the solution of Laplace's equation. For this, if polynomial solutions are selected as interpolation functions in the elements, the condition in the interior is satisfied exactly. The energy integral method can no longer be used, but an alternative functional is proposed that consists solely of surface integrals. The method offers a possibility of satisfying both natural and essential boundary conditions for both the unknown function and its derivative. The chapter discusses the possibilities of solving biharmonic problems and also a nonlinear system as in the elasto-plastic problem.