Keywords: least squares, integral equations, external force, body force, displacement, plate.

The finite element method has been broadly based on energy principles. An exception is the stiffness matrix development for the constant strain triangle where nodal forces were determined directly from static's. The overwhelming majority of the developments have used energy formulations [1]. A rich variety of energy types used in these formulation have included potential energy, complementary energy, and various mixed energy forms. In usual energy methods an inner product of stress and strain terms expressed as differentials are integrated over the volume of the finite element. The mathematical foundation, i.e., completeness and convergence, of the various energy formulations has been a major concern since 1969.

A broader view than structural analysis founded on energy principles is possible provided that a squared functional with a minimum can be constructed. The implementation functionals can be different from the usual energy development. A logical choice is a functional that directly uses the field equations. Here we can work with equilibrium, geometry of deformation, and constitutive laws. A functional that fits these requirements is based on the method of least squares.

The concept that has not been tried concerns the use of integral operators when dealing with the geometry of deformation. Usually, the displacement field is assumed and then differentiated to recover the stain. In the mixed methods independent stress and displacement fields are assumed for integration [2]. Overlooked are the integral equations which relate the strains on the interior to the displacement on the boundary. In beams, these integral equation are often expressed by the moment area theorems. These theorems relate curvatures on the interior to displacement coordinates on the boundary [3]. There is a significant benefit which accrues from the use of the integral equations. The interpolation functions can be in the form of stresses on moment. When converted to strains, these may be integrated to find the exterior displacement.

The resisting force intensities through matrix transformations will be linked to the transverse displacement quantities. The terms when expressed in term of will be subject to a variation for the purpose of minimizing the square error. This will be on the second of variation. The second variation has loads and displacements. Two examples are tried with the method. The first has the following on four sides:

1. zero deflection and free rotation;
2. free deflection and zero rotation;
3. free deflection and zero rotation; and
4. zero deflection and free rotation.

The next has edge conditions which are different. The edges are:

1. zero deflection and zero rotation;
2. free deflection and zero rotation;
3. free deflection and zero rotation; and
4. zero deflection and free rotation.

Reference [4] agrees with these two studies.

1

Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, v. 1, McGraw-Hill, New York, N. Y., 1989, 110-149.

2

Xue, W-M. and Atluri, S.N., "Existence and Stability, and Discrete BB and Rank Conditions, for General Mixed-hybrid Finite Element in Elasticity," Proc. ASME Conf. Hybrid and Mixed Finite Element Methods, Miami Beach, FL., 1985, 91-112.

3

Selna, L.G. and Hakam, M.A., "Least Square Methods in Structural Analysis", Developments in Engeering Computational Technology, Topping, B.H.V. (Editor), Civil-Comp Press, Edinburgh, Scotland, 2000. doi:10.4203/ccp.71.4.1

4

Timoshenko, S.P. and Woinowsky-Krieger, S., Theory of Plates and Shells, Second Edition, McGraw-Hill, New York, N.Y., 1959, 269-277.

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