On the development of generalized compatible elements in two dimensional elasticity (1987)

The derivations of two generalized compatible elements for use in solving two dimensional elasticity problems are presented. In the first element, generalized parameters for partitioning lines are used as unknowns. In the second, a mixture of both generalized nodal point parameters and generalized parameters for partitioning lines is used. For improvement of the agreement of stresses and strains for adjacent elements along their common boundaries, the method of Lagrangian multipliers is used. These formulations of elements are derived by using compatible displacement functions for interior points, and satisfying the agreement between deflections, stresses, and strains for adjacent elements along the partition lines between the elements in the average sense. The advantages and disadvantages of the method are discussed at the end of the paper. Compatible Displacement Functions in 2-Dimensional Elasticity Using linear combinations of biharmonic polynomials of second to fifth orders, an Airy's stress function of 15 undetermined coefficients is constructed: Because this function has the fewest possible parameters, it is also the smoothest one. The stress components ( a a T ) and strain x' y' xy components (E x' Ey' ?xy ) at any point of an element can be expressed in terms of these 15 unknown coefficients. The integration of E with respect X to x yields the expression for the displacement of the x-component where f(y) is an arbitrary function of y. Similarly, the integration of € with respect to y Y gives the y-component of the displacement function g(x) is another arbitrary function to be determined. f(y) and g(x) are obtained by requiring the shear strains calculated from u and v to match those calculated from @, with the additional requirement that the rigid body motion must be included. 1) Professor of Civil Engineering 2) Professor of Electrical and Computer Engineering The compatible displacement functions for a finite element include 18 unknown parameters, and are given in matrix form as: where L F ~ I = 1-VX. a ~ . X, (Y~+w~)/~, -WY. {A) = a vector of 18 unknown coefficients, and a = 1+p, b = 2+p. c = 3+2p, d = 1+2p and p = Poisson's ratio. ( 7 ) Note that the stress and strain fields calculated by the use of the expressions shown in equations (4) to (7) can satisfy both equilibrium and compatibility conditions in 2-dimensional elasticity. Therefore, they are called compatible displacement functions. Shape functions constructed from these functions using the customary pointwise nodal parameters cannot satisfy compatibility conditions along the partitioning lines between elements and along the boundaries of the domain.' Methods for improving the agreement between elements are presented in the following paragraphs. Generalized Nodal Parameters of Partitioning Lines It is well known that arbitrary functions along a line can be characterized by the moments the function with respect to a reference point. such as the mid-point of the line element. For solutions in 2-dimensional elasticity problems, i is advantageous to use the following quantities defined along the j-th partitioning line or nodal 1 ine : i) Normal and Tangential Displacements and Rotations the displacement component which is normal to the displacement component which is parallel to 5: the rate of change of v with ,respect to E Copvright 'E American In\titute of Aeronautic5 and Astronautics, I n c . , 1987. Al l rights reserved. ii) Strain Components with Respect to c and 7) Axes the axial strain normal to E the axial strain parallel to e the shearing strain with .respect to E and q I iii) Stress Components Related to Tractions on the Boundary the normal stress component along the nodal line the shearing stress component along the nodal line the bending moment around the midpoint of the nodal line Once the direction of the j-th nodal line has been chosen, this is taken to be the local axis with an orthogonal q axis. Let the local unit normal vector (to the E axis) be given by q x and q v in global coordinates. Then the functions defined above can be calculated as follows: Let the mid-point of the j-th partitioning line be the origin of the local axis. Define the linear operators: The m-th moment of the functions listed in equations (8) to (10) can be calculated as The integration can be carried out either analytically or by the use of numerical quadrature over the entire length of the partitioning line. The operator L used in the integration may be any one of L6, LE, or L u . Note that the zeroth moment is the product of the average value of the functions over the line and the length of the line. The use of these moments to establish equations for the displacement field of each element is presented below. Generalized Elements Usinv Nodal Line -