Disc Under Bi-axial Tension

24 unknowns are used in all cases. Results are shown along a radial line of y=0 and along the arc from -15 degrees to +15 degrees on the boundary (radius =1). Linear Lagrange polynomials (non-conforming) and cubic Hermite polynomials (conforming) demostrate several facets of the study.

Results

• Linear Lagrange interpolation, stresses along the radial line,displacement boundary conditions

  Results show that the solution by direct method is inbetween the two indirect methods. The errors in the center of the disc are: 0.04% for indirect with force singularity; 0.8% with indirect displacement singularity; and 0.31% with the direct method.

• Linear Lagrange interpolation, stresses along the arc on the boundary, displacement boundary conditions.

  Results show that there is a spike at each element end. The spike is most severe for indirect displacement discontinuity that has a 2nd order singularity in the stress expression. Spike is smallest for indirect with force singularity which has only a first order singularity. The direct method also shows a severe spike but its magnitude is inbetween the two indirect method. Boundedness of stresses with second order singularity require that the deivative of the function should be continuous at each element end, which is a condition not satisfied by Lagrange equations. The errors along the arc vary for : Indirect Force Singularity: 0.43% ; Indirect Displacement singularity 37.7% ; Direct BEM: 13.5%

• Linear Lagrange interpolation, stresses along the arcs that move inward from the boundary, displacement boundary conditions.

  Results show that the effect of spikes propogate into the body. The error decreases as one moves inward but a permenant error is intorduced into the solution by use of non-conforming elements.

• Cubic Hermite interpolation, stresses along the radial line, Displacement Boundary Conditions

  The error in the center for inderect force singularity is 0.005% and for indirect displacement singularity the error is 0.006%. Though the accuracy for indirect force singularity improved by order of magnitude, the accuracy for indirect displacement improved by 3 orders of magnitude. Note the number of unknowns are the same as in linear representation and the mesh is same because the Hermite element length was taken as twice that of Linear Lagrange element. No results are given for direct BEM due to programing difficulties related to extacting slope information from given displacement values at collocation points. This problem will be rectified in the near future.

• Cubic Hermite interpolation, stresses along the arc on boundary, displacement boundary conditions.

  Results show that the spikes at element end have diapppeared if the graph scales are kept the same. The error at element end for indirect force singularity is 0.025% and for indirect displacement singularity the error is 0.006