Products: ABAQUS/Standard ABAQUS/Design
The design sensitivity analysis capability in ABAQUS is illustrated for a uniaxially loaded plate with a hole. In particular, the sensitivity of the stress concentration to a change in the shape of the hole from circular to elliptical is studied. Since an analytical solution exists, the results are easily verified.
Taking advantage of symmetry, a quarter of a linear elastic square plate with a circular hole is modeled with CPS4 elements. The radius of the hole is 10 units. To simulate an infinite sheet, the plate width is set to 40 times the radius of the hole. A uniaxial tensile pressure P equal to 100 units is applied to the plate in the 2-direction during a linear static step in ABAQUS/Standard. The finite element model is shown in Figure 1.17.2–1. The linear static step is followed by a static perturbation step where the load is perturbed by 50 units. Since this problem is linear, the perturbation should produce a sensitivity that is scaled by half.
We wish to study the effect of changing the shape of the hole from circular to elliptical on the maximum absolute stress, 
; therefore, the nodal coordinates need to be related to the shape of the hole. Based on experience, the effect of a perturbation in the shape of the hole on the nodal coordinates lying outside the shaded region (shown in Figure 1.17.2–1) can be neglected. The circular hole can be regarded as a special case of an ellipse with its major axis lying along the x-axis. Let a and b represent the semi-major and semi-minor axes, respectively; and consider a point with coordinates 
lying on a circle concentric to the hole. The coordinates of this point can be parameterized as 
and 
, where 
and 
are constants and 
represents the angle (measured from the positive x-axis) of the position vector to the point from the center of the circle. If the point lies on the boundary of the hole, 
. Positive values of and move the point into the interior of the plate. The dependence of the nodal coordinates on the parameter a is specified by providing the gradients of x and y with respect to a on the *PARAMETER SHAPE VARIATION option. The gradients are given as
The maximum absolute stress, , in an infinite plate with an elliptical hole subjected to uniaxial tension 
perpendicular to the major axis is given by 
and is the 22-component at the end of the major axis of the ellipse (point M in Figure 1.17.2–2). The contour of 
in the vicinity of the hole is shown in Figure 1.17.2–3 (shaded elements in Figure 1.17.2–1). The finite element model gives a maximum stress of 299.3 units, which is close to the expected value of 300 units.
The sensitivity of at M with respect to a is given by 
. The value of 
obtained from ABAQUS/Design is 19.99 units and compares well with the analytical result of 20 units. Figure 1.17.2–4 shows the contours of 
near the hole. As expected, is most sensitive to the variation in the semi-major axis at points nearest to the region of stress concentration. The value of in the perturbation step is 9.995 units, exactly half the value in the linear static step as expected.
near the circular hole.