Structural & Thermal Analysis Using the ANSYS Release 12.1 Environment
Kent L. Lawrence
Mechanical and Aerospace Engineering
University of Texas at Arlington
Schroff Development Corporation
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Plane stress and plane strain problems are an important subclass of general threedimensional problems. The tutorials in this lesson demonstrate: ♦Solving planar stress concentration problems.
♦Evaluating potential inaccuracies in the solutions.
♦Using the various ANSYS 2D element formulations.
It is possible for an object such as the one on the cover of this book to have six components of stress when subjected to arbitrary three-dimensional loadings. When referenced to a Cartesian coordinate system these components of stress are: Normal Stresses
σx, σy, σz
τxy, τyz, τzx
Figure 2-1 Stresses in 3 dimensions.
In general, the analysis of such objects requires three-dimensional modeling as discussed in Lesson 4. However, two-dimensional models are often easier to develop, easier to solve and can be employed in many situations if they can accurately represent the behavior of the object under loading.
A state of Plane Stress exists in a thin object loaded in the plane of its largest dimensions. Let the X-Y plane be the plane of analysis. The non-zero stresses σx, σy, and τxy lie in the X - Y plane and do not vary in the Z direction. Further, the other stresses (σz,τyz , and τzx ) are all zero for this kind of geometry and loading. A thin beam loaded in its plane and a spur gear tooth are good examples of plane stress problems. ANSYS provides a 6-node planar triangular element along with 4-node and 8-node quadrilateral elements for use in the development of plane stress models. We will use both triangles and quads in solution of the example problems that follow. 2-3 PLATE WITH CENTRAL HOLE
To start off, let’s solve a problem with a known solution so that we can check our computed results as well as our understanding of the FEM process. The problem is that of a tensile-loaded thin plate with a central hole as shown in Figure 2-2.
Figure 2-2 Plate with central hole.
The 1.0 m x 0.4 m plate has a thickness of 0.01 m, and a central hole 0.2 m in diameter. It is made of steel with material properties; elastic modulus, E = 2.07 x 1011 N/m2 and Poisson’s ratio, ν = 0.29. We apply a horizontal tensile loading in the form of a pressure p = -1.0 N/m2 along the vertical edges of the plate. Because holes are necessary for fasteners such as bolts, rivets, etc, the need to know stresses and deformations near them occurs very often and has received a great deal of study. The results of these studies are widely published, and we can look up the stress concentration factor for the case shown above. Before the advent of suitable computation methods, the effect of most complex stress concentration geometries had to be evaluated experimentally, and many available charts were developed from experimental results. The uniform, homogeneous plate above is symmetric about horizontal axes in both geometry and loading. This means that the state of stress and deformation below a
Plane Stress / Plane Strain
horizontal centerline is a mirror image of that above the centerline, and likewise for a vertical centerline. We can take advantage of the symmetry and, by applying the correct boundary conditions, use only a quarter of the plate for the finite element model. For small problems using symmetry may not be too important; for large problems it can save modeling and solution efforts by eliminating one-half or a quarter or more of the work. Place the origin of X-Y coordinates at the center of the hole. If we pull on both ends of the plate, points on...