# 2D Stress Analysis (Draft 1, 10/10/06)

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1 2D Stress Analysis (Draft 1, 10/10/06) Plane stress analysis Generally you will be forced to utilize the solid elements in CosmosWorks due to a complicated solid geometry. To learn how to utilize local mesh control for the solid elements it is useful to review some two-dimensional (2D) problems employing the triangular elements. Historically, 2D analytic applications were developed to represent, or bound, some classic solid objects. Those special cases include plane stress analysis, plane strain analysis, axisymmetric analysis, flat plate analysis, and general shell analysis. After completing the following 2D approximation you should go back and solve the much larger 3D version of the problem and verify that you get essentially the same results for both the stresses and deflections. Plane stress analysis is the 2D stress state that is usually covered in undergraduate courses on mechanics of materials. It is based on a thin flat object that is loaded, and supported in a single flat plane. The stresses normal to the plane are zero (but not the strain). There are two normal stresses and one shear stress component at each point (σ x, σ y, and τ). The displacement vector has two translational components (u_x, and u_y). Therefore, any load (point, line, or area) has two corresponding components. The CosmosWorks shell elements can be used for plane stress analysis. However, only their in-plane, or membrane, behavior is utilized. That means that only the elements inplane displacements are active and available to be restrained. To create such a study you need to construct the 2D shape and extrude it with a constant thickness that is small compared to the other two dimensions of the part. Then begin a mid-surface shell study. Before solid elements became easy to generate it was not unusual to model some shapes as 2.5D. That is, they were plane stress in nature but had regions of different constant thickness. This concept can be useful in validating the results of a solid study if you have no analytic approximation to use. Since the mid-surface shells extract their thickness automatically from the solid body you should use a mid-plane extrude when you are building such a part. One use of a plane stress model here is to illustrate the number of elements that are needed through the depth of a region, which is mainly in a state of bending, in order to capture a good approximation of the flexural stresses. Elementary beam theory and 2D elasticity theory both show that the longitudinal normal stress (σ x ) varies linearly through the depth. For pure bending it is tension at one depth extreme, compression at the other, and zero at the center of its depth (also known as the neutral-axis). When the bending is due, in part, to a transverse force then the shear stress (τ) is maximum at the neutral axis and zero at the top and bottom fibers. For a rectangular cross-section the shear stress varies parabolically through the depth. Page 1 of 18. Copyright J.E. Akin. All rights reserved.

5 Figure 4 Since the stresses through the depth are going to be examined here, you should plan ahead and insert some split lines on the front surface to be used to list and/or graph selected stress and deflection components. Here splits were inserted near the interior quarter points (Figure 5) so, including the end lines five graphing sections will be available. Having finished the geometric model the restraints and loads will be applied. Figure 5 Edge restraints and loads Remember that shells defined by selected surfaces must have their restraints and loads applied directly to the edges of the selected surface. First the symmetry and anti-symmetry restraints will be applied. Since the shell mesh will be flat it is easy to use its edges to define directions for loads, or restraints. In Figure 6, the zero horizontal (x) deflection is applied as a symmetry condition on the edge corresponding to the beam vertical centerline, and then as the anti-symmetry condition along the edge of the neutral axis. At the simply supported end it is necessary to assume how that support will be accomplished. Beam theory treats it as a point support, but in 2D or 3D that causes a false infinite stress at the point. Another split line was introduced, in Figure 7, and about one-third of that end was Page 5 of 18. Copyright J.E. Akin. All rights reserved.

6 Figure 6 Figure 7 picked to provide the vertical restraint required. This serves as a reminder that where, and how, parts are restrained is an assumption. So it is wise to investigate more than one such assumption. Software tutorials are intended to illustrate specific features of the software, and usually do not have the space for, or intention of, presenting the best engineering judgment. Immovable restraints are often used in tutorials, but they are unusual in real applications. Rigid body motion restraint Since a general shell element is being used in a plane stress application it still has the ability to translate normal to its plane and to rotate about the in-plane axes (x and y). Those three rigid body motions must also be eliminated in any plane stress analysis. That is done simply Page 6 of 18. Copyright J.E. Akin. All rights reserved.

7 by zeroing the z-translation along the top edge of the beam, as in Figure 8. The combination of all the plane stress supports and the rigid body motion restraint is seen in Figure 9. Figure 8 Figure 9 Moment application as a non-uniform pressure A linear variation of equal and opposite pressures, relative to the neutral axis, can be used to apply a statically equivalent moment to a continuum body that does not have rotational degrees of freedom. It also has the side benefit of matching the normal stress distribution in a beam subjected to a state of pure bending. That usually requires the user to define a local coordinate system at the axis about which the moment acts. In this case, it must be located at the neutral axis of the beam: 1. Select Insert Reference Geometry Coordinate System to open the Coordinate System panel. 2. Right click one end of the neutral axis to set the origin of Coordinate System If the y-axis is vertical pick OK, else pick part edges to orientate the y-axis (Figure 10). Page 7 of 18. Copyright J.E. Akin. All rights reserved.

8 Figure 10 The application of the non-uniform pressure is applied at the front vertical edge at the simple support in the Pressure panel of Figure 11. A unit pressure value is used to set the units and the magnitude is defined by multiplying that value by a non-dimensional polynomial of the spatial coordinates of a point, relative to local Coordinate System 1 defined above. Figure 11 Page 8 of 18. Copyright J.E. Akin. All rights reserved.

9 Mesh and run the study Having completed the restraints and loads the default names in the manager menu have been changed (by slow double clicks) to reflect what they are intended to accomplish (left of Figure 12). Also, the mesh has been designed to be crude so as to illustrate how mesh control is needed in regions of solids subjected to local bending. Post-processing and result validation Figure 12 The maximum vertical deflection and the maximum horizontal fiber stress will be recovered and compared to a beam theory estimate in order to try to validate the FEA study. For this simple geometry and pure bending moment the beam theory results should be much more accurate than is usually true. As stated above, the maximum vertical deflection at the centerline is predicted by beam theory to be v max = 1.36e-3 m. The resultant displacement vectors are seen in Figure 13. They are seen to become vertical at the centerline. A probe Figure 13 Page 9 of 18. Copyright J.E. Akin. All rights reserved.

10 displacement result at the bottom point of the vertical centerline line gives v FEA = 1.36e-3 m, which agrees to three significant figures with the elementary theory. A detail view of the support region (bottom of Figure 13) shows that the displacement vectors close to the restraint are basically rotating about that support. For bending by end couples only, the elementary theory states that the horizontal fiber stress is constant along the length of the beam and is equal to the applied end pressure. That is, the top fiber is predicted to be in compression with a stress value σ x = p max = 3.75e7 N/m 2. That seems to agree with the contour range in Figure 14 and indeed, a stress probe there gives a value of σ x = -3.77e7 N/m 2. Beam theory gives a linear variation, through the depth, from that maximum to zero at the neutral axis. To compare with that, a graph of along the quarter point split line is given in Figure 15. It shows that the seven nodes along the edges of the three quadratic elements have picked up the predicted linear graph quite well. For the next load case of a full span line load the shear stress (that is zero here) will be parabolic and the corresponding graph will be less accurate for such a crude mesh. Figure 14 Figure 15 Page 10 of 18. Copyright J.E. Akin. All rights reserved.

13 Figure 18 Figure 19 simple beam theory value of -3.75e7 N/m 2, being about a 7% difference. Since the mesh is so crude the beam stress is probably the most accurate and the plane stress value will match it as a reasonably fine mesh is introduced. The purpose of the crude mesh is to illustrate the need for mesh control is solids undergoing mainly flexural stresses. To illustrate that point, Figure 20 presents the normal stress and shear stress, through the depth, at the L/4 and L/8 positions. Beam theory says the normal stress is linear while the shear stress is parabolic. Page 13 of 18. Copyright J.E. Akin. All rights reserved.

14 The beam theory shear stress should be zero at the top fiber and, for a rectangular crosssection, has a maximum value at the neutral axis of 1.88e6 N/m 2. The detail graph values in Figure 21 shows a plane stress maximum shear of 1.84e6 N/m 2 and a minimum of 0.3 e6 N/m 2. That is quite good agreement, but it took six quadratic elements through the full depth to capture the shear. The top parabolic segment is approximated by three linear segments. L / 4 L / 8 Figure 20 Figure 21 Page 14 of 18. Copyright J.E. Akin. All rights reserved.

16 quick spot check verifies the expected results. The reaction force components were verified (Figure 23 right) before listing the maximum deflection and fiber stress (Figure 24). Figure 24 What remains to be done is to examine the likely failure criteria that could be applied to this material. They include the Von Mises effective stress, the maximum principle shear stress, and the maximum principle normal stress. The Von Mises contour values are shown at the top of Figure 25, twice the maximum shear stress (the stress intensity) is given in the middle of that figure while the bottom displays the maximum principle stress (P3). Actually, P3 is compressive here but it corresponds to the mirror image tension on the bottom fiber of the actual beam. All three need to be compared to the yield point stress of 2.8e7 N/m 2. The arrow in the top part of the figure highlights where that falls on the color bar. All of the criteria exceed that value, so the part will have to be revised. At this point failure is determined even before a Factor of Safety (FOS) has been assigned. For ductile materials, the common values for the FOS range from 1.3 to 5, or more [1]. Assume a FOS = 3. The current design is a factor of about 3.3 over the yield stress. Combining that with the FOS means that the stresses need to be reduced by about a factor of 10. The cross-sectional moment of inertia, I = t h 3 /12, is proportional to the thickness, t, so doubling the thickness cuts the deflections and stresses in half. Changing the depth, h, is more effective for bending loads. It reduces the deflection by 1/ h 3 and the stresses by a factor of 1/(2 h 2 ). The desired reduction of stresses could be obtained by increasing the depth by a factor of The above discussion assumed that buckling has been eliminated by a buckling eigenanalysis. Since buckling is usually sudden and catastrophic it would require a much higher FOS. Very high precision machines are usually governed by deflection limits. That case is not considered here. Advanced output options There are times when the software will not provide the graphical output you desire. For example, you may wish to graph the plane stress deflection against experimentally measured deflections. The CosmosWorks List Selected feature for any contoured value allows the data on selected edges, split lines, or surfaces to be saved to a file in a comma separated value format (*.csv). Such a file can be opened in an Excel spreadsheet, or Matlab, to be plotted and/or combined with other data. To illustrate the point, when the beam deflection values were contoured the bottom edge was selected to place its deflections in a table. Page 16 of 18. Copyright J.E. Akin. All rights reserved.

17 Figure 25 Then the Report Options display was used to Save those data (node number, deflection value, and x-, y-, z-coordinates) to a named file (Figure 26). Next, the data were opened in Excel, sorted by x-coordinate, and graphed as deflection versus position (Figure 27). You Page 17 of 18. Copyright J.E. Akin. All rights reserved.

18 could add experimental deflection values to the same file and add a second curve for comparison purposes. Figure 26 Figure 27 References [1] R.L. Norton, Machine Design; An Intergrated Approach, Prentice Hall, [2] E.P. Popov, Engineering Mechanics of Solids, Prentice Hall, Page 18 of 18. Copyright J.E. Akin. All rights reserved.

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