Stresses in an Elliptical Beam

Transcription

1 Problem: An offset tensile link is shaped to clear an obstruction with a geometry as shown in the figure. The cross section at the critical location is elliptical, with a major axis of 4 in and a minor axis of 2 in. For a load of 20 kip, estimate the stresses at the inner and outer surfaces of the critical section Joseph Shigley and Charles Mischke. Mechanical Engineering Design 5th ed. New York: McGraw Hill, May 2002.

2 Overview Anticipated time to complete this tutorial: 30 minutes Tutorial Overview This tutorial is divided into five parts: 1) Tutorial Basics 2) Preprocessing 3) Solution 4) Post Processing 5) Hand Calculations Audience This tutorial assumes familiarity of ANSYS 8.0; therefore, it does not go into step by step detail. Prerequisites 1) ANSYS 8.0 in house Structural Tutorial 2) Completion of all Basic Machine Design Tutorials 3) Completion of three or more Guided Machine Design Tutorials 4) Completion of Guided Utility Hook Tutorial Objectives 1) Create an elliptical area 2) Create volumes by sweeping areas 3) Use symmetry boundary constraints to simplify the problem 4) Find the inner and outer stresses of a curved beam Outcomes 1) Explore possibilities with the graphical user interface (GUI) 2) Learn how to create and mesh simple geometries 3) Increase efficiency in problem set up and solving speed 2

3 In this tutorial: Instructions appear on the left. Visual aids corresponding to the text appear on the right. Tutorial Basics All commands on the toolbars are labeled. However, only operations applicable to the tutorial are explained. The instructions should be used as follows: Bold > Example: Italics MB1 MB2 MB3 Text in bold are buttons, options, or selections that the user needs to click on > Preprocessor > Element Type > Add/Edit/DeleteFile would mean to follow the options as shown to the right to get you to the Element Types window Text in italics are hints and notes Click on the left mouse button Click on the middle mouse button Click on the right mouse button Some basic ANSYS functions are: To rotate the models use Ctrl and MB3. To zoom use Ctrl and MB2 and move the mouse up and down. To translate the models use Ctrl and MB1. 3

4 Preprocessing 1) Setup the necessary preprocessor parameters. Add an element. Add a material model. Note: Use Solid 185 and assume steel 2) Create a sufficient number of keypoints that represents the geometry of an ellipse. To create the spline: > Preprocessor > Modeling > Create > Lines > Splines > Spline thru KPs Select half of the keypoints at first, then select the other half. Create the elliptical area by arbitrary lines. Add 3 more keypoints to complete half the geometry of the problem. Connect keypoints as shown to the right. Note: Use Arcs by end KPs & Rad and straight line Add a Line fillet where the two lines connect. Note: Use a radius of 10. Use the extrude areas along lines command to create the volume of the link. Before meshing the model, add a hard point in the center of the left edge cross section where the force will be applied 4

5 Solution 3) Mesh the model. If possible, use the volume sweep command. Note: Select Smart size and reduce the fine mesh to 3. Doing this should allow you not to have the error shown below. 4) Apply a symmetric boundary condition. > Solution > Define Loads > Apply > Structural > Displacement > Symmetry B.C. > On Areas Select the area shown to the right in purple. 5) Apply the lb load. 6) Solve the model. 5

6 Post Processing 7) Use the post processing tools to get the inner and outer stress values. > General Postproc > Results Viewer Then select the Von Mises stress. To better read the results, > General Postproc > List Results > Nodal Solution > Stress > Principal For this model, node 1 & 2 represent the outer and inner nodes of the critical section. The inner stress is about kpsi and the outer was about kpsi. The answers are relatively close to the analytical solution (about 3 % error). Try resolving the problem using different meshes and mesh densities to see if you can get solutions that are closer to the analytical ones. 6

7 Hand Calculations The numerical integration was carried out using Simpson s rule. r b br b/r A = ( ) = in 3.4( ) R = = in. 3( ) r n = = in..4( ) 3 The eccentricity e is then e = R rn = = in. For comparison, using 21 ordinates; 2 A = in R = 12in r n = e = in Continuing with the 11 -ordinate analysis, c = = in c i o = = in M = F(2 + 2) = 20000(4) = 80000inlb σ = σ i o = ( ) ( ) ( ) ( )14 = psi = psi 7

8 Hand Calculations Note that A, R, r n were evaluated by numerical integration even though it is possible to I 2 write A= ðab and R=12 by inspection. This is because R= where integral errors are I1 I1 correlated. Similarly r n =. This approach leads to more accurate R and r n and also I3 improves the estimate of e, something to keep in mind when writing computer programs. 8