Lab#2 Mohr s Circle and state of stress By C. Daley

Transcription

1 Engineering Ship Structures I Lab#2 Mohr s Circle and state of stress By C. Daley Overview Consider a steel plate subject to some type of in-plane loading, for example as in the case in lab#1. Any freebody taken from the web would have some form of normal and tangential forces along the edges. For most cases the edge stresses will not be constant. However, if we consider a very small block, and assume that the material is a smooth continuum, the limit stresses at a point will be constant on each edge. General free body with varying stresses For very small free-bodies the stresses are uniform, where all points in the bodies have the same set of stresses. In such cases, there are different (but constant) stresses on every plane cut through the body. The Mohr s circle is a graphical way of representing the stresses on all planes Lab#1: Introduction to ANSYS, page 1

2 The figure below illustrates how a Mohr s circle can help determine the stresses. ANSYS Model #2 biaxial stress block Step 1: describe and sketch the problem: In this lab we will construct a model of a 100mm x 100mm x 20 mm block of steel. A vertical stress of 20 Pa (compression) and horizontal stress of 10 Pa (tension) will be applied. There is no applied shear stress, which means that we have applied the principal stresses. A principal stress is one on a plane that has no shear. There are always two planes (at right angles to each other, that have no shear). If we model this correctly, every point in the body will be in exactly the same state of stress. In such a case, every plane will have a uniform normal stress and a uniform shear stress. Also, any two parallel planes will have the same stresses. And all stresses will be able to be determined by a single mohr s circle. We wish to find the stresses on an inclined plane through the body. To do this we need to create two bodies that part of one part Lab#1: Introduction to ANSYS, page 2

3 Step 2: estimate expected results (analytically): The bar has the following properties: Max principal stress: sy = 20 MPa (compression) Min principal stress: sx = -10MPa (tension) Mean stress = 5 MPa Circle radius = 15 MPa Angle of cut plan from s1 plane : = ATAN(100/80) = deg Normal stress on cut plane: sn = x (cos2 ) = 1.71 MPa (compression) Shear stress on cut plane: tau = 15 x (sin2 ) = MPa slope 1.25 theta radians degrees sy 20 Mpa sx -10 Mpa (sy-sx)/2 15 Mpa (sy+sx)/2 5 Mpa tau Mpa sig 1.71 Mpa Step 3: open ANSYS Workbench ) First, save the (empty) project as Block1.wbpj 2) The left hand window shows a set of analysis type options. Select Static Structural and drag the icon to the right, placing it in the Project Schematic window Lab#1: Introduction to ANSYS, page 3

4 Step4: open Geometry and create the CAD model 1) By Clicking Geometry in the Project window, ANSYS will open a CAD modeling program called Design Modeler. 2) Select Millimeter length unit for the model. And click [ OK ] 3) You now see the main window where the CAD model will be displayed, just as in Lab# Lab#1: Introduction to ANSYS, page 4

5 4) Select the rectangle tool and sketch a rectangle, approximately square. (you may click the LOOK AT icon ); 5) Use the Dimension tab (below Modify), and using the General dimension tool, select the right side of the rectangle and drag a dimension a bit to the right. Do the same for the bottom. You can directly type the dimensions you want in the Details View at the lower left and update the height and width of the rectangle to 100 x 100mm. Finally draw a line from the top (about 20mm to the right of the upper left corner) to the lower right corner Lab#1: Introduction to ANSYS, page 5

6 6) Next use the Trim command in the Modify tab to trim the top right and right sides of the block. 7) Use the General dimension tool to update the width of short top to 20mm Lab#1: Introduction to ANSYS, page 6

7 8) To create a solid body, hit the extrude icon and extrude sketch1 20mm. Then hit Generate, and change view to isometric. You will see the solid body in grey as shown below. 9) At this point we need to create a 2 nd body that will be the upper right part of our block. Use the add sketch button (in the menu area at the top) to add a second sketch on the XY Plane. Use the sketching tools to add at triangle, which when extruded will fill in the block. The polyline tool will work best. Start at the top, snap to the top right point on sketch 1, draw a horizontal line to a point nearly above the lower right point, then a vertical line down to snap to the lower right point. Next right-click and select closed-end. This will have created a right angle triangle that perfectly fills in the missing part of the body Lab#1: Introduction to ANSYS, page 7

8 10) Next use the extrude tool to extrude this part 20mm. It is important at this point to select the add frozen option for this extrude. If you use the add Material option, you will merge this material and not create a 2 nd body. We need a 2 nd body to have a cut plane for later stress analysis. 11) We now have 2 bodies. The last thing to do is to merge the 2 bodies into one part. This action fuses the two bodies together (like welding) but leave them as distinct bodies. Do this by selecting the two solids and right-click to select Form New Part. The tree changes from 2 Parts, 2 Bodies to 1 Part, 2 Bodies Lab#1: Introduction to ANSYS, page 8

9 Step5: open Model and create the Finite Element model 1) Return to the ANSYS window, and click on the Model feature in the Project window. This will start the ANSYS Mechanical program, to setup the actual finite element model. 2) The Mechanical window shows the 1 part and two solids. At first the model is shown with no mesh or loads yet. On the left is a list of the model features that have to be set. By default, the material to be used will be structural steel. Select the Mesh icon in the Project and right-click Generate Mesh. The mesh on the two bodies is in common (see below). (Note: you can see what happens when you go back to Design Modeler and explode the part. The two bodies will not have a common mesh and this problem won t solve at all, because the boundary conditions we will use leave the triangle able to translate as a rigid body.) 5003 Lab#1: Introduction to ANSYS, page 9

10 3) Next we will set the applied loads and support conditions on the block. We will need to use a special set of support conditions to create the desired stress state. We will use 3 symmetry planes to hold the block and also allow the block to maintain a uniform state of stress. You will need to bring the side and bottom of the block into view. You can use these tools: Rotate, pan, zoom smooth, zoom select and zoom all: Select the front, bottom and left sides and apply a frictionless support: This has the effect of creating 3 planes of symmetry, so that the stress on the bottom will be only a vertical stress (no shear). Likewise the left side will only have a horizontal stress. And the front will have no stress, but is supported. (Note: using fixed supports would create stress non-uniformity, and attempting to apply balancing loads would leave the block free to translate as a rigid body.) The two applied stresses can be applied as pressures of 20Pa to the top and -10 Pa to the right side. To insert a pressure load, right-click on Static Structural, select Insert and Pressure. Then select the two top planes of the block and click Apply. Set the pressures to 20 and -10 respectively Lab#1: Introduction to ANSYS, page 10

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12 4) To specify output, right click on Solution in the tree, and select Insert, then Stress, then Equivalent Stress. Hit the button in the menu at the top of the screen. (The solve command can be selected before or after selecting the various types of output). The solution for the equivalent stress is shown below. In this case 2 nd solid (the triangular part) has been hidden (right click on the solid icon and select hide) The whole block shows a single color with the stress of Pa Lab#1: Introduction to ANSYS, page 12

13 While there are no actual stresses of Pa in the block, the biaxial stress of 20 and -10 will bring the steel to the same % of yield as if a uniaxial stress of Pa had been applied. (see sketch below). In this way, the equivalent stress can be compared to the yield stress to see if the material is at risk of becoming plastic. 5) Next we need to create a new coordinate system that will permit the determination of the stresses on the cut plane. Make sure the cut plane is visible (hide the upper triangular block). In the Outline window, select the Coordinate systems icon, right-click and insert a new Coordinate system. Rename the new coordinate system cut coordinate system. In the details select Define By for both the Origin and Principal Axis of the new coordinate system, and select Geometry 5003 Lab#1: Introduction to ANSYS, page 13

14 Selection. You will need to pick the cut plane in order to set the values correctly. The new plane will be as shown below. 6) With the new coordinate system defined, you can now insert solution results for both shear and normal stresses, and get the results as shown below. As expected, the normal stress on the cut plane is 1.71 Pa, and the shear stress is Pa Lab#1: Introduction to ANSYS, page 14

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16 Self Study Exercises: Student: For each of these exercises, modify the model that you have developed above to explore the model behavior and answer the questions given. Show the instructor your results and make sure that it is recorded that you have completed the exercises. Exercise #1 Change the Support Conditions. Change the bottom and left side support conditions to fixed and see what happens to the model. First try deleting the support on the front face and see what happens. Tell the instructor what happened. Then replace the frictionless support on the front face and show the instructor the stresses that result. Ex#1 Initials of Instructor Exercise #2 Redo the analysis to create pure shear. Re-sketch the block to be two 100 x 100 right triangles. Apply tangential (shear) stresses to the surfaces to create perfect shear. (hint: in this case you will need to fix the bottom,) Ex#2 Initials of Instructor 5003 Lab#1: Introduction to ANSYS, page 16