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Strand7 Tutorial: 2D Plane Stress Concentration Problem Description Introduction In this tutorial you will investigate the stress concentration at the root of a slot in a steel strip, which has been placed under tension. Description of Geometry and Loading The geometry, loading and material properties of the strip are described in Figure 2.1 below.

Figure 2.1 – Problem Geometry and Loading Required Results We need to calculate the stress concentration factor. From ESDU (Engineering Science Data Unit) sheets compiled by the Royal Aeronautical Society, the stress concentration factor for the given geometry is a function of the following ratios.

For these values the stress concentration factor K’ = 2.81, where:

Note This tutorial assumes you have a solid understanding of all the material covered in the preceding tutorial and familiarity with some standard Strand7 functions.

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Creating the FE Model Before creating a model of the strip, you must decide what type of elements to use. In this case either plates or bricks would be adequate. Due to the inherent two-dimensional nature of the problem, and the fact that the width and length are far greater than the thickness, plates are the best choice. Symmetry Considerations One of the most common methods of reducing the size of a FE model is through the use of symmetry conditions. In this case you can see that the geometry and loading of the plate has two distinct lines of symmetry as shown in Figure 2.2 below. It is not difficult to see that the resultant stress/strain/displacement field in each quarter will be the same. With this in mind, it makes little sense to build a model of the entire strip and then wait while the same data is calculated four times over. Therefore only one quarter of the strip needs to be modelled. Boundary conditions are applied along the edges where symmetry conditions exist to make the quarter model behave as it would in the complete strip.

Figure 2.2 – Lines of Symmetry Planning the Mesh Although the accuracy of the results depends on the quality of the mesh, there are usually a number of equally appropriate meshes you could construct for a given problem. The solution provided here is one of these. One of the problems faced in modelling this strip is the assumption that it is of infinite length. Use of a finite length strip to approximate the situation can be justified if, once the problem has been solved, little stress variation near the end of the strip is observed. If the stress varies considerably, then the strip must be lengthened and the problem solved again. For this problem try using a length of 300 millimetres. A good starting point is to break down the geometry of the strip into smaller, easier to manage areas as shown in Figure 2.3 below (remember that only one-quarter needs to be modelled).

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Figure 2.3 – Break Down of ¼ Model

Snap Grid Having a basic plan, you can now begin modelling. 1 Start Strand7; 2 Select FILE, New; 3 The Model Units dialog box will appear; 4 Under Length select ‘mm’ ; 5 Under Modulus/Stress select ‘MPa’ ; 6 Click here to view the Model Units dialog box; 7 Click OK; You should now save the model as ‘Tutorial 2.ST7’ . We will make use of the Snap Grid function of Strand7 to construct the model. Creating a Snap Grid The Snap Grid function is a tool provided to make the model building process easier. By defining a snap grid of suitable dimensions and spacing, you can quickly create nodes and elements by clicking on the appropriate grid points. Referring to Figure 2.3 , the most suitable grid for this problem is to have limits of 150x100 mm (X, Y) with a grid spacing of 5x5 mm (X, Y). The following outlines how to set up the snap grid. 1 Right-Click the Show Snap Grid button; 2 The Grid Settings dialog box will appear; 3 Under ‘Number of grids’ in the X box, type or select 30; 4 Under ‘Number of grids’ in the Y box, type or select 20; 5 Under ‘Grid Limits’ in the X Minimum box enter 0; 6 Under ‘Grid Limits’ in the X Maximum box enter 150; 7 Under ‘Grid Limits’ in the Y Minimum box enter 0; 8 Under ‘Grid Limits’ in the Y Maximum box enter 100; 9 Click OK; Your screen should look like Figure 2.4 below.

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Figure 2.4 – Snap Grid

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Plate Elements When constructing a model using plate elements, you can choose from five types: 3 and 6 node triangular elements and 4, 8 and 9 node quadrilateral elements. These are shown in Figure 2.5 below.

Figure 2.5 – Plate Element Types The 3 and 4 node elements have linear shape functions, whereas the 6, 8 and 9 node elements have quadratic shape functions. This means linear elements always have straight edges and the quadratic elements can have curved edges as a quadratic curve is fitted through the three nodes along an edge. Curves can be approximated by using many small linear elements, however for this model use the quadratic (Quad8) elements. For a more detailed description of plate elements, refer to the Reference Manual. When using quadratic Quad8 elements, each plate must have four corner nodes and four side nodes. Referring to Figure 2.3, there are three regions where Quad8 elements can be used (regions 1,2 and 3), however the fourth region has five ‘corners’ . This problem is overcome by dividing the region into 2 as shown in Figure 2.6 below. This plate layout will later be subdivided into smaller plates to give a more accurate result.

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Figure 2.6 – Plate Layout Creating Quad8 Plate Elements Plates are connected in a similar manner to beams. Initially we will create three Quad8 plate elements at the locations shown in Figure 2.7 below.

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Figure 2.7 – Initial Plates and Coordinates of the Corners To define a Quad8 plate element, you must first define the four corner nodes in a clockwise or anti-clockwise manner (It is good practice to create elements consistently). Creating Plates 1, 2 and 3 1 From the CREATE menu select Element (alternatively, press CTRL+E); 2 Under ‘Type’ select Quad8; 3 Click the four nodes defining the corners of Plate 1 in an anti-clockwise manner. The

coordinates of the grid points can be viewed during the plate connection operation, by pressing the SHIFT key and pointing to a grid point;

4 After defining the corner nodes, an outline of the plate will be drawn as shown in Figure 2.8 below. The Quad8 element is not complete as you still need to define the four mid-side nodes;

5 To do this, click All in the Connections dialog box to fully define the Quad8 element. This automatically places the mid-side nodes at points on a straight line between the corner nodes;

6 Follow the same procedure to create Plates 2 and 3.

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Figure 2.8 – Not yet Complete Quad8 plate Your model should look like Figure 2.9 below.

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Figure 2.9 – The model after creating the first thr ee Quad8 plates

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Creating the Circular Slot There are a number of ways of creating the curved edges of the circular slot. The simplest method is to use one of the Grading tools available in Strand7. Grading Grading performs the transition from a fine mesh to a coarse mesh, allowing specific areas with high stress gradients to have a finer mesh. Additional grading tools are available that allow you to transform plates with straight edges to plates with curved edges. This is the tool that we will use to create the geometry of the slot. 1 Select TOOLS, Grade Plates and Bricks; 2 The Grades dialog box as shown in Figure 2.10 below will appear.

Figure 2.10 – Grades dialog box This dialog box contains a number of buttons corresponding to the different grading types available in Strand7. When an element is graded, it is subdivided in the same way as displayed on the button.

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When performing grades, Strand7 must not only know which element to grade, but also the orientation of grading. As you can see, most of the buttons in the figure have a highlighted edge (or edges) marked by a dashed line that aids in the orientation of the grade. This requires marking the edge of the element to be graded in a manner that corresponds to the highlighted edge shown on the button. To create the radius of the slot, follow these steps. 1 Open the Grades dialog box (if not already open); 2 Set the appropriate Grade button; 3 Select the appropriate plate edges ; 4 Under ‘Radius Ratio’ enter 20/70 (this is the ratio of the radius of the slot to the length of the

edge of the plate); 5 Click Apply; 6 Close the Grades dialog box. Now that the radius of the slot has been defined, create the final Quad8 plate element to complete the geometry of the model. The model should now look like Figure 2.11 below.

Figure 2.11 – Completed model geometry The Snap Grid is no longer required and can be hidden by clicking on the Snap Grid button.

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Subdividing Elements The mesh of the current model is too coarse to produce accurate results. The Subdivide tool can be used to divide the plates into a number of smaller plates resulting in a finer mesh and more accurate results. Subdivide Select TOOLS, Subdivide, to display the Subdivide dialog box shown in Figure 2.12 below. As you can see, there are two sections that are accessible: ‘Divisions’ and ‘Targets’ . Divisions This option has three parameters, A, B and C. These parameters define the number of subdivisions along the A, B and C directions. For plate elements, only the A and B parameters are required while bricks require all three parameters. The directions are displayed on the element once the element is selected. Beams can also be subdivided, and they require only the A parameter. Targets This option sets the element to be generated as a result of the subdivision. It is only required when subdividing plate and brick elements since these elements have various types.

Figure 2.12 – Subdivide Dialog Box Example As an example, to subdivide the Quad8 plate element shown in Figure 2.13 below, into 4 vertical sections and 3 horizontal sections. 1 Select TOOLS, Subdivide; 2 Select the element you wish to subdivide; 3 The A and B directions will be indicated on the plate. 4 Referring to Figure 2.13 below, we need 3 ‘A’ divisions and 4 ‘B’ divisions. 5 Under ‘Target’ , Plate select Quad8; 6 Click Apply.

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Figure 2.13 – Example Subdivision Subdividing the Strip The five plate elements used for modelling the quarter strip will now be subdivided to create a finer mesh. When subdividing a mesh, you must ensure that adjacent elements are subdivided consistently such that they join at common nodes. Subdivide plates 1 to 5, as shown in Figure 2.14 below, ensuring that under ‘Target’ , Plate, Quad8 is selected. If you make any errors, use Undo.

Figure 2.14 – Subdivided Mesh Mesh Cleaning As the mesh is now complete, it is good practice to Clean the mesh. This sorts and renumbers the nodes and elements, removing any unused nodes or null elements from the mesh. 1 Select TOOLS, Clean, Mesh (alternatively, press CTRL+ALT+C); 2 Select the zip tolerance tab and change the value to 1.0e-4 and then click Apply; 3 After the mesh cleaning has finished, a Confirm dialog box will appear asking if you want to

view the log file. Click Yes; 4 The log shows what was carried out during the mesh cleaning; 5 Click Close. You are now ready to apply the load to the model.

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Applying the Loads The load in this tutorial is a plate Edge Pressure. Plate elements can have loads applied to them through Edge Pressure, Edge Shear, Edge Normal Shear, Surface Pressure and Surface Shear as shown in Figure 2.15 below. For a detailed description of plate loads and the settings of their parameters, see the Reference Manual.

Figure 2.15 – Plate Element Load Types For this tutorial, an edge pressure is applied along the left edge of the quarter model. Referring to Figure 2.1 , the edge pressure is σ*, which is arbitrary in this tutorial (we shall set it to 100MPa). To apply an edge pressure to a plate element, you have to first select the plate edges that you want to apply the edge pressure to. Figure 2.16 below shows where the edge pressure is applied on the model. There are two ways you can select the left edge. One way is to select each outer plate edge individually (five in total). Alternatively, and certainly more efficiently in larger models, you can use the Select By Region function. Select by Region Function This function allows you to select all elements within a specified edge, surface or solid. To access this function you can either select EDIT, Select, by Region, or click on the Select by Region button on the Strand7 toolbar. 1 Select ATTRIBUTES, Plate, Edge Pressure ; 2 Click the Select by Region button; 3 From the dropdown menu select Global XYZ:[Cartesian]; 4 Click the top node on the left edge; 5 Click the bottom node on the left edge; 6 The edge should now be highlighted with a blue line; 7 Click Apply; 8 5 dashed lines on the plate edges will appear indicating they have been selected; 9 In the Select by Region dialog box, click OK; 10 In the Plate Attributes dialog box, under ‘Edge Pressure’ in the Value box enter 100; 11 Click Apply; 12 Close the Plate Attributes dialog box. When you have finished applying the edge pressure, your model should look like Figure 2.16 below.

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Figure 2.16 – Edge Pressure Applied to Model

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Freedom Conditions and Restraints Global Freedom Condition For a two-dimensional plane stress problem, the global freedom condition require only two active degrees of freedom, i.e. translation in the X and Y directions. This means that the nodes have zero Z translations and no rotations. This condition can be entered manually, or by using the Auto Set function. 1 Select GLOBAL, Load & Freedom Cases; 2 Click the Freedom Cases tab; 3 Click 2D Plane; 4 Click OK; The global freedom condition is now set. Boundary Restraint Conditions Knowing the global freedom condition makes calculating the boundary restraint conditions much easier as there are only two degrees of freedom to consider, DX and DY. For this problem, restraint conditions need to be applied to the symmetry axes. Displacement on one side of a line of symmetry is reflected on the other side. This means that on a line of symmetry, there is no displacement normal to the line, only displacement along the symmetry line is allowed. This is shown in Figure 2.17 below. Therefore all nodes along the X symmetry line must have DY fixed, and all nodes along the Y symmetry line must have DX fixed. This leaves only one degree of freedom: displacement along the symmetry line. In this case, applying these freedom conditions also restricts rigid body motion as no rotation or translation of the body as a whole is possible.

Figure 2.17 – Symmetry Restraint Conditions . 1 Select ATTRIBUTES, Nodes, Restraint; 2 Click X sym; 3 Click the Select by Region icon and click on the two end nodes of the top edge; 4 In Select by Region dialog box, click Apply; 5 In Node Attributes dialog box, click Apply; 6 Similarly, assign symmetry conditions to the right edge (Hint, use Y sym). When the restraints are applied to both symmetry axes, the model is complete, and should look like Figure 2.18 below.

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Figure 2.18 – Complete Model

Property Input For a two dimensional plane stress analysis, Strand7 needs the Modulus of Elasticity, Poisson’s Ratio and the Membrane Thickness (plate thickness). Plate properties are entered in the same manner as beam properties. Plate elements are used for several solution types and the appropriate element type must be selected. 1 Select PROPERTY, Plate; 2 Under ‘Type’ select 2D Plane Stress; 3 Click the Materials button; 4 Double click Mil Hdbk 5G – Carbon Steel; 5 Select AISI 1025 Cold Rolled Sheet and Strip; 6 Click OK; 7 Click the Geometry tab; 8 In the Membrane Thickness text box enter 5; 9 Click Close; The model is now fully defined. You can now solve the model.

Solving the Model As explained earlier, this tutorial is a 2D plane stress analysis of a linear stress concentration factor. Therefore the Linear Static Solver will be used. 1 Select SOLVER, Linear Static; 2 Click Solve; 3 When the solution is complete, close the solver panel.

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Post Processing You can now begin to process the results of the analysis. The required result is the stress at the tip of the slot. Once you have this value, the stress concentration factor can be calculated using the formulas given previously . You need to investigate the stress distribution throughout the strip. In this tutorial we are concerned with the maximum principal stress and the best way of viewing this is to display a contour plot. 1 Select RESULTS, Open Results File; 2 Open the appropriate ‘ .LSA’ file; 3 Select RESULTS, Results Settings; 4 Under ‘Draw as’ select Contour; 5 Under ‘Quantity’ select Stress; 6 Under ‘Axis System’ select Combined; 7 Click the 11 option (i.e. maximum principal stress); 8 Click OK. Your screen should look like Figure 2.19 below.

Figure 2.19 – Contour Plot of Maximum Principal Str ess

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By referring to the colour legend, you can see that the left end of the strip has an even stress distribution. Recall that the model is approximating an infinitely long strip; this even stress distribution indicates that the approximation is valid. The contour plot shows the greatest stress occurring at the tip of the slot (indicated by a purple colour), and the value of this stress is given at the top of the legend. You can zoom into this area for a closer look at the stress distribution. The Peek function can be used to obtain a precise value of the stress at locations that are not at the upper or lower end of the legend. 1 Click Zoom In; 2 Zoom into the area around the stress concentration; 3 Select RESULTS, Peek, 4 Click the Plate tab; 5 Under ‘Axis System’ select Combined; 6 Make sure Nodes at the bottom of the dialog box is checked; 7 Click on the plate at the tip of the slot. Given a maximum principal stress of 516 MPa, you can calculate the reference stress using the equations given earlier:

The stress concentration factor can now be calculated:

The error between this value and the target value is around 8%, which is due to the mesh around the slot being too coarse. By examining the element at the stress concentration, you can see the stress gradient is very high (i.e. the stress variation across the element is more than half the scale of the legend). Without having to rebuild the entire model, you can refine the mesh around the stress concentration to obtain a more accurate representation of the stress field.

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Mesh Refinement To refine the mesh, you must close the result file by selecting RESULTS, Close Results File. Before you begin the mesh refinement, zoom in to the area shown in Figure 2.20 below.

Figure 2.20 – Zoom Area for Mesh Refinement To begin refining the mesh, select TOOLS, Subdivide and divide the plate with the stress concentration into nine smaller plates (i.e. 3x3 subdivision) as shown in Figure 2.21 below. Once this is done, the mesh is finer around the stress concentration, however the adjacent plates are not consistently divided as there are now three plate edges connecting to one plate edge. To avoid having to subdivide the entire mesh, the surrounding elements can be graded.

Figure 2.21 – Refinement of Mesh Grading Grading performs the transition from a fine to a coarse mesh allowing specific areas with high stress gradients to have a finer mesh. To begin grading, select TOOLS, Grade Plates and Bricks to display the dialog box shown in Figure 2.22 below. This displays the various grading options available.

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Figure 2.22 – Grades Dialog Box To ensure mesh uniformity, we need to define a grading element that allows grading from three edges to one edge. The best element to use is the one in the third row of the first column of Figure 2.22 above. When an element is graded, it is subdivided in the same fashion as shown by the button. Hence there are two elements adjacent to the subdivided element that need to be graded. Strand7 must not only know which element to grade, but also the orientation of the grading. As you can see, most of the icons in Figure 2.22 have a highlighted edge (white dashed line) that aids in the orientation of the grade. This requires marking the edge of the element to be graded in a manner that corresponds to the highlighted edge shown in Figure 2.22. The following describes the procedures for grading the elements. Figure 2.23 below illustrates the process. 1 Select TOOLS, Grade Plates and Bricks; 2 Click the grade icon in the first column of the third row; 3 Select the two plate edges as shown in Figure 2.23 below; 4 Click Apply; 5 Close the Grades dialog box.

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Figure 2.23 – Plate Grading Procedure Note The Strand7 Subdivide and Grade functions automatically generate plates with the same property and loads applied to them as the parent plate. In addition, new nodes generated also have forces and restraint conditions automatically applied to them. When refinement of the mesh around the slot has been completed, clean the mesh as before (with a zip tolerance of 1.0e-4) and once again run the Linear Static Solver. Post Processing the Refined Mesh The post processing to be carried out is the same as that described previously. A summary of the results of the analysis is listed in Table 2.1 below. To get an even more accurate result, subdivide all the plates in the mesh by 3x3. This produces the ‘Fine Mesh’ results in Table 2.1 below.

Mesh σref σmax K’ K’ (target)

Coarse Mesh 200 516.0 2.58 2.81

Graded Mesh 200 558.0 2.79 2.81

Fine Mesh 200 561 2.81 2.81

Table 2.1 – Results of Analysis