Workshop 12

Nonlinear Analysis of a Skew Plate

IntroductionYou have been asked to model the plate shown in Figure W12–1. It is skewed 30° to theglobal 1-axis, is built-in at one end, and is constrained to move on rails parallel to theplate axis at the other end. You are to determine the midspan deflection when the platecarries a uniform pressure. You are to perform both linear and nonlinear static anddynamic analyses.

Figure W12–1. Sketch of the skewed plate

Defining the model geometry Start ABAQUS/CAE in the workshops/skewPlate directory, and enter the Partmodule. A default model name is assigned by ABAQUS/CAE (Model-1). However,since the linear analysis model will later form the basis of the nonlinear analysis model,you should rename the current model to give it a more descriptive name.

· From the main menu bar, select ModelRenameModel-1. Rename themodel linear.

Create a three-dimensional, deformable body with a planar shell base feature. Name thepart Plate, and specify an approximate part size of 4.0. A suggested approach tocreating the part geometry is outlined in the following procedure:

To sketch the plate geometry:1. In the Sketcher create a vertical line of length 0.4 m using the Create Lines:

Connected tool.

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2. Using the Create Construction: Line at an Angle tool, create a constructionline oriented 30° with respect to the horizontal through each of the line’sendpoints.

3. Using the Create Isolated Point tool, create an isolated point at a horizontaldistance of 1.0 m to the right of the vertical line. Create a vertical construction linethrough this point.

4. Using the Create Lines: Connected tool, draw the skewed rectangle using thepreselection points at the intersections of the construction lines to position thecorner vertices.

The final sketch is shown in Figure W12–2.

Figure W12–2. Sketch of the plate geometry

5. In the prompt area, click Done to finish the sketch.

Defining the material and section properties and the localmaterial directions The plate is made of an isotropic, linear elastic material with a Young's modulusE = 30E9 Pa and a Poisson's ratio = 0.3. Enter the Property module, and create thematerial definition; name the material Steel.The orientation of the structure in the global coordinate system is shown in Figure W12–1. The global Cartesian coordinate system defines the default materialdirections, but the plate is skewed relative to this system. It will not be easy to interpretthe results of the simulation if you use the default material directions because the directstress in the material 1-direction, 11 , will contain contributions from both the axialstress, produced by the bending of the plate, and the stress transverse to the axis of theplate. It will be easier to interpret the results if the material directions are aligned with theaxis of the plate and the transverse direction. Therefore, a local rectangular coordinatesystem is needed in which the local x’-direction lies along the axis of the plate (i.e., at30° to the global 1-axis) and the local y’-direction is also in the plane of the plate.

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To define shell section properties and local material directions:1. Define a homogeneous shell section named PlateSection. Assign a shell

thickness of 0.8E-2 and the Steel material definition to the section. Specifythat section integration be performed before the analysis since the material islinear elastic.

6. Define a rectangular datum coordinate system as shown in Figure W12–3 using

the Create Datum CSYS: 2 Lines tool .

Figure W12–3. Datum coordinate system used to define local material directions

7. From the main menu bar, select AssignMaterial Orientation and select theentire part as the region to which local material directions will be applied. In theviewport, select the datum coordinate system created earlier. Select Axis-3 for thedirection of the approximate shell normal. No additional rotation is needed aboutthis axis.

Tip: To verify that the local material directions have been assigned correctly, selectToolsQuery from the main menu bar and perform a property query on the materialorientations.Once the part has been meshed and elements have been created in the model, all elementvariables will be defined in this local coordinate system.

8. Assign the section definition to the plate.

Creating an assembly, defining an analysis step, and specifyingoutput requests Instance the plate in the Assembly module. Before leaving the Assembly module, definegeometry sets to facilitate output request and boundary condition definitions. You willfirst need to partition the plate in half to create a geometry set at the plate midspan.

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Select this edge to be along the local x’-direction

Select this edge to be in the local x’-y’ plane

To partition the plate and define geometry sets:1. Partition the plate in half using the Partition Face: Shortest Path Between 2

Points tool . Use the midpoints of the skewed edges of the plate to create thepartition shown in Figure W12–4.

Figure W12–4. Partition used to define a geometry set at the plate midspan

9. Select ToolsSetCreate to create a geometry set for the midspan namedMidSpan. Similarly, create sets for the left and right edges of the plate and namethem EndA and EndB, respectively.

Next, create a single static, general step in the Step module. Name the step ApplyPressure, and specify the following step description: Uniform pressure (20kPa) load. Accept all the default settings for the step.Among the output you will need are the nodal displacements and element stresses as fielddata. These data will be used to create deformed shape plots and contour plots inABAQUS/Viewer. You will also want to write the displacements at the midspan ashistory data to create X–Y plots in ABAQUS/Viewer.

To change the default output requests:1. Edit the field output request so that only the nodal displacements and element

stresses for the whole model are written as field data to the output database(.odb) file.

10. Edit the history output request so that only nodal displacements for the MidSpangeometry set are written as history data to the output database file.

Prescribing boundary conditions and applied loads As shown in Figure W12–1, the left end of the plate is completely fixed; the right end isconstrained to move on rails that are parallel to the axis of the plate. Since the latterboundary condition direction does not coincide with the global axes, you must define alocal coordinate system that has an axis aligned with the plate. You can use the datumcoordinate system that you created earlier to define the local material directions.

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EndB

EndAMidSpan

To assign boundary conditions in a local coordinate system:1. Switch to the Load module.11. Select BCCreate, and define a Displacement/Rotation mechanical

boundary condition named Rail boundary condition in the Initialstep.

In this example you will assign boundary conditions to sets rather than to regions selecteddirectly in the viewport. Thus, when prompted for the regions to which the boundarycondition will be applied, click Sets in the prompt area of the viewport.

12. From the Region Selection dialog box that appears, select set EndB. Toggle onHighlight selections in viewport to make sure the correct set is selected. Theright edge of the plate should be highlighted. Click Continue.

13. In the Edit Boundary Condition dialog box, click Edit to specify the localcoordinate system in which the boundary condition will be applied. In theviewport, select the datum coordinate system that was created earlier to define thelocal directions. The local 1-direction is aligned with the plate axis.

14. In the Edit Boundary Condition dialog box, fix all degrees of freedom exceptfor U1.

The right edge of the plate is now constrained to move only in the direction of the plateaxis. Once the plate has been meshed and nodes have been generated in the model, allprinted nodal output quantities associated with this region (displacements, velocities,reaction forces, etc.) will be defined in this local coordinate system.Complete the boundary condition definition by fixing all degrees of freedom at the leftedge of the plate (set EndA). Name this boundary condition Fix left end. Use thedefault global directions for this boundary condition.Finally, define a uniform pressure load named Pressure across the top of the shell inthe Apply Pressure step. Select both regions of the part using [Shift]+Click, andchoose the top side of the shell (Magenta) as the surface to which the pressure load willbe applied. You may need to rotate the view to more clearly distinguish the top side of theplate. Specify a load magnitude of 2.0E4 Pa.

Creating the mesh and defining a job Figure W12–5 shows the suggested mesh for this simulation.

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Figure W12–5. Suggested mesh design for the skewed plate simulation

You must answer the following questions before selecting an element type: Is the platethin or thick? Are the strains small or large? The plate is quite thin, with a thickness-to-minimum span ratio of 0.02. (The thickness is 0.8 cm, and the minimum span is 40 cm.)While we cannot readily predict the magnitude of the strains in the structure, we think thatthe strains will be small. Based on this information, choose quadratic shell elements(S8R5) because they give accurate results for thin shells in small-strain simulations. Forfurther details on shell element selection, refer to Section 15.6.2 of theABAQUS/Standard User’s Manual.Enter the Mesh module, and seed the part using a global element size of 0.1. From themain menu bar, select MeshControls to specify the structured mesh technique for thismodel. Create a quadrilateral mesh using quadratic, reduced-integration shell elementswith five degrees of freedom per node (S8R5).Enter the Job module, and define a job named SkewPlate with the followingdescription: Linear Elastic Skew Plate. 20 kPa Load.Save your model in a model database file named SkewPlate.cae. Submit the job for analysis, and monitor the solution progress; correct any modelingerrors detected by the solver, and investigate the cause of any warnings.

Postprocessing the linear analysis resultsSwitch to the Visualization module to postprocess the analysis results. By default,ABAQUS/Viewer plots the fast representation of the model. Plot the undeformed modelshape by selecting PlotUndeformed Shape from the main menu bar or by clicking

the tool in the toolbox.

Element normals Use the undeformed shape plot to check the model definition. Check that the elementnormals for the skew-plate model were defined correctly and point in the positive 3-direction.

To display the element normals:1. In the prompt area, click Undeformed Shape Options.

The Undeformed Shape Plot Options dialog box appears. 15. Set the render style to Shaded. 16. Click the Normals tab. 17. Toggle on Show normals. 18. Click OK to apply the settings and to close the dialog box.

The default view is isometric. You can change the view using the options in the view

menu or the view tools (such as ) from the toolbar.

To change the view:1. From the main menu bar, select ViewSpecify.

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The Specify View dialog box appears. 19. From the list of available methods, select Viewpoint. 20. Enter the X-, Y- and Z-coordinates of the viewpoint vector as -0.2, -1, 0.8

and the coordinates of the up vector as 0, 0, 1. 21. Click OK.22. From the main menu bar, select ViewParallel to turn perspective off.

ABAQUS/Viewer displays your model in the specified view, as shown in Figure W12–6.

Figure W12–6. Shell element normals in skewed plate model

Symbol plots Symbol plots display the specified variable as a vector originating from the node orelement integration points. You can produce symbol plots of most tensor- and vector-valued variables. The exceptions are mainly nonmechanical output variables and elementresults stored at nodes, such as nodal forces. The relative sizes of the arrows indicate therelative magnitude of the results, and the vectors are oriented along the global direction ofthe results. You can plot results for the resultant of variables such as displacement (U),reaction force (RF), etc.; or you can plot individual components of these variables.

To generate a symbol plot of the displacement:1. From the main menu bar, select ResultField Output.

The Field Output dialog box appears; by default, the Primary Variable tab is selected.23. From the list of output variables, select U. 24. Click OK.

The Select Plot Mode dialog box appears. 25. Choose Symbol, and click OK.

ABAQUS/Viewer displays a symbol plot of the displacements in the 3-direction on thedeformed model shape.

26. To modify the attributes of the symbol plot, click Symbol Options in theprompt area.

The Symbol Plot Options dialog box appears; by default, the Basic tab is selected.

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27. To plot the symbols on the undeformed model shape, click the Shape tab andtoggle on Undeformed shape.

28. Click OK to apply the settings and to close the dialog box. A symbol plot on the undeformed model shape appears, as shown in Figure W12–7.

Figure W12–7. Symbol plot of displacement

Material directions ABAQUS/Viewer also lets you visualize the element material directions. This feature isparticularly helpful, allowing you to ensure the correctness of the material directions.Material directions are associated with element integration points. Consequently, to viewmaterial directions the current field output variable must be an element-based variable.

To plot the material directions:1. From the main menu bar, select ResultField Output.

The Field Output dialog box appears; by default, the Primary Variable tab is selected.29. From the list of output variables, select S. 30. Click OK.

The current primary field output variable changes to stress at integration points.

2. From the main menu bar, select PlotMaterial Orientations; or click the tool in the toolbox.

The material orientation directions are plotted on the deformed shape. By default, thetriads that represent the material orientation directions are plotted without arrowheads.

31. To display the triads with arrowheads, click Material Orientation Options inthe prompt area.

The Material Orientation Plot Options dialog box appears. 32. Click the Color & Style tab; then click the Triad tab. 33. Set the Arrowhead option to use filled arrowheads in the triad. 34. Click OK to apply the settings and to close the dialog box.

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35. From the main menu bar, select ViewViews Toolbox; or click the tool inthe toolbar.

The Views toolbox appears. 36. Use the predefined views available in the toolbox to display the plate as shown in

Figure W12–8. In this figure, perspective is turned off. In Figure W12–8 nondefault material direction colors have been used: the material 1-direction is colored red, and the material 2-direction is colored blue.

Figure W12–8. Plot of material orientation directions in the plate

Adding geometric nonlinearityNow perform the simulation considering geometrically nonlinear effects. From the mainmenu bar, select ModelCopy, and copy the model named linear to a new modelnamed nonlinear. The changes required for this model are described next.

Step definitionEnter the Step module. From the main menu bar, select StepEditApply Pressureto edit the step definition. In the Basic tabbed page of the Edit Step dialog box, toggleon Nlgeom to include geometric nonlinearity effects and set the time period for the stepto 1.0. In the Incrementation tabbed page, set the initial increment size to 0.1. Thedefault maximum number of increments is 100; ABAQUS may use fewer incrementsthan this upper limit, but it will stop the analysis if it needs more.You may wish to change the description of the step to reflect that it is now a nonlinearanalysis step.

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Output controlIn a linear analysis ABAQUS solves the equilibrium equations once and calculates theresults for this one solution. A nonlinear analysis can produce much more output becauseresults can be requested at the end of each converged increment. If you do not select theoutput requests carefully, the output files become very large, potentially filling the diskspace on your computer. If selected carefully, data can be saved frequently during thesimulation without using excessive disk space.

1. From the main menu bar, select OutputField Output RequestsManagerto open the Field Output Requests Manager.

37. On the right side of the dialog box, click Edit to open the field output editor. 38. Remove the field output requests defined for the linear analysis model, and specify

the default field output requests by selecting Preselected defaults underOutput Variables.

This preselected set of output variables is the most commonly used set of field variablesfor a general static procedure.

39. To reduce the size of the output database file, write field output every secondincrement. Note that if you were simply interested in the final results, you couldselect The last increment.

40. The history output request for the displacements of the nodes at the midspan canbe kept from the previous analysis.

Running and monitoring the job In the Job module, create a job named NlSkewPlate and give it the descriptionNonlinear Elastic Skew Plate. Remember to save your model database file.Submit the job for analysis, and monitor the solution progress. If any errors areencountered, correct them; if any warning messages are issued, investigate their sourceand take corrective action as necessary.The Job Monitor is particularly useful in nonlinear analyses. It gives a brief summary ofthe automatic time incrementation used in the analysis for each increment. Theinformation is written as soon as the increment is completed, so you can monitor theanalysis as it is running. This facility is useful in large, complex problems. Theinformation given in the Job Monitor is the same as that given in the status file(NlSkewPlate.sta).

Comparing the linear and nonlinear analysis resultsWhen the job has completed, enter the Visualization module and plot the deformed modelshape. The final deformed shape is shown in Figure W12–9.

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Figure W12–9. Final deformed shape

Next, create an X–Y plot of the displacement history of a midspan node. Use the nodeindicated in Figure W12–10.

Figure W12–10. Midspan node

1. In the Labels tabbed page of the Undeformed Shape Plot Options dialogbox, toggle on Show node labels and click Apply to identify the node’s label.

41. From the main menu bar, select ResultHistory Output. 42. Select the output variable U3 for the midspan node, and click Save As. Give the

curve the name nl-20kPa.43. From the main menu bar, select FileOpen and open the ODB file for the linear

skew plate job.44. Repeat steps 2 and 3 above to create a curve named lin-20kPa based on the

linear analysis results.

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45. From the main menu bar, select ToolsXY DataManager to open the XYData Manager.

46. In the dialog box, select both curves and click Plot.The nonlinear effects are relatively mild at this load level, as shown in Figure W12–11.

Figure W12–11. Midspan displacement history (pressure = 20kPa)

Query the plot to identify the value of the midspan displacement at the end of eachanalysis job.

47. From the main menu bar, select ToolsQuery.48. In the Query dialog box, select Probe values and click OK.49. Drag the mouse across the curve to obtain the values of the midspan

displacements under full loading (this state corresponds to a time equal to 1.0).50. Enter the vertical displacement (U3) of the midspan node from each analysis in

Table W12–1.Load (Pa) Linear (m) NLGEOM (m)2.0E46.0E4

Table W12–1. Midspan displacements

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nonlinear analysis

linear analysis

Triple the load in both the linear and nonlinear analysis models, and rerun each of thejobs.

1. Switch to the Load module and for each model do the following:A. From the main menu bar, select LoadEditPressure. B. In the Edit Load dialog box, enter a value of 6.0E4 for the pressure.

51. Switch to the Job module, and resubmit each analysis job.

Create and plot displacement history curves from each analysis as described earlier; namethe curves lin-60kPa and nl-60kPa. Probe the X–Y plot, and enter the verticaldisplacement (U3) of the midspan node from each analysis in Table W12–1. Thenonlinear effects under the larger load are clearly evident, as shown in Figure W12–12.

Figure W12–12. Midspan displacement history (pressure = 60kPa)

How does tripling the load affect the midspan displacement in each analysis?

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nonlinear analysis

linear analysis

Optional modifications to the modelIf time permits, perform the analyses that are described next.

1. Adding material nonlinearityYou will specify the post-yield behavior of the material using the Mises (or classical)metal plasticity model. Note that ABAQUS requires the use of true stress and logarithmicplastic strain when defining plasticity data. The data for this problem are shown in FigureW12–13 and are plotted using this strain measure.

Figure W12–13. Stress versus strain curveHint: The total stain tot at any point on the curve is equal to the sum of the elastic strainel and plastic strain pl . The elastic strain at any point on the curve can be evaluated from

Young’s modulus and the true stress .trueel E

= Use the following relationship to

determine the plastic strains:.pl tot el tot E = - = -

The changes to the nonlinear model are described next.

Material properties1. Enter the Property module; and, if necessary, select the nonlinear model. From

the main menu bar, select MaterialEditSteel. 52. Select MechanicalPlasticityPlastic to invoke the classical metal

plasticity model. Enter data corresponding points A and B on the stress-straincurve shown in Figure W12–13.

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pl

A

B

Slope = E

106

10-3

Tip: You can use the message area of ABAQUS/CAE as a simple calculator. Forexample, to compute the plastic strain at B, enter 0.02-(3e7/3e10) in the messagearea and hit [Enter]. The value of the plastic strain is printed in the message area.

Section definition1. From the main menu bar, select SectionEditPlateSection.53. In the Edit Section dialog box, toggle on During analysis to indicate that

section integration will be performed during the analysis. This is required since thematerial is no longer linear elastic.

LoadsEnter the Load module. Change the magnitude of the applied pressure load to 1.E4 Pa.

Job definition 1. In the Job module, create a job named PlSkewPlate and enter the following job

description: Elastic-Plastic Skew Plate. Remember to save yourmodel database file.

54. Submit the job for analysis, and monitor the solution progress. Correct anymodeling errors, and investigate the source of any warning messages.

PostprocessingTo postprocess the results, contour the S11 stress component in the plate:

1. From the main menu bar, select PlotContours. 55. From the main menu bar, select ResultField Output.56. In the Field Output dialog box, select S11 as the stress component.

The contour plot appears as shown in Figure W12–14.

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Figure W12–14. Contour plot of S11

Create and plot of the displacement history of the midspan displacement node. The plotappears as shown in Figure W12–15.

Figure W12–15. Midspan displacement history (elastic-plastic analysis)

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Yield first occurs here.

2. Dynamic analysisYou will now investigate the response of the structure to a sudden loading. This requiresthe simulation of a dynamic event. You will use the explicit dynamics solver in thisanalysis. Copy the nonlinear model to a model named dynamic and make the following changesto the dynamic model.

Material properties1. Enter the Property module. From the main menu bar, select

MaterialEditSteel. 57. Select GeneralDensity to define the material density. Enter a density value of

7800 kg/m3.

Step definition and output requests1. Enter the Step module. From the main menu bar, select StepDeleteApply

pressure to delete the general static step.Note this will delete all step-dependent objects such as loads and output requests. Thesemust be redefined.

58. From the main menu bar, select StepCreate to create a Dynamic, Explicitstep. Name the step Apply pressure. In the Edit Step dialog box, enter thefollowing description for the step: Dynamic analysis and prescribe a timeperiod of 1.0 second.

59. Create a new history output. Request that displacements for the set MidSpan bewritten to the output database (.odb) file as history data.

Load definition1. Enter the Load module. From the main menu bar, select LoadCreate to create

a pressure load in the Apply pressure step.60. Name the load Pressure. Select both regions of the part using [Shift]+Click,

and choose the top side of the shell (Magenta) as the surfaces to which thepressure load will be applied. You may need to rotate the view to more clearlydistinguish the top side of the plate. Specify a load magnitude of 1.0E4 Pa.

In a dynamic analysis, the load is applied instantaneously by default. Thus, while thestatic models simulated the response of the plate to gradually applied load, this modelsimulates the response of the plate to a suddenly applied load.

Mesh1. Enter the Mesh module. From the main menu bar, select MeshElement Type

to modify the element type of all regions in the model.61. In the Element Type dialog box, select Explicit as the element library, Shell as

the element family, and Linear as the geometric order. Among the availableelement controls, choose Finite membrane strains and the Relax stiffnesshourglass control. The selected element type is S4R.

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Job definition1. In the Job module, create a job named DynSkewPlate and enter the following

job description: Dynamic Skew Plate. Remember to save your modeldatabase file.

62. Submit the job for analysis, and monitor the solution progress. Correct anymodeling errors, and investigate the source of any warning messages.

PostprocessingPlot the time history of the vertical displacement of the midspan and the model energiesALLKE, ALLIE, and ALLAE. The results are shown in Figure W12–16 and Figure W12–17, respectively. The early transient response is depicted in these figures. Thedisplacements show progressively smaller oscillations about a steady state. In fact, if thetime period of the analysis is increased and the simulation is rerun, these oscillationsdisappear completely and the displacement solution converges to a steady-state solution.Note that the steady-state behavior is different from the static behavior due to the path-and history-dependent effects induced by the plasticity material model.

Figure W12–16. Midspan displacement history (dynamic analysis)

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Figure W12–17. Model energy history (dynamic analysis)

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