Chapter 5: Analysis of a Truss

Transcription

1 Application of the Finite Element Method Using MARC and Mentat 5-1 Chapter 5: Analysis of a Truss 5.1 Problem Statement and Objectives A truss will be analyzed in order to predict whether any members will fail due to either material yield or buckling. The geometrical, material, and loading specifications for the truss are given in Figure 5.1. Each member of the truss has a solid circular cross section. Geometry: Material: Steel Area of members 1 and 2: 30 cm 2 Yield Strength: 250 MPa Area of members 3 and 4: 20 cm 2 Modulus of Elasticity: 200 GPa Area of member 5: 25 cm 2 Poisson s Ratio: 0.3 Loading: Vertical Load: P=18kN P 1 m m o 3 m 3 m Figure 5.1 Geometry, material, and loading specifications for the truss.

2 Application of the Finite Element Method Using MARC and Mentat Analysis Assumptions 1. No friction is present in any of the truss pin joints. Thus, each truss member is an ideal two-force member. Also, there is no friction between the ground and the rollers. 2. Deflections are small enough that geometrically linear analysis is valid. 3. Because the geometry, material properties, and loading conditions are all symmetric about the vertical plane of symmetry that passes through member 5, the response of the structure (i.e., displacements, strains, and stresses) will also be symmetric about this plane. Therefore, a symmetric model may be used. 5.3 Mathematical Idealization Other than the assumptions above, no additional simplifications need be made for a truss. Each truss member can be represented by a two-noded linear truss finite element. This model should yield the correct analytical values for displacements and stresses. 5.4 Finite Element Model The finite element model of this structure will be developed using 3D linear two-noded truss finite elements. The present analysis can be greatly simplified by taking advantage of the vertical plane of symmetry in the truss. Hence, it is necessary to model only one-half of the truss, as shown in Figure 5.2. The boundary conditions on the symmetry plane are those that occur naturally on this plane, as can be verified by obtaining a solution using the entire truss. Taking advantage of symmetry reduces the modeling effort, the amount of computer memory, and the amount of CPU time required to obtain a solution. Admittedly, the savings are small in this simple problem. When a reduced model is developed due to symmetry, note that loads and truss members that lie within the plane of symmetry should be treated in a special way. If a load is within a plane of symmetry, then only one-half that load should be applied to the symmetric model. Similarly, if a truss member lies in a plane of symmetry, then only one-half of the cross-sectional area of that member should be assigned to the element representing the member. In other words, member 5 will be modeled as having one-half of its actual cross-sectional area.

3 Application of the Finite Element Method Using MARC and Mentat 5-3 P/2 1 m m 3 30 o 3 m Figure 5.2 One-half domain to be modeled. 5.4 Model Validation The structure under consideration is statically determinate. Therefore, simple hand calculations can be performed to calculate the reactions and internal forces in the truss members. These results should be almost identical to the finite element results, with the only difference being due to round off errors. Additionally, it is always good practice to examine the predicted deformed shape of the structure to ensure that boundary conditions are properly satisfied and that the structural deflections are in the expected directions. In the present model, the deflection of the center joints should be downward, in the direction of the point load. At the same time, the joints on rollers should move along the rolling direction. A good way to verify these motions is to superpose a properly scaled deformed mesh on the undeformed mesh. 5.5 Post Processing The axial stress in any member can be estimated by assuming that the axial stress is uniformly distributed over the cross-section. The resultant force at any cross-section is known from the analysis, so the axial stress in the i th member can be estimated using the relation: σ i = Pi A i where A i is the cross-sectional area and P i is the axial force in the i th member.

4 Application of the Finite Element Method Using MARC and Mentat 5-4 Alternatively, σ i can be calculated using the one-dimensional Hooke s law σ i = Eε i, where Li ε i = is the strain in the i th member. The change in length Li can be calculated using the Li distance formula: where x j, y j and j v j {[ ( x2 + u2 ) ( x1 + u1 )] + [( y2 + v2 ) ( y1 + v1 )] } Li Li = 2 u, (j = 1, 2) are the nodal coordinates and nodal displacements, respectively, of the i th member. It follows that ( x u y + v ) j +, is the position of the j th node of the i th member after deformation. Note that j = 1, 2 refers to the local (element) node number. j j The axial stress in each member should be compared to the yield stress to determine if material failure will occur due to the applied loading. Note that this is a uniaxial state of stress. j When a truss member carries compressive load, the possibility of buckling should be examined. For the purpose of buckling analysis, each member in the truss can be considered a pinned column. Hence, Euler s formula can be used to predict if a member will buckle: EI P = π 2 cr 2 L where I is the second moment of the cross-sectional area and L is the original length of the member.

5 Application of the Finite Element Method Using MARC and Mentat 5-5 PROCEDURE FOR ANALYSIS OF THE TRUSS 1. Add points to define geometry. 1a. Add points. <ML> MAIN MENU / MESH GENERATION <ML> MAIN MENU / MESH GENERATION / PTS ADD Enter the coordinates at the command line, one point per line with a space separating each coordinate. > > > The points may not appear in the Graphics window because Mentat does not yet know the size of the model being built. When the FILL command command in the static menu is executed, Mentat calculates a bounding box for the model and fits the model inside the Grapics window. <ML> STATIC MENU / FILL The points should now be visible in the Graphics window. 1b. Display point labels. <ML> STATIC MENU / PLOT <ML> STATIC MENU / PLOT / LABEL POINTS <ML> STATIC MENU / PLOT / REDRAW 1c. Return to MESH GENERATION menu. <MR> or <ML> RETURN The result of this step is shown in Figure5.3

6 Application of the Finite Element Method Using MARC and Mentat 5-6 Figure 5.3. If the steps above were not followed precisely (e.g., if the points were entered in an order different than the order in which they appear in the above list), then the point labels will differ from those shown in Figure 5.3. These labels are simply used as identifiers in the following step, and do not affect the model. As long as the correct coordinates were entered, do not worry if the labels are not exactly as shown in Figure 5.3. Just keep track of the differences between the labels so that the appropriate procedures will be followed in the steps below. 2. Add three 2-noded line elements. 2a. Select ELEMENT CLASS. In the MESH GENERATION menu, the currently selected type of element that can be generated is displayed to the immediate right of the ELEMENT CLASS button. Change the element type to LINE (2): <ML> MAIN MENU / MESH GENERATION / ELEMENT CLASS <ML> MAIN MENU / MESH GENERATION / ELEMENT CLASS / LINE (2) <ML> MAIN MENU / MESH GENERATION / ELEMENT CLASS / RETURN 2b. Create the line elements to form the truss. <ML> MAIN MENU / MESH GENERATION / ELEMS ADD

7 Application of the Finite Element Method Using MARC and Mentat 5-7 <ML> to select point 1 and then point 2 to create an element from point 1 to point 2. <ML> to select point 2 and then point 3 to create an element from point 2 to point 3. <ML> to select point 1 and then point 3 to create an element from point 1 to point 3. 2c. Turn off point labels. <ML> STATIC MENU / PLOT / LABEL POINTS <ML> STATIC MENU / PLOT / REDRAW 2d. Turn on element labels. <ML> STATIC MENU / PLOT / LABEL ELEMENTS <ML> STATIC MENU / PLOT / REDRAW 2e. Turn on node labels. <ML> STATIC MENU / PLOT / LABEL NODES <ML> STATIC MENU / PLOT / REDRAW 2f. Return to MESH GENERATION menu. <MR> or <ML> RETURN The result of this step is shown in Figure 5.4. Figure 5.4

8 Application of the Finite Element Method Using MARC and Mentat Sweep the mesh to insure that all elements are properly connected. <ML> MAIN MENU / MESH GENERATION / SWEEP <ML> MAIN MENU / MESH GENERATION / SWEEP / ALL Note: Duplicate geometrical and mesh entities will be deleted so that proper mesh connectivity is achieved. Return to MESH GENERATION menu. <MR> or <ML> RETURN 4. Add boundary conditions. 4a. Specify the symmetry conditions on the right edge of the model. 4a1. Set up a new boundary condition set. <ML> MAIN MENU / BOUNDARY CONDITIONS <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / NEW <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / NAME At the command line, enter a name for this boundary condition set. > Symmetry 4a2. Define the nature of the boundary condition. <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / FIXED DISPLACEMENT Recall: Truss elements have three displacement DOFs at each node. Here, displacement in the x- and z-directions are to be constrained. <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / FIXED DISPLACEMENT / ON X DISPLACE <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / FIXED DISPLACEMENT / ON Z DISPLACE The small box to the immediate left of the ON button for X DISPLACE and Z DISPLACE should now be highlighted. <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / FIXED DISPLACEMENT / OK 4a3. Apply the condition to nodes 2 and 3. <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / NODES ADD

9 Application of the Finite Element Method Using MARC and Mentat 5-9 <ML> to select node 2. <ML> to select node 3. <MR> or END LIST The result of this step is shown below in Figure 5.5. Figure 5.5 4b. Specify the vertical load at node 3. 4b1. Set up a new boundary condition set. <ML> MAIN MENU / BOUNDARY CONDITIONS <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / NEW <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / NAME At the command line, enter a name for this boundary condition set. > PointLoad 4b2. Define the nature of the boundary condition. <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / POINT LOAD <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / POINT LOAD / ON Y FORCE The small box to the immediate left of the ON button for X FORCE should now be highlighted.

10 Application of the Finite Element Method Using MARC and Mentat 5-10 <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / POINT LOAD / Y FORCE > -9.0e3 Note: Because the load acts along a plane of symmetry, only half the load is applied to the model. <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / POINT LOAD / OK 4b3. Apply the load to node 3. ADD <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / NODES <ML> to select node 3. <MR> or END LIST The result of this step is shown in Figure 5.6. Figure 5.6

11 Application of the Finite Element Method Using MARC and Mentat c. Specify the sliding condition at node 1. Note: The sliding condition will be imposed by defining a local coordinate system at node 1 and applying the conditions with respect to this new local system. 4c1. Set up a new boundary condition set. <ML> MAIN MENU / BOUNDARY CONDITIONS <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / NEW <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / NAME At the command line, enter a name for this boundary condition set. > Sliding 4c2. Define the local coordinate system. Note: The new local coordinate system is defined by specifying three points: the origin, a point on the new x' axis, and a point on the new y' axis. <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / TRANSFORMS <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / TRANSFORMS / ALIGN At the command line, enter the coordinates for the origin of the new coordinate system. This point does not have to be the nodal coordinates. > At the command line, enter the coordinates for a point on the x'-axis. > At the command line, enter the coordinates for a point on the y'-axis. > c3. Apply the new local coordinate system to node 1. <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / TRANSFORMS / NODES ADD <ML> to select node 1. <MR> or END LIST

12 Application of the Finite Element Method Using MARC and Mentat c4. Define the nature of the boundary condition. <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / FIXED DISPLACEMENT <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / FIXED DISPLACEMENT / ON Y DISPLACE <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / FIXED DISPLACEMENT / ON Z DISPLACE The small box to the immediate left of the ON button for Y DISPLACE and Z DISPLACE should now be highlighted. <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / FIXED DISPLACEMENT / OK 4c5. Apply the condition to node 1. ADD <ML> MAIN MENU / BOUNDARY CONDITIONS / MECHANICAL / NODES <ML> to select node 1. <MR> or END LIST The Result of this step is shown in Figure 5.7 Figure 5.7.

13 Application of the Finite Element Method Using MARC and Mentat d. Display all boundary conditions for verification. <ML> MAIN MENU / BOUNDARY CONDITIONS / ID BOUNDARY CONDS After verifying that boundary conditions have been applied properly, turn off the boundary condition ID's by repeating the last command. 4e. Return to the MAIN menu. <ML> MAIN MENU / BOUNDARY CONDITIONS / MAIN 5. Specify the material properties of each element. 5a. Set up a new material property set. <ML> MAIN MENU / MATERIAL PROPERTIES <ML> MAIN MENU / MATERIAL PROPERTIES / NEW <ML> MAIN MENU / MATERIAL PROPERTIES / NAME At the command line, enter a name for this material property set. > Steel 5b. Define the nature of the material. <ML> MAIN MENU / MATERIAL PROPERTIES / ISOTROPIC <ML> MAIN MENU / MATERIAL PROPERTIES / ISOTROPIC / YOUNG'S MODULUS > 200.0e9 Note: In this 1D elasticity problem, only Young's modulus needs to be specified. <ML> MAIN MENU / MATERIAL PROPERTIES / ISOTROPIC / OK 5c. Apply the material properties to all elements. <ML> MAIN MENU / MATERIAL PROPERTIES / ELEMENTS ADD Since the properties are being applied to all elements in the model, the simplest way to select the elements is to use the ALL EXISTING option. <ML> ALL: EXIST. 5d. Display all material properties for verification. <ML> MAIN MENU / MATERIAL PROPERTIES / ID MATERIALS

14 Application of the Finite Element Method Using MARC and Mentat 5-14 After verifying that material properties have been applied properly, turn off the material property ID's by repeating the last command. 5e. Return to the MAIN menu. <ML> MAIN MENU / MATERIAL PROPERTIES / MAIN 6. Specify the cross-sectional area of each element. 6a. Specify cross-sectional area for element one. 6a1. Set up a new geometric property set. <ML> MAIN MENU / GEOMETRIC PROPERTIES <ML> MAIN MENU / GEOMETRIC PROPERTIES / NEW <ML> MAIN MENU / GEOMETRIC PROPERTIES / NAME At the command line, enter a name for this geometric property set. > X1 6a2. Define the nature of the geometric property. <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D / TRUSS <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D / TRUSS / AREA > <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D / TRUSS / OK 6a3. Apply the geometric property to element one. <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D / ELEMENTS ADD <ML> on element 1 <MR> or END LIST 6b. Specify cross-sectional area for element two. 6b1. Set up a new geometric property set. <ML> MAIN MENU / GEOMETRIC PROPERTIES <ML> MAIN MENU / GEOMETRIC PROPERTIES / NEW <ML> MAIN MENU / GEOMETRIC PROPERTIES / NAME At the command line, enter a name for this geometric property set. > X2

15 Application of the Finite Element Method Using MARC and Mentat b2. Define the nature of the geometric property. <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D / TRUSS <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D / TRUSS / AREA The cross-sectional area of element two is taken as one-half of the cross-sectional area of the bar because the element lies along the plane of symmetry. > <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D / TRUSS / OK 6b3. Apply the geometric property to element two. <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D / ELEMENTS ADD <ML> on element 2 <MR> or END LIST 6c. Specify cross-sectional area for element three. 6c1. Set up a new geometric property set. <ML> MAIN MENU / GEOMETRIC PROPERTIES <ML> MAIN MENU / GEOMETRIC PROPERTIES / NEW <ML> MAIN MENU / GEOMETRIC PROPERTIES / NAME At the command line, enter a name for this geometric property set. > X3 6c2. Define the nature of the geometric property. <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D / TRUSS <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D / TRUSS / AREA The cross-sectional area of element two is taken as one-half of the cross-sectional area of the bar because the element lies along the plane of symmetry. > <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D / TRUSS / OK 6c3. Apply the geometric property to element three. <ML> MAIN MENU / GEOMETRIC PROPERTIES / 3D / ELEMENTS ADD <ML> on element 3

16 Application of the Finite Element Method Using MARC and Mentat 5-16 <MR> or END LIST 6d. Display all geometric properties for verification. <ML> MAIN MENU / GEOMETRIC PROPERTIES / ID GEOMETRIES After verifying that geometric properties have been applied properly, turn off the geometric property ID's by repeating the last command. 6e. Return to the MAIN menu. <ML> MAIN MENU / GEOMETRIC PROPERTIES / MAIN 7. Prepare the loadcase. <ML> MAIN MENU / LOADCASES <ML> MAIN MENU / LOADCASES / MECHANICAL <ML> MAIN MENU / LOADCASES / MECHANICAL / STATIC <ML> MAIN MENU / LOADCASES / MECHANICAL / STATIC / LOADS Verify that all loads (i.e., boundary constraints and point load) created in step 4 are selected. The small box to the immediate left of all selected loads will be highlighted. If they are not already selected, then select them using the <ML>. OK <ML> MAIN MENU / LOADCASES / MECHANICAL / STATIC / LOADS / <ML> MAIN MENU / LOADCASES / MECHANICAL / STATIC / OK <ML> MAIN MENU / LOADCASES / MECHANICAL / MAIN 8. Prepare the job for execution. 8a. Specify the analysis class and select loadcases. <ML> MAIN MENU / JOBS <ML> MAIN MENU / JOBS / MECHANICAL <ML> MAIN MENU / JOBS / MECHANICAL / lcase1 8b. Select the analysis dimension. <ML> MAIN MENU / JOBS / MECHANICAL / 3D 8c. Select the desired output variables. <ML> MAIN MENU / JOBS / MECHANICAL / JOB RESULTS <ML> MAIN MENU / JOBS / MECHANICAL / JOB RESULTS / stress <ML> MAIN MENU / JOBS / MECHANICAL / JOB RESULTS / OK <ML> MAIN MENU / JOBS / MECHANICAL / OK 8d. Select the element to use in the analysis.

17 Application of the Finite Element Method Using MARC and Mentat 5-17 <ML> MAIN MENU / JOBS / ELEMENT TYPES <ML> MAIN MENU / JOBS / ELEMENT TYPES / MECHANICAL <ML> MAIN MENU / JOBS / ELEMENT TYPES / MECHANICAL / 3D TRUSS/BEAM Select element number 9, a two-noded line truss element. <ML> MAIN MENU / JOBS / ELEMENT TYPES / 3D TRUSS/BEAM / 9 <ML> MAIN MENU / JOBS / ELEMENT TYPES / 3D TRUSS/BEAM / OK 8e. Apply the element selection to all elements. Since the element type is being applied to all elements in the model, the simplest way to select the elements is to use the ALL EXISTING option. <ML> ALL: EXIST. 8f. Display all element types for verification. <ML> MAIN MENU / JOBS / ELEMENT TYPES / ID TYPES After verifying that element types have been applied properly, turn off the element type ID's by repeating the last command. <ML> MAIN MENU / JOBS / ELEMENT TYPES / RETURN 8g. SAVE THE MODEL! <ML> STATIC MENU / FILES <ML> STATIC MENU / FILES / SAVE AS In the box to the right side of the SELECTION heading, type in the name of the file that you want to create. The name should be of the form FILENAME.mud, where FILENAME is a name that you choose. <ML> STATIC MENU / FILES / SAVE AS / OK <ML> STATIC MENU / FILES / RETURN 8h. Execute the analysis. <ML> MAIN MENU / JOBS / RUN <ML> MAIN MENU / JOBS / RUN / SUBMIT 1 8i. Monitor the status of the job. <ML> MAIN MENU / JOBS / RUN / MONITOR When the job has completed, the STATUS will read: Complete.

18 Application of the Finite Element Method Using MARC and Mentat 5-18 A successful run will have an EXIT NUMBER of Any other exit number indicates that an error occurred during the analysis, probably due to an error in the model. 8j. Return to the main menu. <ML> MAIN MENU / JOBS / RUN / OK <ML> MAIN MENU / JOBS / RETURN 9. Postprocess the results. 9a. Open the results file and display the results. <ML> MAIN MENU / RESULTS <ML> MAIN MENU / RESULTS / OPEN DEFAULT <ML> MAIN MENU / RESULTS / NEXT INCREMENT <ML> MAIN MENU / RESULTS / BEAM CONTOUR A contour plot of the X-displacement should appear. Other output variables can also be plotted, if desired. 9b. Display a different output variable. <ML> MAIN MENU / RESULTS / SCALAR <ML> MAIN MENU / RESULTS / SCALAR / Comp 11 of Stress <ML> MAIN MENU / RESULTS / SCALAR / OK A contour plot of the stress in the X-direction should appear. 9c. Display both the original and the deformed geometry of the truss. <ML> MAIN MENU / RESULTS / DEF & ORIG If it is difficult to distinguish between the deformed and the original mesh, then it may be necessary to scale up (i.e., exaggerate) the deformations. This is done as follows: <ML> MAIN MENU / RESULTS / DEFORMED SHAPE SETTINGS <ML> MAIN MENU / RESULTS / DEFORMED SHAPE SETTINGS / AUTOMATIC In the above step, Mentat will automatically choose a scaling factor for the displacements that it thinks is appropriate. If the result of this step is not satisfactory, then the user may choose a scaling factor by selecting MANUAL instead of AUTOMATIC under the DEFORMATION SCALING heading. After selecting MANUAL, the user will be prompted to input a real number to be used as the scaling factor. Trial and error is often needed to identify a suitable factor.

19 Application of the Finite Element Method Using MARC and Mentat d. Display nodal values of the output variable. <ML> MAIN MENU / RESULTS / NUMERICS It is sometimes difficult to read the values when the entire model is displayed. To more easily view the nodal values, zoom in on the region of interest using the zoom box on the static menu. To view the entire model again, use the FILL command on the static menu. 10. Using the analysis results, determine if any of the truss members is expected to experience material failure or to buckle due to the applied loading. FOR THE MODEL ANALYZED, TURN IN: (1) A "numerics" plot for: (a) x-displacement (b) y-displacement (c) x-component of stress For the above plots, show both the undeformed and the deformed geometry. (2) A discussion of failure or buckling in the members.