Prescribed Deformations

Transcription

1 u Prescribed Deformations

2 Outline 1 Description 2 Finite Element Model 2.1 Geometry Definition 2.2 Properties 2.3 Boundary Conditions Constraints Prescribed Deformation 2.4 Loads Dead Weight Load Combination Table 2.5 Meshing 3 Structural Linear Static Analysis 3.1 Analysis Commands 3.2 Vertical Displacement DtY Appendix A Additional Information Prescribed Deformations 2/17

3 1 Description In this tutorial we show how to apply an imposed displacement in. The model in Figure 1 represents a simply supported beam subjected to gravity load and a prescribed vertical displacement u = 0.01 m in the mid-section. The material is assumed linear elastic with a Young s modulus equal to 10 MPa and zero Poisson s ratio. The mass density is 700 kg/m 3. The cross section of the beam, that is constant throughout its length, has a rectangular shape: the height and width are equal to and 0.01 m, respectively. In Figure 2 we show the discretized model that we will create in. It is worth noting that to apply the imposed displacement u we first need to support the vertical displacement of the node at the mid-section. This will have some implications in the set up of the analysis. u = 0.01 m u = 0.01 m 0.5 m 0.5 m m Figure 1: Characteristic dimensions of the model in meters Figure 2: Discrete model Prescribed Deformations 3/17

4 2 Finite Element Model For the modeling session we start a new project for structural analysis [Fig. 3]. The dimensions of the domain for the 2D model are set equal to 10 m. Main menu File New project [Fig. 3] Figure 3: New project dialog Prescribed Deformations 4/17

5 2.1 Geometry Definition We create a first half of the beam. Then, by means of the command Array copy, we create the second half. Accordingly, the full geometry of the model will have a midspan node where we apply the vertical deformation u. Main Menu Geometry Create Add line [Fig. 4] Main Menu Geometry Modify Array copy [Fig. 5] Viewer Viewpoints Top View [Fig. 6] Figure 4: Add straight line Figure 5: Array copy beam 1 Figure 6: Top view Prescribed Deformations 5/17

6 2.2 Properties We assign the element class and the material and geometrical properties to the beams. We use Euler-Bernoulli beam finite elements (Class I beam). The material is linear elastic. Moreover, we need to specify the height and depth of the rectangular cross section of the beam (0.025 and 0.01 m, respectively). Main Menu Geometry Analysis Property assignments [Fig. 7] Property assignments Add new material [Fig. 8] [Fig. 9] Figure 7: Property assignments Figure 8: Add new material Figure 9: Edit material properties Prescribed Deformations 6/17

7 Property assignments Add new geometry [Fig. 10] Figure 10: Edit geometrical properties Prescribed Deformations 7/17

8 2.3 Boundary Conditions Constraints We support the left and right nodes of the beam in X- and Y-direction. Main Menu Geometry Analysis Attach support [Fig. 11] [Fig. 12] Figure 11: Apply constraints Figure 12: View constraints The correct set of constraints for a simply supported beam should have either the left or the right node constrained only in the Y-direction. Nevertheless, since the Poisson s ratio of the beam is null and geometry nonlinearities are not considered, the set constraints in Figure 11 is acceptable. Prescribed Deformations 8/17

9 2.3.2 Prescribed Deformation We apply a vertical displacement u = 0.01 m in the midspan node. As mentioned in Section 1, this is achieved by: applying a vertical constraint to the midspan node (Figure 13) and assigning a vertical prescribed deformation u at the same location (Figure 14). Main Menu Geometry Analysis Attach support [Fig. 13] Main menu Geometry Analysis Attach load [Fig. 14] [Fig. 15] Figure 13: Apply vertical constraint Figure 14: Apply vertical nodal deformation Figure 15: Vertical prescribed deformation Prescribed Deformations 9/17

10 2.4 Loads Dead Weight We include the effect of the dead weight in the analysis. Main menu Geometry Analysis Global load [Fig. 16] Figure 16: Attach global load - Dead weight Prescribed Deformations 10/17

11 2.4.2 Load Combination Table Since the material properties are linear elastic and geometric nonlinearities are not considered, the total deformation of the structure is derived by superposition of the results from the applied loads. Namely, we will create a load combination for the summation of the effects from dead weight and vertical displacement. Geometry Loads Open load combination table [Fig. 17] Figure 17: Load combination table Prescribed Deformations 11/17

12 2.5 Meshing We use the default discretization settings (i.e., all geometries will be discretized in 4 elements). Therefore, we just need to generate the mesh. Main Menu Geometry Analysis Generate mesh [Fig. 18] Figure 18: Finite element mesh Prescribed Deformations 12/17

13 3 Structural Linear Static Analysis 3.1 Analysis Commands We will perform a linear structural analysis. Main Menu Analysis New Analysis Analysis browser Analysis ( ) Rename Linsta [Fig. 19] Analysis browser LinSta ( ) Add command Structural linear static [Fig. 20] [Fig. 21] Main Menu Analysis Run Analysis Figure 19: Analysis window Figure 20: Add command Figure 21: Analysis tree Prescribed Deformations 13/17

14 3.2 Vertical Displacement DtY We inspect the vertical displacement DtY of the beam. To compare the contour plots from different load combinations, we set the minimum value of the color scale to -1.2e-2 m. Results browser Output linear static analysis Nodal results Displacements DtY [Fig. 22] Property panel Result Contour plot settings Color scale limits Specified value [Fig. 23] Property panel Result Contour plot settings Specified values Minimum value -1.2e-2 [Fig. 24] Figure 22: Results browser DtY Figure 23: Specified values for color scale limits Figure 24: Minimum value for color scale Prescribed Deformations 14/17

15 Finally, we display the deformed configurations and the corresponding contour plots for the three load combinations. Results browser Case Load-combination 1 [Fig. 25] Results browser Case Load-combination 2 [Fig. 26] Results browser Case Load-combination 3 [Fig. 27] Figure 25: Vertical displacement DtY - dead weight Figure 26: Vertical displacement DtY - vertical dispalcement Figure 27: Vertical displacement DtY - Load combination 3 Figure 25 shows the structural response of the beam subjected to gravity load: the displacement at the mid-section is equal to zero due to the vertical support. For the second load combination (see Figure 26), the vertical displacement at the mid-section is equal to 1 cm as enforced by the applied deformation. Finally, the results in Figure 26 have derived by superposition of the previous two results as defined the load combination table in Figure 17. Prescribed Deformations 15/17

16 Appendix A Additional Information Folder: Tutorials/PrescribedDef Number of elements 10 Keywords: analys: linear static. constr: suppor. elemen: beam class1 l6ben rectan. load: deform weight. materi: elasti isotro. option: direct. post: binary ndiana. pre: dianai. result: cauchy displa extern force green moment reacti strain stress total. Prescribed Deformations 16/17

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