Elastic Analysis of a Bending Plate

Transcription

1 analys: linear static. constr: suppor. elemen: plate q12pl. load: elemen face force. materi: elasti isotro. option: direct units. post: binary ndiana. pre: dianai. result: cauchy displa extern force green moment reacti strain stress total. Elastic Analysis of a Bending Plate

2 Outline 1 Description 2 Finite Element Model 2.1 Units 2.2 Geometry definition 2.3 Properties 2.4 Boundary conditions Constraints Load 2.5 Meshing 3 Structural linear static analysis 3.1 Analysis commands 4 Results 4.1 Reaction forces FBZ 4.2 Contour plot Vertical displacements DtZ 4.3 Contour plot Shear force Qxz 4.4 Contour plot Bending moments Mxx 4.5 Diagram Bending moment Mxx 4.6 Diagram Bending moment Mxy 4.7 Diagram Distributed forces Qxz Elastic Analysis of a Bending Plate 2/23

3 1 Description This tutorial presents the linear-elastic analysis of the plate structure presented by Blaauwendraad 1. 2 m 2 m The plate has the shape presented in [Fig. 1], with two long edges of 4 m and two short edges of 2 m. The thickness of the plate is 160 mm. The material properties are: Modulus of elasticity E = N/m 2 Poisson s ration ν = 0.2 A distributed load of 1000 N/m 2 is applied orthogonally the plate. The orthogonal axes x and y are aligned with the bottom and left long edges, respectively. The support in the corner restricts displacement normal to the plate u z and free the rotations in x and y. The two short edges are simply-supported. All the other edges are free. y 2 m 2 m We will use four-node quadrilateral plate bending Mindlin-Reissner elements. z x Figure 1: Schematic representation of the model (Blaauwendraad1). 1 J. Blaauwendraad, Plates and FEM Surprises and Pitfalls, pp , Springer, 2010,. Elastic Analysis of a Bending Plate 3/23

4 2 Finite Element Model For the modeling session we start a new project for structural analysis [Fig. 2]. The dimensions of the domain for the 3D model are set equal to 10 km. We will use linear hexagonal finite elements. Main menu File New [Fig. 2] Figure 2: New project dialog Elastic Analysis of a Bending Plate 4/23

5 2.1 Units We choose meter for the unit Length, kilogram for Mass, Newton for Force (SI units). Geometry browser Reference system Units [Fig. 3] Property Panel [Fig. 4] Figure 3: Geometry browser Figure 4: Property Panel - Units Elastic Analysis of a Bending Plate 5/23

6 2.2 Geometry definition We create a polygon sheet with the size and the dimensions of the plate shown in [Fig. 1]. The set of coordinets detoniting the plate nodes is listed in [Table 1]. Main Menu Geometry Create Add polygon sheet [Fig. 5] Viewer Hide workingplane Viewer Viewpoints Top View Viewer Fit all [Fig. 6] x y z [m] [m] [m] Table 1: Node coordinates of the polygon sheet Figure 5: Geometry - Add polygon sheet Figure 6: Top view Plate sheet Elastic Analysis of a Bending Plate 6/23

7 2.3 Properties We assign the material and geometrical properties to the plate. Regarding the geometry, we need to specify a unique direction for the plate elements local axis [Fig. 10] (this will lead congruent contour plots of the element results, such as distributed moments and forces). Main Menu Geometry Analysis Property assignments [Fig. 7] Property assignments Add new material [Fig. 8] [Fig. 9] Property assignments Add new geometry [Fig. 10] Figure 8: Add new material Figure 7: Property assignments Figure 9: Material properties Figure 10: Geometrical properties Elastic Analysis of a Bending Plate 7/23

8 2.4 Boundary conditions Constraints Since the plate elements do not have membrane strains, we constraine the model only to vertical displacements. We first define the support in the vertex in the left bottom corner of the plate as illustrated in [Fig. 1]. Main menu Geometry Analysis Attach support [Fig. 11] [Fig. 12] Figure 11: Attach supports Figure 12: Constrained vertex Elastic Analysis of a Bending Plate 8/23

9 Then, we define the supports along the edges indicated in [Fig. 1]. Main menu Geometry Analysis Attach support [Fig. 13] [Fig. 14] Figure 13: Attach supports Figure 14: Constrained edges Elastic Analysis of a Bending Plate 9/23

10 2.4.2 Load We create the distributed load applied perpendicularly to the plate. Main menu Geometry Analysis Attach load [Fig. 15] [Fig. 16] Figure 15: Attach load Figure 16: Applied distributed load Elastic Analysis of a Bending Plate 10/23

11 2.5 Meshing We set the element size as m and generate the mesh. Main Menu Geometry Analysis Set mesh properties [Fig. 17] Main Menu Geometry Analysis Generate mesh Viewer Viewpoints Isometric view 1 [Fig. 18] Figure 17: Mesh properties Figure 18: Finite element mesh Elastic Analysis of a Bending Plate 11/23

12 3 Structural linear static analysis 3.1 Analysis commands We will perform a linear structural analysis. Main Menu Analysis New Analysis Analysis browser Right click ( ) Analysis1 Rename Linsta [Fig. 19] Analysis browser Right click ( ) 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 Elastic Analysis of a Bending Plate 12/23

13 4 Results 4.1 Reaction forces FBZ To validate the results we check that the reaction forces are in equilibrium with the applied load. Since the load has only the vertical component and the plate elements do not have membrane strain, we check only the reaction forces along the Z-axis (FBZ). Results browser Output linear static analysis Nodal results Reaction Forces FBZ [Fig. 22] [Fig. 23] Figure 22: Results browser - show table Figure 23: Nodal reaction forces FBZ The summation of all nodal reaction forces is equal to N, in equilibrium with the total applied load: q A plate = 1000 N/m 2 12 m 2 = N Elastic Analysis of a Bending Plate 13/23

14 4.2 Contour plot Vertical displacements DtZ We will make a contour plot of the vertical displacements DtZ [Fig. 26]. In order to have a smooth contour plot, we choose continuous color scale. Results browser Output linear static analysis Nodal results Displacements DtZ [Fig. 24] Property panel Result view settings Contour plot settings [Fig. 25] [Fig. 26] Figure 24: Results browser Figure 25: Results settings Figure 26: Vertical displacement (DtZ) Elastic Analysis of a Bending Plate 14/23

15 4.3 Contour plot Shear force Qxz We make the contour plot of the shear force Qxz [Fig. 28]. Results browser Output linear static analysis Element results Distributed Forces Qxz [Fig. 27] [Fig. 28] Figure 27: Results browser Figure 28: Shear force (Qxz) Elastic Analysis of a Bending Plate 15/23

16 4.4 Contour plot Bending moments Mxx Similarly, we produce the contour plot of the bending moments Mxx [Fig. 30]. Results browser Output linear static analysis Element results Distributed Moments Mxx [Fig. 29] [Fig. 30] Figure 29: Results browser Figure 30: Bending moment (Mxx) Elastic Analysis of a Bending Plate 16/23

17 4.5 Diagram Bending moment Mxx We now create a diagram plot for the distributed bending moment Mxx along the lines AB, CD, EF and GH depicted in [Fig. 31]. 2 Lines AB, CD and EF are used to compare our results with those in [1]. Since these lines are coincident with the mesh grid, by plotting the results along line FG we show that can make the diagram of the element results along any arbitrary line. F B D 2 H node coordinates A (0.5, 0, 0) B (0.5, 4, 0) C (2, 0, 0) D (2, 4, 0) E (0, 0, 0) F (0, 4, 0) G (4, 0, 0) H (0, 2, 0) y z E x A 0.5 C 2 G Table 2: Nodes coordinates (in meters) Figure 31: Schematic representation of the lines where to plot the diagram of the distributed moments Mxx (mesures are in meters) 2 In this case, the lines coincides with mesh lines. Nevertheless, this is not mandatory (the axis of the diagram can pass through a generic pair of coordinates on the plate surface). Elastic Analysis of a Bending Plate 17/23

18 We define the axis of the lines presented in [Fig. 31] in. We will start with line AB. Results browser Output linear static analysis Element results Distributed Moments Mxx [Fig. 29] [Fig. 30] View setting Apply view setting Property panel Result Probing curve setting Add Curve [Fig. 32] Property panel Result Probing curve setting Double click probe-curve Rename Line AB [Fig. 33] Property panel Result Probing curve setting Line AB Number of intervals between points 16 [Fig. 33] Property panel Result Probing curve setting Line AB add Point coordinates [Fig. 34] Property panel Result Probing curve setting Line AB Point coordinates [Fig. 35] Figure 32: Add curve Figure 33: Rename curve and interval number Figure 34: Add coordinates for Line AB Figure 35: Line AB point coordinates Elastic Analysis of a Bending Plate 18/23

19 Similarly, we add the probing curves for lines CD, EF and GH. For all lines we set the number of intervals between points equal to 16. Then, we increase the scale factor of the diagrams from 1 to 10 to improve the visualization of the diagram. View setting Apply view setting Property panel Result Probing curve setting Scale factor 10 [Fig. 36] Results browser Show absolute deformation [Fig. 37] Figure 36: Scale factor Figure 37: Diagrams of Mxx Elastic Analysis of a Bending Plate 19/23

20 For a detailed representation of these results, it is possible to visualize each single diagram on a Cartesian plane as shown in [Fig. 39]. Results browser Output linear static analysis Element results Distributed Moments Right click ( ) Mxx Show contour probe [Fig. 38] Figure 38: Show contour probe Figure 39: Bending moment Mxx along line CD These results are in agreement with those in [Figure15.3, pag. 292, 1]. Elastic Analysis of a Bending Plate 20/23

21 4.6 Diagram Bending moment Mxy We can use the same lines to show other results diagrams. Here, for example, we show the variation of Mxy along line EF [Fig. 41]. Results browser Output linear static analysis Element results Distributed Moments Right click ( ) Mxy [Fig. 40] Figure 40: Show contour probe Figure 41: Bending moment Mxy along line EF These results are in agreement with those in [Figure15.4, pag. 293, 1]. Elastic Analysis of a Bending Plate 21/23

22 4.7 Diagram Distributed forces Qxz We show the variation of Qxz along line AB [Fig. 43]. Results browser Output linear static analysis Element results Distributed Forces Right click ( ) Qxz [Fig. 42] Figure 42: Show contour probe Figure 43: Bending moment Qxz along line AB These results are in agreement with those in [Figure15.5, pag. 293, 1]. Elastic Analysis of a Bending Plate 22/23

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