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Beyond Difference: From Canonical Geography to Hybrid Geographies

10 pages
  • mémoire
  • exposé - matière : philosophy
  • exposé
  • expression écrite
CENTENN IAL FORUM Beyond Difference: From Canonical Geography to Hybrid Geographies Mei-Po Kwan Department of Geography, The Ohio State University Geography as a discipline has evolved into a fieldof enormous breadth in the last century. Whiledistinctive theoretical perspectives have emerged in different periods, the ebbs and flows of new approaches are often marked by vitriolic contestations. In response to articulations of new visions of what ge- ography is or should be, debates often turned into an- tagonistic discourses that are surprisingly tenacious once being set in motion (Martin 1989).
  • fluid identities
  • structure of scientific revolutions
  • feminist geography
  • geographical analysis
  • human geography
  • geography
  • nature
  • discipline
  • science
  • research
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Problems taken from “ANSYS Tutorial11.0” by Kent L. Lawrence(Worked With Femap and NeiNastran) by John R. Carr 10/2/2010 Lesson 1TrussesThe first model worked from the ANSYS Tutorial was a shelf model that was 3D originally, but with applied symmetry for both the geometry and loads was modeled as a 2D model with rod elements. The total downward load was 1200 lb in the original physical model, but was reduced to 600 lb with application of ½ symmetry which meant that bending for the shelf was neglected. This 600 lb load was shared at the two remaining nodes, applying 300 lb to each node in the ydirection. A generic steel was used as the material for the rods with the following material properties: E = 30e6 psi and poison’s ratio of0.27. No tensile or compressive yield strength was given for the model, but for maximum stress results comparison a value of 215,000 psi will be used, which was taken from the AISI 4340 Steel Femap material library. The rod elements had a rectangular cross sectional area of 0.50 x 0.25 (inches) so the area was inputted as 0.125 2 in . Nodes 1 and 3 were modeled as pinned with no displacement in any of the three linear directions. The model with nodes, elements, constraints and loading is given below:
Figure 1FEA model of the shelf truss with 3 nodes, 2 elements, constraints and loads. A linear elastic static analysis was performed giving the following results shown in Figure 2:
Figure 2Contour plot of maximum rod equivalent stress of 4000 psi and a total translation of 0.00867 inch. ANSYS results: maximum total translation of 0.008667 inch downward and max element stress of 4000 psi. With the above NEi Nastran results of maximum stress of 4000 psi compared to a yield strength of 215,000, no stress magnitudes or deflections are a safety issue for this model and its intended and declared usage. The second truss model worked was a modified truss where the steel rod of element 2 is replaced by a composite tube. The material properties for the composite material used werethe following: E = 1.2e7 psi and poison’s ratio of 0.3, along with an area of 0.35 2 in . We also include a prescribed displacement of 0.01 inch to the left of node 3 (in thex direction). This model with all constraints and loads prior to analysis is shown below:
Figure 3Modified truss model with a composite tube used for element 2 and a prescribed displacement of 0.01 inch at node 3.
A linear elastic static analysis run produced the following results:
Figure 4Modified truss contour plot giving a maximum stress of 3200 psi and a maximum displacement equal to the prescribed displacement at node 3 of 0.01 inch. ANSYS results: No results were given for the modified truss. Lesson 2Plane Stress Plane Strain Model 1Plate With Central Hole A plate with dimensions of 1.0 m x 0.4 m (length x width) and a thickness of 0.01 m has a centrally located hole with a diameter of 0.2 m. It has a load applied to each end (RHS and LHS) and due to geometry and loading symmetry can be modeled as a ¼ plate with symmetry BCs applied at the cut planes The material is steel with the following 2 properties: E = 2.07e11 N/mA load equivalent to aand a poison’s ratio of 0.29. 2 pressure load of p = 1.0 N/m is applied to the RHS end of the ¼ symmetry model. Because of problems loading a pressure load to a curve in Femap, the static loading of F = p * A = (1.0) * (0.01) * (0.2) = 0.002 N was applied to the 9 nodes of the RHS, thus each load was F/9 = 0.0002222 N for an equivalent static loading. On the LHS of the model xsymmetry constraints were applied (T1= R5= R6= 0) and along the bottom edge ysymmetry constraints were applied (T2= R4= R6= 0). To prevent a singularity error while running NEi Nastran further constrains were added by setting T3= 0 for all the nodes, insuring a plane stress problem. The model with loads and constraints is given below with a mesh density of esize = 0.025 m:
Figure 5Plate with central hole ¼ symmetry model with constraints and loading prior to solution. A linear elastic static analysis was performed giving the following results:
Figure 6: Plate with central hole FEA static analysis results (esize = 0.025) with give a maximum total translation of 3.438e12 m and a maximum shell von mises top stress of 4.029 Pa. ANSYS results with a refined mesh give a total translation of 3.21e12 and a maximum stress of 4.386 Pa. The mesh was refined to be esize = 0.0250/2 = 0.0125 m which gave 17 nodes and 16 elements around the central hole. This model, prior to analysis, is shown below:
Figure 7Refined mesh (esize = 0.0125 m) for the plate with central hole ¼ symmetry model.
Figure 8Contour plot of the plate with central hole with a refined mesh of esize = 0.0125, giving total translation of 3.363e12 m and maximum shell von mises top stress of 4.236 Pa which agrees better with the ANSYS results. Model 2Seatbelt Component A seatbelt “tongue” was modeled with ½ symmetry with a horizontal cut plane giving the upper portion of the tongue. Keypoints were designated to represent the geometry of the model and curvelines were drawn for the straight lines and arcs were drawn to represent the 2 fillets on the inner portion of the slot. The left edge was constrained such that Uxwas zero and the lower two curves where constrained such that Uy= 0 was used to represent yaxis symmetry. An element size of esize = 0.05 inch was used for the initial mesh. The load on the RHS of the slot was 1000 lb for the full model and 500 lb for the ½ symmetry model. Using the 9 nodes along the slot face to evenly distribute the load gave a nodal force of 500/9 = 55.56 lb for each of the 9 nodes. Also, to insure a planar
problem and avoid singularity errors with NEi Nastran, all nodes were constrained such that Tz= 0. The model prior to the results is shown below:
Figure 9FEA model with esize = 0.05 inch used to model the seatbelt component tongue. A linear elastic static analysis was performed and gave the following results:
Figure 10Contour results for esize = 0.05 for the seatbelt tongue model giving a total translation of 0.00174 inch and a max shell von mises top stress of 95631 psi. Mesh refinement was performed on the model with an esize of 0.025 inch being used and an equivalent nodal static load of 500/16 = 31.25 lb for each node on the slot. The model, without constraints, is shown below:
Figure 11The seatbelt component tongue model with esize = 0.025 inch. The results of a linear elastic static analysis for an esize of 0.025 inch is shown below:
Figure 12Contour plot of refined mesh (esize = 0.025) FEA model for the seatbelt component tongue given a total translation of 0.00178 inch and max shell von mises top stress of 111315 psi. The mesh was further refined to esize = 0.0125 with the equivalent nodal static force on the slot being = 500/31 = 16.129 lb and the following results from the second mesh refinement are shown below:
Figure 13Contour plot of the seatbelt component tongue with esize = 0.0125 giving total translation of 0.00178 inch and max shell von mises top stress of 138,090 psi. The results shown in Figure 13 represent the converged results and agree well with the ANSYS results of 140,000 psi. Lesson 3Axisymmetric Problems A steel pressure vessel with planar ends is subjected to an internal pressure of 35 MPa. The vessel has an outer diameter of 200 mm with an overall length of 400 mm and a wall thickness of 25 mm with an inside radius of 25 mm at the ends. Only the top or bottom half need be considered and either half has axisymmetric symmetry in geometry and loading, thus an axisymmetric “slice” is modeled. The cross section is created with keypoints and lines along with a single arc with the appropriate dimensions. A very fine mesh of 0.0025 was initially used to prevent the need for mesh refinement. The model prior to solution is shown below:
Figure 14Cylindrical pressure vessel section cut that was modeling using a mesh of esize = 0.0025 m. A linear elastic static analysis was run and produced the following results.
Figure 15A full view of the contour plot for the cylindrical pressure vessel FEA results giving a total translation of 0.0000997 m and max axisym von mises stress of 205.4 Pa.
Figure 15A zoom view of the contour plot for the cylindrical pressure vessel FEA results giving a total translation of 0.0000997 m and max axisym von mises stress of 205.4 Pa. Lesson 43D Problems Model filepressure_vessel_lesson4.mod A 3D version of the previous axisymmetric model was created by revolving the cross section 90°The same material was used with steelto create a solid. SI units having the
following properties: E = 200 GPa, poison’s ratio is 0.3, and the yield strength = 330 MPa. A mesh size of 0.0025 was used in a tri mesh with midside nodes creating 1959 elements and 3397 nodes. Symmetry BCs were applied to the appropriate faces or surfaces of the 3D model and an internal pressure load of 35e6 Pa was applied to the three internal faces. The model prior to solution is shown below:
Figure 16FEA model for the 3D pressure vessel with loads and constraints. The analysis was run for a linear elastic static load case and the following results were obtained:
Figure 17Analysis results for the 3D pressure vessel model giving a total translation of 0.0000929 m and a max solid von mises stress of 210.2 MPa well within the yield strength limit of the material. Note that the 3D results are in good agreement with the axisymmetric results given in Lesson 3.