2.31 Assignment 8 Due Wed, Oct 31 at 9:30 am Ok, I know all the MIT undergraduates thought that they had had enough of the hole in the plate story in the 2.002 lab to last the rest of their lives, and instead here we are again. We have a 6X6 square plate (H=3m W=3 m, unit thickness) with a hole of radius a =1m right in the middle. For the nostalgic, the graduate students, and the people that have burned their 2.002 handouts, I have attached the corresponding 2.002 document at the end of the assignment. We are going to compare FE and theoretical evaluation of stress concentrations for different meshes and element types. The plate is loaded with an axial stress acting on the top edge of the plate σσσσtop= 100 MPa. And yes you guessed it, the material is steel. 1) Pen and paper work (no FE) (aka: The Return of 2.002) For the case of a circular hole in an infinitely wide plate, using equation 1b in the 2.002 handout to derive an analytical expression for the distribution of axial stress along the equatorial ligament (θ=π/2) as a function of distance (r) from the center of the hole. Plot the curve for r =aÆW For the actual case of a circular hole in the finite width plate above, obtain the value of the stress concentration factor, Ktg , from the graph in Figure 3 of the 2.002 handout. Compare the max level of stress σσσσmax [=axial stress at the edge of the hole] in the finite plate, to that obtained above for the infinite width plate. Explain the reasons for the discrepancy between the two stress concentration values. 2) FE model Create FE models of the component using plane stress elements. The objective is to investigate cost/benefits of mesh refinement, mesh structuring as well as the effects of various element choices. Please construct and run the following FE models. Make a table where for each model, you give the max s22 stress and the CPU time it took to run the model (it is at the bottom of the .dat file: USER TIME (SEC)). For all models partition the plate in 4 quadrants (top-bottom, left-right) before you seed, so as to have symmetric meshes. 2W2Haσσσσtop r θ1) Coarse quadrilateral meshes : seed=1.0. Try four types of quadrilateral elements: (1a) linear, reduced integration elements (CPS4R), (1b): linear, full integration (CPS4), (1c): quadratic, reduced integration (CPS8R), (1d): quadratic full integration(CPS8). 2) Refined quadrilateral meshes uniform seed, seed= 0.3. Try the same 4 types of elements: 2a:CPS4R, 2b:CPS4, 2c:CPS8R, 2d:CPS8. 3) Very refined quadrilateral meshes uniform seed, seed= 0.1. Try the same 4 types of elements: 3a:CPS4R, 3b:CPS4, 3c:CPS8R, 3d:CPS8. 4) Refined quadrilateral meshes with biased seeds. Give a general seed of 0.3, and then give local (edge) seeds around the hole (8 elements for each of the four quarters), and biased seeds on the 4 ligaments on the x,y axes. Bias with a factor 6 and put 10 elements along these edges. I get a mesh that looks like this Try the same 4 types of elements: 4a:CPS4R, 4b:CPS4, 4c:CPS8R, 4d:CPS8. 5) One last set of models using triangular elements. Get rid of the edge seed, and of the global seeds. Reseed with a global seed of 0.3. Use the MeshÆControl tool : select the entire model and assign Tri element shape to the whole plate. Now go on as usual: assign element type (Tri) linear, and mesh the part. Run a job with linear elements (5a: CPS3). Change the elements to quadratic standard formulation (CPS6) and run your last job: 5b:CPS6. Comment on the results of this parametric study. What did you learn in terms of cost/benefits of using different discretizations/formulations? The stress contours are symmetric about the 2-axis (left-right) but not about the 1-axis (top-bottom): why? Where do you think that the inconsistency with the theoretical σσσσmax comes from? I will demonstrate the following in class: you do not need to do this Compare the estimated stress fields for models 2a, 2b, 2c, 2d, 5a, 5b, by superposing the FE profiles of axial stress along the ligament with the theoretical estimate of the stress profile (for the hole in the infinite plate). Also on the same plot, mark the theoretical σσσσmax for the finite plate obtained using the stress concentration factor, Ktg. Comment on the consistencies/inconstistencies of the stress field for different FE models, as opposed to just looking at the max stress level.1Massachusetts Institute of TechnologyDepartment of Mechanical EngineeringCambridge, MA 021392.002 Mechanics and Materials IIJust when you thought you could forget about this....The stress distribution around a hole in an infinite plateThe stress distributions around a central hole can be estimated for the simple case ofan infinitely wide plate subjected to elastic tensile loading. The overall stress distributionsin the plate are given by (Figure 1):σrr=σ21−a2r2+σ21+ 3a4r4− 4a2r2cos2θ (1a)σθθ=σ21+a2r2−σ21+ 3a4r4cos2θ (1b)τrθ=−σ21−3a4r4+2a2r2sin2θ (1c)where σ is the magnitude of the applied uniaxial far-field stress.2Figure 1 Stress distribution around a circular hole.For r = a, the stress distribution around the hole is given by:σrr = 0 (2a)σθθ = σ (1 - 2cos2θ) (2b)τrθ = 0 (2c)For θ = π/2, σθθ =σmax= 3σ. This corresponds to the peak stress of the stress distribution(Figure 2). Hence, we may state that the stress concentration factor (the ratio of themaximum stress to the far field stress) for this notch geometry is equal to 3. The conceptof a stress concentration factor will be further discussed in the following section.3However, it is important to note that the stresses in the immediate vicinity of the hole aremuch higher than the far field stresses. Failure may therefore initiate prematurely from theedge of hole.Figure 2 Distribution of σθθ: (a) around the hole, and (b) along the ligament (θ = π/2).The stress along the ligament , where θ = π/2 (Figure 2), decreases rapidly withincreasing distance from the notch. This is a clear example of the St. Venant principlewhich states that the perturbations in the stress field due to a small geometrical discontinuity4(with size d) are localized within regions with ~ 3d from the discontinuity. The stresslevels outside this region are therefore close to the nominal applied stress levels.Stress Concentration Factors For Different