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FEA - Stress Levels & Concentration A change in the shape of a component carrying a load will have the effect of increasing the stress, nearly always at a concave corner. The degree of this stress concentration will depend primarily on the proportions of the change in shape and the type of load. Here - the effect on tension and bending is shown. Torsion will have a different level of concentration. This figure comes from Norton. Peterson has the most comprehensive collection of such graphs and formulae. 1

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The above strap is subjected to tension from the ends. The tensile stress in the strap should be inversely proportional to the cross sectional area of the strap. Since the thin middle of the strap is ½ the height of the ends, the stress at the middle could be expected to be about 2 times that at the fat ends, provided the shape plays no part ! Convex edge, external corner Concave edge, internal corner Tensile force 2

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Stress concentration of ~ 1.5 Stress concentration of ~ 3 3

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The concave corner has a stress level of about 1&1/2 that of the thin middle of the strap (the left end). The fat end has about half the stress of the thin middle, as expected. The convex corner has about 1/200 the stress of the corner !! This filleted corner can be said to generate a stress concentration of about 1.5. Had we analysed the strap with perfectly sharp convex corners we would have found a stress concentration of infinity ! Hence convex corners should be filleted to as large a radius as practical and convex corner may be chamfered A sizeable fillet has been added to the concave corner Convex corner left sharp 4

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Quarter ellipses come close to being ideal fillets, generating stress concentration of just over 1. Ideally components should be uniformly stressed, which would result in smoothly and continuously sculptured parts. Such parts are typically so costly that they turn up in military planes and F1 cars, The outline of a quarter of a long ellipse The under-stressed convex corner could be chamfered or a second quarter ellipse could be blended into the first. 5

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We can retain a shoulder if it is required in the machine assembly while reducing stress concentration by adopting geometries that reduce the stiffness of the shoulder. Here a hole is used, which is a little too large and not in the best of locations. Most tables of stress concentration factors for different loadings and geometric shapes come from polarised light studies 6

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FEA of Welded and Bolted bases 7

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L Brackets Thick & Thin There are in-out forces at the hole A on the 100 mm leg. This generates a bending moment on this leg. It can be resisted by a couple (two forces) at holes on the 60 mm leg Unfortunately bending moments tend to create the largest stresses and deflection in parts that we make using the sort of materials and sections we find useful. Consequently we try to get rid of the bending moment in the 100 mm leg by ‘triangulating’, ie adding a web that goes from the load to the reaction in straight line. 16 A

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The web welded to one side of the narrow bracket deals with the biggest bending moment, about the X axis but not the lesser moment about the Y axis. Of course the deflection about Y may be tolerable to you, but it will require a relatively heavier bracket, now for satellites and FSAE cars that is not good enough ! X Y 17

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Now if the boss is not going to sack you, could consider a web that removes both moments about X and Y axes at the same time. I have added bosses (very thick washers welded in place) at the bolt locations, because no matter what you do the highest stresses will be almost certainly around the bolt holes. These bosses are optional if you can tolerate distortion around the bolt holes Note this web is aligned with all the forces and reactions and has cut-outs intended to provide access with nuts, washers and tools. This solution may seem a little over the top but in many situations it is the design of choice. 18

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But the forces of evil will not rest … The not so smart graduate from the other university, will insist that two webs will be better than one and with sufficient holes access is not denied to real toolies. 19

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20 a) b) c) MR P Orlov, Mir publishing, Moscow, has the above figure in one of his books. We have a triangulated frame above and a cantilever beam below it. In a) the cantilever beam is of the same section as those making up the frame, its highest tress is about 10 000 times the stresses in the frame. In b) the diameter of the cantilever beam is increased so that its maximum stress is the same as that in the frame. Finally in c) the diameter is increased so that the deflection is the same as that in the frame. Please note what happened to the mass increase of the cantilever beam, X 64 X 120. But But mass may not be the whole consideration, because you may not have the room for a frame.

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