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Unformatted text preview: 1IIIIIIStress Stress transformationstransformationsPrepared by: NasserPrepared by: NasserEddinEddinM., M., Ph.DPh.D2Stress transformationsStress transformations3OUTLINESOUTLINES• Problem Statement• Formulation of the Problem• Applying Equilibrium• Transforming the Shear Stress• Transforming the Normal Stress• The Stress Transformation Equations4A BalloonImagine that we isolate a smallelement of an inflated balloon.Most of the stresses found in a balloon run parallel to the surface of the balloon. Therefore, for this problem we will ignore any stress that occurs normal to the surface of the balloon. Assume that we can calculate these stresses in the plane of the balloon.5With Known StressesSpecifically, assume that we can calculate the stresses σx, σy, and τxyon the surface.Remember, τxyalways equals τyx, so we only need to calculate one of them6StressStress(Cont.)(Cont.)Just to remind you :The stresses σxand τxyare actually the normal and tangential components of the traction vector TxSimilarly, the stresses σyand τyxare the normal and tangential components of the traction vector Ty7Element orientationElement orientationLet's formalize the orientationof the element we isolated fromthe balloon. A normal vector hasbeen drawn on one side of theelement. By observation it shouldbe clear that this normal vector isequivalent to the unit vector i.Now let's rotate the element...What is the new orientationof the element?8Calculating a Normal Vectorfor the Rotated Element9Stress transformationStress transformationThe question we must answer whendealing with stress transformation is:"Knowing the value of the stress components for the original orientation of the element, what are the stresses on the element after it has been rotated?"10Stress transformation (Cont.)Stress transformation (Cont.)In our derivation of the stresstransformation equations, we use thetraction vector Tx'instead of its...
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 Spring '09
 D.A
 Shear, Stress

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