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- Theory of Elasticity and Plasticity - TEP
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Module2/Lesson2 Module 2: Analysis of Stress 2.2.1 PRINCIPAL STRESS IN THREE DIMENSIONS For the three-dimensional case, for principal stresses it is required that three planes of zero shear stress exist, that these planes are mutually perpendicular, and that on these planes the normal stresses have maximum or minimum values. As discussed earlier, these normal stresses are referred to as principal stresses, usually denoted by s1, s2 and s3. The largest stress is represented by s1 and the smallest by s3. Again considering an oblique plane Equation (2.25). x ¢ , the normal stress acting on this plane is given by the s x¢ = sx l2 + sy m2 + sz n2 + 2 (txy lm + tyz mn + txz ln) (2.27) The problem here is to determine the extreme or stationary values of s x¢ . To accomplish this, we examine the variation of s x¢ relative to the direction cosines. As l, m and n are not independent, but connected by l2 + m2 + n2 = 1, only l and m may be regarded as independent variables. Thus, ¶s x ' = 0, ¶l ¶s x ' =0 ¶m (2.27a) Differentiating Equation (2.27), in terms of the quantities in Equations (2.22a), (2.22b), (2.22c), we obtain ¶n = 0, ¶l ¶n Ty + Tz = 0, ¶m T x+ T z (2.27b) From n2 = 1 - l2 - m2, we have ¶n l =¶l n and ¶n m =¶m n Introducing the above into Equation (2.27b), the following relationship between the components of T and n is determined 1 Applied Elasticity for Engineers T.G.Sitharam & L.GovindaRaju

Module2/Lesson2 T x T y Tz = = l m n (2.27c) These proportionalities indicate that the stress resultant must be parallel to the unit normal and therefore contains no shear component. Therefore from Equations (2.22a), (2.22b), (2.22c) we can write as below denoting the principal stress by s P Tx = s P l Ty = s P m T z = sP n (2.27d) These expressions together with Equations (2.22a), (2.22b), (2.22c) lead to (sx - sP)l + txy m + txz n = 0 txy l+(sy - sP) m + tyz n = 0 (2.28) txz l + tyz m + (sz - sP) n = 0 A non-trivial solution for the direction cosines requires that the characteristic determinant should vanish. (s x - s P ) t xy t xz t xy t xz (s y - s P ) t yz t yz (s z - s P ) =0 (2.29) Expanding (2.29) leads to s P3 - I 1s P2 + I 2s P - I 3 = 0 (2.30) where I1 = sx + sy + sz (2.30a) I2 = sx sy + sy sz + szsx - t 2xy - t 2yz -t 2xz (2.30b) s x t xy t xz I3 = t xy s y t yz t xz t yz s z (2.30c) The three roots of Equation (2.30) are the principal stresses, corresponding to which are three sets of direction cosines that establish the relationship of the principal planes to the origin of the non-principal axes. 2.2.2 STRESS INVARIANTS Invariants mean those quantities that are unexchangeable and do not vary under different conditions. In the context of stress tensor, invariants are such quantities that do not change with rotation of axes or which remain unaffected under transformation, from one set of axes 2 Applied Elasticity for Engineers T.G.Sitharam & L.GovindaRaju

Module2/Lesson2 to another. Therefore, the combination of stresses at a point that do not change with the orientation of co-ordinate axes is called stress-invariants. Hence, from Equation (2.30) sx + sy + sz = I1 = First invariant of stress sxsy + sysz + szsx - t 2xy - t 2yz - t 2zx = I2 = Second invariant of stress sxsysz - sxt 2yz - syt 2xz - szt 2xy + 2txy tyz txz = I3 = Third invariant of stress 2.2.3 EQUILIBRIUM OF A DIFFERENTIAL ELEMENT Figure 2.11(a) Stress components acting on a plane element When a body is in equilibrium, any isolated part of the body is acted upon by an equilibrium set of forces. The small element with unit thickness shown in Figure 2.11(a) represents part 3 Applied Elasticity for Engineers T.G.Sitharam & L.GovindaRaju

Module2/Lesson2 of a body and therefore must be in equilibrium if the entire body is to be in equilibrium. It is to be noted that the components of stress generally vary from point to point in a stressed body. These variations are governed by the conditions of equilibrium of statics. Fulfillment of these conditions establishes certain relationships, known as the differential equations of equilibrium. These involve the derivatives of the stress components. Assume that sx, sy, txy, tyx are functions of X, Y but do not vary throughout the thickness (are independent of Z) and that the other stress components are zero. Also assume that the X and Y components of the body forces per unit volume, Fx and Fy, are independent of Z, and that the Z component of the body force Fz = 0. As the element is very small, the stress components may be considered to be distributed uniformly over each face. Now, taking moments of force about the lower left corner and equating to zero, ¶s y ö Dx æ ¶t ö Dy 1 æ + (t xy Dy ) - çç s y + Dy ÷÷Dx + ççt yx + yx Dy ÷÷DxDy 2 2 è ¶y 2 è ¶y ø ø ¶t æ ö ¶s x ö Dy Dx 1 æ - ççt xy + xy Dx ÷÷DxDy + ç s x + Dx ÷ Dy + s y Dx - t yx Dx + ¶x ¶x 2 2 2 è ø è ø (Fx DyDx ) Dy - ( Fy DxDy ) Dx = 0 2 2 - (s x Dy ) Neglecting the higher terms involving Dx, and Dy and simplifying, the above expression is reduced to txy Dx Dy = tyx Dx Dy or txy = tyx In a like manner, it may be shown that tyz = tzy and txz = tzx Now, from the equilibrium of forces in x-direction, we obtain ¶t yx ö æ ¶s x ö æ -sx Dy + ç s x + Dx ÷Dy + ççt yx + Dy ÷÷Dx - t yx Dx + Fx DxDy = 0 ¶x ¶y è ø è ø Simplifying, we get or ¶s x ¶t yx + + Fx = 0 ¶x ¶y ¶s x ¶t xy + + Fx = 0 ¶x ¶y 4 Applied Elasticity for Engineers T.G.Sitharam & L.GovindaRaju

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