Lausn
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BYG301G – CONTINUUM MECHANICS 1 Homeworks 4 & 5 – Fall 2012 Assigned: Thursday, 20 September 2012 Due: Thursday, 4 October 2012 8:20 Problem 1 A two-dimensional state of stress in the x, y plane is given by đđ§ = đđ§đ„ = đđ§đŠ = 0: đđ„ [đđđ ] = [đ đ„đŠ đđ„đŠ đđŠ ] a) Using general principal value theory, establish the eigenvalue problem and the characteristic equation, the stress invariants, and derive the following equations for the principal stresses đ1,2 = đđ„ + đđŠ đđ„ − đđŠ 2 ± √( ) + đđ„đŠ 2 2 2 Lausn: b) Use the stress transformation equations to show that the maximum shear stress equals đđ„ − đđŠ 2 đđđđ„ = ±√( ) + đđ„đŠ 2 2 Lausn: c) Using the invariants from (a) demonstrate the invariant nature of the principal stresses and maximum shear stresses by showing that 1 1 đ1,2 = đŒ1 ± √đŒ1 2 − 4đŒ2 2 2 and 1 đđđđ„ = √đŒ1 2 − 4đŒ2 2 Lausn: d) Uncouple the stress tensor into isotropic and deviatoric components both for the general đ„, đŠ state and principal state đ„P, đŠP. Show how the results in (c) relate to the components of the uncoupled stress tensor Lausn: First we uncouple for general state in isotropic and deviatoric: đđ„ + đđŠ đđ„ − đđŠ 0 đđ„đŠ đđ„ đđ„đŠ đ đ đđ 0 đ„ đ„đŠ 2 2 = + = + [đđđ ] = [đ ] [ ] [ ] [ ] [ đđ„ − đđŠ ] đđŠ đđ„ + đđŠ 0 đđ đđ„đŠ đđŠ đ„đŠ đ − 0 đ„đŠ 2 2 Next to uncouple for principal state: đđ„ [đđđ ] = [đ đ„đŠ đđ„đŠ đđ đđŠ ] = [ 0 đđ„ 0 ]+[0 đđ đđ„ + đđŠ 0 2 đđŠ ] = [ 0 0 đđ„ + đđŠ 2 đđ„ − đđŠ ]+[ 2 0 0 đđ„ − đđŠ ] − 2 Now to see how the results in (c) relate to the components of these uncoupled stress tensor. 2 Start with general stress. We use invariants that we found in (a) đŒ2 = đđ„ + đđŠ and đŒ2 = đđ„ đđŠ − đđ„đŠ If we exchange đđ = đđ„ +đđŠ 2 1 = 2 đŒ1 and then we have the first part of the equation of (c), the second 2 part is a little bit difficaulter, rewrite for đŒ2 and have đđ„ đđŠ = đŒ2 + đđ„đŠ also we can rewrite to Then đđ„ +đđŠ 2 √( 1 √đŒ12 2 2 đđ„2 −2đđ„ đđŠ +đđŠ2 ) =√ 4 1 1 đđ„ +đđŠ 2 in = 2 √đđ„2 + đđŠ2 − 2đđ„ đđŠ and change đđ„2 + đđŠ2 = đŒ12 − đđ„ đđŠ 2 )= − 4đđ„ đđŠ = 2 √đŒ12 − 4(đŒ2 + đđ„đŠ 1 √đŒ12 2 2 so − 4đŒ2 − 4đđ„đŠ 1 1 2 đ1,2 = đŒ1 ± √đŒ1 2 − 4đŒ2 − 4đđ„đŠ 2 2 We do exactly the same for principal stress but there will đđ„đŠ = 0 so the answer is 1 1 đ1,2 = đŒ1 ± √đŒ1 2 − 4đŒ2 2 2 e) Use the stress transformation equations to calculate the normal stresses and shear stresses in the quadrilateral state (i.e., the stresses on a plane normal to an axis that makes an equal angle to both principal axes xp , yp). Lausn: Snúum principal state um 45° til að fá quadrilatral state. Byrjum á normal stresses og notum jöfnur 4.11 bls 44 í nótum Ragnars: đ′đ„ = đđ„ + đđŠ đđ„ − đđŠ − cos(2 × 45) + đđ„đŠ sin(2 × 45) 2 2 đ′đ„ = đ′đŠ = đđ„ + đđŠ + đđ„đŠ 2 đđ„ + đđŠ đđ„ − đđŠ − cos(2 × 45) − đđ„đŠ sin(2 × 45) 2 2 đ′đŠ = đđ„ + đđŠ − đđ„đŠ 2 Að lokum finnum við lausn fyrir shear stresses: đ′đ„đŠ = − đđ„ − đđŠ sin(2 × 45) + đđ„đŠ cos(2 × 45) 2 đ′đ„đŠ = đđŠ − đđ„ 2 Problem 2 The stresses acting on a rectangular element along the đ„, đŠ axes are given as đđ„ [đđđ ] = [đ đ„đŠ đđ„đŠ 80 đđŠ ] = [30 30 ] đđđ 40 Part 1: Carry out calculations using equations resulting from the stress transformation equations, the principal stress eigenvalue problem, and the quadrilateral stress state Part 2: Carry out calculations using Mohr’s circle with the following convention 1. đ axis is positive down. 2. Point X comprising of stress components on the đ„-side of the element is plotted as (đx, đxy) while point Y is plotted as đy, −đxy . In all cases, draw up the elements and the stress components in the new coordinate system, relative to the original configuration in the đ„, đŠ coordinate system. a) Determine the principal stresses and the directions of principal axes. Lausn: b) stresses. Determine the maximum shear stresses and axis direction, and calculate the normal Lausn: c) Calculate the quadrilateral normal and shear stresses i.e., those that occur on a plane normal to an axis that makes an equal angle to both principal axes đ„P, đŠP. Lausn: d) Resolve both the original and the principal tensor into their isotropic and deviatoric components (For Part 2, show the isotropic and deviatoric components of normal stress on the Mohr’s Circle). Lausn: e) (For Part 1): Calculate the invariants of the stress tensor for both the original đ„, đŠ orientation and the principal đ„P, đŠP orientation (checking to make sure they are actually the same in either system). Lausn:
