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634_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

# 634_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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Unformatted text preview: 07Ch07.qxd 628 9/27/08 1:24 PM Page 628 CHAPTER 7 Analysis of Stress and Strain Problem 7.7-9 A element of material in plane strain (see figure) is subjected to strains Px 480 * 10 6 6 , Py 70 * 10 and gxy 420 * 10 . Determine the following quantities: (a) the strains for an element oriented at an angle u 75°, (b) the principal strains, and (c) the maximum shear strains. Show the results on sketches of properly oriented element. Solution 7.7-9 Element in plane strain âx 480 * 10 6 gxy 420 * 10 âx + ây âx For up âx1 gx1y1 + 2 2 ây âx 2 2 ây1 For u ây cos 2u + sin 2u + âx + ây 70 * 10 gxy 2 gxy 2 6 22.85°: 6 ây âx + ây âx1 2 sin 2u 2 â1 22.8° ° up2 cos 2u + gxy 2 sin 2u 6 568 * 10 ‹ up1 cos 2u ây âx + 568 * 10 â2 112.8 18 * 10 6 ; 6 ; âx1 75°: âx1 202 * 10 6 ây1 348 * 10 6 gx1y1 569 * 10 6 gmax 2 A gmax 587 * 10 MAXIMUM SHEAR STRAINS us1 PRINCIPAL STRAINS â1,2 âx + ây ; 2 A 275 * 10 â1 tan 2up 6 568 * 10 gxy 6 âx ây a ây âx up1 2 us2 âx â2 1.0244 2up 45.69° and 225.69° up 22.85° and 112.85° âaver 6 18 * 10 6 2 ; 67.8° 587 * 10 âx + ây 2 b 2 293 * 10 22.2° or 157.8° 6 us1 + 90° gmin ây 2 gxy 2 b +a b 2 2 ; 293 * 10 587 * 10 gxy 6 45° gmax a 2 b +a 6 ; 275 * 10 6 6 6 , ...
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