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600_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

600_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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594 CHAPTER 7 Analysis of Stress and Strain Problem 7.3-20 An element in plane stress is subjected to stresses s x 50 MPa and t xy 42 MPa (see figure). It is known that one of the principal stresses equals 33 MPa in tension. (a) Determine the stress s y . (b) Determine the other principal stress and the orientation of the principal planes, then show the principal stresses on a sketch of a properly oriented element. y x O 42 MPa s y 50 MPa Solution 7.3-20 (a) S TRESS Because is smaller than a given principal stress, we know that the given stress is the larger principal stress. Solve for (b) P RINCIPAL STRESSES tan(2 u p ) 2 t xy s x s y u p 2 atan a 2 t xy s x s y b 2 s y s y 11.7 MPa ; s 1 s x + s y 2 + A a s x s y 2 b 2 + t
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Unformatted text preview: unknown t xy ± 42 MPa Therefore, For For u p 2 ±² 26.8° : s 2 ± ² 71.3 MPa ; u p 1 ± 63.2° : s 1 ± 33.0 MPa ; + t xy sin 1 2 u p 2 2 s 2 ± s x + s y 2 + s x ² s y 2 cos 1 2 u p 2 2 + t xy sin 1 2 u p 1 2 s 1 ± s x + s y 2 + s x ² s y 2 cos 1 2 u p 1 2 u p 1 ± 90° + u p 2 u p 1 ± 63.15° u p 2 ± ² 26.85° + t xy sin 1 2 u p 2 2 s 2 ± s x + s y 2 + s x ² s y 2 cos 1 2 u p 2 2 + t xy sin 1 2 u p 1 2 s 1 ± s x + s y 2 + s x ² s y 2 cos 1 2 u p 1 2 Therefore, For For u p 2 ± 66.5° : s 2 ± 410 psi ; u p 1 ± ² 23.5° : s 1 ± 6700 psi ; 07Ch07.qxd 9/27/08 1:18 PM Page 594...
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