07-03 Chap Gere - 452 CHAPTER 7 Analysis of Stress and...

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Problem 7.4-7 An element in pure shear is subjected to stresses t xy 5 3000 psi, as shown in the figure. Using Mohr’s circle, determine (a) the stresses acting on an element oriented at a counterclockwise angle u5 70° from the x axis and (b) the principal stresses. Show all results on sketches of properly oriented elements. Solution 7.4-7 Pure shear 452 CHAPTER 7 Analysis of Stress and Strain y x O 3000 psi s x 5 0 s y 5 0 t xy 5 3000 psi (a) E LEMENT AT u 5 70 8 (All stresses in psi) 2 u 5 140 8 u 5 70 8 R 5 3000 psi Origin O is at center of circle. Point D : Point D 9 : t x 1 y 1 5 R sin 50 85 2298 psi s x 1 52 R cos 50 852 1928 psi t x 1 y 1 R sin 50 2298 psi s x 1 5 R cos 50 1928 psi (b) P RINCIPAL STRESSES Point P 1 : s 1 5 R 5 3000 psi Point P 2 : s 2 R 3000 psi u p 2 45 8 2 u p 2 90 8 u p 1 5 45 8 2 u p 1 5 90 8 D' D A ( u 5 0) 3000 psi 2 u p 1 2 u = 140 8 P 1 P 2 R R O s x 1 B ( u 5 90 8 ) t x 1 y 1 50 8 x O 2300 psi u 5 70 8 1930 psi 1930 psi y D D' x O u p 1 5 45 8 3000 psi 3000 psi y P 2 P 1 Problem 7.4-8 An element in pure shear is subjected to stresses t xy 16 MPa, as shown in the figure. Using Mohr’s circle, determine (a) the stresses acting on an element oriented at a counterclockwise angle 20° from the x axis and (b) the principal stresses. Show all results on sketches of properly oriented elements. y x O 16 MPa
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Solution 7.4-8 Pure shear SECTION 7.4 Mohr’s Circles 453 s x 5 0 s y 5 0 t xy 52 16 MPa (a) E LEMENT AT u 5 20 8 (All stresses in MPa) 2 u 5 40 8 u 5 20 8 R 5 16 MPa Origin O is at center of circle. Point D : Point D 9 : t x 1 y 1 5 R cos 2 u 5 12.26 MPa s x 1 5 R sin 2 u 5 10.28 MPa t x 1 y 1 R cos 2 u 12.26 MPa s x 1 R sin 2 u 10.28 MPa (b) P RINCIPAL STRESSES Point P 1 : s 1 5 R 5 16 MPa Point P 2 : s 2 R 16 MPa u p 2 5 45 8 2 u p 2 5 90 8 u p 1 5 135 8 2 u p 1 5 270 8 D' D A ( u 5 0) 2 u 2 u p 2 2 u p 1 P 1 P 2 R R O s x 1 ( u 5 20 8 ) B ( u 5 90 8 ) t x 1 y 1 16 x O 12.3 MPa u 5 20 8 10.3 MPa 10.3 MPa y D D' x O u p 2 5 45 8 16 MPa 16 MPa y P 2 P 1 Problem 7.4-9 An element in pure shear is subjected to stresses t xy 5 4000 psi, as shown in the figure. Using Mohr’s circle, determine (a) the stresses acting on an element oriented at a slope of 3 on 4 (see figure) and (b) the principal stresses. Show all results on sketches of properly oriented elements. O 3 4 y x 4000 psi Solution 7.4-9 Pure shear s x 5 0 s y 5 0 t xy 5 4000 psi (a) E LEMENT AT A SLOPE OF 3 ON 4 (All stresses in psi) 2 u 5 73.740 8 u 5 36.870 8 R 5 4000 psi Origin O is at center of circle. u 5 arctan 3 4 5 36.870 8 u 3 4 D' D A ( u 5 0) 2 u 2 u p 1 P 1 P 2 R R O s x 1 B ( u 5 90 8 ) t x 1 y 1 4000 16.260 8
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454 CHAPTER 7 Analysis of Stress and Strain Point D : Point D 9 : t x 1 y 1 52 R sin 16.260 852 1120 psi s x 1 R cos 16.260 3840 psi t x 1 y 1 5 R sin 16.260 85 1120 psi s x 1 5 R cos 16.260 3840 psi (b) P RINCIPAL STRESSES Point P 1 : s 1 5 R 5 4000 psi Point P 2 : s 2 R 4000 psi u p 2 45 8 2 u p 2 90 8 u p 1 5 45 8 2 u p 1 5 90 8 x O 1120 psi u 5 36.87 8 3840 psi 3840 psi y D D' x O u p 1 5 45 8 4000 psi 4000 psi y P 2 P 1 Problems 7.4-10 through 7.4-15 An element in plane stress is subjected to stresses s x , s y , and t xy (see figure).
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07-03 Chap Gere - 452 CHAPTER 7 Analysis of Stress and...

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