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# ch7-3 - 452 CHAPTER 7 Analysis of Stress and Strain Problem...

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Problem 7.4-7 An element in pure shear is subjected to stresses xy 3000 psi, as shown in the figure. Using Mohr’s circle, determine (a) the stresses acting on an element oriented at a counterclockwise angle 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 x 0 y 0 xy 3000 psi (a) E LEMENT AT 70 (All stresses in psi) 2 140 70 R 3000 psi Origin O is at center of circle. Point D : Point D : t x 1 y 1 R sin 50 2298 psi s x 1 R cos 50 1928 psi t x 1 y 1 R sin 50 2298 psi s x 1 R cos 50 1928 psi (b) P RINCIPAL STRESSES Point P 1 : 1 R 3000 psi Point P 2 : 2 R 3000 psi u p 2 45 2 u p 2 90 u p 1 45 2 u p 1 90 D' D A ( 0) 3000 psi 2 p 1 2 = 140 P 1 P 2 R R O x 1 B ( 90 ) x 1 y 1 50 x O 2300 psi 70 1930 psi 1930 psi y D D' x O p 1 45 3000 psi 3000 psi y P 2 P 1 Problem 7.4-8 An element in pure shear is subjected to stresses 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 Circle for Plane Stress 453 x 0 y 0 xy 16 MPa (a) E LEMENT AT 20 (All stresses in MPa) 2 40 20 R 16 MPa Origin O is at center of circle. Point D : Point D : t x 1 y 1 R cos 2 u 12.26 MPa s x 1 R sin 2 u 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 : 1 R 16 MPa Point P 2 : 2 R 16 MPa u p 2 45 2 u p 2 90 u p 1 135 2 u p 1 270 D' D A ( 0) 2 2 p 2 2 p 1 P 1 P 2 R R O x 1 ( 20 ) B ( 90 ) x 1 y 1 16 x O 12.3 MPa 20 10.3 MPa 10.3 MPa y D D' x O p 2 45 16 MPa 16 MPa y P 2 P 1 Problem 7.4-9 An element in pure shear is subjected to stresses xy 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 x 0 y 0 xy 4000 psi (a) E LEMENT AT A SLOPE OF 3 ON 4 (All stresses in psi) 2 73.740 36.870 R 4000 psi Origin O is at center of circle. u arctan 3 4 36.870 3 4 D' D A ( 0) 2 2 p 1 P 1 P 2 R R O x 1 B ( 90 ) x 1 y 1 4000 16.260 R
454 CHAPTER 7 Analysis of Stress and Strain Point D : Point D : t x 1 y 1 R sin 16.260 1120 psi s x 1 R cos 16.260 3840 psi t x 1 y 1 R sin 16.260 1120 psi s x 1 R cos 16.260 3840 psi (b) P RINCIPAL STRESSES Point P 1 : 1 R 4000 psi Point P 2 : 2 R 4000 psi u p 2 45 2 u p 2 90 u p 1 45 2 u p 1 90 x O 1120 psi 36.87 3840 psi 3840 psi y D D' x O p 1 45 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 x , y , and xy (see figure). Using Mohr’s circle, determine the stresses acting on an element oriented at an angle from the x axis. Show these stresses on a sketch of an element oriented at the angle . ( Note: The angle is positive when counterclockwise and negative when clockwise.) y x O xy x y Solution 7.4-10 Plane stress (angle ) x 21 MPa y 11 MPa xy 8 MPa 50 (All stresses in MPa) arctan 8 5 57.99 R (5) 2 (8) 2 9.4340 MPa 2 100 42.01 Point D ( 50 ): Point D ( 40 ): t x 1 y 1 R sin b 6.31 MPa s x 1 16 R cos b 8.99 MPa t x 1 y 1 R sin b 6.31 MPa s x 1

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ch7-3 - 452 CHAPTER 7 Analysis of Stress and Strain Problem...

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