# chapter 12 answers - Problem 12.1 The components of plain...

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Problem 12.1 The components of plain stress at a point p of a material are σ x =20MPa , σ y =0 and τ xy . If θ =45 , what are the stresses σ 0 x , σ 0 y and τ 0 xy at point p ? Solution: Using Equation (12-7) to Fnd σ 0 x : σ 0 x = σ x + σ y 2 + σ x σ y 2 (cos 2 θ )+ τ xy (sin 2 θ ) σ 0 x == 20 MPa+0 2 + 20 MPa 0 2 (cos 90 )+0 ANS: σ 0 x =10MPa Using Equation (12-9) to Fnd σ 0 y : σ 0 y = σ x + σ y 2 σ x σ y 2 (cos 2 θ ) τ xy (sin 2 θ ) σ 0 y = 20 MPa+0 2 20 MPa 0 2 (cos 90 ) 0 ANS: σ 0 y Using Equation (12-8) to Fnd τ 0 xy : τ 0 xy = σ x σ y 2 (sin 2 θ τ xy (cos 2 θ ) τ 0 xy = 20 MPa 0 2 (sin 90 ANS: τ 0 xy = 10 MPa Problem 12.2 The components of plane stress at a point p of a material are σ x , σ y and τ xy = 25 ksi .I f θ , what are the stresses σ 0 x , σ 0 y and τ 0 xy at point p ? Solution: Using Equation (12-7) to Fnd σ 0 x : σ 0 x = σ x + σ y 2 + σ x σ y 2 (cos 2 θ τ xy (sin 2 θ ) σ 0 x = 0+0 2 + 0 0 2 (cos 90 ) + (25 ksi) (sin 90 ) ANS: σ 0 x = 25 ksi Using Equation (12-9): σ 0 y = σ x + σ y 2 σ x σ y 2 cos 2 θ τ xy sin 2 θ σ 0 y = 0+0 2 0 0 2 cos(90 ) 25 sin(90 ) ANS: σ 0 y = 25 ksi Using Equation (12-8) to Fnd τ 0 xy : τ 0 xy = σ x σ y 2 (sin 2 θ τ xy (cos 2 θ ) τ 0 xy = 0 0 2 (sin 90 ) + (25 ksi) (cos 90 ) ANS: τ 0 xy

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Problem 12.3 The components of plane stress at a point p of a material are σ x = 8 ksi , σ y = 6 ksi and τ xy = 6 ksi .I f θ =30 , what are the stresses σ 0 x , σ 0 y and τ 0 xy at point p ? Solution: Using Equation (12-7) to Fnd σ 0 x : σ 0 x = σ x + σ y 2 + σ x σ y 2 (cos 2 θ )+ τ xy (sin 2 θ ) σ 0 x = 8 ksi+6 ksi 2 + 8 ksi 6 ksi 2 (cos 60 )+( 6 ksi) (sin 60 ) ANS: σ 0 x = 9 . 7 ksi Using Equation (12-9): σ 0 y = σ x + σ y 2 σ x σ y 2 (cos 2 θ ) τ xy (sin 2 θ ) σ 0 y = 8 ksi+6 ksi 2 8 ksi 6 ksi 2 (cos 60 ) ( 6 ksi) (sin 60 ) ANS: σ 0 y =7 . 7 ksi Using Equation (12-8) to Fnd τ 0 xy : τ 0 xy = σ x σ y 2 (sin 2 θ τ xy (cos 2 θ ) τ 0 xy = ( 8 6) 2 sin 60 +( 6) cos 60 ANS: τ 0 xy =3 . 062 ksi Problem 12.4 During liftoff, strain gauges attached to one of the Space Shuttle main engine nozzles determine that the components of plane stress σ 0 x =66 . 46 MPa , σ 0 y =82 . 54 MPa , and τ 0 xy =6 . 75 MPa at θ =20 . What are the stresses σ x , σ y and τ xy at that point? Solution: Adding Equations (12-7) and (12-9): σ 0 x + σ 0 y = σ x + σ y σ x + σ y = 149 MPa [1] Using Equation [1] in Equation (12-9): 82 . 54 MPa = 149 MPa 2 σ x σ y 2 [cos (40 )] τ xy [sin (40 )] 8 . 04 MPa = ( σ x σ y )(0 . 383) (0 . 643) τ xy τ xy = 12 . 5MPa ( σ x σ y )(0 . 596) [2] Substituting the given information into Equation (12-8): 6 . 75 MPa = σ x σ y 2 [sin (40 )] + τ xy [cos (40 )] 6 . 75 MPa = (0 . 321)( σ x σ y τ xy (0 . 766) τ xy =8 . 81 MPa + (0 . 419)( σ x σ y ) [3] Subtracting Equations [2] and [3]: 0= 21 . 31 1 . 015( σ x σ y ) σ x σ y = 20 . 99 MPa [4] Adding Equations [1] and [4]: 2 σ x = 128 . 01 MPa ANS: σ x =64MPa σ y =85MPa τ xy = 0 MPa
Problem 12.5 The components of plane stress at a point p of a material are σ x = 240 MPa , σ y = 120 MPa , and τ xy = 240 MPa and the components referred to x 0 y 0 z 0 coordinate system are σ 0 x = 347 MPa , σ 0 y = 227 MPa and τ 0 xy = 87 MPa . What is the an-

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chapter 12 answers - Problem 12.1 The components of plain...

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