chapter 13 answers - Problem 13.1 The components of plane...

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Problem 13.1 The components of plane strain at point p are ε x = 0 . 003 , ε y = 0, and γ xy = 0 . If θ = 45 , what are the strains ε x , ε y , and γ xy at point p ? Solution: From Equation (13-7): ε x = ε x + ε y 2 + ε x ε y 2 (cos 2 θ )+ γ xy 2 (sin 2 θ ) = 0 . 003 + 0 2 + 0 . 003 0 2 (cos 90 )+ 0 2 (sin 90 ) ANS: ε x = 0 . 0015 From Equation (13-8): ε y = ε x + ε y 2 ε x ε y 2 (cos 2 θ ) γ xy 2 (sin 2 θ ) = 0 . 003 + 0 2 0 . 003 0 2 (cos 90 ) 0 2 (sin 90 ) ANS: ε y = 0 . 0015 From Equation (13-9): γ xy 2 = ε x ε y 2 (sin 2 θ )+ γ xy 2 (cos 2 θ ) = 0 . 003 0 2 (sin 90 )+ 0 2 (cos 90 ) ANS: γ xy = 0 . 003 Problem 13.2 The components of plane strain at point p are ε x = 0 , ε y = 0 , and γ xy = 0 . 004 . If θ = 45 , what are the strains ε x , ε y , and γ xy at point p ? Solution: From Equation (13-7): ε x = ε x + ε y 2 + ε x ε y 2 (cos 2 θ )+ γ xy 2 (sin 2 θ ) = 0 + 0 2 + 0 0 2 (cos 90 )+ 0 . 004 2 (sin 90 ) ANS: ε x = 0 . 002 From Equation (13-8): ε y = ε x + ε y 2 ε x ε y 2 (cos 2 θ ) γ xy 2 (sin 2 θ ) = 0 + 0 2 0 0 2 (cos 90 ) 0 . 004 2 (sin 90 ) ANS: ε y = 0 . 002 From Equation (13-9): γ xy 2 = ε x ε y 2 (sin 2 θ )+ γ xy 2 (cos 2 θ ) = 0 0 2 (sin 90 )+ 0 . 004 2 (cos 90 ) ANS: γ xy = 0 Problem 13.3 The components of plane strain at point p are ε x = 0 . 0024 , ε y = 0 . 0012 , and γ xy = 0 . 0012 , If θ = 25 , what are the strains ε x , ε y , and γ xy at point p ? Solution: From Equation (13-7): ε x = ε x + ε y 2 + ε x ε y 2 (cos 2 θ )+ γ xy 2 (sin 2 θ ) = 0 . 0024 + 0 . 0012 2 + 0 . 0024 0 . 0012 2 (cos 50 )+ 0 . 0012 2 (sin 50 ) ANS: ε x = 0 . 00222 From Equation (13-8): ε y = ε x + ε y 2 ε x ε y 2 (cos 2 θ ) γ xy 2 (sin 2 θ ) = 0 . 0024 + 0 . 0012 2 0 . 0024 0 . 0012 2 (cos 50 ) 0 . 0012 2 (sin 50 ) ANS: ε y = 0 . 00102 From Equation (13-9): γ xy 2 = ε x ε y 2 (sin 2 θ )+ γ xy 2 (cos 2 θ ) = 0 . 0024 0 . 0012 2 (sin 50 )+ 0 . 0012 2 (cos 50 ) ANS: γ xy = 0 . 00099

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Problem 13.4 The components of plane strain at point p of a bit during a drilling operation are ε x = 0 . 00400 , ε y = 0 . 00300 , and γ xy = 0 . 00600 , and the com- ponents referred to the x y z coordinate system are ε x = 0 . 00125 , ε y = 0 . 00025 , and γ xy = 0 . 00910 . What is the angle θ ? Solution: From Equation (13-8): 0 . 00125 = 0 . 004 + ( 0 . 003) 2 + 0 . 004 ( 0 . 003) 2 (cos 2 θ )+ 0 . 006 2 (sin 2 θ ) Using a graphing calculator to determine the value of θ : 2 θ = 40 ANS: θ = 20 Problem 13.5 The components of plane strain at point p are ε x = 0 . 0024 , ε y = 0 . 0012 , and γ xy = 0 . 0048 . The extensional strains ε x = 0 . 00347 , and ε y = 0 . 00227 . Determine γ xy and the angle θ . Solution: From Equation (13-7): ε x = ε x + ε y 2 + ε x ε y 2 (cos 2 θ ) + γ xy 2 (sin 2 θ ) 0 . 00347 = 0 . 0024 + ( 0 . 0012) 2 + 0 . 0024 ( 0 . 0012) 2 (cos 2 θ )+ 0 . 0048 2 (sin 2 θ ) Using a graphing calculator: θ = 35 Equation (13-9): γ xy 2 = ε x ε y 2 (sin 2 θ )+ γ xy 2 (cos 2 θ ) = 0 . 0024 ( 0 . 0012) 2 (sin 70 )+ 0 . 0048 2 (cos 70 ) ANS: γ xy = 0 . 00174
Problem 13.6 During liftoff, strain gauges attached to one of the Space Shuttle main engine nozzles determine that the components of plane strain at point p are strains ε x = 0 . 00665 , and ε y = 0 . 00825 , and γ xy = 0 . 00135 for a coordinate system oriented at θ = 20 . What are the strains ε x , ε y , and γ xy at that point?

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