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Unformatted text preview: 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 (137): ε x = ε x + ε y 2 + ε x − ε y 2 (cos 2 θ )+ γ xy 2 (sin 2 θ ) = . 003 + 0 2 + . 003 − 2 (cos 90 ◦ )+ 2 (sin 90 ◦ ) ANS: ε x = 0 . 0015 From Equation (138): ε y = ε x + ε y 2 − ε x − ε y 2 (cos 2 θ ) − γ xy 2 (sin 2 θ ) = . 003 + 0 2 − . 003 − 2 (cos 90 ◦ ) − 2 (sin 90 ◦ ) ANS: ε y = 0 . 0015 From Equation (139): γ xy 2 = − ε x − ε y 2 (sin 2 θ )+ γ xy 2 (cos 2 θ ) = − . 003 − 2 (sin 90 ◦ )+ 2 (cos 90 ◦ ) ANS: γ xy = − . 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 (137): ε x = ε x + ε y 2 + ε x − ε y 2 (cos 2 θ )+ γ xy 2 (sin 2 θ ) = 0 + 0 2 + − 2 (cos 90 ◦ )+ . 004 2 (sin 90 ◦ ) ANS: ε x = 0 . 002 From Equation (138): ε y = ε x + ε y 2 − ε x − ε y 2 (cos 2 θ ) − γ xy 2 (sin 2 θ ) = 0 + 0 2 − − 2 (cos 90 ◦ ) − . 004 2 (sin 90 ◦ ) ANS: ε y = − . 002 From Equation (139): γ xy 2 = − ε x − ε y 2 (sin 2 θ )+ γ xy 2 (cos 2 θ ) = − − 2 (sin 90 ◦ )+ . 004 2 (cos 90 ◦ ) ANS: γ xy = 0 Problem 13.3 The components of plane strain at point p are ε x = − . 0024 , ε y = 0 . 0012 , and γ xy = − . 0012 , If θ = 25 ◦ , what are the strains ε x , ε y , and γ xy at point p ? Solution: From Equation (137): ε x = ε x + ε y 2 + ε x − ε y 2 (cos 2 θ )+ γ xy 2 (sin 2 θ ) = − . 0024 + 0 . 0012 2 + − . 0024 − . 0012 2 (cos 50 ◦ )+ − . 0012 2 (sin 50 ◦ ) ANS: ε x = − . 00222 From Equation (138): ε y = ε x + ε y 2 − ε x − ε y 2 (cos 2 θ ) − γ xy 2 (sin 2 θ ) = − . 0024 + 0 . 0012 2 − − . 0024 − . 0012 2 (cos 50 ◦ ) − − . 0012 2 (sin 50 ◦ ) ANS: ε y = 0 . 00102 From Equation (139): γ xy 2 = − ε x − ε y 2 (sin 2 θ )+ γ xy 2 (cos 2 θ ) = − − . 0024 − . 0012 2 (sin 50 ◦ )+ − . 0012 2 (cos 50 ◦ ) ANS: γ xy = 0 . 00099 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 = − . 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 = − . 00025 , and γ xy = 0 . 00910 . What is the angle θ ? Solution: From Equation (138): . 00125 = . 004 + ( − . 003) 2 + . 004 − ( − . 003) 2 (cos 2 θ )+ . 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 = − . 0012 , and γ xy = . 0048 . The extensional strains ε x = 0 . 00347 , and ε y = − . 00227 . Determine γ xy and the angle θ ....
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This note was uploaded on 04/03/2008 for the course MAE 101 taught by Professor Orient during the Winter '08 term at UCLA.
 Winter '08
 ORIENT
 Statics, Strain

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