Explanation Given information: The state of strain at a point on the bracket has components of ε x = 150 × 10 − 6 , ε y = 200 × 10 − 6 , and γ x y = − 700 × 10 − 6 . The element oriented at an angle of 30 ° clockwise from the original position. Calculation: Apply the strain transformation Equations as shown below. Find the normal strain component in x ′ direction as shown below. ε x ′ = ε x + ε y 2 + ε x − ε y 2 cos 2 θ + γ x y 2 sin 2 θ Here, ε x is the normal strain in x direction, ε y is the normal strain in y direction, and γ x y is the shear strain of the element. Substitute 150 × 10 − 6 for ε x , 200 × 10 − 6 for ε y , − 700 × 10 − 6 for γ x y , and − 30 ° for θ . ε x ′ = [ ( 150 × 10 − 6 ) + ( 200 × 10 − 6 ) 2 + ( 150 × 10 − 6 ) − ( 200 × 10 − 6 ) 2 cos 2 ( − 30 ° ) + ( − 700 × 10 − 6 ) 2 sin 2 ( − 30 ° ) ] = ( 175 × 10 − 6 ) − ( 25 × 10 − 6 × 0.5 ) − ( 350 × 10 − 6 × ( − 0.866 ) ) = 466 × 10 − 6 Hence, the normal strain component in x ′ direction is 466 × 10 − 6 _ . Find the normal strain component in y ′ direction as shown below. ε y ′ = ε x + ε y 2 − ε x − ε y 2 cos 2 θ − γ x y 2 sin 2 θ Substitute 150 × 10 − 6 for ε x , 200 × 10 − 6 for ε y , − 700 × 10 − 6 for γ x y , and − 30 ° for θ . ε y ′ = [ ( 150 × 10 − 6 ) + ( 200 × 10 − 6 ) 2 − ( 150 × 10 − 6 ) − ( 200 × 10 − 6 ) 2 cos 2 ( − 30 ° ) − ( − 700 × 10 − 6 ) 2 sin 2 ( − 30 ° ) ] = ( 175 × 10 − 6 ) + ( 25 × 10 − 6 × 0

Statics and Mechanics of Materials (5th Edition)

5th Edition

ISBN: 9780134382593

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 14.8, Problem 90P

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Find the equivalent in-plane strains on an element.

Sketch the deformed element due to the equivalent in-plane strains within x-y plane.

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Chapter 14.8, Problem 89P

Chapter 14.8, Problem 91P

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Statics and Mechanics of Materials (5th Edition)

Ch. 14.3 - In each ease, the state of stress x, y, xy...Ch. 14.3 - Given the state of stress shown on the element,...Ch. 14.3 - Determine the normal stress and shear stress...Ch. 14.3 - Prob. 2FP Ch. 14.3 - Determine the equivalent state of stress on an...Ch. 14.3 - Prob. 4FP Ch. 14.3 - The beam is subjected to the load at its end....Ch. 14.3 - Prob. 6FP Ch. 14.3 - Prove that the sum of the normal stresses x+y=x+y...Ch. 14.3 - Determine the stress components acting on the...

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Find the equivalent in-plane strains on an element. Sketch the deformed element due to the equivalent in-plane strains within x-y plane.

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Find the equivalent in-plane strains on an element. Sketch the deformed element due to the equivalent in-plane strains within x-y plane.

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Find the equivalent in-plane strains on an element.

Sketch the deformed element due to the equivalent in-plane strains within x-y plane.