Explanation Given information: The state of strain at the point on the leaf of the caster assembly has components of ε x = − 400 × 10 − 6 , ε y = 860 × 10 − 6 , and γ x y = 375 × 10 − 6 . The element oriented at an angle of 30 ° counterclockwise 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 − 400 × 10 − 6 for ε x , 860 × 10 − 6 for ε y , 375 × 10 − 6 for γ x y , and 30 ° for θ . ε x ′ = [ ( − 400 × 10 − 6 ) + ( 860 × 10 − 6 ) 2 + ( − 400 × 10 − 6 ) − ( 860 × 10 − 6 ) 2 cos 2 ( 30 ° ) + ( 375 × 10 − 6 ) 2 sin 2 ( 30 ° ) ] = ( 230 × 10 − 6 ) − ( 630 × 10 − 6 × 0.5 ) + ( 187.5 × 10 − 6 × 0.866 ) = 77.4 × 10 − 6 Hence, the normal strain component in x ′ direction is 77.4 × 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 − 400 × 10 − 6 for ε x , 860 × 10 − 6 for ε y , 375 × 10 − 6 for γ x y , and 30 ° for θ . ε y ′ = [ ( − 400 × 10 − 6 ) + ( 860 × 10 − 6 ) 2 − ( − 400 × 10 − 6 ) − ( 860 × 10 − 6 ) 2 cos 2 ( 30 ° ) − ( 375 × 10 − 6 ) 2 sin 2 ( 30 ° ) ] = ( 230 × 10 − 6 ) + ( 630 × 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 88P

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Find the equivalent in-plane strains on an element.

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

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Chapter 14.8, Problem 87P

Chapter 14.8, Problem 89P

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Chapter 14 Solutions

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

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Chapter 14 Solutions

Find the equivalent in-plane strains on an element.

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