(a) Explanation Given information: The state of strain at the point on the member has components of ε x = 180 × 10 − 6 , ε y = − 120 × 10 − 6 , and γ x y = − 100 × 10 − 6 . Calculation: Apply the strain transformation Equations as shown below. Find the in-plane principal strains as shown below. ε 1 , 2 = ε x + ε y 2 ± ( ε x − ε y 2 ) 2 + ( γ x y 2 ) 2 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 180 × 10 − 6 for ε x , − 120 × 10 − 6 for ε y , and − 100 × 10 − 6 for γ x y . ε 1 , 2 = [ ( 180 × 10 − 6 ) + ( − 120 × 10 − 6 ) 2 ± ( ( 180 × 10 − 6 ) − ( − 120 × 10 − 6 ) 2 ) 2 + ( − 100 × 10 − 6 2 ) 2 ] = ( 30 × 10 − 6 ) ± ( 25 , 000 × 10 − 12 ) = ( 30 × 10 − 6 ) ± ( 158.11 × 10 − 6 ) Find the principal strains ε 1 and ε 2 . ε 1 = ( 30 × 10 − 6 ) + ( 158.11 × 10 − 6 ) = 188 × 10 − 6 ε 2 = ( 30 × 10 − 6 ) − ( 158.11 × 10 − 6 ) = − 128 × 10 − 6 Hence, the principal strains are ε 1 = 188 × 10 − 6 _ and ε 2 = − 128 × 10 − 6 _ . Find the orientation of principal strain as shown below. tan 2 θ p = γ x y ε x − ε y Substitute 180 × 10 − 6 for ε x , − 120 × 10 − 6 for ε y , and − 100 × 10 − 6 for γ x y (b) To determine Find the maximum in-plane shear strain and average normal strain. Sketch the deformed element due to the equivalent in-plane strains within x-y plane.

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 92P

(a)

To determine

Find the in-plane principal strains on an element and specify the orientation of the element.

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

(b)

To determine

Find the maximum in-plane shear strain and average normal strain.

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

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Chapter 14.8, Problem 91P

Chapter 14.8, Problem 93P

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Chapter 14 Solutions

Statics and Mechanics of Materials (5th Edition)

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Find the in-plane principal strains on an element and specify the orientation of the element. Sketch the deformed element due to the equivalent in-plane strains within x-y plane.

Solution Summary: The author explains how to find the in-plane principal strains on an element and specify the orientation of the element.