Explanation Given information: The normal strain in Rosette 1 is ε 1 = − 50 × 10 − 6 in ./in . . The Rosette 1 makes an angle with the x -axis is θ 1 = 45 ° . The normal strain in Rosette 2 is ε 2 = + 360 × 10 − 6 in ./in . . The Rosette 2 makes an angle with the x -axis is θ 2 = − 45 ° . The normal strain in Rosette 3 is ε 3 = + 315 × 10 − 6 in ./in . . The Rosette 3 makes an angle with the x -axis is θ 3 = 0 ° . Calculation: Write the equation for the normal strain 1 as follows; ε 1 = ε x cos 2 θ 1 + ε y sin 2 θ 1 + γ x y sin θ 1 cos θ 1 Substitute − 50 × 10 − 6 in ./in . for ε 1 and 45 ° for θ 1 . − 50 × 10 − 6 = ε x cos 2 ( 45 ° ) + ε y sin 2 ( 45 ° ) + γ x y sin ( 45 ° ) cos ( 45 ° ) (1) Write the equation for the normal strain 2 as follows; ε 2 = ε x cos 2 θ 2 + ε y sin 2 θ 2 + γ x y sin θ 2 cos θ 2 Substitute + 360 × 10 − 6 in ./in . for ε 2 and − 45 ° for θ 2 . + 360 × 10 − 6 = ε x cos 2 ( − 45 ° ) + ε y sin 2 ( − 45 ° ) + γ x y sin ( − 45 ° ) cos ( − 45 ° ) (2) Write the equation for the normal strain 3 as follows; ε 3 = ε x cos 2 θ 3 + ε y sin 2 θ 3 + γ x y sin θ 3 cos θ 3 Substitute + 315 × 10 − 6 in ./in . for ε 3 and 0 ° for θ 3 . + 315 × 10 − 6 = ε x cos 2 ( 0 ° ) + ε y sin 2 ( 0 ° ) + γ x y sin ( 0 ° ) cos ( 0 ° ) (3) Solve the Equations (1), (2), and (3) simultaneously. ε x = 315 × 10 − 6 in ./in . ε y = − 5 × 10 − 6 in ./in . γ x y = − 410 × 10 − 6 in

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Chapter 7, Problem 169RP

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Find the maximum in-plane normal strain.

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