Explanation Given information: A material is subjected to stresses σ x and σ y . The material constants are E and ν . Calculation: Apply the normal stress formula for x shown below. σ x = σ x + σ y 2 + σ x − σ y 2 cos 2 θ + τ x y sin 2 θ Here, σ x and σ y are the normal stresses in the x and y direction, τ x y is the shear stress, and θ is the orientation. The shear stress τ x y = 0 . Substitute 0 for τ x y . σ x = σ x + σ y 2 + σ x − σ y 2 cos 2 θ (1) Consider cos 2 θ = 2 cos 2 θ − 1 . For the normal direction. Substitute 2 cos 2 θ − 1 for cos 2 θ in Equation (1). σ n = σ x + σ y 2 + σ x − σ y 2 ( 2 cos 2 θ − 1 ) = σ x + σ y 2 + ( σ x − σ y ) cos 2 θ − σ x − σ y 2 = σ x 2 + σ y 2 + σ x cos 2 θ − σ y cos 2 θ − σ x 2 + σ y 2 = σ y + σ x cos 2 θ − σ y cos 2 θ = σ x cos 2 θ + σ y ( 1 − cos 2 θ ) = σ x cos 2 θ + σ y sin 2 θ Apply the normal stress formula for y shown below. σ y = σ x + σ y 2 − σ x − σ y 2 cos 2 θ − τ x y sin 2 θ The shear stress τ x y = 0 . Substitute 0 for τ x y . σ y = σ x + σ y 2 − σ x − σ y 2 cos 2 θ (2) For y direction. Substitute 2 cos 2 θ − 1 for cos 2 θ in Equation (1)

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Chapter 10.6, Problem 10.45P

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The orientation θ of the strain gage.

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Chapter 10.6, Problem 10.44P

Chapter 10.6, Problem 10.46P

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Chapter 10 Solutions

MECHANICS OF MATERIAL IN SI UNITS

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The orientation θ of the strain gage.

Solution Summary: The author explains that the strain gage's orientation is underset_mathrmtan-1left.

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