Explanation Assumption: The sign conventions for the stress components are explained below. The outward normal on the right-hand face of the element is representing as the positive x and x ' axes. The normal stress σ x and σ x ' are positive if it acts along the positive x and x ' axes. The normal stress σ y and σ y ' are positive if it acts along the positive y and y ' axes. The shear stress τ x y and τ x ' y ' are positive if it acts along the positive x and x ' and y and y ' axes. Calculation: Sketch the stress components on the element as shown in Figure 1. The angle θ is positive if it curls in the counterclockwise direction. Apply sign conventions to element a in Figure 1. The value of normal stresses are σ x = 85 MPa and σ y = 85 MPa . The value of shear stress is τ x y = 0 and the angle θ = − 45 ° . Calculate the normal stress in the element a for the x ' plane ( σ x ' ) a using the stress transformation equation: ( σ x ' ) a = σ x + σ y 2 + σ x − σ y 2 cos 2 θ + τ x y sin 2 θ (1) Substitute 85 MPa for σ x , 85 MPa for σ y , − 45 ° for θ , and 0 for τ x y in Equation (1). ( σ x ' ) a = 85 + 85 2 + 85 − 85 2 cos 2 ( − 45 ° ) + 0 = 85 + 0 + 0 = 85 MPa Calculate the normal stress in the element a for the y ' plane ( σ y ' ) a using the stress transformation equation: ( σ y ' ) a = σ x + σ y 2 − σ x − σ y 2 cos 2 θ − τ x y sin 2 θ Substitute 85 MPa for σ x , 85 MPa for σ y , − 45 ° for θ , and 0 for τ x y . ( σ y ' ) a = 85 + 85 2 − 85 − 85 2 cos 2 ( − 45 ° ) − 0 = 85 − 0 − 0 = 85 MPa Calculate the shear stress for element a oriented in the x ' and y ' plane ( τ x ' y ' ) a using the stress transformation equation: ( τ x ' y ' ) a = − σ x − σ y 2 sin 2 θ + τ x y cos 2 θ Substitute 85 MPa for σ x , 85 MPa for σ y , − 45 ° for θ , and 0 for τ x y . ( τ x ' y ' ) a = − 85 − 85 2 sin 2 ( − 45 ° ) + 0 = 0 Apply sign conventions to element b in Figure 1. The value of normal stresses are σ x = 0 and σ y = 0 . The value of shear stress is τ x y = 60 MPa and the angle θ = − 60 ° . Calculate the normal stress in the element b for the x ' plane ( σ x ' ) b using the stress transformation equation: Substitute 0 for σ x , 0 for σ y , − 60 ° for θ , and 60 MPa for τ x y in Equation (1)

Statics and Mechanics of Materials (5th Edition)

5th Edition

ISBN: 9780134382593

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 14.3, Problem 18P

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Find the resultant state of stress on the element.

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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 resultant state of stress on the element.

Solution Summary: The author explains the sign conventions for the stress components on the element as shown in Figure 1.

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