Explanation Given information: The element is oriented 60 ° in the counterclockwise direction. Calculation: 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 ' axes. Sketch the orientation of the element as shown in Figure 1. The angle θ is positive if it curls in the counterclockwise direction. Refer to Figure 1. The value of θ is 60 ° . Apply sign conventions to the given Figure. The value of normal stress are σ x = 150 MPa and σ y = 100 MPa . The value of shear stress is τ x y = 75 MPa . Calculate the normal stress for the x ' plane ( σ x ' ) using the stress transformation equation: σ x ' = σ x + σ y 2 + σ x − σ y 2 cos 2 θ + τ x y sin 2 θ Substitute 150 MPa for σ x , 100 MPa for σ y , 60 ° for θ , and 75 MPa for τ x y . σ x ' = 150 + 100 2 + 150 − 100 2 cos 2 ( 60 ° ) + 75 sin 2 ( 60 ° ) = 125 − 12.5 + 64.95 = 177.45 MPa ≈ 177 MPa Therefore, the normal stress acting on the x ' plane ( σ x ' ) is 177 MPa _ . Calculate the normal stress for the y ' plane ( σ y ' ) using the stress transformation equation: σ y ' = σ x + σ y 2 − σ x − σ y 2 cos 2 θ − τ x y sin 2 θ Substitute 150 MPa for σ x , 100 MPa for σ y , 60 ° for θ , and 75 MPa for τ x y

Statics and Mechanics of Materials (5th Edition)

5th Edition

ISBN: 9780134382593

Author: Russell C. Hibbeler

Publisher: PEARSON

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Stress transformation

The term stress indicates the measure of internal forces that develop due to an external force's application on an object. When a specific external force is applied to an object, internal stress generates in the material. The value of stress can obtain b…

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Chapter 14.3, Problem 12P

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Find the equivalent state of stress on an element for orientation of 60° in the counterclockwise direction.

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Chapter 14.3, Problem 11P

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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 state of stress on an element for orientation of 60 ° in the counterclockwise direction.

Solution Summary: The author calculates the equivalent state of stress on an element for orientation of 60° in the counterclockwise direction.

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Find the equivalent state of stress on an element for orientation of 60° in the counterclockwise direction.

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Chapter 14 Solutions