Explanation Given information: The element is oriented 40 ° in the clockwise 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 ' and y and y ' axes. Apply sign conventions to the given Figure. The value of normal stresses are σ x = 6 ksi and σ y = − 10 ksi . The value of shear stress is τ x y = 0 . The angle θ is positive if it curls in the counterclockwise direction. Therefore, the value of θ is − 40 ° . The procedure to construct Mohr’s circle is shown below. The coordinate systems are the horizontal axis represents the normal stress with positive towards right and the vertical axis represents shear stress with positive downwards. Plot the center of the circle C which is located σ avg = σ x + σ y 2 from the origin. Plot the reference point A having coordinates A ( σ x , τ x y ) . This point represents that the x ′ axis coincides with the x axis. Hence, θ = 0 ° . Plot the reference point B having coordinates B ( σ y , − τ x y ) Connect the point A with the centre C and find CA by trigonometry which represents the radius of the circle R . Sketch the circle. Find the average normal stress ( σ avg ) using the relation: σ avg = σ x + σ y 2 Substitute 6 ksi for σ x and − 10 ksi for σ y . σ avg = 6 − 10 2 = − 2 ksi Therefore, the coordinates of the reference point C is ( − 2 , 0 ) . The coordinates of the reference point A ( σ x , τ x y ) = ( 6 , 0 ) . The coordinates of the reference point B ( σ y , − τ x y ) = ( − 10 , 0 ) . Find the radius of the circle using the relation: R = ( σ x − σ avg ) 2 + ( τ x y ) 2 Substitute − 2 ksi for σ avg , 6 ksi for σ x , and 0 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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Find the equivalent state of stress on an element for orientation of 40° in the clockwise direction.

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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 40 ° in the clockwise direction.

Solution Summary: The author calculates the equivalent state of stress on an element for orientation of 40° in the clockwise direction.

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