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 ' & y and y ' axes. Calculation: Apply sign conventions to the given Figure. The value of normal stresses are σ x = 80 kPa and σ y = 0 . The value of shear stress is τ x y = 30 kPa . 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 points A having coordinates A ( σ x , τ x y ) . This point represents that the x ′ axis coincides with the x axis. Hence, θ = 0 ° . 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 (1) Substitute 80 kPa for σ x and 0 for σ y in Equation (1). σ avg = 80 + 0 2 = 40 kPa Therefore, the coordinates of the reference point C is ( 40 , 0 ) . The coordinates of the reference point A ( σ x , τ x y ) = ( 80 , 0 ) . Find the radius of the circle using the relation: R = ( σ x − σ avg ) 2 + ( τ x y ) 2 (2) Substitute 40 kPa for σ avg , 80 kPa for σ x , and 30 kPa for τ x y in Equation (2)

Statics and Mechanics of Materials (5th Edition)

5th Edition

ISBN: 9780134382593

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 14.4, Problem 8FP

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Find the principal stresses.

Find the orientation of the principal stresses.

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Find the principal stresses. Find the orientation of the principal stresses.

Solution Summary: The author explains the sign conventions for the stress components and the procedure to construct Mohr's circle.