Explanation Given information: The element is oriented 20 ° 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 ' axes. Calculation: Apply sign conventions to the given Figure. The value of normal stresses are σ x = 3 ksi and σ y = − 2 ksi . The value of shear stress is τ x y = − 4 ksi . 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 3 ksi for σ x and − 2 ksi for σ y in Equation (1). σ avg = 3 − 2 2 = 0.5 ksi Therefore, the coordinates of the reference point C is ( 0.5 ksi , 0 ) . The coordinates of the reference point A ( σ x , τ x y ) = ( 3 , − 4 ) . Find the radius of the circle using the relation: R = ( σ x − σ avg ) 2 + ( τ x y ) 2 (2) Substitute 0.5 ksi for σ avg , 3 ksi for σ x , and − 4 ksi for τ x y in Equation (2). R = ( 3 − 0.5 ) 2 + ( − 4 ) 2 = 4.717 ksi Let the orientation angle ( θ ) be measured from the positive x axis to the positive x ' axis. Sketch the Mohr’s circle as shown in Figure 1. The angle θ is positive if it curls in the counterclockwise direction. The value of θ is − 20 ° . Find the coordinates of the rotated element P ( σ x ' , τ x ' y ' )

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 9.4, Problem 9.52P

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The equivalent state of stress on an element oriented 20° in the clockwise direction.

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Chapter 9 Solutions

Mechanics of Materials

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The equivalent state of stress on an element oriented 20 ° in the clockwise direction.

The equivalent state of stress on an element oriented 20° in the clockwise direction.

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Chapter 9 Solutions