To determine The equivalent state of stress on an element oriented 30 ° in the clockwise direction. Explanation Given information: The element is oriented 30 ° 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 = 350 MPa and σ y = 230 MPa . The value of shear stress is τ x y = − 480 MPa . 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 C having coordinates C ( 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 (1) Substitute 350 MPa for σ x and 230 MPa for σ y in Equation (1). σ avg = 350 + 230 2 = 290 MPa Therefore, the coordinates of the reference point C is ( 290 , 0 ) . The coordinates of the reference point A ( σ x , τ x y ) = ( 350 , − 480 ) . The coordinates of the reference point C ( σ y , − τ x y ) = ( 230 , 480 ) . Find the radius of the circle using the relation: R = ( σ x − σ avg ) 2 + ( τ x y ) 2 (2) Substitute 290 MPa for σ avg , 350 MPa for σ x , and − 480 MPa for τ x y in Equation (2). R = ( 350 − 290 ) 2 + ( − 480 ) 2 = 483.73 MPa 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.

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 9.4, Problem 9.53P

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The equivalent state of stress on an element oriented 30° in the clockwise direction.

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Chapter 9.4, Problem 9.52P

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