To determine The stress components acting on the plane AB . Explanation 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. Calculation: Apply sign conventions to the given Figure. The value of normal stresses are σ x = − 50 MPa and σ y = − 100 MPa . The value of shear stress is τ x y = − 28 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 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 center C . 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 − 50 MPa for σ x and − 100 MPa for σ y in Equation (1). σ avg = − 50 − 100 2 = − 75 MPa Therefore, the coordinates of the reference point C is ( − 75 MPa , 0 ) . The coordinates of the reference point A ( σ x , τ x y ) = ( − 50 , − 28 ) . Find the radius of the circle using the relation: R = ( σ x − σ avg ) 2 + ( τ x y ) 2 (2) Substitute − 75 MPa for σ avg , − 50 MPa for σ x , and − 28 MPa for τ x y in Equation (2). R = ( − 50 + 75 ) 2 + ( − 28 ) 2 = 37

9th Edition

ISBN: 9789810694364

Author: Russell C Hibbeler

Publisher: Pearson Education

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Chapter 9.3, Problem 9.30P

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The stress components acting on the plane AB.

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Chapter 9 Solutions

Mechanics Of Materials, Si Edition

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