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 = 350 MPa and σ y = − 200 MPa . The value of shear stress is τ x y = 500 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 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 − 200 MPa for σ y in Equation (1). σ avg = 350 − 200 2 = 75 MPa Therefore, the average normal stress is 75 MPa _ . Therefore, the coordinates of the reference point C is ( 75 , 0 ) . The coordinates of the reference point A ( σ x , τ x y ) = ( 350 , 500 ) . 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 , 350 MPa for σ x , and 500 MPa for τ x y in Equation (2). R = ( 350 − 75 ) 2 + ( 500 ) 2 = 570.64 MPa Sketch the Mohr’s circle as shown in Figure 1. The coordinates of B and D represents the principal stress σ 1 and σ 2 . Find the principal stress ( σ 1 ) using the relation: σ 1 = R + σ avg (3) Substitute 570

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Any functioning component, such as machine parts, structural members, pressure vessels, and so on, are all under the influence of more than one force. When anybody is under the influence of more than one force such as shear forces, bending moments, axial…

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Chapter 9.4, Problem 9.60P

To determine

The principal stresses.

The maximum in-plane shear stress.

The average normal stress.

The orientation of the principal stresses.

The orientation of the maximum in-plane shear stress.

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Chapter 9.4, Problem 9.59P

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Chapter 9 Solutions

Mechanics of Materials

Ch. 9.3 - In each case, the state of stress x, y, xy...Ch. 9.3 - Given the state of stress shown on the element,...Ch. 9.3 - Determine the normal stress and shear stress...Ch. 9.3 - Determine the equivalent state of stress on an...Ch. 9.3 - Also, find the corresponding orientation of the...Ch. 9.3 - Determine the equivalent state of stress on an...Ch. 9.3 - Determine the maximum principal stress at point B.Ch. 9.3 - Determine the principal stress at point C.Ch. 9.3 - Prove that the sum of the normal stresses x + y =...Ch. 9.3 - 9-2. The state of stress at a point in a member is...

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The principal stresses. The maximum in-plane shear stress. The average normal stress. The orientation of the principal stresses. The orientation of the maximum in-plane shear stress.

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The principal stresses. The maximum in-plane shear stress. The average normal stress. The orientation of the principal stresses. The orientation of the maximum in-plane shear stress.

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Chapter 9 Solutions

The principal stresses.

The maximum in-plane shear stress.

The average normal stress.

The orientation of the principal stresses.

The orientation of the maximum in-plane shear stress.