(a) 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 = − 12 ksi and σ y = − 8 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 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 B having coordinates B ( 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 − 12 ksi for σ x and − 8 ksi for σ y in Equation (1). σ avg = − 12 − 8 2 = − 10 ksi Therefore, the average normal stress is − 10 ksi _ . Therefore, the coordinates of the reference point C is ( − 10 , 0 ) . The coordinates of the reference point A ( σ x , τ x y ) = ( − 12 , 4 ) (b) To determine 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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ISBN: 2810014920922

Author: HIBBELER

Publisher: PEARSON

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Chapter 9.4, Problem 9.59P

(a)

To determine

The principal stresses.

(b)

To determine

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 9.58P

Chapter 9.4, Problem 9.60P

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MECHANICS OF MATERIALS-TEXT

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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