Explanation Consider the state of stress on the right element. The state of stress on the element is σ x = σ 0 , σ y = 0 , and τ x y = 0 . Apply the procedure for the construction of Mohr’s circle ass shown below. Find the centre of the circle C located σ avg = σ x + σ y 2 from the origin. Plot the reference points A having coordinates A ( σ x , τ A ) . Connect the point A with C and from the shaded triangle and find the radius R of the circle. Sketch the circle once R has been determined. Construct the Mohr’s circle as shown below. Calculate the centre of the circle ( σ avg ) using average normal strain as shown below. σ avg = σ x + σ y 2 (1) Substitute σ 0 for σ x and 0 for σ y in Equation (1). σ avg = σ 0 2 The center of the circle is σ 0 2 . Coordinates of the reference point X . X = ( σ x , − τ x y ) (2) Substitute σ 0 for σ x and 0 for τ x y in Equation (2). X = ( σ 0 , 0 ) Coordinates of the reference point Y . Y = ( σ y , τ x y ) (3) Substitute 0 for σ y and 0 for τ x y in Equation (3). Y = ( 0 , 0 ) Calculate the radius ( R ) of the circle as shown below. R = ( σ x − σ avg ) 2 + ( τ x y ) 2 (4) Substitute σ 0 for σ x , σ 0 2 for σ avg , and 0 for τ x y in Equation (4). R = ( σ 0 − σ 0 2 ) 2 + 0 = ( σ 0 2 ) 2 = σ 0 2 Sketch the Mohr’s circle for the resolved components of the state of stress of the right element as shown in Figure 1. Refer to Figure 1. The state of stress of the element along x direction as follows: σ x = 1 2 σ 0 + 1 2 σ 0 cos 2 θ The state of stress of the element along y direction as follows: σ y = 1 2 σ 0 − 1 2 σ 0 cos 2 θ The shear stress of the element τ x y = 1 2 σ 0 sin 2 θ Calculate the addition of the state of stress of the elements by superposition as shown below. The state of stress of the element along x direction as follows: σ x = σ 0 + 1 2 σ 0 + 1 2 σ 0 cos 2 θ = 3 2 σ 0 + 1 2 σ 0 cos 2 θ The state of stress of the element along y direction as follows: σ y = 0 + 1 2 σ 0 − 1 2 σ 0 cos 2 θ = 1 2 σ 0 − 1 2 σ 0 cos 2 θ The shear stress of the element τ x y = 1 2 σ 0 sin 2 θ Construct the Mohr’s circle as shown below. Calculate the centre of the circle ( σ avg ) using average normal strain as shown below. Substitute 3 2 σ 0 + 1 2 σ 0 cos 2 θ for σ x and 1 2 σ 0 − 1 2 σ 0 cos 2 θ for σ y in Equation (1). σ avg = 3 2 σ 0 + 1 2 σ 0 cos 2 θ + 1 2 σ 0 − 1 2 σ 0 cos 2 θ 2 = σ 0 The center of the circle is C = σ 0

7th Edition

ISBN: 9780073398235

Author: Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, David F. Mazurek

Publisher: McGraw-Hill Education

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Stress transformation

The term stress indicates the measure of internal forces that develop due to an external force's application on an object. When a specific external force is applied to an object, internal stress generates in the material. The value of stress can obtain b…

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Chapter 7, Problem 161RP

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Find the principal planes and the principal stresses for the state of plane stress.

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Find the principal planes and the principal stresses for the state of plane stress.

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