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: 9780100257061

Author: BEER

Publisher: YUZU

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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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Chapter 7, Problem 160RP

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60 MPa PROBLEM 7.3 45 MPa For the given state of stress, determine the normal and shearing stresses exerted on the oblique face of the shaded triangular element shown. Use a method of 120 MPA analysis based on the equilibrium of that element, as was done in the 70° derivations of Sec. 7.2. O=14.19 MPa T = 15.19 MPa

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

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