To determine The principal stresses acting at the point. The stress in the plane BC ( σ B C ) . Explanation Given information: Planes AB and BC are subjected to the stresses are shown in the given Figure. Calculation: 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 ' axes. Sketch the forces and the plane on the sectioned element as shown in Figure 1. The angle θ is positive if it curls in the counterclockwise direction. Therefore, The value of θ is − 135 ° . Apply sign conventions to Figure 1. The value of normal stress are σ x = − 15 ksi and σ y = σ A C . The value of shear stress are τ x y = 5 ksi and τ x ' y ' = 6 ksi . The expression for shear stress for the plane oriented ( τ x ' y ' ) using the stress transformation equation is shown below: τ x ' y ' = − σ x − σ y 2 sin 2 θ + τ x y cos 2 θ (1) Substitute 6 ksi for τ x ' y ' , − 15 ksi for σ x , σ A C for σ y , − 135 ° for θ , and 5 ksi for τ x y in Equation (1). 6 = − − 15 + σ A C 2 sin 2 ( − 135 ° ) + 5 cos 2 ( − 135 ° ) 6 = 7.5 − 0.5 σ A C + 0 σ A C = − 3 ksi Find the in-plane principal stresses ( σ 1 ) and ( σ 2 ) using the relation: σ 1 , 2 = σ x + σ y 2 ± ( σ x − σ y 2 ) 2 + τ x y 2 (2) Substitute − 15 ksi for σ x , − 3 ksi for σ y , and 5 ksi for τ x y in Equation (2)

9th Edition

ISBN: 9789810694364

Author: Russell C Hibbeler

Publisher: Pearson Education

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Chapter 9.3, Problem 9.20P

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The principal stresses acting at the point.

The stress in the plane BC (σBC) .

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Chapter 9.3, Problem 9.19P

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The principal stresses acting at the point.

The stress in the plane BC (σBC) .