To determine The principal stresses. The maximum in-plane shear stress at point A . The orientation of the state of stress. Explanation Given information: The bracket is subjected to a force of 3 kip. Calculation: Let N be the normal force and M be the moment at the section. Sketch the free body diagram of the bracket’s left cut segment as shown in Figure 1. Apply equations of equilibrium. Sum of forces in the x -direction is equal to 0. Σ F x = 0 N − 3 kip = 0 N = 3 kip Sum of moments at O is equal to 0. Σ M O = 0 3 kip ( 4 in . ) − M = 0 M = 12 kip ⋅ in . Find the area ( A ) of the I section: A = ( 1 in . × 2 in . ) − ( 0.75 in . × 1.5 in . ) = 0.875 in 2 Find the moment of inertia of the I section ( I ) : I = 1 12 ( 1 in . ) ( 2 in . ) 3 − 1 12 ( 0.75 in . ) ( 1.5 in . ) 3 = 0.45573 in 4 Refer the cross section: The centroid distance of the section ( y ) is 1 in. Find the normal stress at point A using the relation: σ A = N A − M y I = 3 kip 0.875 in 2 − ( − 12 kip ⋅ in . ) ( 1 in . ) 0.45573 in 4 = 29.76 ksi There is no shear force in the section. Therefore, the shear stress at point A is τ A = 0 . The value of normal stresses are σ x = 29.76 ksi and σ y = 0 . The value of shear stress is τ x y = 0 . Find the in-plane principal stresses: σ 1 = σ x = 29.76 ksi ≈ 29.8 ksi σ 2 = σ y = 0 Therefore, the in-plane principal stresses are 29.8 ksi _ and 0. Represent the state of stress on the elements as shown in Figure 2. Find the value of maximum in-plane shear stress ( τ max in-plane ) using the relation: τ max in-plane = ( σ x − σ y 2 ) 2 + τ x y 2 (1) Substitute 29.76 ksi for σ x , 0 for σ y , and 0 for τ x y in Equation (1). τ max in-plane = ( 29.76 − 0 2 ) 2 + ( 0 ) 2 = 14

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 9.3, Problem 9.26P

To determine

The principal stresses.

The maximum in-plane shear stress at point A.

The orientation of the state of stress.

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Chapter 9.3, Problem 9.25P

Chapter 9.3, Problem 9.27P

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The principal stresses.

The maximum in-plane shear stress at point A.

The orientation of the state of stress.