Explanation Given information: The bending moment is applied alone to a solid bar with a circular cross section. It causes the distortion energy as the combination of an applied bending moment M and torque T . Calculation: Consider that the distortion energy is the bending moment applied to the solid bar. The normal stress applied to the solid bar. σ = M e c I Here, M e is the equivalent bending moment, c is the radius of the bar, and I is the moment of inertia. Find the principal stresses in complex system as shown below. σ 1 , 2 = σ x + σ y 2 ± ( σ x − σ y 2 ) 2 + τ x y 2 (1) Here, σ x is the normal stress in x direction, σ y is the normal stress in y direction, and τ x y is the shear stress. Substitute M e c I for σ x , 0 for σ y , and 0 for τ x y in Equation (1). σ 1 , 2 = M e c I + 0 2 ± ( M e c I − 0 2 ) 2 + 0 = M e c 2 I ± M e c 2 I σ 1 = M e c 2 I + M e c 2 I = M e c I σ 2 = M e c 2 I − M e c 2 I = 0 Apply the maximum distortion theory for plane stress as shown below. σ 1 2 − σ 1 σ 2 + σ 2 2 = σ Y 2 Here, σ 1 and σ 2 are the principal stresses and σ Y is the principal yield stress. Apply the strain-energy density as shown below. u d = 1 + ν 3 E ( σ 1 2 − σ 1 σ 2 + σ 2 2 ) (2) Here, E is the modulus of elasticity and ν is the Poisson’s ratio. Substitute M e c I for σ 1 and 0 for σ 2 in Equation (2). ( u d ) 1 = 1 + ν 3 E ( ( M e c I ) 2 − ( M e c I ) ( 0 ) + ( 0 ) 2 ) = 1 + ν 3 E ( M e 2 c 2 I 2 ) (3) Consider that the distortion energy is the combination of the bending moment and torque applied to the solid bar. Consider the normal stress and shear stress applied to the solid bar. σ = M c I Here, M is the bending moment. τ = T c J Here, T is the torque and J is the polar moment of inertia. Substitute σ for σ x , 0 for σ y , and τ for τ x y in Equation (1). σ 1 , 2 = σ + 0 2 ± ( σ − 0 2 ) 2 + τ 2 = σ 2 ± σ 2 4 + τ 2 σ 1 = σ 2 + σ 2 4 + τ 2 σ 2 = σ 2 − σ 2 4 + τ 2 Consider a = σ 2 and b = σ 2 4 + τ 2 . Hence, σ 1 = a + b and σ 2 = a − b . Substitute ( a + b ) for σ 1 and ( a − b ) for σ 2 in Equation (2)

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Any functioning component, such as machine parts, structural members, pressure vessels, and so on, are all under the influence of more than one force. When anybody is under the influence of more than one force such as shear forces, bending moments, axial…

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Chapter 10.7, Problem 10.67P

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To derive: An expression for an equivalent bending moment Me.

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To derive: An expression for an equivalent bending moment M e .

Solution Summary: The author explains that the distortion energy is the bending moment applied to the solid bar with a circular cross section.

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