Explanation Given information: The maximum and minimum moment of inertia are I min = 0.23 × 10 − 6 mm 4 and I max = 2.28 × 10 − 6 mm 4 . The angle made by the principal axis are 25.7 ° and 64.3 ° . Calculation: Calculate the moment of inertia ( I u ) and ( I v ) as follows: I u = I min = 0.23 × 10 − 6 mm 4 I v = I max = 2.28 × 10 − 6 mm 4 Show the cross-section of the cross-section as shown in Figure 1. Refer Figure 1. Consider the moment about the principal axes are M u and M v . Calculate the moment M u about the principal axes as using the relation: M u = M 0 sin θ Substitute 64.3 ° for θ . M u = M 0 sin 64.3 ° Calculate the moment M v about the principal axes as using the relation: M v = M 0 cos θ Substitute 64.3 ° for θ . M v = M 0 cos 64.3 ° Consider the angle made by the neutral axis with the principal axis is denoted by ϕ . Calculate the value of the angle ϕ using the relation: tan ϕ = I v I u tan θ (1) Substitute 0.23 × 10 − 6 mm 4 for I u , 2.28 × 10 − 6 mm 4 for I v , and 64.3 ° for θ in Equation (1). tan ϕ = 2.28 × 10 − 6 0.23 × 10 − 6 tan ( 64.3 ° ) = 9.91304 × 2.077 tan ϕ = 20.59 ϕ = tan − 1 ( 20.59 ) = 87.22 ° Consider the points A and B are the farthest points from neutral axis. Calculate the distance y B and z B as follows: y B = − 45 mm z B = − 35 mm Calculate the perpendicular distance ( u B ) from principal axis to point B using the relation: u B = y B cos θ + z B sin θ (2) Substitute − 45 mm for y B , − 35 mm for z B , and 64.3 ° for θ in Equation (2). u B = − 45 cos ( 64.3 ° ) + ( − 35 ) sin ( 64.3 ° ) = − 45 × 0.4336 − 35 × 0.9010 = − 19.512 − 31.535 = − 51.5 mm Calculate the perpendicular distance ( v B ) from principal axis to point B using the relation: v B = z B cos θ − y B sin θ (3) Substitute − 45 mm for y B , − 35 mm for z B , and 64

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ISBN: 9780073398235

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

Publisher: McGraw-Hill Education

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Chapter 4.9, Problem 152P

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The largest permissible value of moment M0 such that the maximum stress in the beam is 80 MPa.

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Chapter 4.9, Problem 151P

Chapter 4.9, Problem 153P

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The largest permissible value of moment M 0 such that the maximum stress in the beam is 80 MPa .

Solution Summary: The author calculates the moment of inertia about the principal axes as using the relation: M_0mathrm

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The largest permissible value of moment M0 such that the maximum stress in the beam is 80 MPa.

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