Explanation Given information: The yield strength ( σ y ) of the beam is 240 MPa . Calculation: Consider the width and depth of the beam are b = 60 mm and d = 90 mm . Calculate the distance ( c ) between the neutral axis and the outer fiber as follows: c = d 2 (1) Substitute 90 mm for d in Equation (1). c = 90 2 = 45 mm × 1 m 1 , 000 mm = 0.045 m Calculate the moment of inertia of cross-section of the beam as follows: I = 1 12 b d 3 (2) Substitute 60 mm for b and 90 mm for d in Equation (2). I = 1 12 × 60 × 90 3 = 3 , 645 , 000 mm 4 × 1 m 4 10 12 mm 4 = 3.645 × 10 − 6 m 4 Consider the value of y is 45 mm . Consider the width and depth of the shaded area are b 1 = 60 mm and d 1 = 45 mm . Calculate the shaded area ( A 1 ) of the cross-section of the beam as follows: A 1 = b 1 d 1 (1) Substitute 60 mm for b 1 and 45 mm for d 1 in Equation (1). A 1 = 60 × 45 = 2 , 700 mm 2 × 1 m 2 10 6 mm 2 = 0.0027 m 2 Calculate the force ( R ) using the relation: R = σ Y A 1 (2) Substitute 240 MPa for σ Y and 0.0027 m 2 for A 1 in Equation (2). R = 240 MPa × ( 10 6 N m 2 1 MPa ) × 0.0027 m 2 = 648 × 10 3 N The value of d is 45 mm

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Chapter 4.6, Problem 87P

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Find the residual stress at y=45 mm.

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Chapter 4.6, Problem 86P

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5.86 The cast iron inverted T-section supports two concentrated loads of magnitude P. The working stresses are 48 MPa in tension, 140 MPa in compression, and 30 MPa in shear. (a) Show that the neutral axis of the cross section is located at d ¼ 48:75 mm and that the moment of inertia of the cross-sectional area about this axis is I ¼ 11:918 106 mm4. (b) Find the maximum allowable value of P.

5.86 The cast iron inverted T-section supports two concentrated loads of magni- tude P. The working stresses are 48 MPa in tension, 140 MPa in compression, and 30 MPa in shear. (a) Show that the neutral axis of the cross section is located at d = 48.75 mm and that the moment of inertia of the cross-sectional area about this axis is I = 11.918 x 106 mm“. (b) Find the maximum allowable value of P. 1.0 m 1.0 m 15 mm 3 m 150 mm NA- d 15 mm 150 mm FIG. P5.86

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Find the residual stress at y = 45 mm .

Solution Summary: The author calculates the moment of inertia of cross-section of the beam as follows: I=112bd3.

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