(a) Explanation Given information: The width and depth of the cross-section of the beam are b = 25 mm and d = 90 mm . The strains at A and B are ε A = + 600 μ and ε B = + 420 μ . The modulus of elasticity ( E ) of the beam is 200 GPa . Calculation: Calculate the cross-sectional area of the beam using the relation: A = b h (1) Substitute 25 mm for b and 90 mm for h in Equation (1). A = 25 × 90 = 2 , 250 mm 2 × ( 1 m 2 10 6 mm 2 ) = 2.25 × 10 − 3 m 2 Calculate the moment of inertia ( I ) using the relation: I = 1 12 b h 3 (2) Substitute 25 mm for b and 90 mm for h in Equation (2). I = 1 12 × 25 × 90 3 = 729 , 000 12 = 1 , 518 , 750 mm 4 × ( 1 m 4 10 12 mm 4 ) = 1.51875 × 10 − 6 m 4 Consider the distance between the neutral axis and the points A and B are denoted by y A and y B . Calculate the value of y A and y B as follows: y A = 60 − 45 mm =15 mm × ( 1 m 1000 mm ) = 0.015 m y B = 15 − 45 mm = − 30 mm × ( 1 m 1000 mm ) = − 0.030 m Calculate the stress σ A from the strain guage at A using the relation: σ A = E ε A (3) Substitute 200 GPa for E and + 600 μ for ε A in Equation (3). σ A = 200 GPa × ( 10 9 N m 2 1 GPa ) × 600 μ × ( 10 − 6 1 μ ) = 200 × 10 9 × 600 × 10 − 6 = 120 × 10 6 N m 2 Calculate the stress σ B from the strain gage at A using the relation: σ B = E ε B (4) Substitute 200 GPa for E and + 420 μ for ε B in Equation (3). σ B = 200 GPa × ( 10 9 N m 2 1 GPa ) × ( 420 μ ) × ( 10 − 6 1 μ ) = − 200 × 10 9 × 420 × 10 − 6 = 84 × 10 6 N m 2 Show the expression for the stress at point A and B as follows: σ A = P A − M y A I (5) σ B = P A − M y B I (6) Subtract Equation (6) from Equation (5) (b) To determine The value of eccentric force P .

7th Edition

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 4.7, Problem 122P

(a)

To determine

The value of distance d.

(b)

To determine

The value of eccentric force P.

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Chapter 4.7, Problem 121P

Chapter 4.7, Problem 123P

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The value of distance d .

Solution Summary: The author calculates the cross-sectional area of the beam using the relation: A=bh and d=90mm.

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