Explanation Consider w be the width as a function of r . w = c 0 + c 1 r (1) Sketch the trapezoidal cross section as shown in Figure 1. Refer to Figure 1. The width w = b 1 at r = r 1 . The width w = b 2 at r = r 2 . The height of the trapezoid is h = r 2 − r 1 . Calculate the width b 1 as shown below. Substitute b 1 for w and r 1 for r in Equation (1). b 1 = c 0 + c 1 r 1 (2) Calculate the width b 2 as shown below. Substitute b 2 for w and r 2 for r in Equation (1). b 2 = c 0 + c 1 r 2 (3) Reduce Equations (1) and (2) as shown below. b 1 − b 2 = c 0 + c 1 r 1 − ( c 0 + c 1 r 2 ) b 1 − b 2 = c 1 ( r 1 − r 2 ) c 1 = b 1 − b 2 − ( r 2 − r 1 ) Substitute h for r 2 − r 1 . c 1 = − b 1 − b 2 h Calculate the value of c 0 as shown below. Substitute b 1 − b 2 − ( r 2 − r 1 ) for c 1 in Equation (2). b 1 = c 0 − b 1 − b 2 ( r 2 − r 1 ) r 1 b 1 r 2 − b 1 r 1 = c 0 ( r 2 − r 1 ) − b 1 r 1 + b 2 r 1 b 1 r 2 − b 2 r 1 = c 0 ( r 2 − r 1 ) c 0 = b 1 r 2 − b 2 r 1 r 2 − r 1 Substitute h for r 2 − r 1 . c 0 = b 1 r 2 − b 2 r 1 h Sketch the cross section of trapezoid with a strip of thickness dr as shown in Figure 2. Refer to Figure 2. Calculate the area of the strip ( d A ) as shown below. d A = w d r Substitute c 0 + c 1 r for w . d A = ( c 0 + c 1 r ) d r d A r = ( c 0 + c 1 r ) d r r Integrate both sides of the Equation

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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The term stress indicates the measure of internal forces that develop due to an external force's application on an object. When a specific external force is applied to an object, internal stress generates in the material. The value of stress can obtain b…

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Chapter 4.10, Problem 188P

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The expression of R for a trapezoidal cross section.

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Chapter 4.10, Problem 187P

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