Eng Mechanics Static Hibbeler 12th Edition Chapter 10 Problems Solution

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1. 1. Engineering Mechanics - Statics Chapter 10 Problem 10-1 Determine the moment of inertia for the shaded area about the x axis. Given: a = 2m b = 4m b ⌠ Ix = 2 ⎮ y a 1 − d y 2 y 4 Solution: Ix = 39.0 m ⎮ b ⌡ 0 Problem 10-2 Determine the moment of inertia for the shaded area about the y axis. Given: a = 2m b = 4m a ⌠ ⎮ 2 ⎡ ⎛x⎞ ⎤ 2 Iy = 2 ⎮ x b⎢1 − ⎜ ⎟ ⎥ dx 4 Solution: Iy = 8.53 m ⌡ ⎣ ⎝ a⎠ ⎦ 0 993 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
2. 2. Engineering Mechanics - Statics Chapter 10 Problem 10-3 Determine the moment of inertia for the thin strip of area about the x axis.The strip is oriented at an angle θ from the x axis. Assume that t << l. Solution: ⌠ l ⌠ 2 2 ⎮ 2 Ix = ⎮ y d A = ⎮ s sin ( θ ) t ds ⌡ ⌡0 A t l sin ( θ ) 1 3 2 Ix = 3 Problem 10-4 Determine the moment for inertia of the shaded area about the x axis. Given: a = 4 in b = 2 in Solution: a ⌠ ⎮ 3 1 ⎡ ⎛x⎞ 3⎤ Ix = ⎮ ⎢b ⎜ ⎟ ⎥ dx ⎮ 3 ⎣ ⎝ a⎠ ⎦ ⌡ 0 4 Ix = 1.07 in 994 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
3. 3. Engineering Mechanics - Statics Chapter 10 Problem 10-5 Determine the moment for inertia of the shaded area about the y axis. Given: a = 4 in b = 2 in Solution: a ⌠ 3 ⎮ 2 ⎛x⎞ I y = ⎮ x b ⎜ ⎟ dx ⌡ ⎝ a⎠ 0 4 Iy = 21.33 in Problem 10-6 Determine the moment of inertia for the shaded area about the x axis. Solution: b ⌠ 3 ⎮ ⎛ x⎞ ⎮ ⎜h ⎟ Ix = ⎮ ⎝ b ⎠ dx = 2 b h3 Ix = 2 bh 3 ⎮ 3 15 15 ⌡ 0 995 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
4. 4. Engineering Mechanics - Statics Chapter 10 Alternatively h ⌠ ⎮ 2⎛ 2⎞ Ix = ⎮ y ⎜ b − b y ⎟ d y = 2 b h3 Ix = 2 bh 3 ⎮ ⎜ 2⎟ 15 15 ⌡0 ⎝ h ⎠ Problem 10-7 Determine the moment of inertia for the shaded area about the x axis. Solution: b ⌠ ⎮ ⎡ 1⎤ ⎮ ⎢ n⎥ 3 Ix = ⎜ ⎞ ⎮ A y2 ⎢a − a ⎛ y ⎟ ⎥ d y Ix = ab ⎮ ⌡ ⎣ ⎝ b⎠ ⎦ 3( 1 + 3n) 0 Problem 10-8 Determine the moment of inertia for the shaded area about the y axis. 996 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
5. 5. Engineering Mechanics - Statics Chapter 10 Solution: ⌠ a ⌠ 2 ⎮ 2 I y = ⎮ x d A = ⎮ x y dx ⌡ ⌡0 a b ⌠ n+ 2 a ⎡⎛ b ⎞ xn + 3 ⎤ ⎢ ⎥ Iy = ⎮ x dx = n⌡ ⎢⎜ an ⎟ n + 3⎥ a 0 ⎣⎝ ⎠ ⎦ 0 3 ba Iy = n+3 Problem 10-9 Determine the moment of inertia for the shaded area about the x axis. Given: a = 4 in b = 2 in Solution: b ⌠ ⎮ 2⎡ y⎞ ⎤ 2 Ix = ⎮ y ⎢a − a ⎛ ⎟ ⎥ d y ⎜ ⌡ ⎣ ⎝ b⎠ ⎦ 0 4 Ix = 4.27 in Problem 10-10 Determine the moment of inertia for the shaded area about the y axis. 997 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
6. 6. Engineering Mechanics - Statics Chapter 10 Given: a = 4 in b = 2 in Solution: a ⌠ Iy = ⎮ x b 2 x dx ⎮ a ⌡ 0 4 Iy = 36.6 in Problem 10-11 Determine the moment of inertia for the shaded area about the x axis Given: a = 8 in b = 2 in Solution: b ⌠ ⎮ 2⎛ ⎜a − a y ⎞ d y 3 Ix = ⎮ y ⎟ Ix = 10.67 in 4 ⎮ ⎜ 3⎟ ⌡0 ⎝ b ⎠ Problem 10-12 Determine the moment of inertia for the shaded area about the x axis 998 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
7. 7. Engineering Mechanics - Statics Chapter 10 Given: a = 2m b = 1m Solution: b ⌠ 2 ⎛ ⎮ 2⎞ Ix = ⎮ y a⎜ 1 − ⎟ dy y 4 Ix = 0.53 m ⎮ ⎜ 2⎟ ⌡− b ⎝ b ⎠ Problem 10-13 Determine the moment of inertia for the shaded area about the y axis Given: a = 2m b = 1m Solution: a ⌠ Iy = ⎮ x 2b 1 − dx 2 x 4 Iy = 2.44 m ⎮ a ⌡ 0 Problem 10-14 Determine the moment of inertia for the shaded area about the x axis. Given: a = 4 in b = 4 in 999 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
8. 8. Engineering Mechanics - Statics Chapter 10 Solution: b ⌠ ⎮ 2⎡ 2⎤ ⎛ y⎞ ⎥ dy Ix = ⎮ y ⎢a − a ⎜ ⎟ ⌡ ⎣ ⎝ b⎠ ⎦ 0 4 Ix = 34.1 in Problem 10-15 Determine the moment of inertia for the shaded area about the y axis. Given: a = 4 in b = 4 in Solution: a ⌠ Iy = ⎮ x b 2 x dx ⎮ a ⌡ 0 4 Iy = 73.1 in 1000 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
9. 9. Engineering Mechanics - Statics Chapter 10 Problem 10-16 Determine the moment of inertia of the shaded area about the x axis. Given: a = 2 in b = 4 in Solution: a ⌠ 3 ⎮ 1⎛ ⎛ π x ⎞⎞ Ix = ⎮ ⎜ b cos ⎜ ⎟ ⎟ dx 3⎝ ⎝ 2a ⎠ ⎠ ⌡ −a 4 Ix = 36.2 in Problem 10-17 Determine the moment of inertia for the shaded area about the y axis. Given: a = 2 in b = 4 in Solution: a ⌠ Iy = ⎮ 2 x b cos ⎜ ⎛ π x ⎞ dx ⎮ ⎟ ⌡ ⎝ 2a ⎠ −a 4 Iy = 7.72 in Problem 10-18 Determine the moment of inertia for the shaded area about the x axis. 1001 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10. 10. Engineering Mechanics - Statics Chapter 10 Given: a = 4 in b = 2 in Solution: Solutiona ⌠ ⎮ ⎛ ⎛ π x ⎞⎞ 3 ⎮ ⎜b cos ⎜ ⎟⎟ Ix = ⎮ ⎝ ⎝ 2a ⎠⎠ dx Ix = 9.05 in 4 ⎮ 3 ⌡ −a Problem 10-19 Determine the moment of inertia for the shaded area about the y axis. Given: a = 4 in b = 2 in Solution: a ⌠ Iy = ⎮ 2 x b cos ⎜ ⎛ π x ⎞ dx 4 ⎮ ⎟ Iy = 30.9 in ⌡ ⎝ 2a ⎠ −a Problem 10-20 Determine the moment for inertia of the shaded area about the x axis. 1002 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
11. 11. Engineering Mechanics - Statics Chapter 10 Given: a = 2 in b = 4 in c = 12 in Solution: a+ b ⌠ ⎮ 3 1 ⎛c 2⎞ Ix = ⎮ ⎜ ⎟ dx ⎮ 3⎝ x ⎠ ⌡ a 4 Ix = 64.0 in Problem 10-21 Determine the moment of inertia of the shaded area about the y axis. Given: a = 2 in b = 4 in c = 12 in Solution: a+ b ⌠ 2⎛ c ⎞ 2 ⎮ Iy = ⎮ x ⎜ ⎟ dx ⌡ ⎝x⎠ a 1003 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
12. 12. Engineering Mechanics - Statics Chapter 10 4 Iy = 192.00 in Problem 10-22 Determine the moment of inertia for the shaded area about the x axis. Given: a = 2m b = 2m Solution: b ⌠ ⎮ 2 ⎛ y2 ⎞ I x = ⎮ y a⎜ ⎟ d y 4 Ix = 3.20 m ⎮ ⎜ b2 ⎟ ⌡0 ⎝ ⎠ Problem 10-23 Determine the moment of inertia for the shaded area about the y axis. Use Simpson's rule to evaluate the integral. Given: a = 1m b = 1m 1004 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
13. 13. Engineering Mechanics - Statics Chapter 10 Solution: a ⌠ 2 ⎮ ⎛x⎞ ⎮ ⎜a⎟ Iy = ⎮ x b e⎝ ⎠ dx 2 4 Iy = 0.628 m ⌡0 Problem 10-24 Determine the moment of inertia for the shaded area about the x axis. Use Simpson's rule to evaluate the integral. Given: a = 1m b = 1m Solution: a ⌠ ⎮ 3 ⎮ ⎡ ⎛ x ⎞ 2⎥ ⎢ ⎜ ⎟⎤ ⎮ ⎢b e⎝ a ⎠ ⎥ Iy = ⎮ ⎣ ⎦ dx Iy = 1.41 m 4 ⎮ 3 ⌡ 0 Problem 10-25 The polar moment of inertia for the area is IC about the z axis passing through the centroid C. The moment of inertia about the x axis is Ix and the moment of inertia about the y' axis is Iy'. Determine the area A. Given: 4 IC = 28 in 1005 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
14. 14. Engineering Mechanics - Statics Chapter 10 4 Ix = 17 in 4 Iy' = 56 in a = 3 in Solution: IC = Ix + Iy Iy = IC − Ix 2 Iy' = Iy + A a Iy' − Iy 2 A = A = 5.00 in 2 a Problem 10-26 The polar moment of inertia for the area is Jcc about the z' axis passing through the centroid C. If the moment of inertia about the y' axis is Iy' and the moment of inertia about the x axis is Ix. Determine the area A. Given: 6 4 Jcc = 548 × 10 mm 6 4 Iy' = 383 × 10 mm 6 4 Ix = 856 × 10 mm h = 250 mm Solution: 2 Ix' = Ix − A h Jcc = Ix' + Iy' 2 Jcc = Ix − A h + Iy' 1006 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
15. 15. Engineering Mechanics - Statics Chapter 10 Ix + Iy' − Jcc A = 2 h 3 2 A = 11.1 × 10 mm Problem 10-27 Determine the radius of gyration kx of the column's cross-sectional area. Given: a = 100 mm b = 75 mm c = 90 mm d = 65 mm Solution: Cross-sectional area: A = ( 2b) ( 2a) − ( 2d) ( 2c) Moment of inertia about the x axis: 1 3 1 3 Ix = ( 2b) ( 2a) − ( 2d) ( 2c) 12 12 Radius of gyration about the x axis: Ix kx = kx = 74.7 mm A Problem 10-28 Determine the radius of gyration ky of the column's cross-sectional area. Given: a = 100 mm b = 75 mm 1007 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
16. 16. Engineering Mechanics - Statics Chapter 10 c = 90 mm d = 65 mm Solution: Cross-sectional area: A = ( 2b) ( 2a) − ( 2d) ( 2c) Moment of inertia about the y axis: 1 3 1 3 Iy = ( 2a) ( 2b) − ( 2c) ( 2d) 12 12 Radius of gyration about the y axis: Iy ky = ky = 59.4 mm A Problem 10-29 Determine the moment of inertia for the beam's cross-sectional area with respect to the x' centroidal axis. Neglect the size of all the rivet heads, R, for the calculation. Handbook values for the area, moment of inertia, and location of the centroid C of one of the angles are listed in the figure. Solution: ⎡ 2⎤ IE = 1 3 ( 6) ( 15 mm) ( 275 mm) + 4⎢1.32 10 mm + 1.36 10 mm 4 ( 3) 2 ⎛ 275 mm ⎜ ⎞ − 28 mm⎟ ⎥ ... 12 ⎣ ⎝ 2 ⎠⎦ ⎡1 2⎤ ⎛ 275 mm + 10mm⎞ ⎥ + 2⎢ 3 ( 75 mm) ( 20 mm) + ( 75 mm) ( 20 mm) ⎜ ⎟ ⎣12 ⎝ 2 ⎠⎦ 1008 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
17. 17. Engineering Mechanics - Statics Chapter 10 6 4 IE = 162 × 10 mm Problem 10-30 Locate the centroid yc of the cross-sectional area for the angle. Then find the moment of inertia Ix' about the x' centroidal axis. Given: a = 2 in b = 6 in c = 6 in d = 2 in Solution: a c⎜ ⎛ c ⎞ + b d⎛ d ⎞ ⎟ ⎜ ⎟ yc = ⎝ 2⎠ ⎝ 2⎠ yc = 2.00 in ac + bd 2 2 ac + ac⎛ − yc⎞ + b d + b d ⎛ yc − 1 3 c 1 3 d⎞ 4 Ix' = ⎜ ⎟ ⎜ ⎟ Ix' = 64.00 in 12 ⎝2 ⎠ 12 ⎝ 2⎠ Problem 10-31 Locate the centroid xc of the cross-sectional area for the angle. Then find the moment 1009 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
18. 18. Engineering Mechanics - Statics Chapter 10 of inertia Iy' about the centroidal y' axis. Given: a = 2 in b = 6 in c = 6 in d = 2 in Solution: a c⎜ ⎛ a ⎞ + b d⎛ a + b⎞ ⎟ ⎜ ⎟ xc = ⎝ 2⎠ ⎝ 2⎠ xc = 3.00 in ac + bd 2 2 c a + c a ⎛ xc − a⎞ d b + d b ⎛a + − xc⎞ 1 3 1 3 b 4 Iy' = ⎜ ⎟ + ⎜ ⎟ Iy' = 136.00 in 12 ⎝ 2⎠ 12 ⎝ 2 ⎠ Problem 10-32 Determine the distance xc to the centroid of the beam's cross-sectional area: then find the moment of inertia about the y' axis. 1010 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
19. 19. Engineering Mechanics - Statics Chapter 10 Given: a = 40 mm b = 120 mm c = 40 mm d = 40 mm Solution: 2( a + b)c⎜ ⎛ a + b ⎞ + 2a d a ⎟ xc = ⎝ 2 ⎠ 2 xc = 68.00 mm 2( a + b)c + 2d a ⎡1 a+b 2⎤ 2 c ( a + b) + c( a + b) ⎛ − xc⎞ ⎥ + 2d a3 + 2d a ⎛xc − 1 a⎞ Iy' = 2⎢ 3 ⎜ ⎟ ⎜ ⎟ ⎣ 12 ⎝ 2 ⎠ ⎦ 12 ⎝ 2⎠ 6 4 Iy' = 36.9 × 10 mm Problem 10-33 Determine the moment of inertia of the beam's cross-sectional area about the x' axis. 1011 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
20. 20. Engineering Mechanics - Statics Chapter 10 Given: a = 40 mm b = 120 mm c = 40 mm d = 40 mm Solution: 1 3 1 3 6 4 Ix' = ( a + b) ( 2c + 2d) − b ( 2d) Ix' = 49.5 × 10 mm 12 12 Problem 10-34 Determine the moments of inertia for the shaded area about the x and y axes. Given: a = 3 in b = 3 in c = 6 in d = 4 in r = 2 in Solution: 1 ⎡ 1 3 1 ⎛ 2c ⎞ 2⎤ ⎛ π r4 2 2⎞ Ix = ( a + b) ( c + d) − ⎢ b c + b c ⎜ d + ⎟ ⎥ − ⎜ + πr d ⎟ 3 3 ⎣36 2 ⎝ 3⎠⎦ ⎝ 4 ⎠ 1012 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
21. 21. Engineering Mechanics - Statics Chapter 10 4 Ix = 1192 in 3 ⎡1 2⎤ ⎛ π r4 1 3 1 ⎛ 2b ⎞ 2 2⎞ Iy = ( c + d) ( a + b) − ⎢ c b + b c ⎜ a + ⎟ ⎥−⎜ + πr a ⎟ 3 ⎣ 36 2 ⎝ 3⎠ ⎦ ⎝ 4 ⎠ 4 Iy = 364.84 in Problem 10-35 Determine the location of the centroid y' of the beam constructed from the two channels and the cover plate. If each channel has a cross-sectional area A c and a moment of inertia about a horizontal axis passing through its own centroid Cc, of Ix'c , determine the moment of inertia of the beam's cross-sectional area about the x' axis. Given: a = 18 in b = 1.5 in c = 20 in d = 10 in 2 Ac = 11.8 in 4 Ix'c = 349 in Solution: 2A c d + a b⎛ c + b⎞ ⎜ ⎟ yc = ⎝ 2⎠ yc = 15.74 in 2A c + a b 2 Ix' = ⎡Ix'c + A c ( yc − d) ⎤ 2 + 2 1 3 ⎛ b ⎞ a b + a b ⎜ c + − yc⎟ Ix' = 2158 in 4 ⎣ ⎦ 12 ⎝ 2 ⎠ Problem 10-36 Compute the moments of inertia Ix and Iy for the beam's cross-sectional area about 1013 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
22. 22. Engineering Mechanics - Statics Chapter 10 the x and y axes. Given: a = 30 mm b = 170 mm c = 30 mm d = 140 mm e = 30 mm f = 30 mm g = 70 mm Solution: 2 Ix = 1 3 1 3 a ( c + d + e) + b c + 1 3 ⎛ e⎞ g e + g e ⎜c + d + ⎟ Ix = 154 × 10 mm 6 4 3 3 12 ⎝ 2⎠ 1 3 1 3 1 3 6 4 Iy = c ( a + b ) + d f + c ( f + g) Iy = 91.3 × 10 mm 3 3 3 Problem 10-37 Determine the distance yc to the centroid C of the beam's cross-sectional area and then compute the moment of inertia Icx' about the x' axis. Given: a = 30 mm e = 30 mm b = 170 mm f = 30 mm c = 30 mm g = 70 mm d = 140 mm 1014 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
23. 23. Engineering Mechanics - Statics Chapter 10 Solution: ( a + b)c⎛ c⎞ ⎛ d⎞ ⎛ e⎞ ⎜ ⎟ + d f⎜ c + ⎟ + ( f + g)e⎜ c + d + ⎟ yc = ⎝ 2⎠ ⎝ 2⎠ ⎝ 2⎠ ( a + b)c + d f + ( f + g)e yc = 80.7 mm 2 2 Ix' = 1 3 ⎛ c⎞ ( a + b) c + ( a + b)c ⎜ yc − ⎟ + 1 3 ⎛ d ⎞ f d + f d ⎜ c + − yc⎟ ... 12 ⎝ 2⎠ 12 ⎝ 2 ⎠ 2 + 1 3 ⎛ e ⎞ ( f + g) e + ( f + g)e ⎜ c + d + − yc⎟ 12 ⎝ 2 ⎠ 6 4 Ix' = 67.6 × 10 mm Problem 10-38 Determine the distance xc to the centroid C of the beam's cross-sectional area and then compute the moment of inertia Iy' about the y' axis. Given: a = 30 mm b = 170 mm c = 30 mm d = 140 mm e = 30 mm f = 30 mm g = 70 mm Solution: f+g b c⎛ + a⎞ + ( c + d) f⎛ b f⎞ ⎜ ⎟ ⎜ ⎟ + ( f + g)e xc = ⎝2 ⎠ ⎝ 2⎠ 2 b c + b c + ( f + g)e xc = 61.6 mm 1015 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
24. 24. Engineering Mechanics - Statics Chapter 10 2 2 1 3 ⎛ a + b − x ⎞ + 1 d f 3 + d f ⎛ x − f ⎞ ... c ( a + b) + c( a + b) ⎜ Iy' = c⎟ ⎜c ⎟ 12 ⎝ 2 ⎠ 12 ⎝ 2⎠ 2 1 3 ⎛ e ( f + g) + e( f + g) ⎜ xc − f + g⎞ + ⎟ 12 ⎝ 2 ⎠ 6 4 Iy' = 41.2 × 10 mm Problem 10-39 Determine the location yc of the centroid C of the beam's cross-sectional area. Then compute the moment of inertia of the area about the x' axis Given: a = 20 mm b = 125 mm c = 20 mm f = 120 mm g = 20 mm f−c d = 2 f−c e = 2 Solution: a + g⎞ ( a + g) f⎛ ⎛ b⎞ ⎜ ⎟ + c b⎜ a + g + ⎟ yc = ⎝ 2 ⎠ ⎝ 2⎠ ( a + g) f + c b yc = 48.25 mm 2 2 1 3 ⎛ f ( a + g) + ( f) ( a + g) ⎜ yc − a + g⎞ 1 3 ⎛b ⎞ Ix' = ⎟ + c b + c b ⎜ + a + g − yc⎟ 12 ⎝ 2 ⎠ 12 ⎝2 ⎠ 6 4 Ix' = 15.1 × 10 mm 1016 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
25. 25. Engineering Mechanics - Statics Chapter 10 Problem 10-40 Determine yc, which locates the centroidal axis x' for the cross-sectional area of the T-beam, and then find the moments of inertia Ix' and Iy' . Given: a = 25 mm b = 250 mm c = 50 mm d = 150 mm Solutuion: ⎛ b ⎞ b2a + ⎛ b + c⎞ ⎜ ⎟ ⎜ ⎟ 2d c yc = ⎝ 2⎠ ⎝ 2⎠ b2a + c2d yc = 207 mm 2 2 Ix' = 1 3 ⎛ b⎞ 2a b + 2a b ⎜ yc − ⎟ + 1 3 ⎛ c ⎞ 2d c + c2d ⎜ b + − yc⎟ 12 ⎝ 2⎠ 12 ⎝ 2 ⎠ 6 4 Ix' = 222 × 10 mm 1 3 1 3 Iy' = b ( 2a) + c ( 2d) 12 12 6 4 Iy' = 115 × 10 mm Problem 10-41 Determine the centroid y' for the beam's cross-sectional area; then find Ix'. Given: a = 25 mm 1017 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
26. 26. Engineering Mechanics - Statics Chapter 10 b = 100 mm c = 25 mm d = 50 mm e = 75 mm Solution: 2( a + e + d)c⎜ ⎛ c ⎞ + 2a b⎛c + b ⎞ ⎟ ⎜ ⎟ yc = ⎝ 2⎠ ⎝ 2⎠ yc = 37.50 mm 2( a + e + d)c + 2a b 2 ( a + e + d) c + 2( a + e + d)c ⎛ yc − 2 3 c⎞ Ix' = ⎜ ⎟ ... 12 ⎝ 2⎠ ⎡1 2⎤ a b + a b ⎛c + − yc⎞ b + 2⎢ ⎥ 3 ⎜ ⎟ ⎣ 12 ⎝ 2 ⎠⎦ 6 4 Ix' = 16.3 × 10 mm Problem 10-42 Determine the moment of inertia for the beam's cross-sectional area about the y axis. Given: a = 25 mm b = 100 mm c = 25 mm d = 50 mm e = 75 mm 1018 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
27. 27. Engineering Mechanics - Statics Chapter 10 Solution: ⎡1 3 2⎤ 2 ( a + d + e) c + 2⎢ b a + a b ⎛ e + ⎞ 1 3 a ly = 3 ⎜ ⎟ ⎥ 12 ⎣12 ⎝ 2⎠ ⎦ 6 4 ly = 94.8 × 10 mm Problem 10-43 Determine the moment for inertia Ix of the shaded area about the x axis. Given: a = 6 in b = 6 in c = 3 in d = 6 in Solution: 3 ba 1 3 1 3 4 Ix = + ca + ( b + c) d Ix = 648 in 3 12 12 Problem 10-44 Determine the moment for inertia Iy of the shaded area about the y axis. Given: a = 6 in b = 6 in c = 3 in d = 6 in 1019 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
28. 28. Engineering Mechanics - Statics Chapter 10 Solution: 3 2 2 ab ⎛b + c ⎞ + 1 d ( b + c) 3 + 1 d( b + c) ⎡ 2( b + c) ⎤ ac + ac⎜ 1 3 1 Iy = + ⎟ ⎢ ⎥ 3 36 2 ⎝ 3⎠ 36 2 ⎣ 3 ⎦ 4 Iy = 1971 in Problem 10-45 Locate the centroid yc of the channel's cross-sectional area, and then determine the moment of inertia with respect to the x' axis passing through the centroid. Given: a = 2 in b = 12 in c = 2 in d = 4 in Solution: c ⎛ c + d ⎞ ( c + d)a b c + 2⎜ ⎟ yc = 2 ⎝ 2 ⎠ b c + 2( c + d)a yc = 2 in 2 2 1 ⎛ y − c ⎞ + 2 a ( c + d) 3 + 2a( c + d) ⎛ c + d − y ⎞ bc + bc⎜ c 3 Ix = ⎟ ⎜ c⎟ 12 ⎝ 2⎠ 12 ⎝ 2 ⎠ 4 Ix = 128 in Problem 10-46 Determine the moments for inertia Ix and Iy of the shaded area. Given: r1 = 2 in 1020 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
29. 29. Engineering Mechanics - Statics Chapter 10 r2 = 6 in Solution: ⎛ π r24 π r14 ⎞ Ix = ⎜ ⎟ 4 − Ix = 503 in ⎝ 8 8 ⎠ ⎛ π r24 π r14 ⎞ Iy = ⎜ ⎟ 4 − Iy = 503 in ⎝ 8 8 ⎠ Problem 10-47 Determine the moment of inertia for the parallelogram about the x' axis, which passes through the centroid C of the area. Solution: h = ( a)sin ( θ ) b ⎡( a)sin ( θ )⎤ = a b sin ( θ ) 1 3 1 3 1 3 3 Ixc = bh = ⎣ ⎦ 12 12 12 a b sin ( θ ) 1 3 3 Ixc = 12 Problem 10-48 Determine the moment of inertia for the parallelogram about the y' axis, which passes through the centroid C of the area. 1021 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
30. 30. Engineering Mechanics - Statics Chapter 10 Solution: A = b( a) sin ( θ ) 1 ⎡⎡b( a) sin ( θ ) b − 1 ( a) cos ( θ ) ( a) sin ( θ ) ( a)cos ( θ )⎤ ⎤ b + ( a)cos ( θ ) xc = ⎢⎢ b( a) sin ( θ ) ⎣ ⎥ ...⎥ = ⎢ 1 2 2 3 ⎦ ⎥ 2 ⎡b + ( a)cos ( θ )⎤ ⎢+ ( a) cos ( θ ) ( a) sin ( θ ) ⎢ ⎥ ⎥ ⎣ 2 ⎣ 3 ⎦ ⎦ 2 ( a) sin ( θ ) b + ( a)sin ( θ ) b ⎛ ⎞ 1 3 b Iy' = ⎜ − xc⎟ ... 12 ⎝2 ⎠ ⎡1 ⎡ ( a)cos ( θ )⎤ 2⎤ ( a) sin ( θ ) ⎡( a)cos ( θ )⎤ + ( a) sin ( θ ) ( a) cos ( θ ) ⎢xc − 1 + −⎢ ⎥ ... 3 ⎣ ⎦ ⎥ ⎣ 36 2 ⎣ ⎦ 3 ⎦ (θ ) ⎤ 2 ⎡ ( a)cos − x ( a) sin ( θ ) ⎡( a)cos ( θ )⎤ + ( a) sin ( θ ) ( a) cos ( θ ) ⎢b + 1 3 1 + ⎣ ⎦ c⎥ 36 2 ⎣ 3 ⎦ Simplifying we find. Iy' = ab 2 ( b + a cos ( θ ) sin ( θ ) 2 2 ) 12 Problem 10-49 Determine the moments of inertia for the triangular area about the x' and y' axes, which pass through the centroid C of the area. 1022 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
31. 31. Engineering Mechanics - Statics Chapter 10 Solution: 1 3 Ix' = bh 36 2 1 ⎛ b − a ⎞ 1 h ( b − a) a h a + ⎜a + ⎟ xc = 3 2 ⎝ 3 ⎠2 = b+a 1 1 3 h a + h ( b − a) 2 2 2 2 b+a ha⎛ a⎞ + b−a b + a⎞ h( b − a) ⎛ a + 1 3 1 2 1 3 1 Iy' = ha + ⎜ − ⎟ h ( b − a) + ⎜ − ⎟ 36 2 ⎝ 3 3 ⎠ 36 2 ⎝ 3 3 ⎠ Iy' = 1 (2 hb b − ab + a 2 ) 36 Problem 10-50 Determine the moment of inertia for the beam's cross-sectional area about the x' axis passing through the centroid C of the cross section. Given: a = 100 mm b = 25 mm c = 200 mm θ = 45 deg 1023 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
32. 32. Engineering Mechanics - Statics Chapter 10 Solution: Ix' = 1 ⎡2a ⎡2( c sin ( θ ) + b)⎤ 3⎤ ... ⎣ ⎣ ⎦⎦ 12 + 4⎢ ⎡ 1 ( c cos ( θ ) ) ( c sin ( θ ) ) 3⎤ − 2⎡1 c4⎛ θ − 1 sin ( 2θ )⎞⎤ ⎥ ⎢ ⎜ ⎟⎥ ⎣ 12 ⎦ ⎣4 ⎝ 2 ⎠⎦ 6 4 Ix' = 520 × 10 mm Problem 10-51 Determine the moment of inertia of the composite area about the x axis. Given: a = 2 in b = 4 in c = 1 in d = 4 in Solution: d ⌠ ⎞ ⎮ 1 ⎡ ⎡ ⎛ x ⎞ 2⎤⎤ 3 1 ⎛ π c4 Ix = ( a + b) ( 2a) − ⎜ + πc a ⎟ + ⎮ ⎢2a⎢1 − ⎜ ⎟ ⎥⎥ dx 3 2 2 3 ⎝ 4 ⎠ ⎮ 3 ⎣ ⎣ ⎝ d ⎠ ⎦⎦ ⌡0 4 Ix = 153.7 in 1024 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
33. 33. Engineering Mechanics - Statics Chapter 10 Problem 10-52 Determine the moment of inertia of the composite area about the y axis. Given: a = 2 in b = 4 in c = 1 in d = 4 in Solution: d ⌠ 3 ⎛ πc 2 2⎞ ⎮ 2 ⎡ x⎞ ⎤ 4 2 ⎟ + ⎮ x 2a⎢1 − ⎛ ⎟ ⎥ dx 1 Iy = ( 2a) ( a + b) − ⎜ + πc b ⎜ 3 ⎝ 4 ⎠ ⌡ ⎣ ⎝ d⎠ ⎦ 0 4 Iy = 271.1 in Problem 10-53 Determine the radius of gyration kx for the column's cross-sectional area. Given: a = 200 mm b = 100 mm Solution: ⎡1 3 a b ⎤ 2 ⎢ ba + ba⎛ + ⎞ ⎥ 1 3 Ix = ( 2a + b) b + 2 ⎜ ⎟ 12 ⎣ 12 ⎝ 2 2⎠ ⎦ Ix kx = kx = 109 mm b( 2a + b) + 2a b 1025 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
34. 34. Engineering Mechanics - Statics Chapter 10 Problem 10-54 Determine the product of inertia for the shaded portion of the parabola with respect to the x and y axes. Given: a = 2 in b = 1 in a b ⌠ ⌠ 4 Ixy = ⎮ ⎮ x y d y dx Ixy = 0.00 m ⎮ ⎮ 2 ⌡− a ⌡ ⎛x⎞ b ⎜a⎟ ⎝ ⎠ Also because the area is symmetric about the y axis, the product of inertia must be zero. Problem 10-55 Determine the product of inertia for the shaded area with respect to the x and y axes. Solution: 1 3 b ⌠h ⎛ ⎞ x ⌠ ⎮ ⎜ ⎟ ⎮⎮ ⎝b⎠ 3 3 2 2 2 2 Ixy = ⎮ ⎮ x y d y dx = b h Ixy = b h ⌡ ⌡ 16 16 0 0 1026 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
35. 35. Engineering Mechanics - Statics Chapter 10 Problem 10-56 Determine the product of inertia of the shaded area of the ellipse with respect to the x and y axes. Given: a = 4 in b = 2 in Solution: a ⌠ ⎮ ⎡⎢ ⎛x⎞ ⎥ 2⎤ ⎮ ⎮ x ⎢b 1 − ⎜ a ⎟ ⎥ 2 ⎝ ⎠ b 1 − ⎛ x ⎞ dx 4 Ixy = ⎮ ⎢ ⎥ ⎜ ⎟ Ixy = 8.00 in ⌡0 ⎣ ⎦ ⎝ a⎠ 2 Problem 10-57 Determine the product of inertia of the parabolic area with respect to the x and y axes. 1027 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
36. 36. Engineering Mechanics - Statics Chapter 10 Solution: a ⌠ ⎜ a ⎞ ⎮ ⎛b x ⎟ 2 ⎮ x 1 3 b 1 2 2 Ixy = ⎮ x⎜ ⎟b dx = a Ixy = a b ⌡0 ⎝ 2 ⎠ a 6 a 6 Problem 10-58 Determine the product of inertia for the shaded area with respect to the x and y axes. Given: a = 8 in b = 2 in Solution: a ⌠ 1 ⎮ ⎮ 3 1 ⎮ ⎛x⎞ b⎜ ⎟ 3 Ixy = ⎮ x ⎝ a ⎠ b ⎛ x ⎞ dx 4 ⎮ ⎜ ⎟ Ixy = 48.00 in ⌡0 2 ⎝ a⎠ Problem 10-59 Determine the product of inertia for the shaded parabolic area with respect to the x and y axes. Given: a = 4 in b = 2 in Solution: a ⌠ Ixy = ⎮ x b x x b dx ⎮ 2 a a ⌡0 1028 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
37. 37. Engineering Mechanics - Statics Chapter 10 4 Ixy = 10.67 in Problem 10-60 Determine the product of inertia for the shaded area with respect to the x and y axes. Given: a = 2m b = 1m Solution: a ⌠ ⎛b x⎞ Ixy = ⎮ x⎜ x 4 ⎮ ⎝2 1− ⎟ b 1 − dx Ixy = 0.333 m a⎠ a ⌡0 Problem 10-61 Determine the product of inertia for the shaded area with respect to the x and y axes. 1029 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
38. 38. Engineering Mechanics - Statics Chapter 10 Solution: h ⌠ 2 ⎮ ⎡ 1⎤ ⎮ ⎢ 3⎥ ⎮ y 1 ⎢b ⎛ y ⎞ ⎥ d y = 3 b2 h2 3 2 2 Ixy = ⎮ ⎜ ⎟ Ixy = h b 2 ⎣ ⎝ h⎠ ⎦ 16 16 ⌡0 Problem 10-62 Determine the product of inertia of the shaded area with respect to the x and y axes. Given: a = 4 in b = 2 in Solution: a ⌠ 3 3 ⎮ ⎛ b⎞ ⎛ x ⎞ ⎛ x ⎞ Ixy = ⎮ x⎜ ⎟ ⎜ ⎟ b ⎜ ⎟ dx ⌡0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2 a a 4 Ixy = 4.00 in 1030 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
39. 39. Engineering Mechanics - Statics Chapter 10 Problem 10-63 Determine the product of inertia for the shaded area with respect to the x and y axes. Solution: a ⌠ ⎮ ⎛ b xn ⎞ xn 2 2 Ixy = ⎮ x⎜ ⎟ b dx a b Ix = provided n ≠ −1 ⎮ ⎝⎜ 2 an ⎟ an 4( n + 1) ⌡ ⎠ 0 Problem 10-64 Determine the product of inertia for the shaded area with respect to the x and y axes. Given: a = 4 ft Solution: a ⌠ ⎮ ( a − x) 2 Ixy = ⎮ x ( a− x) dx 2 2 ⌡0 4 Ixy = 0.91 ft 1031 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
40. 40. Engineering Mechanics - Statics Chapter 10 Problem 10-65 Determine the product of inertia for the shaded area with respect to the x and y axes. Use Simpson's rule to evaluate the integral. Given: a = 1m b = 0.8 m Solution: a ⌠ ⎮ 2 2 ⎛x⎞ ⎛x⎞ b⎞ ⎜ a ⎟ ⎜a⎟ ⎮ Ixy = ⎮ x⎛ ⎟ e⎝ ⎠ b e⎝ ⎠ dx ⎜ Ixy = ⎮ ⎝ 2⎠ ⌡0 Problem 10-66 Determine the product of inertia for the parabolic area with respect to the x and y axes. Given: a = 1 in b = 2 in Solution: Due to symmetry about y axis Ixy = 0 a ⌠ Also ⎮ 2 x ⎮ b+b ⎮ a 2 ⎛ 2⎞ ⎜ b − b x ⎟ dx 4 Ixy = ⎮ x Ixy = 0.00 m 2 ⎜ a ⎟ 2 ⎮ ⌡ ⎝ ⎠ −a 1032 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
41. 41. Engineering Mechanics - Statics Chapter 10 Problem 10-67 Determine the product of inertia for the cross-sectional area with respect to the x and y axes that have their origin located at the centroid C. Given: a = 20 mm b = 80 mm c = 100 mm Solution: c⎛b a⎞ Ixy = 2b a ⎜ − ⎟ 2 ⎝ 2 2⎠ 4 Ixy = 4800000.00 mm Problem 10-68 Determine the product of inertia for the beam's cross-sectional area with respect to the x and y axes. Given: a = 12 in b = 8 in c = 1 in d = 3 in Solution: ⎛ c ⎞ ⎛ b ⎞ c b + ⎛ a ⎞ ⎛ c ⎞ ( a − 2c)c + d c⎛ a − c ⎞⎛ d ⎞ 4 Ixy = ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ Ixy = 97.75 in ⎝ 2⎠⎝ 2 ⎠ ⎝ 2 ⎠⎝ 2 ⎠ ⎝ 2 ⎠⎝ 2 ⎠ Problem 10-69 Determine the location (xc, yc) of the centroid C of the angle's cross-sectional area, and then 1033 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
42. 42. Engineering Mechanics - Statics Chapter 10 compute the product of inertia with respect to the x' and y' axes. Given: a = 18 mm b = 150 mm Solution: ⎛ a ⎞ a b + a( b − a) ⎛ a + b ⎞ ⎜ ⎟ ⎜ ⎟ xc = ⎝ 2⎠ ⎝ 2 ⎠ a b + a( b − a) xc = 44.1 mm ⎛ b ⎞ a b + ⎛ a ⎞ a( b − a) ⎜ ⎟ ⎜ ⎟ yc = ⎝ 2⎠ ⎝ 2⎠ a b + a( b − a) yc = 44.1 mm ⎛ Ix'y' = a b ⋅ −⎜ xc − a⎞⎛ b ⎞ ⎛ a⎞⎛ b a ⎞ ⎟ ⎜ − yc⎟ + a( b − a) ⋅ −⎜ yc − ⎟ ⎜ + − xc⎟ ⎝ 2⎠⎝ 2 ⎠ ⎝ 2⎠⎝ 2 2 ⎠ 6 4 Ix'y' = −6.26 × 10 mm 1034 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
43. 43. Engineering Mechanics - Statics Chapter 10 Problem 10-70 Determine the product of inertia of the beam's cross-sectional area with respect to the x and y axes that have their origin located at the centroid C. Given: a = 5 mm b = 30 mm c = 50 mm Solution: a + b⎞ a( b − a) ⎛ ⎜ ⎛ a⎞ ⎟ + c a⎜ ⎟ xc = ⎝ 2 ⎠ ⎝ 2⎠ a( b − a) + a c xc = 7.50 mm a( b − a) ⎛ a⎞ ⎛c⎞ ⎜ ⎟ + c a⎜ ⎟ yc = ⎝ 2⎠ ⎝ 2⎠ a( b − a) + c a yc = 17.50 mm Ixy = ( b − a)a⎜ ⎛ a − y ⎞ ⎛ a + b − x ⎞ + a c⎛ a − x ⎞ ⎛ c − y ⎞ c⎟ ⎜ c⎟ ⎜ c⎟ ⎜ c⎟ ⎝2 ⎠⎝ 2 ⎠ ⎝2 ⎠⎝ 2 ⎠ 3 4 Ixy = −28.1 × 10 mm 1035 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
44. 44. Engineering Mechanics - Statics Chapter 10 Problem 10-71 Determine the product of inertia for the shaded area with respect to the x and y axes. Given: a = 2 in b = 1 in c = 2 in d = 4 in Solution: lxy = 2a( c + d)a⎜ ⎛ c + d ⎞ − π b2 a d ⎟ ⎝ 2 ⎠ 4 lxy = 119 in Problem 10-72 Determine the product of inertia for the beam's cross-sectional area with respect to the x and y axes that have their origin located at the centroid C. Given: a = 1 in b = 5 in c = 5 in Solution: Ixy = 2b a⎜ ⎛ a − b ⎞ ⎛c + a ⎞ ⎟⎜ ⎟ ⎝ 2 2⎠⎝ 2 ⎠ 4 Ixy = −110 in 1036 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
45. 45. Engineering Mechanics - Statics Chapter 10 Problem 10-73 Determine the product of inertia for the cross-sec-tional area with respect to the x and y axes. Given: a = 4 in b = 1 in c = 6 in Solution: lxy = b a⎜ ⎛ a ⎞ ⎛ c + 3b ⎞ + c b⎛b + c ⎞ ⎛ b ⎞ 4 ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ lxy = 72 in ⎝ 2 ⎠⎝ 2⎠ ⎝ 2⎠⎝ 2 ⎠ Problem 10-74 Determine the product of inertia for the beam's cross-sectional area with respect to the u and v axes. 1037 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
46. 46. Engineering Mechanics - Statics Chapter 10 Given: a = 150 mm b = 200 mm t = 20 mm θ = 20 deg Solution: Moments of inertia Ix and Iy: 1 3 1 3 6 4 Ix = 2a ( 2b) − ( 2a − t) ( 2b − 2t) Ix = 511.36 × 10 mm 12 12 2 3 2 3 4 Iy = t ( 2a) + ( b − t) t Iy = 90240000.00 mm 12 12 4 The section is symmetric about both x and y axes; Ixy = 0mm therefore Ixy = 0. ⎛ Ix − Iy ⎞ ⎟ sin ( 2θ ) + Ixy cos ( 2θ ) 6 4 Iuv = ⎜ Iuv = 135 × 10 mm ⎝ 2 ⎠ Problem 10-75 Determine the moments of inertia Iu and Iv and the product of inertia Iuv for the rectangular area.The u and v axes pass through the centroid C. Given: a = 40 mm b = 160 mm θ = 30 deg 1038 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
47. 47. Engineering Mechanics - Statics Chapter 10 Solution: 1 3 1 3 4 Ix = ab Iy = ba Ixy = 0 mm 12 12 Ix + Iy ⎛ Ix − Iy ⎞ Iu = +⎜ ⎟ cos ( 2θ ) − Ixy sin ( 2θ ) 2 ⎝ 2 ⎠ 6 4 Iu = 10.5 × 10 mm ⎛ Ix + Iy ⎞ ⎛ Ix − Iy ⎞ Iv = ⎜ ⎟−⎜ ⎟ cos ( 2θ ) − Ixy sin ( 2θ ) ⎝ 2 ⎠ ⎝ 2 ⎠ 6 4 Iv = 4.05 × 10 mm ⎛ Ix − Iy ⎞ Iuv = ⎜ ⎟ sin ( 2θ ) + Ixy cos ( 2θ ) ⎝ 2 ⎠ 6 4 Iuv = 5.54 × 10 mm Problem 10-76 Determine the distance yc to the centroid of the area and then calculate the moments of inertia Iu and Iv for the channel`s cross-sectional area. The u and v axes have their origin at the centroid C. For the calculation, assume all corners to be square. Given: a = 150 mm b = 10 mm c = 50 mm θ = 20 deg 1039 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
48. 48. Engineering Mechanics - Statics Chapter 10 Solution: 2a b b ⎛ c⎞ + 2c b⎜ b + ⎟ yc = 2 ⎝ 2⎠ yc = 12.50 mm 2a b + 2c b 2 ⎡1 2⎤ 2a b + 2a b ⎛ yc − b⎞ b c + b c ⎛b + ⎞ 1 c + 2⎢ ⎥ 3 3 Ix = ⎜ ⎟ ⎜ − yc⎟ 12 ⎝ 2⎠ ⎣12 ⎝ 2 ⎠⎦ 3 4 Ix = 908.3 × 10 mm ⎡1 3 b ⎤ 2 ⎢ c b + c b ⎛a − ⎞ ⎥ 1 3 6 4 Iy = b ( 2a) + 2 ⎜ ⎟ Iy = 43.53 × 10 mm 12 ⎣12 ⎝ 2⎠ ⎦ 4 Ixy = 0 mm (By symmetry) ⎛ Ix + Iy ⎞ ⎛ Ix − Iy ⎞ ⎟ cos ( 2θ ) − Ixy sin ( 2θ ) 6 4 Iu = ⎜ ⎟+⎜ Iu = 5.89 × 10 mm ⎝ 2 ⎠ ⎝ 2 ⎠ ⎛ Ix + Iy ⎞ ⎛ Ix − Iy ⎞ ⎟ cos ( 2θ ) + Ixy sin ( 2θ ) 6 4 Iv = ⎜ ⎟−⎜ Iv = 38.5 × 10 mm ⎝ 2 ⎠ ⎝ 2 ⎠ Problem 10-77 Determine the moments of inertia for the shaded area with respect to the u and v axes. Given: a = 0.5 in b = 4 in c = 5 in θ = 30 deg 1040 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
49. 49. Engineering Mechanics - Statics Chapter 10 Solution: 4 Moment and Product of Inertia about x and y Axes: Since the Ixy = 0 in shaded area is symmetrical about the x axis, 1 3 1 3 4 Ix = 2a c + b ( 2a) Ix = 10.75 in 12 12 2 1 3⎛a + b ⎞ + 1 c ( 2a) 3 2a b + 2a b ⎜ 4 Iy = ⎟ Iy = 30.75 in 12 ⎝ 2⎠ 12 Moment of Inertia about the Inclined u and v Axes ⎛ Ix + Iy ⎞ ⎛ Ix − Iy ⎞ ⎟ cos ( 2θ ) − Ixy sin ( 2θ ) 4 Iu = ⎜ ⎟+⎜ Iu = 15.75 in ⎝ 2 ⎠ ⎝ 2 ⎠ ⎛ Ix + Iy ⎞ ⎛ Ix − Iy ⎞ ⎟ cos ( 2θ ) + Ixy sin ( 2θ ) 4 Iv = ⎜ ⎟−⎜ Iv = 25.75 in ⎝ 2 ⎠ ⎝ 2 ⎠ Problem 10-78 Determine the directions of the principal axes with origin located at point O, and the principal moments of inertia for the rectangular area about these axes. Given: a = 6 in b = 3 in Solution: 1 3 4 Ix = ba Ix = 216 in 3 1 3 4 Iy = ab Iy = 54 in 3 a b 4 Ixy = ab Ixy = 81 in 2 2 1041 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
50. 50. Engineering Mechanics - Statics Chapter 10 −2Ixy ⎛ Ixy ⎞ tan ( 2θ ) = 1 θ = atan ⎜ 2 ⎟ θ = −22.5 deg Ix − Iy 2 ⎝ −Ix + Iy ⎠ 2 Ix + Iy ⎛ Ix − Iy ⎞ 2 4 Imax = + ⎜ ⎟ + Ixy Imax = 250 in 2 ⎝ 2 ⎠ 2 Ix + Iy ⎛ Ix − Iy ⎞ 2 4 Imin = − ⎜ ⎟ + Ixy Imin = 20.4 in 2 ⎝ 2 ⎠ Problem 10-79 Determine the moments of inertia Iu , Iv and the product of inertia Iuv for the beam's cross-sectional area. Given: θ = 45 deg a = 8 in b = 2 in c = 2 in d = 16 in Solution: 2 2 3 I x = ( a + b) c + 1 3 ⎛ d⎞ 2b d + 2b d ⎜ ⎟ Ix = 5.515 × 10 in 3 4 3 12 ⎝ 2⎠ 1 3 1 3 3 4 Iy = [ 2( a + b) ] c + ( 2b) d Iy = 1.419 × 10 in 12 12 4 Ixy = 0 in Ix + Iy Ix − Iy cos ( 2θ ) − Ixy sin ( 2θ ) 3 4 Iu = + Iu = 3.47 × 10 in 2 2 Ix + Iy Ix − Iy cos ( 2θ ) + Ixy sin ( 2θ ) 3 4 Iv = − Iv = 3.47 × 10 in 2 2 1042 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
51. 51. Engineering Mechanics - Statics Chapter 10 Ix − Iy sin ( 2θ ) + Ixy cos ( 2θ ) 3 4 Iuv = Iuv = 2.05 × 10 in 2 Problem 10-80 Determine the directions of the principal axes with origin located at point O, and the principal moments of inertia for the area about these axes. Given: a = 4 in b = 2 in c = 2 in d = 2 in r = 1 in Solution: 3 ⎛ πr 2 2⎞ 4 1 ( c + d) ( a + b) − ⎜ + πr a ⎟ 4 Ix = Ix = 236.95 in 3 ⎝ 4 ⎠ 3 ⎛ πr 2 2⎞ 4 1 ( a + b) ( c + d) − ⎜ + πr d ⎟ 4 Iy = Iy = 114.65 in 3 ⎝ 4 ⎠ ⎛ a + b ⎞ ⎛ d + c ⎞ ( a + b) ( d + c) − d aπ r2 4 Ixy = ⎜ ⎟⎜ ⎟ Ixy = 118.87 in ⎝ 2 ⎠⎝ 2 ⎠ −Ixy ⎛ Ixy ⎞ tan ( 2 θ p ) = 1 θp = atan ⎜ 2 ⎟ θ p = −31.39 deg Ix− Iy 2 ⎝ −Ix + Iy ⎠ 2 θ p1 = θ p θ p1 = −31.39 deg θ p2 = 90 deg + θ p1 θ p2 = 58.61 deg 2 Ix + Iy ⎛ Ix − Iy ⎞ 2 4 Imax = + ⎜ ⎟ + Ixy Imax = 309 in 2 ⎝ 2 ⎠ 1043 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

Eng Mechanics Static Hibbeler 12th Edition Chapter 10 Problems Solution

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