FM_TOC 46060 6/22/10 11:26 AM Page i Instructor’s Solutions Manual MECHANICS OF MATERIALS EIGHTH EDITION R. C. HIBBELER Prentice Hall Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo FM_TOC 46060 6/22/10 11:26 AM Page ii Vice President and Editorial Director, ECS: Marcia Horton Senior Acquisitions Editor: Tacy Quinn Editorial Assistant: Coleen McDonald Executive Marketing Manager: Tim Galligan Senior Managing Editor: Scott Disanno Project Manager: Rose Kernan Senior Operations Supervisor: Alan Fischer Operations Specialist: Lisa McDowell Art Director: Kenny Beck Text and Cover Designer: Kenny Beck Photo Researcher: Marta Samsel Cover Images: High rise crane: Martin Mette/Shutterstock; close up of crane with heavy load: Mack7777/Shutterstock; close up of hoisting rig and telescopic arm of mobile crane: 36clicks/Shutterstock Media Director: Daniel Sandin Copyright © 2011, 2008, 2005, 2003, 2001 by R. C. Hibbeler. Published by Pearson Prentice Hall. All rights reserved. Manufactured in the United States of America. This publication is protected by Copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, 1 Lake Street, Upper Saddle River, NJ 07458. Many of the designations by manufacturers and seller to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps. 10 9 8 7 6 5 4 3 2 1 ISBN 10: 0-13-602312-6 ISBN 13: 978-0-13-602312-8 FM_TOC 46060 6/22/10 11:26 AM Page iii CONTENTS To the Instructor iv 1 Stress 1 2 Strain 73 3 Mechanical Properties of Materials 92 4 Axial Load 122 5 Torsion 214 6 Bending 329 7 Transverse Shear 472 8 Combined Loadings 532 9 Stress Transformation 619 10 Strain Transformation 738 11 Design of Beams and Shafts 830 12 Deflection of Beams and Shafts 883 13 Buckling of Columns 1038 14 Energy Methods 1159 FM_TOC 46060 6/22/10 11:26 AM Page iv To the Instructor This manual contains the solutions to all the problems in Mechanics of Materials, Eighth Edition. As stated in the preface of the text, the problems in every section are arranged in an approximate order of increasing difficulty. Be aware that answers to all but every fourth problem, which is indicated by an asterisk (*), are listed in the back of the book. Also, every fourth problem has an additional hint for the solution and is indicated with a bullet (•). Finally, those problems indicated by a square (■) will require additional numerical work. You may wish to use one of the lists of homework problems given on the following pages. Here you will find three lists for which the answers are in the back of the book, a fourth list for problems without answers, and a fifth sheet which can be used to develop your own personal syllabus. The prepared lists generally represent assignments with an easy, a moderate, and sometimes a more challenging problem. If you have any questions regarding the solutions in this manual, I would greatly appreciate hearing from you. R. C. Hibbeler [email protected] FM_TOC 46060 6/22/10 11:26 AM Page v ANSWER ASSIGNMENT Section Title Assignment 1 with Hints 1.1–1.2 1.3–1.5 1.6–1.7 2.1–2.2 3.1–3.5 3.6–3.8 4.1–4.2 4.3–4.5 4.6 4.7 4.8–4.9 5.1–5.3 5.4 5.5 5.6 5.7 5.8 5.9–5.10 6.1–6.2 6.3–6.4 6.5 6.6–6.7 6.8 6.9 6.10–6.11 7.1–7.3 7.4 7.5–7.6 8.1 8.2 9.1–9.2 9.3 9.4–9.6 9.7 10.1–10.2 10.3 10.4–10.5 10.6 10.7 11.1–11.2 11.3 11.4 12.1–12.2 12.3 12.4 12.5 12.6–12.7 12.8 12.10 13.1–13.3 13.4–13.5 13.6 13.7 14.1–14.2 14.3 14.4 14.5–14.6 14.7 14.8 14.9 Equilibrium of a Deformable Body Average Normal and Shear Stress Design of Simple Connections Strain The Stress Strain Diagram Poisson’s ratio, Shear Stress-Strain Diagram Deformation of an Axially Loaded Member Statically Indeterminate Member Thermal Stresses Stress Concentrations Inelastic Deformation and Residual Stresses Torsion Stress and Power Angle of Twist Statically Indeterminate Members Noncircular Shafts Thin-Walled Tubes Stress Concentrations Inelastic Torsion and Residual Stresses Shear and Moment Diagrams Bending Stress Unsymmetric Bending Composite Beams Curved Beams Stress Concentrations Inelastic Bending Shear Stress Shear Flow in Built-up Members Shear Center Thin-Walled Pressure Vessels Stress Due to Combined Loadings Stress Transformation Princ. Stress and Max. In-Plane Shear Stress Mohr’s Circle Absolute Maximum Shear Stress Strain Transformation Mohr’s Circle Abs. Maximum Shear Strain, Strain Rosettes Material Property Relations Theories of Failure Prismatic Beam Design Fully Stressed Beams Shaft Design Slope and Displacement by Integration Discontinuity Functions Moment-Area Theorems Method of Superposition Indet. Beams-Method of Integration Indet. Beams-Mom. Area Theorems Indet. Beams-Method of Superposition Buckling of an Ideal Column The Secant Formula, Inelastic Buckling Design of Columns for Concentric Loading Design of Columns for Eccentric Loading Elastic Strain Energy Conservation of Energy Impact Principle of Virtual Forces-Trusses Principle of Virtual Forces-Beams Castigliano’s Theorem-Trusses Castigliano’s Theorem-Beams 1–5, 1–9, 1–21 1–37, 1–45, 1–53, 1–61 1–73, 1–81, 1–85, 1–97 2–5, 2–13, 2–21 3–5, 3–9, 3–17 3–25, 3–29, 3–33 4–5, 4–13, 4–17, 4–21 4–33, 4–41, 4–49, 4–61 4–69, 4–73, 4–77 4–89, 4–93 4–101, 4–105, 4–113 5–5, 5–13, 5–25, 5–29, 5–37 5–49, 5–57, 5–61, 5–69 5–81, 5–85, 5–93 5–101, 5–105 5–109, 5–117 5–121, 5–125 5–133, 5–137 6–5, 6–9, 6–17, 6–29, 6–41, 6–45 6–53, 6–57, 6–69, 6–85 6–113, 6–117, 6–121, 6–125 6–129, 6–133, 6–137 6–145, 6–149, 6–153 6–157, 6–161 6–169, 6–173, 6–177 7–5, 7–13, 7–25 7–33, 7–41, 7–45 7–53, 7–57, 7–69 8–5, 8–13 8–21, 8–33, 8–45, 8–53, 8–57 9–5, 9–9, 9–21 9–17, 9–25, 9–33 9–61, 9–65, 9–73 9–85, 9–93 10–5, 10–9 10–21, 10–17 10–25, 10–29 10–33, 10–41, 10–49 10–61, 10–73, 10–81, 10–89 11–5, 11–9, 11–17, 11–25 11–33, 11–37 11–41, 11–45 12–5, 12–9, 12–17, 12–25 12–37, 12–41, 12–49 12–57, 12–61, 12–73 12–93, 12–97, 12–101 12–105, 12–113 12–117, 12–120 12–121, 12–125, 12–129 13–5, 13–13, 13–25 13–49, 13–57, 13–65 13–89, 13–97, 13–105 13–109, 13–121, 13–125 14–5, 14–13, 14–21 14–29, 14–33, 14–41 14–45, 14–49, 14–57 14–73, 14–81, 14–85 14–89, 14–101, 14–109 14–125, 14–129, 14–133 14–137, 14–141, 14–145 V FM_TOC 46060 VI 6/22/10 11:26 AM Page vi ANSWER ASSIGNMENT Section Title Assignment 2 1.1–1.2 1.3–1.5 1.6–1.7 2.1–2.2 3.1–3.5 3.6–3.8 4.1–4.2 4.3–4.5 4.6 4.7 4.8–4.9 5.1–5.3 5.4 5.5 5.6 5.7 5.8 5.9–5.10 6.1–6.2 6.3–6.4 6.5 6.6–6.7 6.8 6.9 6.10–6.11 7.1–7.3 7.4 7.5–7.6 8.1 8.2 9.1–9.2 9.3 9.4–9.6 9.7 10.1–10.2 10.3 10.4–10.5 10.6 10.7 Equilibrium of a Deformable Body Average Normal and Shear Stress Design of Simple Connections Strain The Stress Strain Diagram Poisson’s ratio, Shear Stress-Strain Diagram Deformation of an Axially Loaded Member Statically Indeterminate Member Thermal Stresses Stress Concentrations Inelastic Deformation and Residual Stresses Torsion Stress and Power Angle of Twist Statically Indeterminate Members Noncircular Shafts Thin-Walled Tubes Stress Concentrations Inelastic Torsion and Residual Stresses Shear and Moment Diagrams Bending Stress Unsymmetric Bending Composite Beams Curved Beams Stress Concentrations Inelastic Bending Shear Stress Shear Flow in Built-up Members Shear Center Thin-Walled Pressure Vessels Stress Due to Combined Loadings Stress Transformation Princ. Stress and Max. In-Plane Shear Stress Mohr’s Circle Absolute Maximum Shear Stress Strain Transformation Mohr’s Circle Abs. Maximum Shear Strain, Strain Rosettes Material Property Relations Theories of Failure 1–1, 1–7, 1–17, 1–25 1–31, 1–42, 1–51, 1–67 1–74, 1–82, 1–86, 1–90 2–2, 2–11, 2–17, 2–26, 2–31 3–1, 3–10, 3–18, 3–22 3–26, 3–30, 3–34 4–6, 4–7, 4–15, 4–25 4–34, 4–42, 4–45, 4–55 4–70, 4–74, 4–75 4–90, 4–95 4–97, 4–103, 4–111 5–3, 5–9, 5–27, 5–39 5–50, 5–53, 5–63, 5–67 5–77, 5–87, 5–91 5–95, 5–99 5–113, 5–118 5–122, 5–123 5–127, 5–135, 5–139 6–1, 6–2, 6–6, 6–10, 6–19, 6–22, 6–27, 6–35 6–50, 6–54, 6–63, 6–70, 6–94 6–109, 6–114, 6–126 6–127, 6–134, 6–141 6–146, 6–150, 6–154 6–158, 6–162 6–165, 6–171, 6–178 7–1, 7–14, 7–23 7–34, 7–42, 7–47 7–54, 7–63, 7–66 8–1, 8–11 8–18, 8–26, 8–43, 8–55, 8–70 9–2, 9–6, 9–18 9–14, 9–26, 9–30, 9–42 9–59, 9–67, 9–82 9–86, 9–94 10–2, 10–10 10–18, 10–19 10–22, 10–26 10–31, 10–43, 10–50 10–63, 10–69, 10–77, 10–86 11.1–11.2 11.3 11.4 12.1–12.2 12.3 12.4 12.5 12.6–12.7 12.8 12.10 13.1–13.3 13.4–13.5 13.6 13.7 14.1–14.2 14.3 14.4 14.5–14.6 14.7 14.8 14.9 Prismatic Beam Design Fully Stressed Beams Shaft Design Slope and Displacement by Integration Discontinuity Functions Moment-Area Theorems Method of Superposition Indet. Beams-Method of Integration Indet. Beams-Mom. Area Theorems Indet. Beams-Method of Superposition Buckling of an Ideal Column The Secant Formula, Inelastic Buckling Design of Columns for Concentric Loading Design of Columns for Eccentric Loading Elastic Strain Energy Conservation of Energy Impact Principle of Virtual Forces-Trusses Principle of Virtual Forces-Beams Castigliano’s Theorem-Trusses Castigliano’s Theorem-Beams 11–1, 11–7, 11–13, 11–23 11–31, 11–38 11–39, 11–42 12–6, 12–11, 12–15, 12–23 12–38, 12–47, 12–50 12–58, 12–66, 12–69 12–87, 12–91, 12–95 12–103, 12–110 12–115, 12–119 12–122, 12–127, 12–134 13–1, 13–7, 13–17, 13–31 13–50, 13–55, 13–63, 13–67 13–82, 13–95, 13–106 13–107, 13–111, 13–119 14–6, 14–10, 14–15 14–25, 14–30, 14–35 14–50, 14–54, 14–63 14–73, 14–79, 14–86 14–90, 14–103, 14–113 14–123, 14–126, 14–134 14–135, 14–138, 14–142 FM_TOC 46060 6/22/10 11:26 AM Page vii ANSWER ASSIGNMENT Section Title Assignment 3 1.1–1.2 1.3–1.5 1.6–1.7 2.1–2.2 3.1–3.5 3.6–3.8 4.1–4.2 4.3–4.5 4.6 4.7 4.8–4.9 5.1–5.3 5.4 5.5 5.6 5.7 5.8 5.9–5.10 6.1–6.2 6.3–6.4 6.5 6.6–6.7 6.8 6.9 6.10–6.11 7.1–7.3 7.4 7.5–7.6 8.1 8.2 9.1–9.2 9.3 9.4–9.6 9.7 10.1–10.2 10.3 10.4–10.5 10.6 10.7 Equilibrium of a Deformable Body Average Normal and Shear Stress Design of Simple Connections Strain The Stress Strain Diagram Poisson’s ratio, Shear Stress-Strain Diagram Deformation of an Axially Loaded Member Statically Indeterminate Member Thermal Stresses Stress Concentrations Inelastic Deformation and Residual Stresses Torsion Stress and Power Angle of Twist Statically Indeterminate Members Noncircular Shafts Thin-Walled Tubes Stress Concentrations Inelastic Torsion and Residual Stresses Shear and Moment Diagrams Bending Stress Unsymmetric Bending Composite Beams Curved Beams Stress Concentrations Inelastic Bending Shear Stress Shear Flow in Built-up Members Shear Center Thin-Walled Pressure Vessels Stress Due to Combined Loadings Stress Transformation Princ. Stress and Max. In-Plane Shear Stress Mohr’s Circle Absolute Maximum Shear Stress Strain Transformation Mohr’s Circle Abs. Maximum Shear Strain, Strain Rosettes Material Property Relations Theories of Failure 1–2, 1–11, 1–18, 1–22 1–34, 1–46, 1–55, 1–62 1–77, 1–83, 1–89, 1–99 2–6, 2–10, 2–18, 2–22 3–3, 3–11, 3–19, 3–23 3–27, 3–31 4–2, 4–11, 4–18, 4–22 4–31, 4–46, 4–53, 4–58 4–71, 4–78, 4–85 4–87, 4–91, 4–94 4–106, 4–109, 4–110 5–6, 5–11, 5–22, 5–31 5–47, 5–54, 5–62, 5–66 5–79, 5–82, 5–83 5–93, 5–102 5–114, 5–117 5–120, 5–123 5–130, 5–134, 5–139 6–7, 6–13, 6–21, 6–23, 6–24, 6–31, 6–37, 6–42 6–51, 6–58, 6–66, 6–82, 6–99 6–111, 6–118, 6–122 6–130, 6–135, 6–138 6–147, 6–151, 6–155 6–159, 6–163 6–170, 6–174, 6–182 7–6, 7–11, 7–27 7–35, 7–43, 7–48 7–50, 7–58, 7–61 8–3, 8–7 8–22, 8–35, 8–42, 8–58 9–7, 9–10, 9–13 9–19, 9–22, 9–31, 9–37 9–63, 9–71, 9–83 9–87, 9–95 10–3, 10–11 10–18, 10–19 10–23, 10–27 10–34, 10–39, 10–47 10–66, 10–74, 10–82, 10–91 11.1–11.2 11.3 11.4 12.1–12.2 12.3 12.4 12.5 12.6–12.7 12.8 12.10 13.1–13.3 13.4–13.5 13.6 13.7 14.1–14.2 14.3 14.4 14.5–14.6 14.7 14.8 14.9 Prismatic Beam Design Fully Stressed Beams Shaft Design Slope and Displacement by Integration Discontinuity Functions Moment-Area Theorems Method of Superposition Indet. Beams-Method of Integration Indet. Beams-Mom. Area Theorems Indet. Beams-Method of Superposition Buckling of an Ideal Column The Secant Formula, Inelastic Buckling Design of Columns for Concentric Loading Design of Columns for Eccentric Loading Elastic Strain Energy Conservation of Energy Impact Principle of Virtual Forces-Trusses Principle of Virtual Forces-Beams Castigliano’s Theorem-Trusses Castigliano’s Theorem-Beams 11–3, 11–6, 11–11, 11–22 11–34, 11–35 11–43, 11–46 12–3, 12–7, 12–18, 12–25 12–35, 12–43, 12–53 12–55, 12–63, 12–74 12–89, 12–94, 12–98 12–106, 12–114 12–118, 12–119 12–123, 12–126, 12–130 13–3, 13–9, 13–18, 13–26 13–47, 13–53, 13–59, 13–70 13–83, 13–99, 13–103 13–110, 13–117, 13–126 14–3, 14–11, 14–14 14–27, 14–31, 14–34 14–51, 14–58, 14–67 14–74, 14–77, 14–82 14–87, 14–97, 14–110 14–125, 14–127, 14–129 14–139, 14–141, 14–143 VII FM_TOC 46060 VIII 6/22/10 11:26 AM Page viii ANSWER ASSIGNMENT Section Title Assignment without Answers 1.1–1.2 1.3–1.5 1.6–1.7 2.1–2.2 3.1–3.5 3.6–3.8 4.1–4.2 4.3–4.5 4.6 4.7 4.8–4.9 5.1–5.3 5.4 5.5 5.6 5.7 5.8 5.9–5.10 6.1–6.2 6.3–6.4 6.5 6.6–6.7 6.8 6.9 6.10–6.11 7.1–7.3 7.4 7.5–7.6 8.1 8.2 9.1–9.2 9.3 9.4–9.6 9.7 10.1–10.2 10.3 10.4–10.5 10.6 10.7 Equilibrium of a Deformable Body Average Normal and Shear Stress Design of Simple Connections Strain The Stress Strain Diagram Poisson’s ratio, Shear Stress-Strain Diagram Deformation of an Axially Loaded Member Statically Indeterminate Member Thermal Stresses Stress Concentrations Inelastic Deformation and Residual Stresses Torsion Stress and Power Angle of Twist Statically Indeterminate Members Noncircular Shafts Thin-Walled Tubes Stress Concentrations Inelastic Torsion and Residual Stresses Shear and Moment Diagrams Bending Stress Unsymmetric Bending Composite Beams Curved Beams Stress Concentrations Inelastic Bending Shear Stress Shear Flow in Built-up Members Shear Center Thin-Walled Pressure Vessels Stress Due to Combined Loadings Stress Transformation Princ. Stress and Max. In-Plane Shear Stress Mohr’s Circle Absolute Maximum Shear Stress Strain Transformation Mohr’s Circle Abs. Maximum Shear Strain, Strain Rosettes Material Property Relations Theories of Failure 1–4, 1–12, 1–20, 1–28 1–36, 1–40, 1–52, 1–60 1–76, 1–88, 1–92, 1–100 2–4, 2–8, 2–16, 2–24 3–4, 3–8, 3–16, 3–20 3–28, 3–32 4–4, 4–12, 4–16, 4–20 4–32, 4–40, 4–44, 4–52 4–68, 4–76, 4–84 4–88, 4–92, 4–96 4–100, 4–104, 4–112 5–4, 5–8, 5–20, 5–36 5–52, 5–56, 5–64, 5–72 5–80, 5–88, 5–92 5–96, 5–104 5–108, 5–116 5–120, 5–124 5–132, 5–136 6–4, 6–8, 6–12, 6–18, 6–20, 6–28, 6–36 6–52, 6–56, 6–68, 6–84 6–112, 6–116, 6–120 6–128, 6–132, 6–140 6–144, 6–152, 6–156 6–160, 6–164 6–168, 6–176, 6–184 7–4, 7–12, 7–24 7–32, 7–40, 7–44 7–52, 7–60, 7–68 8–4, 8–8 8–20, 8–28, 8–36, 8–56, 8–68 9–4, 9–8, 9–20 9–16, 9–28, 9–32, 9–36 9–60, 9–68, 9–76 9–84, 9–92 10–4, 10–8 10–16, 10–20 10–24, 10–28 10–32, 10–40, 10–44 10–60, 10–72, 10–76, 10–88 11.1–11.2 11.3 11.4 12.1–12.2 12.3 12.4 12.5 12.6–12.7 12.8 12.10 13.1–13.3 13.4–13.5 13.6 13.7 14.1–14.2 14.3 14.4 14.5–14.6 14.7 14.8 14.9 Prismatic Beam Design Fully Stressed Beams Shaft Design Slope and Displacement by Integration Discontinuity Functions Moment-Area Theorems Method of Superposition Indet. Beams-Method of Integration Indet. Beams-Mom. Area Theorems Indet. Beams-Method of Superposition Buckling of an Ideal Column The Secant Formula, Inelastic Buckling Design of Columns for Concentric Loading Design of Columns for Eccentric Loading Elastic Strain Energy Conservation of Energy Impact Principle of Virtual Forces-Trusses Principle of Virtual Forces-Beams Castigliano’s Theorem-Trusses Castigliano’s Theorem-Beams 11–4, 11–12, 11–20, 11–28 11–32, 11–36 11–40, 11–44 12–8, 12–12, 12–20, 12–24 12–36, 12–44, 12–48 12–56, 12–64, 12–72 12–88, 12–96, 12–100 12–104, 12–112 12–116, 12–120 12–124, 12–128, 12–136 13–4, 13–8, 13–16, 13–24 13–48, 13–56, 13–64, 13–72 13–88, 13–96, 13–104 13–108, 13–116, 13–120 14–4, 14–16, 14–20 14–28, 14–32, 14–40 14–48, 14–52, 14–64 14–72, 14–80, 14–84 14–88, 14–96, 14–104 14–124, 14–128, 14–132 14–136, 14–140, 14–144 FM_TOC 46060 6/22/10 11:26 AM Page ix ANSWER ASSIGNMENT Section Title 1.1–1.2 1.3–1.5 1.6–1.7 2.1–2.2 3.1–3.5 3.6–3.8 4.1–4.2 4.3–4.5 4.6 4.7 4.8–4.9 5.1–5.3 5.4 5.5 5.6 5.7 5.8 5.9–5.10 6.1–6.2 6.3–6.4 6.5 6.6–6.7 6.8 6.9 6.10–6.11 7.1–7.3 7.4 7.5–7.6 8.1 8.2 9.1–9.2 9.3 9.4–9.6 9.7 10.1–10.2 10.3 10.4–10.5 10.6 10.7 Equilibrium of a Deformable Body Average Normal and Shear Stress Design of Simple Connections Strain The Stress Strain Diagram Poisson’s ratio, Shear Stress-Strain Diagram Deformation of an Axially Loaded Member Statically Indeterminate Member Thermal Stresses Stress Concentrations Inelastic Deformation and Residual Stresses Torsion Stress and Power Angle of Twist Statically Indeterminate Members Noncircular Shafts Thin-Walled Tubes Stress Concentrations Inelastic Torsion and Residual Stresses Shear and Moment Diagrams Bending Stress Unsymmetric Bending Composite Beams Curved Beams Stress Concentrations Inelastic Bending Shear Stress Shear Flow in Built-up Members Shear Center Thin-Walled Pressure Vessels Stress Due to Combined Loadings Stress Transformation Princ. Stress and Max. In-Plane Shear Stress Mohr’s Circle Absolute Maximum Shear Stress Strain Transformation Mohr’s Circle Abs. Maximum Shear Strain, Strain Rosettes Material Property Relations Theories of Failure 11.1–11.2 11.3 11.4 12.1–12.2 12.3 12.4 12.5 12.6–12.7 12.8 12.10 13.1–13.3 13.4–13.5 13.6 13.7 14.1–14.2 14.3 14.4 14.5–14.6 14.7 14.8 14.9 Prismatic Beam Design Fully Stressed Beams Shaft Design Slope and Displacement by Integration Discontinuity Functions Moment-Area Theorems Method of Superposition Indet. Beams-Method of Integration Indet. Beams-Mom. Area Theorems Indet. Beams-Method of Superposition Buckling of an Ideal Column The Secant Formula, Inelastic Buckling Design of Columns for Concentric Loading Design of Columns for Eccentric Loading Elastic Strain Energy Conservation of Energy Impact Principle of Virtual Forces-Trusses Principle of Virtual Forces-Beams Castigliano’s Theorem-Trusses Castigliano’s Theorem-Beams Assignment IX