A First Course in the Finite Element Method PDF
Preview A First Course in the Finite Element Method
CONVERSIONFACTORSU.S.CustomaryUnitstoSIUnits QuantityConvertedfromU.S.Customary To SIEquivalent (Acceleration) 1foot/second2(ft/s2) meter/second2(m/s2) 0.3048m/s2 1inch/second2(in./s2) meter/second2(m/s2) 0.0254m/s2 (Area) 1foot2(ft2) meter2(m2) 0.0929m2 1inch2(in.2) meter2(m2) 645.2mm2 (Density,mass) 1poundmass/inch3(lbm/in.3) kilogram/meter3(kg/m3) 27.68Mg/m3 1poundmass/foot3(lbm/ft3) kilogram/meter3(kg/m3) 16.02kg/m3 (Energy,Work) 1Britishthermalunit(BTU) Joule(J) 1055J 1foot-poundforce(ft-lb) Joule(J) 1.356J 1kilowatt-hour Joule(J) 3:60 106 J (cid:31) (Force) 1kip(1000lb) Newton(N) 4.448kN 1poundforce(lb) Newton(N) 4.448N (Length) 1foot(ft) meter(m) 0.3048m 1inch(in.) meter(m) 25.4mm 1mile(mi),(U.S.statute) meter(m) 1.609km 1mile(mi),(internationalnautical) meter(m) 1.852km (Mass) 1poundmass(lbm) kilogram(kg) 0.4536kg 1slug(lb-sec2/ft) kilogram(kg) 14.59kg 1metricton(2000lbm) kilogram(kg) 907.2kg (Momentofforce) 1pound-foot(lb ft) Newton-meter(N m) 1.356N m (cid:30) (cid:30) (cid:30) 1pound-inch(lb in.) Newton-meter(N m) 0.1130N m (cid:30) (cid:30) (cid:30) (Momentofinertiaofanarea) 1inch4 meter4(m4) 0:4162 10 6 m4 (cid:29) (cid:31) (Momentofinertiaofamass) 1pound-foot-second2(lb ft s2) kilogram-meter2(kg m2) 1.356kg m2 (cid:30) (cid:30) (cid:30) (cid:30) (Momentum,linear) 1pound-second(lb s) kilogram-meter/second(kg m/s) 4.448N s (cid:30) (cid:30) (cid:30) (Momentum,angular) pound-foot-second(lb ft s) Newton-meter-second(N m s) 1.356N m s (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. PROPERTIESOFSOLIDSNotes:(cid:31) massdensity,m mass,I massmomentof inertia. ¼ ¼ ¼ 1. Slender Rod y pd2Lr m = 4 d mL2 z L Iy = Iz = 12 x 2. Thin Disk y t pd2tr d m = 4 md2 I = x 8 z x md2 I = I = y z 16 3. Rectangular Prism y m = abcr I = m (a2 + b2) x 12 b I = m (a2 + c2) y 12 z c a x Iz =1m2(b2 + c2) 4. Circular Cylinder y pd2Lr m = d 4 md2 z L Ix = 8 x I = I = m(3d2 + 4L2) y z 48 5. Hollow Cylinder y di m =pL4r(do2 − di2) d o I = m(d2 + d2) z L x 8 o i x Iy = Iz =4m8(3do2 + 3di2 + 4L2) Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. CONVERSIONFACTORSU.S.CustomaryUnitstoSIUnits(Continued) QuantityConvertedfromU.S.Customary To SIEquivalent (Power) 1foot-pound/second(ft lb/s) Watt(W) 1.356W � 1horsepower(550ft lb/s) Watt(W) 745.7W � (Pressure,stress) 1atmosphere(std)(14.7.lb/in.2Þ Newton/meter2(N/m2orPa) 101.3kPa 1pound/foot2(lb/ft2) Newton/meter2(N/m2orPa) 47.88Pa 1pound/inch2(lb/in.2orpsi) Newton/meter2(N/m2orPa) 6.895kPa 1kip/inch2(ksi) Newton/meter2(N/m2orPa) 6.895MPa (Springconstant) 1pound/inch(lb/in.) Newton/meter(N/m) 175.1N/m (Temperature) T( F) 1.8T( C) 32 � � ¼ þ (Velocity) 1foot/second(ft/s) meter/second(m/s) 0.3048m/s 1knot(nauticalmi/h) meter/second(m/s) 0.5144m/s 1mile/hour(mi/h) meter/second(m/s) 0.4470m/s 1mile/hour(mi/h) kilometer/hour(km/h) 1.609km/h (Volume) 1foot3(ft3) meter3(m3) 0.02832m3 1inch3(in.3) meter3(m3) 16:39 10 6 m3 � � Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. A First Course in the Finite Element Method ENHANCED SIXTH EDITION, SI VERSION Daryl L. Logan University of Wisconsin–Platteville Australia • Brazil • Canada • Mexico • Singapore • United Kingdom • United States Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Important Notice: Media content referenced within the product description or the product text may not be available in the eBook version. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. A First Course in the Finite Element Method: © 2017, 2012, 2007 © 2023 Cengage Learning, Inc. Sixth Edition, SI Version WCN: 02-300 Daryl L. Logan ALL RIGHTS RESERVED. 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All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. v CONTENTS Preface to the SI Edition ix Preface x Digital Resource xii Acknowledgments xv Notation xvi 1 Introduction 1 Chapter Objectives 1 Prologue 1 1.1 Brief History 3 1.2 Introduction to Matrix Notation 4 1.3 Role of the Computer 6 1.4 General Steps of the Finite Element Method 7 1.5 Applications of the Finite Element Method 15 1.6 Advantages of the Finite Element Method 21 1.7 Computer Programs for the Finite Element Method 25 References 27 Problems 30 2 Introduction to the Stiffness (Displacement) Method 31 Chapter Objectives 31 Introduction 31 2.1 Definition of the Stiffness Matrix 32 2.2 Derivation of the Stiffness Matrix for a Spring Element 32 2.3 Example of a Spring Assemblage 36 2.4 Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) 38 2.5 Boundary Conditions 40 2.6 Potential Energy Approach to Derive Spring Element Equations 55 Summary Equations 65 References 66 Problems 66 3 Development of Truss Equations 72 Chapter Objectives 72 Introduction 72 3.1 Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates 73 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. vi Contents 3.2 Selecting a Displacement Function in Step 2 of the Derivation of Stiffness Matrix for the One-Dimensional Bar Element 78 3.3 Transformation of Vectors in Two Dimensions 82 3.4 Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane 84 3.5 Computation of Stress for a Bar in the x – y Plane 89 3.6 Solution of a Plane Truss 91 3.7 Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space 100 3.8 Use of Symmetry in Structures 109 3.9 Inclined, or Skewed, Supports 112 3.10 Potential Energy Approach to Derive Bar Element Equations 121 3.11 Comparison of Finite Element Solution to Exact Solution for Bar 132 3.12 Galerkin’s Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations 136 3.13 Other Residual Methods and Their Application to a One-Dimensional Bar Problem 139 3.14 Flowchart for Solution of Three-Dimensional Truss Problems 143 3.15 Computer Program Assisted Step-by-Step Solution for Truss Problem 144 Summary Equations 146 References 147 Problems 147 4 Development of Beam Equations 169 Chapter Objectives 169 Introduction 169 4.1 Beam Stiffness 170 4.2 Example of Assemblage of Beam Stiffness Matrices 180 4.3 Examples of Beam Analysis Using the Direct Stiffness Method 182 4.4 Distributed Loading 195 4.5 Comparison of the Finite Element Solution to the Exact Solution for a Beam 208 4.6 Beam Element with Nodal Hinge 214 4.7 Potential Energy Approach to Derive Beam Element Equations 222 4.8 Galerkin’s Method for Deriving Beam Element Equations 225 Summary Equations 227 References 228 Problems 229 5 Frame and Grid Equations 239 Chapter Objectives 239 Introduction 239 5.1 Two-Dimensional Arbitrarily Oriented Beam Element 239 5.2 Rigid Plane Frame Examples 243 5.3 Inclined or Skewed Supports—Frame Element 261 5.4 Grid Equations 262 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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