Solutions Manual
A First Course In The Finite Element Method
By Daryl L. Logan
6th Edition
, Contents
Chapter 1 .............................................................................................................................. 1
Chapter 2 .............................................................................................................................. 3
Chapter 3 ............................................................................................................................ 25
Chapter 4 .......................................................................................................................... 137
Chapter 5 .......................................................................................................................... 203
Chapter 6 .......................................................................................................................... 315
Chapter 7 .......................................................................................................................... 363
Chapter 8 .......................................................................................................................... 383
Chapter 9 .......................................................................................................................... 397
Chapter 10 ........................................................................................................................ 423
Chapter 11 ........................................................................................................................ 449
Chapter 12 ........................................................................................................................ 477
Chapter 13 ........................................................................................................................ 499
Chapter 14 ........................................................................................................................ 539
Chapter 15 ........................................................................................................................ 561
Chapter 16 ........................................................................................................................ 591
Appendix A ....................................................................................................................... 629
Appendix B: ...................................................................................................................... 635
Appendix D ....................................................................................................................... 641
, Chapter 1
1.1. A Finite Element Is A Small Body Or Unit Interconnected To Other Units To Model
A LargerStructure Or System.
1.2. Discretization Means Dividing The Body (System) Into An Equivalent System Of Finite
ElementsWith Associated Nodes And Elements.
1.3. The Modern Development Of The Finite Element Method Began In 1941 With The
Work OfHrennikoff In The Field Of Structural Engineering.
1.4. The Direct Stiffness Method Was Introduced In 1941 By Hrennikoff. However, It Was
NotCommonly Known As The Direct Stiffness Method Until 1956.
1.5. A Matrix Is A Rectangular Array Of Quantities Arranged In Rows And Columns That Is
Often UsedTo Aid In Expressing And Solving A System Of Algebraic Equations.
1.6. As Computer Developed It Made Possible To Solve Thousands Of Equations In A
Matter OfMinutes.
1.7. The Following Are The General Steps Of The Finite Element
Method.Step 1
Divide The Body Into An Equivalent System Of Finite Elements With Associated
Nodes And Choose The Most Appropriate Element Type.
Step 2
Choose A Displacement Function Within Each Element.
Step 3
Relate The Stresses To The Strains Through The Stress/Strain Law—Generally
CalledThe Constitutive Law.
Step 4
Derive The Element Stiffness Matrix And Equations. Use The Direct
Equilibrium Method, A Work Or Energy Method, Or A Method Of Weighted
Residuals To Relate TheNodal Forces To Nodal Displacements.
Step 5
Assemble The Element Equations To Obtain The Global Or Total Equations And
Introduce Boundary Conditions.
Step 6
Solve For The Unknown Degrees Of Freedom (Or Generalized
Step 7
Displacements).Solve For The Element Strains And Stresses.
Step 8
Interpret And Analyze The Results For Use In The Design/Analysis Process.
1.8. The Displacement Method Assumes Displacements Of The Nodes As The Unknowns Of
The Problem. The Problem Is Formulated Such That A Set Of Simultaneous Equations Is
Solved For Nodal Displacements.
1.9. Four Common Types Of Elements Are: Simple Line Elements, Simple Two-Dimensional
Elements,Simple Three-Dimensional Elements, And Simple Axisymmetric Elements.
1.10 Three Common Methods Used To Derive The Element Stiffness Matrix And Equations Are
(1) Direct Equilibrium Method
(2) Work Or Energy Methods
1
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, (3) Methods Of Weighted Residuals
1.11. The Term ‘Degrees Of Freedom’ Refers To Rotations And Displacements That Are
Associated WithEach Node.
1.12. Five Typical Areas Where The Finite Element Is Applied Are As Follows.
(1) Structural/Stress Analysis
(2) Heat Transfer Analysis
(3) Fluid Flow Analysis
(4) Electric Or Magnetic Potential Distribution Analysis
(5) Biomechanical Engineering
1.13. Five Advantages Of The Finite Element Method Are The Ability To
(1) Model Irregularly Shaped Bodies Quite Easily
(2) Handle General Load Conditions Without Difficulty
(3) Model Bodies Composed Of Several Different Materials Because Element Equations
AreEvaluated Individually
(4) Handle Unlimited Numbers And Kinds Of Boundary Conditions
(5) Vary The Size Of The Elements To Make It Possible To Use Small Elements Where Necessary
2
© 2023
A First Course In The Finite Element Method
By Daryl L. Logan
6th Edition
, Contents
Chapter 1 .............................................................................................................................. 1
Chapter 2 .............................................................................................................................. 3
Chapter 3 ............................................................................................................................ 25
Chapter 4 .......................................................................................................................... 137
Chapter 5 .......................................................................................................................... 203
Chapter 6 .......................................................................................................................... 315
Chapter 7 .......................................................................................................................... 363
Chapter 8 .......................................................................................................................... 383
Chapter 9 .......................................................................................................................... 397
Chapter 10 ........................................................................................................................ 423
Chapter 11 ........................................................................................................................ 449
Chapter 12 ........................................................................................................................ 477
Chapter 13 ........................................................................................................................ 499
Chapter 14 ........................................................................................................................ 539
Chapter 15 ........................................................................................................................ 561
Chapter 16 ........................................................................................................................ 591
Appendix A ....................................................................................................................... 629
Appendix B: ...................................................................................................................... 635
Appendix D ....................................................................................................................... 641
, Chapter 1
1.1. A Finite Element Is A Small Body Or Unit Interconnected To Other Units To Model
A LargerStructure Or System.
1.2. Discretization Means Dividing The Body (System) Into An Equivalent System Of Finite
ElementsWith Associated Nodes And Elements.
1.3. The Modern Development Of The Finite Element Method Began In 1941 With The
Work OfHrennikoff In The Field Of Structural Engineering.
1.4. The Direct Stiffness Method Was Introduced In 1941 By Hrennikoff. However, It Was
NotCommonly Known As The Direct Stiffness Method Until 1956.
1.5. A Matrix Is A Rectangular Array Of Quantities Arranged In Rows And Columns That Is
Often UsedTo Aid In Expressing And Solving A System Of Algebraic Equations.
1.6. As Computer Developed It Made Possible To Solve Thousands Of Equations In A
Matter OfMinutes.
1.7. The Following Are The General Steps Of The Finite Element
Method.Step 1
Divide The Body Into An Equivalent System Of Finite Elements With Associated
Nodes And Choose The Most Appropriate Element Type.
Step 2
Choose A Displacement Function Within Each Element.
Step 3
Relate The Stresses To The Strains Through The Stress/Strain Law—Generally
CalledThe Constitutive Law.
Step 4
Derive The Element Stiffness Matrix And Equations. Use The Direct
Equilibrium Method, A Work Or Energy Method, Or A Method Of Weighted
Residuals To Relate TheNodal Forces To Nodal Displacements.
Step 5
Assemble The Element Equations To Obtain The Global Or Total Equations And
Introduce Boundary Conditions.
Step 6
Solve For The Unknown Degrees Of Freedom (Or Generalized
Step 7
Displacements).Solve For The Element Strains And Stresses.
Step 8
Interpret And Analyze The Results For Use In The Design/Analysis Process.
1.8. The Displacement Method Assumes Displacements Of The Nodes As The Unknowns Of
The Problem. The Problem Is Formulated Such That A Set Of Simultaneous Equations Is
Solved For Nodal Displacements.
1.9. Four Common Types Of Elements Are: Simple Line Elements, Simple Two-Dimensional
Elements,Simple Three-Dimensional Elements, And Simple Axisymmetric Elements.
1.10 Three Common Methods Used To Derive The Element Stiffness Matrix And Equations Are
(1) Direct Equilibrium Method
(2) Work Or Energy Methods
1
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, (3) Methods Of Weighted Residuals
1.11. The Term ‘Degrees Of Freedom’ Refers To Rotations And Displacements That Are
Associated WithEach Node.
1.12. Five Typical Areas Where The Finite Element Is Applied Are As Follows.
(1) Structural/Stress Analysis
(2) Heat Transfer Analysis
(3) Fluid Flow Analysis
(4) Electric Or Magnetic Potential Distribution Analysis
(5) Biomechanical Engineering
1.13. Five Advantages Of The Finite Element Method Are The Ability To
(1) Model Irregularly Shaped Bodies Quite Easily
(2) Handle General Load Conditions Without Difficulty
(3) Model Bodies Composed Of Several Different Materials Because Element Equations
AreEvaluated Individually
(4) Handle Unlimited Numbers And Kinds Of Boundary Conditions
(5) Vary The Size Of The Elements To Make It Possible To Use Small Elements Where Necessary
2
© 2023
