A First Course in the Finite Element
Method – 6th Edition
ST
UV
SOLUTIONS
IA
MANUAL
_A
PP
Daryl L. Logan
RO
Comprehensive Solution Manual for
VE
Instructors and Students
D?
© Daryl L. Logan
All rights reserved. Reproduction or distribution without permission is prohibited.
??
©STUDYSTREAM
, TABLE OF CONTENTS
A First Course in the Finite Element Method – 6th Edition
ST
Daryl L. Logan
UV
1. Introduction
2. Introduction to the Stiffness (Displacement) Method
3. Development of Truss Equations
4. Development of Beam Equations
IA
5. Frame and Grid Equations
6. Development of the Plane Stress and Plane Strain Stiffness Equations
7. Practical Considerations in Modeling; Interpreting Results; and Examples
_A
of Plane Stress/Strain Analysis
8. Development of the Linear-Strain Triangle Equations
9. Axisymmetric Elements
10.Isoparametric Formulation
PP
11.Three-Dimensional Stress Analysis
12. Plate Bending Element
13. Heat Transfer and Mass Transport
14. Fluid Flow in Porous Media and through Hydraulic Networks; and
RO
Electrical Networks and Electrostatics
15. Thermal Stress
16. Structural Dynamics and Time-Dependent Heat Transfer
VE
D?
??
©STUDYSTREAM
, Contents
Chapter 1.............................................................................................................................. 1
ST
Chapter 2.............................................................................................................................. 3
Chapter 3............................................................................................................................ 25
UV
Chapter 4.......................................................................................................................... 137
Chapter 5.......................................................................................................................... 203
IA
Chapter 6.......................................................................................................................... 315
Chapter 7.......................................................................................................................... 363
_A
Chapter 8.......................................................................................................................... 383
Chapter 9.......................................................................................................................... 397
Chapter 10........................................................................................................................ 423
PP
Chapter 11........................................................................................................................ 449
Chapter 12........................................................................................................................ 477
RO
Chapter 13........................................................................................................................ 499
Chapter 14........................................................................................................................ 539
Chapter 15........................................................................................................................ 561
VE
Chapter 16........................................................................................................................ 591
Appendix A........................................................................................................................629
D?
Appendix B:.......................................................................................................................635
Appendix D........................................................................................................................641
??
, C h a p te r 1
ST
1.1. A finite element is a small body or unit interconnected to other units to model a larger
structure or system.
1.2. Discretization means dividing the body (system) into an equivalent system of finite elements
UV
with associated nodes and elements.
1.3. The modern development of the finite element method began in 1941 with the work of
Hrennikoff in the field of structural engineering.
1.4. The direct stiffness method was introduced in 1941 by Hrennikoff. However, it was not
commonly 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 used
IA
to 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 of
minutes.
1.7. The following are the general steps of the finite element method.
_A
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.
PP
Step 3
Relate the stresses to the strains through the stress/strain law—generally called
the constitutive law.
Step 4
Derive the element stiffness matrix and equations. Use the direct equilibrium
RO
method, a work or energy method, or a method of weighted residuals to relate the
nodal forces to nodal displacements.
Step 5
Assemble the element equations to obtain the global or total equations and
introduce boundary conditions.
Step 6
VE
Solve for the unknown degrees of freedom (or generalized displacements).
Step 7
Solve for the element strains and stresses.
Step 8
Interpret and analyze the results for use in the design/analysis process.
D?
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
© 2023
Method – 6th Edition
ST
UV
SOLUTIONS
IA
MANUAL
_A
PP
Daryl L. Logan
RO
Comprehensive Solution Manual for
VE
Instructors and Students
D?
© Daryl L. Logan
All rights reserved. Reproduction or distribution without permission is prohibited.
??
©STUDYSTREAM
, TABLE OF CONTENTS
A First Course in the Finite Element Method – 6th Edition
ST
Daryl L. Logan
UV
1. Introduction
2. Introduction to the Stiffness (Displacement) Method
3. Development of Truss Equations
4. Development of Beam Equations
IA
5. Frame and Grid Equations
6. Development of the Plane Stress and Plane Strain Stiffness Equations
7. Practical Considerations in Modeling; Interpreting Results; and Examples
_A
of Plane Stress/Strain Analysis
8. Development of the Linear-Strain Triangle Equations
9. Axisymmetric Elements
10.Isoparametric Formulation
PP
11.Three-Dimensional Stress Analysis
12. Plate Bending Element
13. Heat Transfer and Mass Transport
14. Fluid Flow in Porous Media and through Hydraulic Networks; and
RO
Electrical Networks and Electrostatics
15. Thermal Stress
16. Structural Dynamics and Time-Dependent Heat Transfer
VE
D?
??
©STUDYSTREAM
, Contents
Chapter 1.............................................................................................................................. 1
ST
Chapter 2.............................................................................................................................. 3
Chapter 3............................................................................................................................ 25
UV
Chapter 4.......................................................................................................................... 137
Chapter 5.......................................................................................................................... 203
IA
Chapter 6.......................................................................................................................... 315
Chapter 7.......................................................................................................................... 363
_A
Chapter 8.......................................................................................................................... 383
Chapter 9.......................................................................................................................... 397
Chapter 10........................................................................................................................ 423
PP
Chapter 11........................................................................................................................ 449
Chapter 12........................................................................................................................ 477
RO
Chapter 13........................................................................................................................ 499
Chapter 14........................................................................................................................ 539
Chapter 15........................................................................................................................ 561
VE
Chapter 16........................................................................................................................ 591
Appendix A........................................................................................................................629
D?
Appendix B:.......................................................................................................................635
Appendix D........................................................................................................................641
??
, C h a p te r 1
ST
1.1. A finite element is a small body or unit interconnected to other units to model a larger
structure or system.
1.2. Discretization means dividing the body (system) into an equivalent system of finite elements
UV
with associated nodes and elements.
1.3. The modern development of the finite element method began in 1941 with the work of
Hrennikoff in the field of structural engineering.
1.4. The direct stiffness method was introduced in 1941 by Hrennikoff. However, it was not
commonly 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 used
IA
to 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 of
minutes.
1.7. The following are the general steps of the finite element method.
_A
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.
PP
Step 3
Relate the stresses to the strains through the stress/strain law—generally called
the constitutive law.
Step 4
Derive the element stiffness matrix and equations. Use the direct equilibrium
RO
method, a work or energy method, or a method of weighted residuals to relate the
nodal forces to nodal displacements.
Step 5
Assemble the element equations to obtain the global or total equations and
introduce boundary conditions.
Step 6
VE
Solve for the unknown degrees of freedom (or generalized displacements).
Step 7
Solve for the element strains and stresses.
Step 8
Interpret and analyze the results for use in the design/analysis process.
D?
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
© 2023
