A First Course in the Finite Element
Method – 6th Edition
ST
UV
SOLUTIONS
IA
MANUAL
?_
AP
Daryl L. Logan
PR
Comprehensive Solution Manual for
OV
Instructors and Students
ED
© Daryl L. Logan
All rights reserved. Reproduction or distribution without permission is prohibited.
?
Created by MedConnoisseur ©2025/2026
, 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
of Plane Stress/Strain Analysis
?_
8. Development of the Linear-Strain Triangle Equations
9. Axisymmetric Elements
10.Isoparametric Formulation
AP
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
PR
Electrical Networks and Electrostatics
15.Thermal Stress
16.Structural Dynamics and Time-Dependent Heat Transfer
OV
ED
?
Created by MedConnoisseur ©2025/2026
, Contents
ST
Chapter 1:............................................................................................................................ 1
Chapter 2:............................................................................................................................ 3
Chapter 3:.......................................................................................................................... 25
UV
Chapter 4:........................................................................................................................ 137
Chapter 5:........................................................................................................................ 203
IA
Chapter 6:........................................................................................................................ 315
Chapter 7:........................................................................................................................ 363
?_
Chapter 8:........................................................................................................................ 383
Chapter 9:........................................................................................................................ 397
Chapter 10:...................................................................................................................... 423
AP
Chapter 11:...................................................................................................................... 449
Chapter 12:...................................................................................................................... 477
PR
Chapter 13:...................................................................................................................... 499
Chapter 14:...................................................................................................................... 539
Chapter 15:...................................................................................................................... 561
OV
Chapter 16:...................................................................................................................... 591
Appendix A: ..................................................................................................................... 629
Appendix B: ..................................................................................................................... 635
ED
Appendix D:..................................................................................................................... 641
?
, Chapter 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.
?_
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.
AP
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
method, a work or energy method, or a method of weighted residuals to relate the
PR
nodal 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 displacements).
OV
Step 7
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
ED
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
?_
AP
Daryl L. Logan
PR
Comprehensive Solution Manual for
OV
Instructors and Students
ED
© Daryl L. Logan
All rights reserved. Reproduction or distribution without permission is prohibited.
?
Created by MedConnoisseur ©2025/2026
, 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
of Plane Stress/Strain Analysis
?_
8. Development of the Linear-Strain Triangle Equations
9. Axisymmetric Elements
10.Isoparametric Formulation
AP
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
PR
Electrical Networks and Electrostatics
15.Thermal Stress
16.Structural Dynamics and Time-Dependent Heat Transfer
OV
ED
?
Created by MedConnoisseur ©2025/2026
, Contents
ST
Chapter 1:............................................................................................................................ 1
Chapter 2:............................................................................................................................ 3
Chapter 3:.......................................................................................................................... 25
UV
Chapter 4:........................................................................................................................ 137
Chapter 5:........................................................................................................................ 203
IA
Chapter 6:........................................................................................................................ 315
Chapter 7:........................................................................................................................ 363
?_
Chapter 8:........................................................................................................................ 383
Chapter 9:........................................................................................................................ 397
Chapter 10:...................................................................................................................... 423
AP
Chapter 11:...................................................................................................................... 449
Chapter 12:...................................................................................................................... 477
PR
Chapter 13:...................................................................................................................... 499
Chapter 14:...................................................................................................................... 539
Chapter 15:...................................................................................................................... 561
OV
Chapter 16:...................................................................................................................... 591
Appendix A: ..................................................................................................................... 629
Appendix B: ..................................................................................................................... 635
ED
Appendix D:..................................................................................................................... 641
?
, Chapter 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.
?_
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.
AP
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
method, a work or energy method, or a method of weighted residuals to relate the
PR
nodal 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 displacements).
OV
Step 7
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
ED
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
