A First Course in the

Solutions Manual (All Chapters Download Link at the end of this file) A First Course in the Finite Element Method 6 th

EDITION

DARYL L. LOGAN 1 / 4

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 2 / 4

1 © 2023 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Chapter 1

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

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

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.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 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).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 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 3 / 4

2 © 2023 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.(3)methods of weighted residuals

1.11. The term ‘ degrees of freedom’ refers to rotations and displacements that are associated with

each 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 are evaluated 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

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