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 withthe 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
Step
3
Step 4
Step 5
Step
6
Choose a displacement function within each element.
Relate the stresses to the strains through the stress/strain law—generally
called the constitutive law.
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.
Assemble the element equations to obtain the global or total equations and
introduce boundary conditions.
Solve for the unknown degrees of freedom (or generalized displacements).
Solve for the element strains and stresses.
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) methods of weighted residuals
1.11. The term ‘degrees of freedom’ refers to rotations and displacements that are associated with each
node.
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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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Chapter 2
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Page 6
By the method of superposition the global stiffness matrix is constructed.
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