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Finite Element Methods (FEM) 1

Finite Element Methods (FEM)

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Finite Element Methods (FEM) 2

Part 2

Finite Element Method

General Procedures

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General concepts The continuum is separated by imaginary

lines or surfaces into a number of finiteelements

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General concepts The elements are assumed to be

connected at discrete number of nodalpoints situated on their boundaries

Generalized displacements are the basicunknowns

A function uniquely defines displacement

field in terms of nodal displacements

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Finite Element Methods (FEM) 5

General concepts The properties of the function:

1. Compatibility between elements

2D 3D elasticity problems, displacementcompatibility

Plates and shells, displacements and their partialderivaties

2. All possible rigid body displacements

included (if not will not converge)3. All uniform strain states included

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Finite Element Methods (FEM) 6

General concepts

The displacement function, uniquely definesstrain within an element in terms of nodal

displacements. These strains with any initial strain, together

with elastic properties define the stress state

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General concepts

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General concepts

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General concepts

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General concepts

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General concepts

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General concepts

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General concepts Apply Ritz procedure on the elements

(minimize total potential energy) Obtain a set of simultaneous equations

with nodal displacements as unknowns Write equations in matrix form, the same

way as of standard stiffness formulation

Solve equations with standard methods

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General concepts

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General concepts

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Finite Element Methods (FEM) 16

General concepts Assemble stiffness matrix of all elements into the

big overall matrix Apply the appropriate boundary conditions

(i) Divide domain into elements

Number all nodes

Number all elements

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General concepts (ii) Assemblage

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General concepts Example

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General concepts

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Finite Element Methods (FEM) 20

General concepts (iii) band width (element numbering sequence

effects bandwidth)

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Finite Element Methods (FEM) 21

General concepts Bandwidth of stiffness matrix is controlled by maximum

difference between two nodal numbers which are directly

connected.

(iv) boundary conditions

(1) Delete corresponding rows and columns (notconvenient in computer applications)

(2) Insert large numbers at corresponding diagonal terms(large number say ) of a stiffness matrix

(3) Insert same large numbers boundary values stiffness coefficient at corresponding entry of the forcevector

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General concepts

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Type of elements

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Type of elements

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Type of elements

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Solution Bound

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