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Lesson 01 Introduction to FEM

Mario Guagliano

FEMLab

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2

FEM: Finite Element Method

It is a numerical method At first it was developed to solve the problem of the computation of

the stresses in loaded structures.

FEM can be applied to many other physical problems, where theunkonwns are field variables (as dispalcement in structural

applications)

It is a useful method to find a numerical solution to a particularproblem.

Thus, it is different from analytical solutions to a class of problems,for istance the fundamental equation that relates bending to

deflection.

d2y

dx2 = M(x) EJ

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3First step: to divide the structure in

elements

The complete geometry is divided in portions called elements, thatform the mesh.

The elements describe the displacement field in a simple manner,basically by means of a polynomial representation.

The nodes and not the sides are the connection points amongelements.

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4The elements and the displacement field.

The displacments of a point in one element are approximated by polynomial

functions:

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5The elements and the displacement field.

The same relation is valid for the nodal displacements

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6The elements and the displacement field.

Then

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7The elements and the displacement field.

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8Strain and stresses

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Stress and strain

Strain field {}=[B]{u}

: area of the undeformed triangular element

The way the node numbers are assigned is free, but the 123 sequence must

be counterclockwiseto obtain a positive element area .

2

2

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10The stiffness matrix

The stiffness matrix can be calculated by equating the

external and the internal work.

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11Stiffness matrix

Kix,jy is the reaction force at node i in x direction when a unit displacement

is applied to node j in direction y.

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12Global Stiffness matrix:

global and local node numbering

Element1

Element 2

1st node 2nd node 3rd node

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13Global Stiffness matrix:

global and local node numbering

Global model number Local (element) number of the node

Element number

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14Solution of the linear elastic problem

After having related the displacement field inside every element to the

nodal displacements at nodes, it is possible to proceed and to solve thelinear elastic problem that is developed in such a way to obtain the value

of the nodal displacements.

By opportunely assigning boundary conditions and loads, so that they

are referred to nodes, a linear system like the following one is solved:

With [K]g is the global stiffness matrix, that gives the reaction forces at

nodes of the model when a unit displacement is assigned to the nodes.

K[ ]g fn{ }g = F{ }gfn{ }g = K[ ]g

1

F{ }g

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15Solution of the linear algeabric system:

Once the nodal displacements are known, it is possible to calculate

the stress and the strain vectors by considering the elements of the

model. Thus, for each element:

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16Congruence of displacements

The adopted expression of the displacements is valid for adiacent

elements, that has one common side (side 1-3)

This garantees that there are no voids or overlaps among elements when

they deform (the displacements are congruent or compatible)

Costant strain + compatibility = convergence of the results byincreasing the mesh density.

1 2

3

4 u1

u3

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17Following points (to be developed)

These are the basis of the FE Method. We will look in depth at

The definition of the stiffness matrix of every element type: [K]el The connection of the stiffness matrices of the different elements to

form the stiffness matrix of the system: [K]Global The schematization of the loads: {f} The representation of boundary conditions: {u}known The methods used to solve the system that gives the nodal

displacements {u}

The computation of the strains {} The computation of the stresses {}