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Linear solid elements in 2D and 3D

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Linear solid elements in 2D and 3D

By the term ”linear element” we mean here the elements with linear approximation of displacement and constant stress/strain distribution over the element.

TRIANGULAR ELEMENT IN 2D

The element has three nodes with six degrees of freedom according to Fig.4-1

Fig.4-1 Triangular element

Components of displacement are approximated by linear shape functions

where are the deformation

parameters and

is the matrix of linear shape functions N1(x,y), N2(x,y), N3(x,y). Their distribution over the triangular element area can be seen in Fig.4-2, the final approximation of displacement over several elements shows Fig.4-3.

Fig.4-2 Linear shape functions of triangular element

N3(x,y)

N2(x,y)

N1(x,y)

3

3

3

2

2

2

1

1

1

Fig.4-3 Continuous approximation of displacement over triangular element

Applying appropriate differential operators on the displacement field we obtain strain and stress fields = [x, y, xy]T , = [x, y, xy]T

= L.N. = B. , = D. = D.B. ,

where L is the matrix of differential operators andDthe material matrix. Very important fact is that the stress and strain fields are constant over the element with discontinuity on its border – see Fig.4-4. The stiffness matrix of the element is then obtained from

Fig.4-4 Discontinuity of stress and strain over triangular elements

Linear triangular element is still used, although it is not very precise and should be used with care especially in bending or in areas with stress concentrations. This is illustrated in examples 4.1 and 4.2. In Ansys, this element can be used as a special (not recommended) version of more general quadrilateral element PLANE42, or PLANE182.

Like all plane elements, the triangle can be used in axisymmetrical analysis. FE mesh then represents the meridian cross section of analysed body. Usually, the global y axis is the axis of symmetry. Illustration of this application is given in Example 4.3.

LINEAR TETRAEDR

The four node 3D linear element (Fig.4-5) is a straightforward application of plane triangular element to three dimensions. It has 12 degrees of freedom, three displacements in each node:

Fig.4-5 Linear tetraedr

Threee components of displacements are approximated in a standard way:

, N contains shape functions:

E is a unit 3x3 matrix.

Like in the triangular element, stress and strain are constant over the whole element:

= L.N. = B. , = D. = D.B.,

but now we have all six components in both tensors :

= [x, y, z, xy,yz,zx]T, = [x, y, z, xy, yz,zx]T.

Element stiffness matrix is then expressed in a standard way:

where V is the element volume.

Like the plane triangular element, the tetraedr is not very good for analysis of stress gradients or bending situations. Nevertheless, in 3D analysis there is a strong argument for using tetraedric mesh, as it can be created simply by automatic mesh generators in 3D bodies of general shapes. To create hexaedral mesh in a body with general shape is much more complicated and the task cannot be done in a fully automatic way yet.

In ANSYS, this element can be again found as a special (not recommended) version of more general hexaedral element SOLID45, or SOLID185