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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 ... – PowerPoint PPT presentation

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1
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
2
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
3
• 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, ?xyT , ? ?x, ?y,
?xyT
• ? L.N.? B.? , ? D.? D.B.? ,
• where L is the matrix of differential operators
and D the 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

4
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
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.
5
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.
6
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, ?zxT, ? ?x, ?y, ?z, ?xy, ?yz,
?zxT. 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