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Topic: FEM: strain-displacement matrix for isoparametric elements
Replies: 2   Last Post: Jul 10, 2012 1:50 AM

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 AMX Posts: 35 Registered: 8/22/09
Re: FEM: strain-displacement matrix for isoparametric elements
Posted: Jul 8, 2012 6:22 PM

On Sun, 8 Jul 2012 03:45:33 -0700 (PDT), Jörg <joerg.meier@vint.de> wrote:
> Hi group,
>
> I'm currently dealing with the basics of the Finite Element Method: I want
> to compute the strains of an element based on the nodal displacements. The
> nodal displacements are given from a FE-Program which is using isoparametric
> elements.
>
> In [1] (section 4.2.5) it is given how to compute the strain vector '{eps}'
> using the strain-displacement matrix '[B]' and the nodal displacement vector
> '{q}':
> {eps} = [B] * {q}
>
> To compute [B] numerically I have assemble it from several sub-matrices
> [B_i] (given at page 23 in [1]). The several [B_i] have to be computed by
> inverting the Jacobian matrix [J]. [J] itself is composed of the derivative
> of the shape function Ni.
>
>
> What I do not understand:
>
> - How many [B_i] I have to provide? Is i ranging from 1 to the count of
> nodes within the element?

Shortly: Yes.
More below.

>
> - I can easily derive the shape functions Ni for the local coordinates 's'
> or 't'. But in general Ni * d/ds or Ni * d/dt remain dependet on 's' and/or
> 't'. Therefore I have to provide a value for 's' and 't' when I numerically
> calculate [J]. But what value of 's' and 't' I have to provide for the
> current [B_i] ?
>

You need to compute [B] in two situations: a) computing stiffness
matrix, b) computing stress at some point

a) you compute some integral expression, compare e.g. (4.20), so
you need to _sum_ values of matrices at (s,t) given by
integration points (cf. Fig. 4.4 filled dots).

b) to compute stress at point you need to know coordinates (x,y) of
this point, so you can compute local coordinates (s,t) and for
those values you can compute [B]

> In other words: for me it is not clear how to assemble numerically the
> matrix [B] for a general isoparametric element. Any help would be much
> appreciated.
>

Matrix [B] has a form (with notation f,_x as partial{f}/partial{x})

| N1,_x 0 N2,_x 0 N3,_x 0 .... |
B=| 0 N1,_y 0 N2,_y 0 N3,y .... |
| N1,_y N1,_x N2,_y N2,_x N3,_y N3,_x .... |

with row lenght equal to number of element DOF and double of
number of shape function (since the same shape functions
approximate two field u and v). Here number of nodes is equal to
number of sahpe functions.

> [1]
> http://homepages.cae.wisc.edu/~suresh/ME964Website/M964Notes/Notes/introfem.pdf

AMX

--
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Date Subject Author
7/8/12 Jörg
7/8/12 AMX
7/10/12 Jörg