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AMX
Posts:
35
Registered:
8/22/09


Re: FEM: straindisplacement 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 FEProgram which is using isoparametric > elements. > > In [1] (section 4.2.5) it is given how to compute the strain vector '{eps}' > using the straindisplacement matrix '[B]' and the nodal displacement vector > '{q}': > {eps} = [B] * {q} > > To compute [B] numerically I have assemble it from several submatrices > [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


