Civil Engineering Reference
In-Depth Information
Because the strains are given by the differentials of the displacements with
respect to
x
and
y
whereas the displacements have now been expressed as
functions of
, the relation between the derivatives in the two co-
ordinate systems has to be established and this is done by the chain rule for
differentiation to give the Jacobian operator or Jacobian transformation
matrix as follows:
x
and
h
(9.27)
or put in matrix form
(9.28)
where
(9.29)
(9.30)
The proper differentials are obtained by inversion of [
J
], which is possible
only when there is a one-to-one correspondence between the natural and the
local co-ordinates. In general [
J
] becomes singular for a quadrilateral with a
re-entrant corner.
(9.31)
Therefore
