44
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
k(X ) = E(X )I Y (X ) ,
(3.17)
and the differential equation (3.15) reads in a more general notation as:
d
dX
k(X )
d
2 u
dX 2
= −Q Z (X ) .
(3.18)
Let us again introduce an auxiliary function v(x) of the form:
v(X ) = k(X )
d
2 u
dX 2 .
(3.19)
Thus, the differential equation (3.18) can be expressed in a much more simpler way
as:
dv(X )
dX
= −Q Z (X ) .
(3.20)
The first-order derivative
dv
dX
is again approximated by the centered difference expression from Table 1.1. Replacing X by
X
2
, we get
dv(X )
dX
i
=
v i+
1
2
− v i−
1
2
X
,
(3.21)
where, for example, the notation ‘i +
1
2
’ refers to the value of the function v in the
middle of node i and i + 1, cf. Fig. 2.6 in Sect. 2.3. Based on the definition of the
auxiliary function v in Eq. (3.19), the functional values of v in Eq. (3.21) at the
intermediate locations can be stated based on centered difference approximations of
second-order accuracy as ((X → X/2):
v i+
1
2
= k i+
1
2
d
2 u
dX 2
i+
1
2
= k i+
1
2
u i+1 − 2u i+
1
2
+ u i
X 2 /4
,
(3.22)
v i−
1
2
= k i−
1
2
d
2 u
dX 2
i−
1
2
= k i−
1
2
u i − 2u i−
1
2
+ u i−1
X 2 /4
.
(3.23)
However, this approach would introduce displacements at intermediate nodes, i.e.,
u i+
1
2
and u i−
1
2
, and we will not proceed with this derivation.
Alternatively, we apply
3 the product rule of differential calculus to Eq. (3.15).
Assuming constant material properties, this gives:
3 Similar to Eq. (2.55) in Sect. 2.3.