Civil Engineering Reference
In-Depth Information
of the cylinder (Figure 1.3b). The uniform bar is a
link
or
bar
finite element
(Chapter 2), so our first approximation is a one-element, finite element model.
The solution is obtained using the strength of materials theory. Next, we model
the tapered cylinder as two uniform bars in series, as in Figure 1.3c. In the two-
element model, each element is of length equal to half the total length of the
cylinder and has a cross-sectional area equal to the average area of the corre-
sponding half-length of the cylinder. The mesh refinement is continued using a
four-element model, as in Figure 1.3d, and so on. For this simple problem, the
displacement of the end of the cylinder for each of the finite element models is as
shown in Figure 1.4a, where the dashed line represents the known solution. Con-
vergence of the finite element solutions to the exact solution is clearly indicated.
r
o
x
r
A
o
A
L
2
L
A
r
L
F
(a)
(b)
Element 1
Element 2
(c)
(d)
Figure 1.3
(a) Tapered circular cylinder subjected to tensile loading:
r
(
x
)
r
L
). (b) Tapered cylinder as a single axial
(bar) element using an average area. Actual tapered cylinder
is shown as dashed lines. (c) Tapered cylinder modeled as
two, equal-length, finite elements. The area of each element
is average over the respective tapered cylinder length.
(d) Tapered circular cylinder modeled as four, equal-length
finite elements. The areas are average over the respective
length of cylinder (element length
r
0
(
x
/
L
)(
r
0
L
4).
