Civil Engineering Reference
In-Depth Information
Substituting the two roots into Equation 3.54, two critical loads can be
obtained as
2
2
0 252
.
π
EI
3 262
.
π
EI
1
and
2
P
=
P
=
(3.56)
cr
cr
2
2
l
l
Comparing
the
irst
critical
loads
with
the
theoretical
solution
.
π ; Zhu 1998), the FEM approach can produce very accu-
rate solutions. It should be noted that the previous solution is based on one
element (two degrees of freedom). If the number of elements in the beam
meshes increases, the accuracy improves accordingly.
T
=
0 250
2
2
(
P
EI l
cr
3.2.9 Applications in bridge analysis
When applying FEM to bridge analysis, there are some common questions
and issues that engineers have to clarify. These issues include (1) what types
of element should be used in a bridge model; (2) when a 2D model is suf-
ficient and when a 3D model is necessary; and (3) how to correctly interpret
FEM results from bridge engineering perspectives, especially when a bridge
is modeled as plate or shell elements.
In Sections 3.2.1 through 3.2.8, only generic principles and procedures
of FEM are briefly illustrated, aiming at helping engineers to understand
the theories behind an FEM package. And, as an example, only 2D frame
element is discussed in detail. In general, truss, frame, and shell elements
can cover most bridge analyses.
Truss element, like a member in a truss bridge, is a line element with only
two nodes. It has only axial strain/stress, and the most important feature
is that its strain/stress is constant over the entire element. Truss element
is also called link element. Bridge bearings, hangers, prestress tendons,
cables, and so on, can be modeled as truss elements.
Frame element, like a member in a frame structure, is a line element with
only two nodes. It behaves as a beam but could be under axial tension/com-
pression or a combination of beam and truss elements. Most FEM pack-
ages combine behaviors of beam, truss, and torsional element into one as a
frame element—the most commonly used element type in bridge analysis.
In line models, girders, stringers, diaphragms, pylons, columns, piers, and
so on are usually modeled as frame elements.
Shell element combines in-plane stress/strain behavior together with bend-
ing of a plate, either as a thin plate or as a thick plate. When a bridge com-
ponent is modeled into the plate level, such as a box girder or steel I-girder,
shell element could be used. Some components that behave in-plane, such as
webs, can be simplified as shells to streamline the modeling.
Nowadays, whether or not to model in 3D is no longer a question because
modern graphical pre- and post-processing tools are widely available.
Search WWH ::
Custom Search