关于斜拉桥的英文
三月三吃什么-上海中招网
Study on nonlinear analysis of a highly
redundant cable-stayed bridge
1．Abstract
A
comparison on nonlinear analysis of a highly
redundant cable-stayed bridge is
performed in
the study. The initial shapes including geometry
and prestress
distribution of the bridge are
determined by using a two-loop iteration method,
i.e., an
equilibrium iteration loop and a
shape iteration loop. For the initial shape
analysis a
linear and a nonlinear computation
procedure are set up. In the former all
nonlinearities of cable-stayed bridges are
disregarded, and the shape iteration is
carried out without considering equilibrium.
In the latter all nonlinearities of the
bridges are taken into consideration and both
the equilibrium and the shape iteration
are
carried out. Based on the convergent initial
shapes determined by the different
procedures,
the natural frequencies and vibration modes are
then examined in details.
Numerical results
show that a convergent initial shape can be found
rapidly by the
two-loop iteration method, a
reasonable initial shape can be determined by
using the
linear computation procedure, and a
lot of computation efforts can thus be saved.
There are only small differences in geometry
and prestress distribution between the
results
determined by linear and nonlinear computation
procedures. However, for the
analysis of
natural frequency and vibration modes, significant
differences in the
fundamental frequencies and
vibration modes will occur, and the nonlinearities
of the
cable-stayed bridge response appear
only in the modes determined on basis of the
initial shape found by the nonlinear
computation.
2. Introduction
Rapid
progress in the analysis and construction of
cable-stayed bridges has been
made over the
last three decades. The progress is mainly due to
developments in the
fields of computer
technology, high strength steel cables,
orthotropic steel decks and
construction
technology. Since the first modern cable-stayed
bridge was built in
Sweden in 1955, their
popularity has rapidly been increasing all over
the world.
Because of its aesthetic appeal,
economic grounds and ease of erection, the
cable-stayed bridge is considered as the most
suitable construction type for spans
ranging
from 200 to about 1000 m. The world’s longest
cable-stayed bridge today is
the Tatara bridge
across the Seto Inland Sea, linking the main
islands Honshu and
Shikoku in Japan.
The Tatara cable-stayed bridge was opened in 1
May, 1999 and has
a center span of 890m and a
total length of 1480m. A cable-stayed bridge
consists of
three principal components, namely
girders, towers and inclined cable stays. The
girder is supported elastically at points
along its length by inclined cable stays so that
the girder can span a much longer distance
without intermediate piers. The dead load
and
traffic load on the girders are transmitted to the
towers by inclined cables. High
tensile forces
exist in cable-stays which induce high compression
forces in towers and
part of girders. The
sources of nonlinearity in cable-stayed bridges
mainly include the
cable sag, beam-column and
large deflection effects. Since high pretension
force
exists in inclined cables before live
loads are applied, the initial geometry and the
prestress of cable-stayed bridges depend on
each other. They cannot be specified
independently as for conventional steel or
reinforced concrete bridges. Therefore the
initial shape has to be determined correctly
prior to analyzing the bridge. Only based
on
the correct initial shape a correct deflection and
vibration analysis can be achieved.
The
purpose of this paper is to present a comparison
on the nonlinear analysis of a
highly
redundant stiff cable-stayed bridge, in which the
initial shape of the bridge will
be determined
iteratively by using both linear and nonlinear
computation procedures.
Based on the initial
shapes evaluated, the vibration frequencies and
modes of the
bridge are examined.
3.
System equations
3.1. General system equation
When only nonlinearities in stiffness are
taken into account, and the system mass
and
damping matrices are considered as constant, the
general system equation of a
finite element
model of structures in nonlinear dynamics can be
derived from the
Lagrange’s virtual work
principle and written as follows:
K
j
b<
br>α
j
-∑S
j
a
j
α
=
M
αβ
q
β
”+ D
αβ
q
β
’
3.2. Linearized system equation
In order
to incrementally solve the large deflection
problem, the linearized
system equations has
to be derived. By taking the first order terms of
the Taylor’s
expansion of the general system
equation, the linearized equation for a small time
(or
load) interval is obtained as follows:
M
αβ
Δq
β
”+ΔD
αβ
q
β
’ +2K
αβ
Δq
β
=Δp
α
-
u
p
α
3.3. Linearized system
equation in statics
In nonlinear statics, the
linearized system equation becomes
2K
αβ
Δq
β
=Δp
α
-
u
p
α
4. Nonlinear analysis
4.1. Initial shape analysis
The initial
shape of a cable-stayed bridge provides the
geometric configuration
as well as the
prestress distribution of the bridge under action
of dead loads of girders
and towers and under
pretension force in inclined cable stays. The
relations for the
equilibrium conditions, the
specified boundary conditions, and the
requirements of
architectural design should be
satisfied. For shape finding computations, only
the
dead load of girders and towers is taken
into account, and the dead load of cables is
neglected, but cable sag nonlinearity is
included. The computation for shape finding is
performed by using the two-loop iteration
method, i.e., equilibrium iteration and
shape
iteration loop. This can start with an arbitrary
small tension force in inclined
cables. Based
on a reference configuration (the architectural
designed form), having
no deflection and zero
prestress in girders and towers, the equilibrium
position of the
cable-stayed bridges under
dead load is first determined iteratively
(equilibrium
iteration). Although this first
determined configuration satisfies the equilibrium
conditions and the boundary conditions, the
requirements of architectural design are,
in
general, not fulfilled. Since the bridge span is
large and no pretension forces exist
in
inclined cables, quite large deflections and very
large bending moments may
appear in the
girders and towers. Another iteration then has to
be carried out in order
to reduce the
deflection and to smooth the bending moments in
the girder and finally
to find the correct
initial shape. Such an iteration procedure is
named here the ‘shape
iteration’. For shape
iteration, the element axial forces determined in
the previous
step will be taken as initial
element forces for the next iteration, and a new
equilibrium configuration under the action of
dead load and such initial forces will be
determined again. During shape iteration,
several control points (nodes intersected by
the girder and the cable) will be chosen for
checking the convergence tolerance. In
each
shape iteration the ratio of the vertical
displacement at control points to the main
span length will be checked, i.e.,
|
vertical displacement at control
points
main span
|
The
shape iteration will be repeated until the
convergence toleranceε, say 10
-4
, is
achieved. When the convergence tolerance is
reached, the computation will stop and
the
initial shape of the cable-stayed bridges is
found. Numerical experiments show
that the
iteration converges monotonously and that all
three nonlinearities have less
influence on
the final geometry of the initial shape. Only the
cable sag effect is
significant for cable
forces determined in the initial shape analysis,
and the
beam-column and large deflection
effects become insignificant.
The initial
analysis can be performed in two different ways: a
linear and a
nonlinear computation procedure.
1. Linear computation procedure: To find the
equilibrium configuration of the bridge, all
nonlinearities of cable stayed bridges are
neglected and only the linear elastic cable,
beam-column elements and linear constant
coordinate transformation coefficients are
used. The shape iteration is carried out
without considering the equilibrium iteration.
A reasonable convergent initial shape is
found, and a lot of computation efforts can be
saved.
2. Nonlinear computation procedure: All
nonlinearities of cable-stayed bridges
are
taken into consideration during the whole
computation process. The nonlinear
cable
element with sag effect and the beam-column
element including stability
coefficients and
nonlinear coordinate transformation coefficients
are used. Both the
shape iteration and the
equilibrium iteration are carried out in the
nonlinear
computation. Newton–Raphson method
is utilized here for equilibrium iteration.
4.2. Static deflection analysis
Based on
the determined initial shape, the nonlinear static
deflection analysis of
cable-stayed bridges
under live load can be performed incrementwise or
iterationwise.
It is well known that the load
increment method leads to large numerical errors.
The
iteration method would be preferred for
the nonlinear computation and a desired
convergence tolerance can be achieved. Newton–
Raphson iteration procedure is
employed. For
nonlinear analysis of large or complex structural
systems, a ‘full’
iteration procedure
(iteration performed for a single full load step)
will often fail. An
increment–iteration
procedure is highly recommended, in which the load
will be
incremented, and the iteration
will be carried out in each load step. The static
deflection analysis of the cable stayed bridge
will start from the initial shape
determined
by the shape finding procedure using a linear or
nonlinear computation.
The algorithm of the
static deflection analysis of cable-stayed bridges
is summarized
in Section 4.4.2.
4.3.
Linearized vibration analysis
When a
structural system is stiff enough and the external
excitation is not too
intensive, the system
may vibrate with small amplitude around a certain
nonlinear
static state, where the change of
the nonlinear static state induced by the
vibration is
very small and negligible. Such
vibration with small amplitude around a certain
nonlinear static state is termed linearized
vibration. The linearized vibration is
different from the linear vibration, where the
system vibrates with small amplitude
around a
linear static state. The nonlinear static state
q
α
can be statically determined
by
nonlinear deflection analysis. After determining
q
α
a
, the system matrices may be
established with respect to such a nonlinear
static state, and the linearized system
equation has the form as follows:
M
αβ
A
q
β
”+
D
αβ
A
q
β
’+
2K
αβ
A
q
β
=p
α
(t)-
T
α
A
where the superscript ‘A’
denotes the quantity calculated at the nonlinear
static state
q
α
a
. This equation
represents a set of linear ordinary differential
equations of second
order with constant
coefficient matrices M
αβ
A
,
D
αβ
A
and 2K
αβ
A
. The
equation can
be solved by the modal
superposition method, the integral transformation
methods or
the direct integration methods.
When damping effect and load terms are
neglected, the system equation becomes
M
αβ
A
q
β
” +
2K
αβ
A
q
β
=0
This equation
represents the natural vibrations of an undamped
system based on
the nonlinear static state
q
α
a
The natural vibration frequencies
and modes can be
obtained from the above
equation by using eigensolution procedures, e.g.,
subspace
iteration methods. For the cable-
stayed bridge, its initial shape is the nonlinear
static
state q
α
a
. When the cable-
stayed bridge vibrates with small amplitude based
on the
initial shape, the natural frequencies
and modes can be found by solving the above
equation.
a
4.4. Computation
algorithms of cable-stayed bridge analysis
The
algorithms for shape finding computation, static
deflection analysis and
vibration analysis of
cable-stayed bridges are briefly summarized in the
following.
4.4.1. Initial shape analysis
1. Input of the geometric and physical data of
the bridge.
2. Input of the dead load of
girders and towers and suitably estimated initial
forces
in cable stays.
3. Find equilibrium
position
(i) Linear procedure
• Linear
cable and beam-column stiffness elements are used.
• Linear constant coordinate transformation
coefficients a
j
α
are used.
•
Establish the linear system stiffness matrix
K
αβ
by assembling element stiffness
matrices.
• Solve the linear system
equation for q
α
(equilibrium position).
• No equilibrium iteration is carried out.
(ii) Nonlinear procedure
• Nonlinear
cables with sag effect and beam-column elements
are used.
• Nonlinear coordinate
transformation coeffi- cients a
j
α
;
a
j
α，β
are used.
• Establish the
tangent system stiffness matrix 2K
αβ
.
• Solve the incremental system equation for
△q
α
.
• Equilibrium iteration is
performed by using the Newton–Raphson method.
4. Shape iteration
5. Output of the
initial shape including geometric shape and
element forces.
6. For linear static
deflection analysis, only linear stiff-ness
elements and
transformation coefficients are
used and no equilibrium iteration is carried out.
4.4.3. Vibration analysis
1. Input of the
geometric and physical data of the bridge. 2.
Input of the initial
shape data including
initial geometry and initial element forces.
3. Set up the linearized system equation of
free vibrations based on the initial
shape.
4. Find vibration frequencies and modes by
sub-space iteration methods, such as
the Rutishauser Method.
5.
Estimation of the trial initial cable forces
In the recent study of Wang and Lin, the shape
finding of small cable-stayed
bridges has been
performed by using arbitrary small or large trial
initial cable forces.
There the iteration
converges monotonously, and the convergent
solutions have
similar results, if different
trial values of initial cable forces are used.
However for
large cable-stayed bridges, shape
finding computations become more difficult to
converge. In nonlinear analysis, the Newton-
type iterative computation can converge,
only
when the estimated values of the solution is
locate in the neighborhood of the
true values.
Difficulties in convergence may appear, when the
shape finding analysis
of cable-stayed bridges
is started by use of arbitrary small initial cable
forces
suggested in the papers of Wang et al.
Therefore, to estimate a suitable trial initial
cable forces in order to get a convergent
solution becomes important for the shape
finding analysis. In the following, several
methods to estimate trial initial cable forces
will be discussed.
5.1. Balance of
vertical loads
5.2. Zero moment control
5.3. Zero displacement control
5.4.
Concept of cable equivalent modulus ratio
5.5.
Consideration of the unsymmetry
If the
estimated initial cable forces are determined
independently for each cable
stay by the
methods mentioned above, there may exist
unbalanced horizontal forces
on the tower in
unsymmetric cable-stayed bridges. Forsymmetric
arrangements of the
cable-stays on the central
(main) span and the side span with respect to the
tower, the
resultant of the horizontal
components of the cable-stays acting on the tower
is zero,
i.e., no unbalanced horizontal forces
exist on the tower. For unsymmetric cable-stayed
bridges, in which the arrangement of cable-
stays on the central (main) span and the
side
span is unsymmetric, and if the forces of cable
stays on the central span and the
side span
are determined independently, evidently unbalanced
horizontal forces will
exist on the tower and
will induce large bending moments and deflections
therein.
Therefore, for unsymmetric cable-
stayed bridges, this problem can be overcome as
follows. The force of cable stays on
the central (main) span T
i
m
can be
determined by
the methods mentioned above
independently, where the superscript m denotes the
main span, the subscript I denotes the ith
cable stay. Then the force of cable stays on
the side span is found by taking the
equilibrium of horizontal force components at the
node on the tower attached with the cable
stays, i.e., T
i
m
cosα
i
=
T
i
s
cosβ
i
, and T
i
s
=
T
i
m
cosα
i
cosβ
i
,
where α
i
is the angle between the ith
cable stay and the girder on
the main span,
andβ
i
, angle between the ith cable stay
and the girder on the side
span.
6.
Examples
In this study, two different types of
small cable-stayed bridges are taken from
literature, and their initial shapes will be
determined by the previously described
shape
finding method using linear and nonlinear
procedures. Finally, a highly
redundant stiff
cable-stayed bridge will be examined. A
convergence tolerance e =10
-4
is used
for both the equilibrium iteration and the shape
iteration. The maximum
number of iteration
cycles is set as 20. The computation is considered
as not
convergent, if the number of the
iteration cycles exceeds 20.
The initial
shapes of the following two small cable stayed
bridges in Sections 6.1
and 6.2 are first
determined by using arbitrary trial initial cable
forces. The iteration
converges monotonously
in these two examples. Their convergent initial
shapes can
be obtained easily without
difficulties. There are only small differences
between the
initial shapes determined by the
linear and the nonlinear computation. Convergent
solutions offer similar results, and they are
independent of the trial initial cable forces.
7. Conclusion
The two-loop iteration with
linear and nonlinear computation is established
for
finding the initial shapes of cable-stayed
bridges. This method can achieve the
architecturally designed form having uniform
prestress distribution, and satisfies all
equilibrium and boundary conditions. The
determination of the initial shape is the
most
important work in the analysis of cable-stayed
bridges. Only with a correct
initial shape, a
meaningful and accurate deflection andor vibration
analysis can be
achieved. Based on numerical
experiments in the study, some conclusions are
summarized as follows:
(1). No
great difficulties appear in convergence of the
shape finding of small
cable-stayed bridges,
where arbitrary initial trial cable forces can be
used to start the
computation. However for
large scale cable-stayed bridges, serious
difficulties
occurred in convergence of
iterations.
(2). Difficulties often occur in
convergence of the shape finding computation of
large cable-stayed bridge, when trial initial
cable forces are given by the methods of
balance of vertical loads, zero moment control
and zero displacement control.
(3). A
converged initial shape can be found rapidly by
the two-loop iteration
method, if the cable
stress corresponding to about 80% of E
eq
=E
value is used for the
trial initial force of
each cable stay in the main span, and the trial
force of the cables
in side spans is
determined by taking horizontal equilibrium of the
cable forces acting
on the tower.
(4).
There are only small differences in geometry and
prestress distributionforces.
The iteration
converges monotonously in these two examples.
Their convergent initial
shapes can be
obtained easily without difficulties. There are
only small differences
between the initial
shapes determined by the linear and the nonlinear
computation.
Convergent solutions offer
similar results, and they are independent of the
trial initial
cable forces.
7. Conclusion
The two-loop iteration with linear and
nonlinear computation is established for
finding the initial shapes of cable-stayed
bridges. This method can achieve the
architecturally designed form having uniform
prestress distribution, and satisfies all
equilibrium and boundary conditions. The
determination of the initial shape is the
most
important work in the analysis of cable-stayed
bridges. Only with a correct
initial shape, a
meaningful and accurate deflection andor vibration
analysis can be
achieved. Based on numerical
experiments in the study, some conclusions are
summarized as follows:
(1). No great
difficulties appear in convergence of the shape
finding of small
cable-stayed bridges, where
arbitrary initial trial cable forces can be used
to start the
computation. However for large
scale cable-stayed bridges, serious difficulties
occurred in convergence of iterations.
(2). Difficulties often occur in
convergence of the shape finding computation of
large cable-stayed bridge, when trial initial
cable forces are given by the methods of
balance of vertical loads, zero moment control
and zero displacement control.
(3). A
converged initial shape can be found rapidly by
the two-loop iteration
method, if the cable
stress corresponding to about 80% of E
eq
=E
value is used for the
trial initial force of
each cable stay in the main span, and the trial
force of the cables
in side spans is
determined by taking horizontal equilibrium of the
cable forces acting
on the tower.
(4).
There are only small differences in geometry and
prestress distribution between
the results of
initial shapes determined by linear and nonlinear
procedures.
(5). The shape finding using
linear computation offers a reasonable initial
shape
and saves a lot of computation efforts,
so that it is highly recommended from the point
of view of engineering practices.
(6). In
small cable-stayed bridges, there are only small
difference in the natural
frequencies based on
initial shapes determined by linear and nonlinear
computation
procedures, and the mode shapes
are the same in both cases.
(7). Significant
differences in the fundamental frequency and in
the mode shapes of
highly redundant stiff
cable stayed bridges is shown in the study. Only
the vibration
modes determined by the initial
shape based on nonlinear procedures exhibit the
nonlinear cable sag and beam-column effects of
cable-stayed bridges, e.g., the first
and
third modes of the bridge are dominated by the
transversal motion of the tower,
not of the
girder. The difference of the fundamental
frequency in both cases is about
12%. Hence a
correct analysis of vibration frequencies and
modes of cable-stayed
bridges can be obtained
only when the ‘correct’ initial shape is
determined by
nonlinear computation, not by
the linear computation.
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