Modeling of the Dynamic State-Vector Probability Distribution
for Long-Span Bridge Buffeting Response
Luca Caracoglia1
1
Assistant Professor of Civil Engineering, Northeastern University, Boston, MA, USA, [email protected]
INTRODUCTION
This paper summarizes some recent research activities in the area of long-span bridge dynamic
response under wind loading. Current research efforts focus on the development of analytical and
numerical methods for the prediction of the dynamic bridge response when uncertainty in the loading is
incorporated. The proposed model is based on the multi-mode formulation for wind loading and dynamic
response of a bridge (e.g., [1]), employs modal analysis and a state-space framework.
In this model the structural dynamic response of the bridge is linear; wind loading characteristics are
linearized about the reference wind mean velocity and direction. Both dynamic equations of motion and
wind loading equations are transformed into equivalent time-domain terms. State augmentation
techniques and stochastic calculus [2] are employed to simulate uncertainty in the characterization of the
aerodynamic loading [3].
In previous studies the use of “compound” random variables in the form of uncertain spatial
correlation was considered both for estimation of buffeting response [4] and for the simulation of the
influence of turbulence on flutter instability [5]. It was shown that, if the dynamic response of the bridge
can be simulated by a reduced set of fundamental structural modes, namely a vertical mode and a
torsional one, the use of second-moment analysis was needed only, while advantages of a representation
based on stochastic calculus were partially unexploited. A limitation was also observed since the direct
physical interpretation of such a compound variable was not possible, thereby possibly limiting the
applicability of the method to real bridges.
In this paper the use compound spatial-correlation variables is avoided by rewriting the extended set
of dynamic equations of motions as a function of each variable (independently). Equations are
subsequently transformed into a Markov-type system. Due to the nature of the state-space equations,
propagation of uncertainty becomes nonlinear and the probability distribution of the response state vector
is no longer Gaussian. Interpretation of the results in terms of probability distribution of the extended
state vector including modal bridge response is provided, based on the numerical solution of the Markovtype equations of motion. Discussion and observations are limited to the stationary response under
stationary turbulent wind conditions.
DESCRIPTION OF THE METHODOLOGY
Wind-induced loading is represented a superposition of aeroelastic forces (drag, lift and moment per
unit length), based on the standard mixed time-frequency flutter-derivative formulation [1] as a function
of the “structural” state variables of the problem, such as displacement, velocity etc., evaluated at the deck
level for each cross section along the bridge longitudinal axis. The dynamic turbulence-induced loading is
equivalently described through drag, lift and moment buffeting components as a function of the vertical
(w) and lateral (u) turbulence. First-order linear expansion of the static force coefficients about the
reference equilibrium position under static wind is employed. Aerodynamic admittance or, equivalently,
joint-acceptance functions were neglected at this time.
Modal equations of motion of the bridge and forcing terms are re-cast in state-space form and converted
into time-domain formulation to allow for the derivation of the solution based on stochastic calculus
techniques. In particular, turbulence terms, u and w, can be simulated by separating the dependency on
time and space (i.e., location along the bridge axis). The time dependency can be simulated as the output
of two uncorrelated auto-regressive processes of input W(s), with W being a scalar Wiener process of unit
variance and s a dimensionless time variable. The spatial dependency can be resolved by defining an
equivalent correlation length as a function of the selected bridge mode j, Lψ,j,, representing the loss in
span-wise correlation of turbulence and buffeting force components. The quantities Lψ,j, dependent on the
selected mode, are assumed as random variables to incorporate the effects of uncertainty by perturbation
about the reference wind turbulence and loading scenario. Details on the modeling can be found in [4].
The equation of motion can be recast into a first-order Itô-type stochastic differential system [2],
where the state vector Z(s) can be partitioned into two sub-vectors: ZAE(s) which includes modal response
and the aerolastic states, and ZTB(s), collecting those terms associated with either turbulence components
or Lψ,j variables. The n-th dimensional stochastic differential system of equations can be derived as
follows,
(1)
dZ(s)  a(Z AE , ZTB )ds  2 c(Z AE , ZTB )  d dW ( s),
with a and c being nonlinear functions of the state variables and d a vector of linear filter coefficients.
The dimension of the system depends on the number of structural modes, aerodynamic states and
turbulence-related terms considered in the simulations.
DISCUSSION OF THE RESULTS
In contrast with previous studies [4], the response of a system contaminated by uncertainty in the
turbulence components described as in Eq. (1) does not lead to a multivariate Gaussian random process
Z(s), even when first-order expansion of buffeting forces, linearly dependent on u and w, is employed.
Inspection of Eq. (1) confirms the presence of nonlinear uncertainty propagation behavior through a and
c. Moreover, the derivation of the reduced Fokker-Planck Equation [2] from Eq. (1) is not possible in “a
suitable form”, in which the joint probability density function, p(z), can be directly represented as an
explicit function of a and c. Numerical approaches were identified for the solution of this deterministic
equation as a function of p(z) and depending on a, c and d. The results of the parametric investigations
conducted on a set of simulated bridge examples, with a limited number of structural modes affecting the
response, will be analyzed and discussed, allowing the selection of an optimal numerical solution
procedure.
ACNOWLEDGEMENTS
This research has been supported by the US National Science Foundation, Grant CMMI 0600575. Initial
stages were supported by Northeastern University, Provost’s Office, Research Development Fund (2007).
REFERENCES
[1] Jones, N.P., Scanlan, R.H., 2001. Theory and full-bridge modeling of wind response of cablesupported bridges, Journal of Bridge Engineering, ASCE, 6 (6), 365-375.
[2] Grigoriu, M., 2002. Stochastic calculus. Applications in Science and Engineering, Birkhäuser, Boston,
MA, USA.
[3] Caracoglia, L., 2008. Influence of uncertainty in selected aerodynamic and structural parameters on
the buffeting response of long-span bridges, Journal of Wind Engineering and Industrial
Aerodynamics, 96 (3), 327-344.
[4] Caracoglia, L., 2008, Recent investigations on long-span bridge aeroelasticity in the presence of
turbulence fields with uncertain span-wise correlation, in: Proceedings of the Sixth International
Colloquium on Bluff Bodies Aerodynamics & Applications (BBAA VI), Polytechnic University of
Milan, Milan, Italy, ISBN 88-901916-3-5, pp. 152-155.
[5] Caracoglia, L., 2008. Some implications of the effects of turbulence on bridge flutter. Fourth
International Conference on 'Advances in Wind and Structures (AWAS'08)', C.-K. Choi, J.D.
Holmes, Y.-D. Kim, H.G. Kwak, eds., Techno-Press, Korea, ISBN 978-89-89693-23-9-98530, Jeju,
South Korea, 1667-1675