In structural engineering, the Bouc–Wen model of hysteresis is used to describe non-linear hysteretic systems. It was introduced by Bouc and extended by Wen, who demonstrated its versatility by producing a variety of hysteretic patterns. This model is able to capture, in analytical form, a range of hysteretic cycle shapes matching the behaviour of a wide class of hysteretical systems. Due to its versatility and mathematical tractability, the Bouc–Wen model has gained popularity. It has been extended and applied to a wide variety of engineering problems, including multi-degree-of-freedom (MDOF) systems, buildings, frames, bidirectional and torsional response of hysteretic systems, two- and three-dimensional continua, soil liquefaction and base isolation systems. The Bouc–Wen model, its variants and extensions have been used in structural control—in particular, in the modeling of behaviour of magneto-rheological dampers, base-isolation devices for buildings and other kinds of damping devices. It has also been used in the modelling and analysis of structures built of reinforced concrete, steel, masonry, and timber.
Contents
- Model formulation
- Definitions
- Absorbed hysteretic energy
- BoucWenBaberNoori model
- Two degree of freedom generalization
- Wang and Wen modification
- Asymmetrical hysteresis
- Calculation of the response based on the excitation time histories
- Analytical calculation of the hysteretic response
- Parameter constraints and identification
- Criticism
- References
Model formulation
Consider the equation of motion of a single-degree-of-freedom (sdof) system:
here,
According to the Bouc–Wen model, the restoring force is expressed as:
where
or simply as:
where
and
therefore, the restoring force can be visualized as two springs connected in parallel.
For small values of the positive exponential parameter
Wen assumed integer values for
Definitions
Some terms are defined below:
Absorbed hysteretic energy
Absorbed hysteretic energy represents the energy dissipated by the hysteretic system, and is quantified as the area of the hysteretic force under total displacement; therefore, the absorbed hysteretic energy (per unit of mass) can be quantified as
that is,
here
Energy dissipation is a good measure of cumulative damage under stress reversals; it mirrors the loading history, and parallels the process of damage evolution. In the Bouc–Wen–Baber–Noori model, this energy is used to quantify system degradation.
Bouc–Wen–Baber–Noori model
An important modification to the original Bouc–Wen model was suggested by Baber and Wen (1981) and Baber and Noori (1985, 1986).
This modification included strength, stiffness and pinching degradation effects, by means of suitable degradation functions:
where the parameters
The pinching function
where:
and
Observe that the new parameters included in the model are:
Foliente (1993) and Heine (2001) slightly altered the pinching function in order to model slack systems. An example of a slack system is a wood structure where displacement occurs with stiffness seemingly null, as the bolt of the structure is pressed into the wood.
Two-degree-of-freedom generalization
A two-degree-of-freedom generalization was defined by Park et al. (1986) to represent the behaviour of a system constituted of a single mass
Wang and Wen (2000) attempted to extend the model of Park et al. (1986) to include cases with varying 'knee' sharpness (i.e.,
Wang and Wen modification
Wang and Wen (1998) suggested the following expression to account for the asymmetric peak restoring force:
where
Asymmetrical hysteresis
Asymmetric hysteretical curves appear due to the asymmetry of the mechanical properties of the tested element, of the imposed cycle motion, or of both. Song and Der Kiureghian (2006) proposed the following function for modelling those asymmetric curves:
where:
and
where
Calculation of the response, based on the excitation time-histories
In displacement-controlled experiments, the time history of the displacement
In force-controlled experiments, Eq. 1, Eq. 2 and Eq. 4 can be transformed in state space form, using the change of variables
and solved using, for example, the Livermore predictor-corrector method, the Rosenbrock methods or the 4th/5th-order Runge–Kutta method. The latter method is more efficient in terms of computational time; the others are slower, but provide a more accurate answer.
The state-space form of the Bouc–Wen–Baber–Noori model is given by:
This is a stiff ordinary differential equation that can be solved, for example, using the function ode15 of MATLAB.
According to Heine (2001), computing time to solve the model and numeric noise is greatly reduced if both force and displacement are the same order of magnitude; for instance, the units kN and mm are good choices.
Analytical calculation of the hysteretic response
The hysteresis produced by the Bouc–Wen model is rate-independent. Eq. 4 can be written as:
where
where,
Parameter constraints and identification
The parameters of the Bouc–Wen model have the following bounds
As noted above, Ma et al.(2004) proved that the parameters of the Bouc–Wen model are functionally redundant; that is, there exist multiple parameter vectors that produce an identical response from a given excitation. Removing this redundancy is best achieved by setting
Constantinou and Adnane (1987) suggested imposing the constraint
Adopting those constraints, the unknown parameters become:
Determination of the model paremeters using experimental input and output data can be accomplished by system identification techniques. The procedures suggested in the literature include:
Once an identification method has been applied to tune the Bouc–Wen model parameters, the resulting model is considered a good approximation of true hysteresis when the error between the experimental data and the output of the model is small enough (from a practical point of view).
Criticism
The hysteretic Bouc–Wen model has received some criticism regarding its ability to accurately describe the phenomenon of hysteresis in materials. Ikhouane and Rodellar (2005) give some insight regarding the behavior of the Bouc–Wen model and provide evidence that the response of the Bouc–Wen model under periodic input is asymptotically periodic.
Charalampakis and Koumousis (2009) propose a modification on the Bouc–Wen model to eliminate displacement drift, force relaxation and nonclosure of hysteretic loops when the material is subjected to short unloading reloading paths resulting to local violation of Drucker's or Ilyushin's postulate of plasticity.