Bouc–Wen model of hysteresis


In structural engineering, the Bouc–Wen model of hysteresis is one of the most used hysteretic models typically employed to describe non-linear hysteretic systems. It was introduced by Robert Bouc and extended by Yi-Kwei 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 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.

Model formulation

Consider the equation of motion of a single-degree-of-freedom system:
here, represents the mass, is the displacement, the linear viscous damping coefficient, the restoring force and the excitation force while the overdot denotes the derivative with respect to time.
According to the Bouc–Wen model, the restoring force is expressed as:
where is the ratio of post-yield to pre-yield stiffness, is the yield force, the yield displacement, and a non-observable hysteretic parameter that obeys the following nonlinear differential equation with zero initial condition, and that has dimensions of length:
or simply as:
where denotes the signum function, and,, and are dimensionless quantities controlling the behaviour of the model. Take into account that in the original paper of Wen, is called, and is called. Nowadays the notation varies from paper to paper and very often the places of and are exchanged. Here the notation used by Song J. and Der Kiureghian A. is implemented. The restoring force can be decomposed into an elastic and a hysteretic part as follows:
and
therefore, the restoring force can be visualized as two springs connected in parallel.
For small values of the positive exponential parameter the transition from elastic to the post-elastic branch is smooth, while for large values that transition is abrupt. Parameters, and control the size and shape of the hysteretic loop. It has been found that the parameters of the Bouc–Wen model are functionally redundant. Removing this redundancy is best achieved by setting.
Wen assumed integer values for ; however, all real positive values of are admissible. The parameter is positive by assumption, while the admissible values for, that is, can be derived from a thermodynamical analysis.

Definitions

Some terms are defined below:
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 can be quantified as
that is,
here is the squared pseudo-natural frequency of the non-linear system; the units of this energy are.
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.

Modifications to the original Bouc–Wen model

Bouc–Wen–Baber–Noori model

An important modification to the original Bouc–Wen model was suggested by Baber and Wen and Baber and Noori.
This modification included strength, stiffness and pinching degradation effects, by means of suitable degradation functions:
where the parameters, and are associated with the strength, stiffness and pinching degradation effects. The, and are defined as linearly-increasing functions of absorbed hysteretic energy :
The pinching function is specified as:
where:
and is the ultimate value of, given by
Observe that the new parameters included in the model are:,,,,,,,,, and. When, or no strength degradation, stiffness degradation or pinching effect is included in the model.
Foliente and Heine 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

Consider a two-degree-of-freedom system subject to two-dimensional excitations. Its equation of motion is given by:
where and stand for the mass and damping matrices, and are the displacements, and are the excitations and and are the restoring forces acting in two orthogonal directions, which are given by
where is the initial stiffness matrix, is the ratio of post-yield to pre-yield stiffness and and represent the hysteretic displacements.
Using this two-degree-of-freedom generalization, Park et al. represented the hysteretic behaviour of the system by:
This model is suited, for instance, to reproduce the geometrically-linear, uncoupled behaviour of a biaxially-loaded, reinforced concrete column. Software like ETABS and SAP2000 use this formulation to model base isolators.
Wang and Wen attempted to extend the model of Park et al. to include cases with varying 'knee' sharpness. However, in so doing, the proposed model was no longer rotationally invariant. Harvey and Gavin proposed an alternative generalization of the Park-Wen model that retained the isotropy and still allowed for, viz.
Take into account that using the change of variables:,,,, the equations reduce to the uniaxial hysteretic relationship with, that is,
since this equation is valid for any value of, the hysteretic restoring displacement is isotropic.

Wang and Wen modification

Wang and Wen suggested the following expression to account for the asymmetric peak restoring force:
where is an additional parameter, to be determined.

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 proposed the following function for modelling those asymmetric curves:
where:
and
where, are six parameters that have to be determined in the identification process. However, according to Ikhouane et al., the coefficients, and should be set to zero. Aloisio et al. extended the formulation presented by Song and Der Kiureghian to reproduce pinching and degradation phenomena. Two additional parameters and lead to the pinched load paths, while eight coefficients determine the strength and stiffness degradation.

Calculation of the response, based on the excitation time-histories

In displacement-controlled experiments, the time history of the displacement and its derivative are known; therefore, the calculation of the hysteretic variable and restoring force is performed directly using equations and.
In force-controlled experiments,, and can be transformed in state space form, using the change of variables,, and as:
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, 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. can be written as:
where within the function serves only as an indicator of the direction of movement. The indefinite integral of can be expressed analytically in terms of the Gauss hypergeometric function. Accounting for initial conditions, the following relation holds:
where, is assumed constant for the transition under examination, and, are the initial values of the displacement and the hysteretic parameter, respectively. is solved analytically for for specific values of the exponential parameter, i.e. for and. For arbitrary values of, can be solved efficiently using e.g. bisection – type methods, such as the Brent's method.

Parameter constraints and identification

The parameters of the Bouc–Wen model have the following bounds,,,,,,,.
As noted above, Ma et al. 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 suggested imposing the constraint in order to reduce the model to a formulation with well-defined properties.
Adopting those constraints, the unknown parameters become:,,, and.
Determination of the model parameters 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.

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 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 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.