Bouc-Wen model of hysteresis
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The Bouc-Wen model is a model that is often used to describe non-linear hysteretic systems. It was introduced by Bouc and extended by Wen , who demonstrated its versatibility by producing a variety of hysteretic patterns.

This model is able to capture in analytical form, a range of shapes of hysteretic cycles which match the behaviour of a wide class of hysteretical systems; therefore, given its versability and mathematical tractability, the Bouc-Wen model has quickly gained popularity and 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 among others.

The Bouc-Wen model and its variants/extensions have been used in applications of structural control, in particular in the modeling of the behaviour of magneto-rheological dampers, base isolation devices for buildings and other kinds of damping devices; it has also been in the modelling and analysis of structures built of reinforced concrete, steel, mansory and timber.

Model formulation

Consider the equation of motion of a single-degree-of-freedom (sdof) 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 (elastic) stiffness, is the yield force, the yield displacement, and a hysteretic parameter (usually called the hysteretic displacement) 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 ( retrieves the elastoplastic hysteresis). Take into account that in the original paper of Wen (1976) , is called , and is called . Nowadays the notation varies from paper to paper and very often the places of and are exchanged; we will stick to the notation used by Ref.. The restoring force can be decomposed into an elastic and a hysteretic part as follows:


and


therefore, the restoring force can be understood 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 parameter is redundant (Ma et al.(2004) ).

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 (Baber and Wen (1981) ).

Softening, hardening, pinching, strength degradation and stiffness degradation

Some terms have to be defined below:
  • Softening: the slope of the hysteresis loop decreases with displacement;
  • Hardening: the slope of the hysteresis loop increases with displacement;
  • Pinched hysteresis loops: thinner loops in the middle than near extreme ends; pinching is a sudden loss of stiffness primarily caused by damage and interaction of structural components under large deformation; it is caused by closing (or unclosed) cracks and yielding of compression reinforcement before closing the cracks in reinforced concrete members, slippong at bolted joints in steel construction, and loosening and slipping of the joints caused by previous cyclic loadings in timber structures with dowel-type fasteners (e.g. nails and bolts).
  • Stiffness degradation: progressive loss of lateral stiffness in each loading cycle;
  • Strength degradation: degradation of strength when cyclically loaded to the same displacement level. Note that the term strength degradation is somewhat misleading since strength degradation can only be modeled if displacement is the input function.

The absorved hysteretic energy

The absorved hysteretic energy represents the energy dissipated by the hysteretic system and is quantified as the area of the hysteretic force under the total displacement; therefore, the absorved hysteretic energy per unit mass, 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 because it mirrors the loading history and parallels the process of damage evolution. As it will be seen in later sections, this energy is used in the Bouc-Wen-Baber-Noori model to quantify the degradation of the system.

Bouc-Wen-Baber-Noori model

An important modification to the original Bouc-Wen model was suggested by Baber and Wen in 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 , and are associated respectively to the strength degradation effect, the stiffness degradation effect and the pinching degradation effect, and they are defined as linearly increasing functions of the absorved hysteretic energy as:




The pinching function is specified as:


where



and is the ultimate value of , and is given by


Observe that the new parameters included in the model are: , , , , , , , , , and . When , or no strength degradation, stiffness degradation, or the pinching effect is included in the model.

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 while stiffness seems to be null while the bolt of a wood structure gets 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 subject to an excitation acting in two orthogonal directions. For instance, this model is suited to reproduce the geometrically linear uncoupled behaviour of a bi-axially loaded reinforced concrete column.

Wang and Wen modification

Wang and Wen (1998) suggested the following model to account for the asymmetric peak restoring force:


where is an additional parameter to be determined in the identification stage.

Asymmetric hysteresis

Asymmetric hysteresis curves appear due to the asymmetry of the mechanical properties of the tested element, of the imposed cycle motion, or of both factors. Song and Der Kiureghian (2006) proposed the following function for modelling those asymmetric curves:


where , are six parameters that have to be determined in the identification process. However, according to Ikhouane et al. (2008) , the coefficients , and should be set to zero.

Calculation of the hysteretic response

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, by substituting , 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 method of the 4th-5th-order Runge-Kutta method. The latter method is more efficient in terms of computational time and 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 are greatly reduced if both force an displacement are in the same order of magnitude. For instance, the units kN and mm are a good choice.

Parameter constraints

The parameters of the Bouc-Wen model have the following bounds , , , , , , , .

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 in order to reduce the model to a formulation with well-defined properties.

Adopting those constraints, the unknown parameters become: , , , and .

Identification of parameters

Determination of the model paremeters using experimental input an output data can be accomplised by system identification techniques. The procedures suggested in the literature are:
  • optimization based on the least-squares method, (using Gauss-Newton methods, evolutionary algorithms, genetic algorithms, etc.): in this case the error difference between the time histories or between the short-time-Fourier transforms of the signals is minimized.
  • extended Kalman filter, unscented Kalman filter, particle filters
  • differential evolution
  • adaptive laws,
  • among others


Once an identification method has been applied to tune the Bouc-Wen model parameters, the resulting model is considered as a good approximation of the true hysteresis when the error between the experimental data and the output of the model is small enough from a practical point of view.
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