In structural engineering, the Bouc–Wen model of hysteresis is a hysteretic model 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 (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.
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 non-observable 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. Here the notation used by Song J. and Der Kiureghian A. (2006)Song J. and Der Kiureghian A. (2006) Generalized Bouc–Wen model for highly asymmetric hysteresis. Journal of Engineering Mechanics. ASCE. Vol 132, No. 6 pp. 610–618 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 foundMa F., Zhang H., Bockstedte A., Foliente G.C. and Paevere P. (2004). Parameter analysis of the differential model of hysteresis. Journal of applied mechanics ASME, 71, pp. 342–349 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, i.e., . The parameter is positive by assumption, while the admissible values for , that is , can be derived from a thermodynamical analysis (Baber and Wen (1981)Baber T.T. and Wen Y.K. (1981). Random vibrations of hysteretic degrading systems. Journal of Engineering Mechanics. ASCE. 107(EM6), pp. 1069–1089).
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.
Definitions
Some terms are defined below: Softening: Slope of hysteresis loop decreases with displacement Hardening: Slope of hysteresis loop increases with displacement Pinched hysteresis loops: Thinner loops in the middle than at the ends. Pinching is a sudden loss of stiffness, primarily caused by damage and interaction of structural components under a large deformation. It is caused by closing (or unclosed) cracks and yielding of compression reinforcement before closing the cracks in reinforced concrete members, slipping 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 stiffness in each loading cycle Strength degradation: Degradation of strength when cyclically loaded to the same displacement level. The term "strength degradation" is somewhat misleading, since strength degradation can only be modeled if displacement is the input function. Rate-independency: The relationship between a system's input and output depends entirely on the history of the input's path, rather than the speed or frequency at which that input changes. If you slow down, speed up, or temporarily freeze the input sequence, the system's output path will trace the exact same hysteresis loop. In other words, the rate-independency implies that the output is insensitive to the time-derivatives of the input . According to the definition of rate-independency, the restoring force is a function of the input and its history, but not of the rate of change of . Mathematically, this can be expressed as:
where is a function that represents the tangential stiffness of the system, which depends on the current input , the direction of change of the input (indicated by ), and the current restoring force . The key point is that the function does not depend on the magnitude of , meaning that the system's response is independent of the rate at which the input changes. Asymmetry: It refers to unequal behavior in the positive and negative directions of loading. In an asymmetric hysteresis loop, the response of the system differs when subjected to positive versus negative loading. Asymmetry can arise due to various factors such as material properties, or geometric configurations that cause the system to respond differently under tension and compression or in different loading directions.
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 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 (1981) and Baber and Noori (1985, 1986).Baber T.T. and Noori M.N. (1985). Random vibration of degrading pinching systems. Journal of Engineering Mechanics. ASCE. 111 (8) p. 1010–1026 .Baber T.T. and Noori M.N. (1986). Modeling general hysteresis behaviour and random vibration applications. Journal of Vibration, Acoustics, Stress, and Reliability in Design. 108 (4) pp. 411–420
This modification included strength, stiffness and pinching degradation effects, by means of suitable degradation functions:
where the parameters , and are associated (respectively) with the strength, stiffness and pinching degradation effects. The , and are defined as linear functions of the 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 , where , p, q, , and are the pinching parameters. When , or no strength degradation, stiffness degradation or pinching effect is included in the model.
Foliente (1993),G. C. Foliente (1993). Stochastic dynamic response of wood structural systems. PhD dissertation. Virginia Polytechnic Institute and State University. Blacksburg, Virginia in collaboration with MP Singh and M. Noori, and later Heine (2001)C. P. Heine (2001). Simulated response of degrading hysteretic joints with slack behavior. PhD dissertation. Virginia Polytechnic Institute and State University. Blacksburg, Virginia URL: http://hdl.handle.net/10919/28576/ 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 biaxial excitations. In this case, the interaction between the restoring forces may considerably change the structural response; for instance, the damage suffered from the excitation in one direction may weaken the stiffness and/or strength degradation in the other direction, and vice versa. The equation of motion that models such interaction 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 (perpendicular) directions, which are given by
where is the initial stiffness matrix, is the ratio of post-yield to pre-yield (elastic) stiffness and and represent the hysteretic displacements.
Using this two-degree-of-freedom generalization, Park et al. (1986)Park Y.J., Ang A.H.S. and Wen Y.K. (1986). Random vibration of hysteretic systems under bi-directional ground motions. Earthquake Engineering Structural Dynamics, 14, 543–557 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 (2000)Wang C.H. and Wen Y.K. (2000). Evaluation of pre-Northridge low-rise steel buildings I: Modeling. Journal of Structural Engineering 126:1160–1168. doi:10.1061/(ASCE)0733-9445(2000)126:10(1160) attempted to extend the model of Park et al. (1986) to include cases with varying 'knee' sharpness (i.e., ). However, in so doing, the proposed model was no longer rotationally invariant (isotropic). Harvey and Gavin (2014)Harvey P.S. Jr. and Gavin H.P. (2014). Truly isotropic biaxial hysteresis with arbitrary knee sharpness. Earthquake Engineering and Structural Dynamics 43, 2051–2057. doi:10.1002/eqe.2436 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 (1998)Wang C.H. and Wen Y.K. (1998) Reliability and redundancy of pre-Northridge low-rise steel building under seismic action. Rep No. UILU-ENG-99-2002, Univ. Illinois at Urbana-Champaign, Champaign, Ill. suggested the following expression to account for the asymmetric peak restoring force:
where is an additional parameter, to be determined.
Asymmetrical hysteresis