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 (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, formula_1 represents the mass, formula_2 is the displacement, formula_3 the linear viscous damping coefficient, formula_4 the restoring force and formula_5 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 formula_6 is the ratio of post-yield formula_7 to pre-yield (elastic) formula_8 stiffness, formula_9 is the yield force, formula_10 the yield displacement, and formula_11 a non-observable hysteretic parameter (usually called the "hysteretic displacement") that obeys the following nonlinear differential equation with zero initial condition (formula_12), and that has  dimensions of length: or simply as: where formula_13 denotes the signum function, and formula_14, formula_15, formula_16 and formula_17 are dimensionless quantities controlling the behaviour of the model (formula_18 retrieves the elastoplastic hysteresis). Take into account that in the original paper of Wen (1976), formula_19 is called formula_20, and formula_16 is called formula_19. Nowadays the notation varies from paper to paper and very often the places of formula_19 and formula_16 are exchanged. Here the notation used by Song J. and Der Kiureghian A. (2006) is implemented. The restoring force formula_4 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 formula_17 the transition from elastic to the post-elastic branch is smooth, while for large values that transition is abrupt. Parameters formula_14, formula_19 and formula_16 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 formula_30. Wen assumed integer values for formula_17; however, all real positive values of formula_17 are admissible. The parameter formula_19 is  positive by assumption, while the admissible values for formula_16, that is formula_35, can be derived from a thermodynamical analysis (Baber and Wen (1981)). 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 formula_36 is the squared pseudo-natural frequency of the non-linear system; the units of this energy are formula_37. 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). This modification included strength, stiffness and pinching degradation effects, by means of suitable degradation functions: where the parameters formula_38, formula_39 and formula_40 are associated (respectively) with the strength, stiffness and pinching degradation effects. The formula_38, formula_42  and formula_39 are defined as linear functions of the absorbed hysteretic energy formula_44: The pinching function formula_40 is specified as: where: and formula_46 is the ultimate value of formula_47, given by Observe that the new parameters included in the model are: formula_48, formula_49, formula_50, formula_51, formula_52, formula_53, formula_54, formula_55, formula_56, formula_57 and formula_58. When formula_59, formula_60 or formula_61 no strength degradation, stiffness degradation or pinching effect is included in the model. Foliente (1993), in collaboration with MP Singh and M. Noori, and later 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. 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 formula_63 and formula_64 stand for the mass and  damping matrices, formula_65 and formula_66 are the displacements, formula_67 and formula_68 are the excitations and formula_69 and formula_70 are the restoring forces acting in two orthogonal (perpendicular) directions, which are given by where formula_72 is the initial stiffness matrix, formula_73 is the ratio of post-yield to pre-yield (elastic) stiffness and formula_74 and formula_75 represent the hysteretic displacements. Using this two-degree-of-freedom generalization, Park "et al." (1986) 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) attempted to extend the model of Park "et al." (1986) to include cases with varying 'knee' sharpness (i.e., formula_76). However, in so doing, the proposed model was no longer rotationally invariant (isotropic). Harvey and Gavin (2014) proposed an alternative generalization of the Park-Wen model that retained the isotropy and still allowed for formula_76, viz. Take into account that using the change of variables: formula_78, formula_79, formula_80, formula_81, the equations reduce to the uniaxial hysteretic relationship with formula_82, that is, since this equation is valid for  any value of formula_83, the hysteretic restoring displacement is isotropic. Wang and Wen modification. Wang and Wen (1998) suggested the following expression to account for the asymmetric peak restoring force: where formula_84 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 geometry or of both. Song and Der Kiureghian (2006) proposed the following function for modelling those asymmetric curves: where: and where formula_85, formula_86 are six parameters that have to be determined in the identification process. However, according to Ikhouane "et al ." (2008), the coefficients formula_87, formula_88 and formula_89 should be set to zero. Aloisio "et al." (2020) extended the formulation presented by Song and Der Kiureghian (2006) to reproduce pinching and degradation phenomena. Two additional parameters formula_90 and formula_91 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 formula_2 and its derivative formula_93 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 formula_94, formula_95, formula_96 and formula_97 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 (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. can be written as: where formula_98 within the formula_99 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 formula_100. Accounting for initial conditions, the following relation holds: where, formula_101 is assumed constant for the (not necessarily  small) transition under examination, formula_102 and formula_103, formula_104 are the initial values of the displacement and the hysteretic parameter, respectively. is solved analytically for formula_105 for specific values of the exponential parameter formula_106, i.e. for formula_107 and formula_82. For arbitrary values of formula_106, 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 formula_110, formula_111, formula_112, formula_113, formula_114, formula_115, formula_15, formula_35. 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 formula_30. Constantinou and Adnane (1987) suggested imposing the constraint formula_119 in order to reduce the model to a formulation with well-defined properties. Adopting those constraints, the unknown parameters become: formula_16, formula_17, formula_122, formula_123 and formula_3. 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: These parameter-tuning algorithms minimize a loss function that are based on one or several  of the following criteria: 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.