Consider the equation of motion of a single-degree-of-freedom (sdof) system:

here,
m
represents the mass,
u
(
t
)
is the displacement,
c
the linear viscous damping coefficient,
F
(
t
)
the restoring force and
f
(
t
)
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
a
:=
k
f
k
i
is the ratio of post-yield
k
f
to pre-yield (elastic)
k
i
:=
F
y
u
y
stiffness,
F
y
is the yield force,
u
y
the yield displacement, and
z
(
t
)
a non-observable hysteretic parameter (usually called the *hysteretic displacement*) that obeys the following nonlinear differential equation with zero initial condition (
z
(
0
)
=
0
), and that has dimensions of length:

or simply as:

where
sign
denotes the signum function, and
A
,
β
>
0
,
γ
and
n
are dimensionless quantities controlling the behaviour of the model (
n
=
∞
retrieves the elastoplastic hysteresis). Take into account that in the original paper of Wen (1976),
β
is called
α
, and
γ
is called
β
. Nowadays the notation varies from paper to paper and very often the places of
β
and
γ
are exchanged. Here the notation used by Ref. is implemented. The restoring force
F
(
t
)
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
n
the transition from elastic to the post-elastic branch is smooth, while for large values that transition is abrupt. Parameters
A
,
β
and
γ
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
A
=
1
.

Wen assumed integer values for
n
; however, all real positive values of
n
are admissible. The parameter
β
is positive by assumption, while the admissible values for
γ
, that is
γ
:=
[
−
β
,
β
]
, can be derived from a thermodynamical analysis (Baber and Wen (1981)).

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.
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
ω
2
:=
k
i
m
is the squared pseudo-natural frequency of the non-linear system; the units of this energy are
J
/
k
g
.

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.

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
ν
(
ε
)
,
η
(
ε
)
and
h
(
z
)
are associated (respectively) with the strength, stiffness and pinching degradation effects. The
ν
(
ε
)
,
A
(
ε
)
and
η
(
ε
)
are defined as linearly-increasing functions of absorbed hysteretic energy
ε
:

The pinching function
h
(
z
)
is specified as:

where:

and
z
u
is the ultimate value of
z
, given by

Observe that the new parameters included in the model are:
δ
ν
>
0
,
δ
A
>
0
,
δ
η
>
0
,
ν
0
,
A
0
,
η
0
,
ψ
0
,
δ
ψ
,
λ
,
p
and
ς
. When
δ
ν
=
0
,
δ
η
=
0
or
h
(
z
)
=
1
no strength degradation, stiffness degradation or 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 with stiffness seemingly null, as the bolt of the structure is pressed into the wood.

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
m
subject to an excitation acting in two orthogonal (perpendicular) directions. For instance, this model is suited to reproduce the geometrically-linear, uncoupled behaviour of a biaxially-loaded, reinforced concrete column.

Wang and Wen (2000) attempted to extend the model of Park *et al.* (1986) to include cases with varying 'knee' sharpness (i.e.,
n
≠
2
). 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
n
≠
2
, viz.

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

where
ϕ
is an additional parameter, to be determined.

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
β
i
,
i
=
1
,
2
,
…
,
6
are six parameters that have to be determined in the identification process. However, according to Ikhouane *et al.* (2008), the coefficients
β
2
,
β
3
and
β
6
should be set to zero.

In *displacement-controlled experiments*, the time history of the displacement
u
(
t
)
and its derivative
u
˙
(
t
)
are known; therefore, the calculation of the hysteretic variable and restoring force is performed directly using equations **Eq. 2** and **Eq. 3**.

In *force-controlled experiments*, **Eq. 1**, **Eq. 2** and **Eq. 4** can be transformed in state space form, using the change of variables
x
1
(
t
)
=
u
(
t
)
,
x
˙
1
(
t
)
=
u
˙
(
t
)
=
x
2
(
t
)
,
x
˙
2
(
t
)
=
u
¨
(
t
)
and
x
3
(
t
)
=
z
(
t
)
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.

The hysteresis produced by the Bouc–Wen model is rate-independent. **Eq. 4** can be written as:

where
u
˙
(
t
)
within the
sign
function serves only as an indicator of the direction of movement. The indefinite integral of Eq.19 can be expressed analytically in terms of the Gauss hypergeometric function
2
F
1
(
a
,
b
,
c
;
w
)
. Accounting for initial conditions, the following relation holds:

where,
q
=
β
sign
(
z
(
t
)
u
˙
(
t
)
)
+
γ
is assumed constant for the (not necessarily small) transition under examination,
A
=
1
and
u
0
,
z
0
are the initial values of the displacement and the hysteretic parameter, respectively. Eq.20 is solved analytically for
z
for specific values of the exponential parameter
n
, i.e. for
n
=
1
and
n
=
2
. For arbitrary values of
n
, Eq.20 can be solved efficiently using e.g. bisection – type methods, such as the Brent’s method.

The parameters of the Bouc–Wen model have the following bounds
a
∈
(
0
,
1
)
,
k
i
>
0
,
k
f
>
0
,
c
>
0
,
A
>
0
,
n
>
1
,
β
>
0
,
γ
∈
[
−
β
,
β
]
.

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
A
=
1
.

Constantinou and Adnane (1987) suggested imposing the constraint
A
β
+
γ
=
1
in order to reduce the model to a formulation with well-defined properties.

Adopting those constraints, the unknown parameters become:
γ
,
n
,
a
,
k
i
and
c
.

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:

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
Genetic algorithms
Particle Swarm Optimization
Adaptive laws
Hybrid methods
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).

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.