In
structural engineering
Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of man-made structures. Structural engineers also must understand and ca ...
, the Bouc–Wen model of hysteresis is one of the most used
hysteretic model
Hysteretic models are mathematical models capable of simulating the complex nonlinear behavior characterizing mechanical systems and materials used in different fields of engineering, such as aerospace, civil, and mechanical engineering. Some exa ...
s
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
Soil liquefaction occurs when a cohesionless saturated or partially saturated soil substantially loses strength and stiffness in response to an applied stress such as shaking during an earthquake or other sudden change in stress condition, ...
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
Masonry is the building of structures from individual units, which are often laid in and bound together by mortar; the term ''masonry'' can also refer to the units themselves. The common materials of masonry construction are bricks, building ...
, 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 found
[Ma 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. The parameter
is positive by assumption, while the admissible values for
, that is