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JOURNAL OF

SOUND AND

VIBRATION

Journal of Sound and Vibration 322 (2009) 954–968

A Bouc–Wen model compatible with plasticity postulates

A.E. Charalampakis, V.K. Koumousis



National Technical University of Athens, Athens, Greece

Received 20 April 2008; received in revised form 7 November 2008; accepted 12 November 2008

Handling Editor: C.L. Morfey

Available online 8 January 2009

Abstract

The versatile Bouc–Wen model has been used extensively to describe hysteretic phenomena in various ﬁelds of

engineering. Nevertheless, it is known that it exhibits displacement drift, force relaxation and nonclosure of hysteretic

loops when subjected to short unloading–reloading paths. Consequently, it locally violates Drucker’s or Ilyushin’s

postulate of plasticity. In this study, an effective modiﬁcation of the model is proposed which eliminates these problems.

A stiffening factor is introduced into the hysteretic differential equation which enables the distinction between virgin

loading and reloading. Appropriate reversal points are utilized effectively to guide the entire process. It is shown that the

proposed modiﬁcation corrects the nonphysical behavior of the model under short unloading–reloading paths without

affecting its response in all other cases. It is further demonstrated that the original and modiﬁed model exhibit signiﬁcantly

different response under seismic excitation.

1. Introduction

The Bouc–Wen model is a smooth phenomenological model that is often used to descri be hysteretic

phenomena. It was introduced by Bouc [1] and further extended by Wen [2], who investigated the random

vibration of hysteretic systems. Although developed independently, it belongs to the class of endochronic

models, ﬁrst introduced by Valanis [3], which use the notion of intrinsic time to describe the inelastic behavior

of materials.

The Bouc–Wen model has been employed successfully in many areas of engineering. Nevertheless, it is

known that it suffers from displacement drift, force relaxation and nonclosure of hysteretic loops when

subjected to short unloading–reloading paths. As a result, it locally violates Drucker’s [4] or Ilyushin’s [5]

postulate of plasticity. Drucker’s postulate states that the work done by an external added stress over a closed

stress loop is nonnegative, while Ilyushin’s postulate states that the total work done over a closed strain loop is

nonnegative. These postulates are of paramount importance in classical elastoplasticity as they imply the

normality rule for the plastic strain rate and the convexity of the yield surface in stress space. Ilyushin’s

postulate is less restrictive and characterizes the behavior of a very large class of materials, while resulting in

the same consequences as Drucker’s [6].

ARTICLE IN PRESS

www.elsevier.com/locate/jsvi

doi:10.1016/j.jsv.2008.11.017



Corresponding author. Tel.: +30 210 772 1657; fax: +30 210 772 1651.

E-mail address: [email protected] (V.K. Koumousis).

The aforementioned deﬁciencies of the Bouc–Wen model have been reported repeatedly in the literature,

e.g., [7–11]. To cope with this issue, a modiﬁcation of the Bouc–Wen model was proposed by Casciati [8].It

involves the introduction of an additional ‘‘counterclockwise’’ hysteretic term which becomes effective when

loading and gives rise to ‘‘negative’’ inelastic displ acements. This modiﬁcation achieves the reduction, yet not

the elimination of the problem [10–12]. Notably, these violations can also be reduced by using a large value of

the exponential parameter of the model. However, this approach results in an almost bilinear behavior and

offers no advantage in comparison to the sim ple bilinear model. In addition, it reduces the accuracy achieved

by using stochastic equivalent linearization techniques [13].

In this study, a simple modiﬁcation is proposed which eliminates the aforementioned nonphysical

behavior of the Bouc–Wen model. Thus, the long-established conclusion that ‘‘when endochronic mod els

are adopted, local viola tions of the Drucker’s stability postulate cannot be avoided’’ (Casciati and Iwan [12])

is reconsidered. The modiﬁcation focuses at the root of the problem, i.e., the reduced reloading stiffness,

by inserting a stiffening factor into the hysteretic differential equation. The modiﬁed model incorporates

the observation that reloading after partial unloading should follow the unloading path up to the reversal

point. Similar remedy was proposed by Riddell and Newmark [14] to correct the nonphysical behavior

of the model by Clough and Johnston [15]. It is shown that the proposed modiﬁcation eliminates

the unrealistic behavior of the Bouc–Wen model with respect to short unloading–reloadi ng paths while

leaving its behavior in full hysteretic loops practically unaffected. It is further shown that, when com-

pared to the original, the modiﬁed model may exhibit signiﬁcantly different response under seismic

excitation.

2. Original model formulation

The restoring force F(t) of a single-degree-of-freedom system can be expressed as:

FðtÞ¼a

uðtÞþð1  aÞF

zðtÞ, (1)

where u(t) is the displacement, F

the yield force, u

the yield displacement, a the ratio of post-yield to pre-yield

(elastic) stiffness and z(t) a dimensionless hysteretic parameter that obeys a single non-linear differential

equation with zero initial condition:

zðtÞ¼

½A jzðtÞj

ðb þ sgnð

uðtÞzðtÞÞgÞ

uðtÞ, (2)

where A, b, g, n are dimensionless quantities controlling the behavior of the model, sgnðÞ is the signum

function and the overdot denotes the derivative with respect to time.

ARTICLE IN PRESS

Fig. 1. Bouc–Wen model.

A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954–968 955

It follows from Eq. (1) that the restoring force F(t) can be analyzed into an elastic and a hysteretic part as

follows:

ðtÞ¼a

uðtÞ (3)

ðtÞ¼ð1  aÞF

zðtÞ (4)

Thus, the model can be visualized as two springs connected in parallel (Fig. 1) where k

¼ F

and

¼ ak

are the initial and post-yielding stiffness of the system.

3. Parameter constraints

The parameters of Bouc–Wen model are functionally redundant; there exists a multiplicity of parameter

vectors that produce an identical response for a given excitation [16]. Removing this redundancy is best

achieved by ﬁxing parameter A to unity [16]. Henceforth, this constraint is assumed to hold.

4. Response

Recently, analytical expressions for the hysteretic response and dissipated energy of Bouc–Wen model were

derived by the authors [17]. These expressions can be employed for the quantiﬁcation of the displacement drift,

the force relaxation and the violation of Ilyushin’s postulate, as demonstrated in the next section. Further,

they form the basis of the proposed modiﬁcat ion as they provide the full unloading path in analytical form.

For sake of completeness, these expressions are reproduced here in brief.

The behavior of Bouc–Wen model can be distinguished into four cases depending on the sign of

u and z.In

illustration, the response under cyclic excitation is sh own in Fig. 2, where the dotted line signiﬁes the path of

the elastic response. Points A and C signify sign reversal of velocity

u whereas points B and D signify sign

reversal of hysteretic force F

or, equivalently, of hysteretic parameter z.

It was shown that the displacement u is associated with the hysteretic parameter z in terms of Gauss’

hypergeometric function

ða; b; c; wÞ [17]. In the non-trivial case of bag, the following relation holds:

u u

¼ z

; 1 þ

; qjzj





, (5)

where q ¼ b þ sgnð

uzÞg and u

, z

are the initial values of the displacement and hysteretic parameter,

respectively. Eq. (5) can be used with arbit rary values of n, b and g provided that q does not change during the

transition under consideration. Proper evaluation techniques for the hypergeometric function are provided in

Appendix A while simpler relations will be produced for speciﬁc values of the exponential parameter. Thus,

for n ¼ 1 Eq. (5) is solved analytically for z as follows [17]:

z ¼

sgnðzÞþðqz

 sgnðzÞÞe

sgnðzÞqðuu

Þ=u

(6)

ARTICLE IN PRESS

Fig. 2. Response of Bouc–Wen model under cyclic excitation.

A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954–968956

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