1
Enhanced Smooth Hysteretic Model with Degrading Properties
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Tathagata Ray1 S.M. ASCE and Andrei M. Reinhorn2 F. ASCE
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Abstract
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The elastic and inelastic nonlinear behavior of structural members use constitutive relations in
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which restoring forces and deformations are not proportional, and often follow different paths in
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loading and unloading using hysteretic functions. There are numerous available models that can
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trace the stiffness and strength changes through yielding, softening and hardening; however,
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models that can address more complex behavior such as degradations, large deformation, bond
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slip and joint gap, do so by complex polygonal rules or smooth continuous functions describing
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momentary (tangent) behavior. There is a need for a unified model based on a combination of
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mechanical springs that can trace the instantaneous combined stiffness used in system analyses.
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In this study, a one-dimensional smooth hysteretic model using series and parallel springs,
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designed for nonlinear structural analysis, is enhanced to incorporate: a) time independent
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properties, b) nonlinear elastic and post-elastic softening and hardening, c) sudden or continuous
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variation of strength, d) degradation of elastic and inelastic stiffness, e) a modified bond-slip
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model, and f) an alternative joint-gap model with a variable gap closing length. Using spring
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analogues or reciprocal structures, the instantaneous force-displacement incremental relations are
1. Ph. D. Candidate, 224 Ketter Hall, Dept. of Civil, Structural and Environmental
Engineering, University at Buffalo, SUNY,
Buffalo, NY 14260, USA. email:
[email protected]
2. Clifford C. Furnas Eminent Professor, 135 Ketter Hall, Dept. of Civil, Structural and
Environmental Engineering, University at Buffalo, SUNY, Buffalo, NY 14260, USA.
email: [email protected]
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formulated, which can be further used to determine the instantaneous tangent-stiffness matrices
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of the elements used in the analyses of complex structures.
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Keywords:-Hysteresis, Nonlinear, Hardening, Degradation, Strength, Stiffness, Pinching,
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Asymmetric, Geometric
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Acknowledgements
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The funding for the research was provided by the National Science Foundation under
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Grants CMMI-NEESR: #0721399 and #0830391. The authors are also grateful to Ki Pung Ryu
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(PhD Candidate, University at Buffalo) for his valuable feedback.
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Introduction
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Structural elements in general show nonlinear force-displacement (or moment-curvature)
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relationships, accompanied with either hardening or softening at larger displacements (or
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curvatures). This nonlinear behavior can be coupled with degradations in terms of strength and
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stiffness, bond-slip-pinching, asymmetric yielding and gap closing. Nonlinear behavior arises
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from (i) material nonlinearities or (ii) geometric changes. Material nonlinearity causes hysteresis
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in the force deformation relationship when sinusoidal or random loading is applied to a structural
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element. Mathematical modeling of this hysteresis behavior is usually done either through a
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Polygonal Hysteretic Model (PHM) or a Smooth Hysteretic Model (SHM), as described by
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Sivaselvan and Reinhorn (1999).
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predefined lines that fit observed behavior. The smooth models usually track the stiffness
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changes through a combination of component springs (reciprocal structures) with
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phenomenological meaning.
The polygonal models track the stiffness changes along
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Clough and Johnston (1966) presented a PHM where the load-displacement path after a
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load-reversal targets the yield point in a negative direction, if the element hasn’t yielded on the
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negative side yet, or aims at the previous maximum displacement point attained in the negative
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direction. A similar rule is applied after load-reversal from a negative displacement increment.
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Takeda et al. (1970) proposed a PHM considering cracking and yielding with a number of load-
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displacement “conditions” and branch stiffness “rules.” Park et al. (1987) and Sivaselvan and
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Reinhorn (1999) extended the Clough and Takeda models to create a generalized PHM with both
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vertex oriented and yield oriented behavior, and coupled with degradation, pinching, and
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asymmetric yielding. Iwan (1966) proposed a PHM for an elastic-perfectly plastic element where
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the elastic branch is piecewise linear. In his model, n number of spring sets (linear spring in
4
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series with slip elements having finite yield force) are considered in parallel. Iwan (1966) and
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Thyagarajan (1989) extended this model by using an infinite number of springs, and thus
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obtaining a curvilinear or smooth hysteretic behavior. Ibarra et al. (2005) proposed a PHM that is
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also capable of simulating strength and stiffness deteriorations, pinching, and residual strength.
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Smooth hysteretic models seem to represent the continuous changes in the materials
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better. A class of smooth hysteretic models (SHM), originally proposed by Bouc (1967) and
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later modified by Wen (1976), served as the basis for many developments of models (for
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example, Park et al., 1987; Kunnath and Reinhorn, 1994) that can address the strain rate
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dependent properties of materials, based on plasticity theory as described by Sivaselvan and
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Reinhorn (2001). Moreover, Sivaselvan and Reinhorn (2001) developed a more versatile model
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that can (i) address degradation of strength and stiffness and (ii) track bond slip and gap opening
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and closing in cross sections of structural members, usually represented by “pinched” hysteretic
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models by changing the properties of component springs. Rodgers et al. (2012) recently
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proposed an SHM, formulated through rational mechanics, that can simulate asymmetric
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yielding, pinching and flag-shaped hysteresis, which can be used to simulate various rate
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dependent dampers. In the present paper, the formulations of Park et al. (1987), Kunnath et al.
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(1997) and Sivaselvan and Reinhorn (2001) is extended, using analog springs, in parallel and in
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series to: (i) simulate rate independent hysteretic behavior; (ii) include nonlinear elastic and
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post-elastic behavior due to kinematic hardening or softening at large deformations due to
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structural confinement, or from coupling between lateral and axial behavior at very large
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deformations, etc.; (iii) incorporate stiffness degradation of a linear elastic spring component,
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similar to Ibarra et al. (2005), who included such a feature in a PHM framework; (iv) modify the
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Gaussian “pinching” model to overcome its various drawbacks and alternatively propose a model
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based on a transcendental function that can address bond slip and variable gap closing; and (v)
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incorporate changes in hysteretic strength in the instantaneous (tangent) stiffness used in the
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global analysis of nonlinear structures. The properties of the component springs can be derived
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from mechanical principles, or validated or calibrated with experimental data. All of the above
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extensions as well as the basic SHM are re-formulated as time independent functions that can be
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used in both nonlinear static and dynamic analyses.
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Formulation of the Smooth Hysteretic Model
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The development of the enhanced model is based on a re-formulation of the original model of
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Sivaselvan and Reinhorn (2001). This formulation is partially repeated here, with minor
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modifications and new notations, in order to define the components used in the new
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developments, which are presented later in the paper.
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The original model was developed along Bouc-Wen's formulation (Bouc 1967, Wen
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1976), which defines the restoring force (F) in a nonlinear spring element for a displacement (u)
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as:
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F  akou  1  a  ko Z
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Here ko is the initial elastic stiffness and a is the post-elastic stiffness ratio. The hysteretic
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variable Z is defined by the temporal differential equation in the second part of Equation (1),
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where the derivatives represented by the dot are with respect to time and A, β, γ are three
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constant parameters. Note that given the constants a, A, β and γ, the two components in Equation
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(1) may be dimensionally inconsistent. Constantinou and Adnane (1987) showed that in order
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to satisfy plasticity theory, the following relations must hold true: A=1 and β+γ=1.
where
Z  Au   uZ Z   uZ 2
(1)
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The model of Sivaselvan and Reinhorn (2001) is reformulated here by re-defining the
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hysteretic variable Z as a time independent non-dimensional variable: Z = Fh/Fhy. Z helps to
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describe the load reversal in the plastic range. Here Fh and Fhy are the force functions of the
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hysteretic spring, which are dependent on displacement and yield strength, respectively. The
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generalized force-displacement relationship, F-u, for a system with yield strength Fy and yield
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displacement uy is:
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F  Fy  a (u / u y )  (1  a) Fh / Fhy 
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Equation (2) shows that the total force F at any deformation can be expressed as the sum of the
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two component springs acting in parallel as shown in Figure 2: (i) the basic elastic spring with
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linear stiffness of the post-elastic component akou, and (ii) the hysteretic elastic ideal plastic
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spring Fy(1-a)Z. Note that Z is the non-dimensional hysteretic variable, which is defined using
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the configuration shown in Figure 3 (a) and from Equation (2) as:
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Z  Fh Fhy   F  akou  [1  a  Fy ]
(2)
(3)
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In the elastic range, Z varies linearly with u, while in the post-elastic range, Z remains
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constant. This dependency of Z vs. u is also shown in Figure 3(b). The variation of Z with the
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deformation u can be expressed as:
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dZ / du  1/ u y  1  Z

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N is the exponent which controls the “smooth” transition between the elastic and post-elastic
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branch, hence the name “Smooth Hysteretic Model.” For smaller values (2<N≤10), a gradual
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smooth transition is produced, while larger values of N (>10) lead to a sharp transition, such as in
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the bilinear model. The signum function sign(x) is defined by sign(x)=x/|x| for x≠0 and sign(x)=1
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for x=0. In the elastic range, it is defined as |Z|<1, |Z|N0 for N>2, as seen in Equation (4).
N
0.5 sign(Z .du)  0.5
(4)
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Hence in the elastic region, dZ/du=1/uy, and is constant. In the post-elastic range, |Z|=1 and
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sign(Z.du)=1, hence dZ/du=0. The signum product Z.du controls the load reversal as shown in
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Figure 3(b). The above model conforms to the kinematic hardening rule described by Chen and
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Han (2007).
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In order to obtain the instantaneous force F during analysis for a given displacement
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history, the first-order differential Equation (4) has to be solved along with Equation (3).
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Equation (4) can be solved in closed form for N=1. For larger values of N (N>2), a direct
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solution of Equation (4) is not possible and numerical integration schemes using iterations or
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semi-implicit integration schemes can be used. The semi-implicit Rosenbrock Integration
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(Nagarajaiah et al. 1991a, b) is shown in detail in Figure 4. The Rosenbrock integration in this
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paper uses Rosenbrock’s coefficients a1 and b1, 0.7886751 and -1.1547005, respectively. The
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coefficients were originally derived by Nagarajaiah et al. (1991a, b) to maintain a fourth order
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truncation error in the Rosenbrock integration method (Rosenbrock 1963). The factors 0.75 and
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0.25 in Equation (i) (see Figure 4) are also selected to achieve the same order of approximation.
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With N being small, i.e., 2 or 3, when the system is loading, unloading and subsequently
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reloading at low force levels, the model may fail to satisfy the Drucker postulate of loop-closing,
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as discussed by Thyagarajan (1989) and Sivaselvan and Reinhorn (2001). For this reason, it is
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prudent to make N larger, i.e., greater than 5 or preferably 10. However, higher values of N may
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create numerical instability in the solution and the force deformation path may overshoot the
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yield surface. Therefore, if the absolute value of Z1 or Zi becomes (numerically) larger than 1, it
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is forced to 1 according to the signum function sign(Z) in order to obtain acceptable results.
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Casciati (1987)
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additional term for N=1.
proposed another solution to satisfy the Drucker postulate by adding an
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For the nonlinear analyses of structures, the continuously changing instantaneous tangent
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stiffness ktangent must be evaluated, which can be obtained by differentiating Equation (2):
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k tan gent  (dF du )  kh 1  Fh Fhy

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Note that Equation (2) includes the value of Z= Fh/Fhy, which is also variable. The tangent
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stiffness, ktangent, can be evaluated at the beginning of any new computation interval (step) using
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the value of Z either a) from the previous step, or b) by solving the first order differential
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Equation (4) for the updated value of Z using a standard semi-implicit solution (as shown in
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Figure 4), therefore providing a better approximation for ktangent.
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Stiffness Degradation of the Hysteretic Spring: Based on experiments and observations, the
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stiffness degradation can be modeled by using the assumption that the unloading branch of the
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hysteretic spring from a positive yield direction at every unloading targets a pre-defined pivot
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point (Park et al. 1987; Sivaselvan and Reinhorn 2001), which is obtained by extending the
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initial elastic slope backwards until it reaches a force of αFhy in the third coordinate plane.
N
0.5  0.5sign  F .du 
h
(5)
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It is observed in Figure 5 that if a load reversal occurs at a force Fc and displacement uc,
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the degraded hysteretic elastic stiffness would be khd, which is the slope of the line joining the
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load reversal point (Fc, uc) and the pivot point. The pivot point is obtained by extending the line
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with initial stiffness on the reverse side until it reaches a force of –(αFhy).
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degradation factor Rk is applied to the initial elastic stiffness kho to obtain the degraded stiffness
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khd. The factor can be obtained from geometry as:
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Rk   Fc   Fhy  /  khouc   Fhy 
The stiffness
(6)
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Stiffness degradation is incorporated into the overall hysteretic model by multiplying the
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hysteretic stiffness by Rk. This ensures that stiffness degradation is applied to the hysteretic
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component of the total tangent stiffness. Although Erlicher and Bursi (2008) recently reported
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some drawbacks of the pivot-point rule, it appears to be acceptable for the macro analysis of
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structural elements.
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Strength Degradation: A change (or deterioration) of the initial yield strength of the hysteretic
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component Fhyo is defined as strength degradation. Such degradation can be a result of loss of
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strength of sections or elements at large deformations (or large ductility demands), and due to
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unrecoverable energy (H) dissipating through cyclic behavior. The degraded yield force (Fhy) can
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therefore be adjusted by a ductility based factor and an energy based factor according to
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Sivaselvan and Reinhorn (2001):
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Fhy  Fhyo 1   umax ucap 
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The expression in the first parenthesis of Equation (7) is the ductility based strength degradation
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factor, where |umax|/ucap is the displacement ductility based on maximum displacement capacity
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ucap, while umax is the latest maximum displacement attained; and β1 is the ductility based
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strength degradation parameter. The expression in the second parenthesis is the energy based
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strength degradation factor, where β2 is the energy based strength degradation parameter. Such
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degradation was observed in experiments and was quantified by numerous researchers. Hult is the
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ultimate unrecoverable work, i.e., energy dissipated if the element is forced by a monotonic
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displacement until it reaches ucap. Note that uy<|umax|<ucap. The functions H and Hult are
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calculated as follows:
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H   Fh .du  Fh 2  2 Rk kh 

1/ 1 
1 
u
0
and
2
1  2   H

H ult 
H ult  Fh you y / 2  Fhyo  ucap  u y 
(7)
(8)
10
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Strength degradation can be incorporated in the global hysteretic model using modified
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Fhy from Equations (7) and (8) to calculate the tangent stiffness ktangent in Equation (5).
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Bond slip and “pinched” hysteretic formulation:
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displacements resulting from loss of bond of section components, or opening and closing of
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cracks, among others. These phenomena lead to engagement and disengagement of the main
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elastic characteristics of the section. Pinching or slip can be incorporated in the model by
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considering an additional stiffness kslip in series with the hysteretic stiffness.
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Gaussian “pinching” model: An expression for kslip developed according to a Gaussian model
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proposed by Baber and Noori (1985) and Reinhorn et al. (1995) is re-expressed here as follows:
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kslip
1/2   Fh / Fhy   Z 






  1/ 2 Rs umax
 umax
H umax
 umax
e


 

Pinching and slip are changes in
Zs 

2

 Z s Fhy  

1
(9)
193


, umax
Here Fh is the current hysteretic force and Fhy is the yield force of the hysteretic spring. umax
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are the maximum positive and negative displacements, respectively. Zs is a parameter denoting
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the sharpness of the slip. Parameter Rs controls the slip-length and Z is the slip-center, mostly
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taken as zero. The above model is suitable for elements with smooth pinching (Zs>0.05) and
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smaller slip-length factors (Rs<0.5). In cases where elements have sharper pinching (Zs<0.01)
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and large slip-length (Rs>0.8), the above model fails to produce the desired hysteresis loop. In
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addition, due to numerical limitations in computing devices, the exponential term in Equation (9)
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may reach a null value for lower Zs and the solution may stop. Moreover, experimental data for
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some structural elements suggest that Rs should not remain constant, and in fact it should
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increase at larger deformations. An improved model is developed in the following sections.
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206
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208
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210
211
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Hysteretic model with asymmetric yielding properties: Some structural elements such as those
made of steel or concrete, may have a single elastic stiffness ko, but may yield at different levels
in the load reversals. In this case, instead of a single Fhyo, there will be two different F+hyo and Fhyo,
associated with two different u+yand u-y. The formulation suggested here is based on the fact
that a “zero” force axis can be defined at the median of the yielding forces F+hyo and F-hyo, and
the behavior is then almost identical to a system with symmetric yielding. This can be achieved
with a translational transformation of the zero axis at each step of the numerical computation.
Therefore, while preserving the principle of kinematic hardening, the case of a system with
different positive and negative yield forces can be reformulated as follows:
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a) Equation (7), used to calculate degraded yield strength, is modified as:
214
1/ 1 
 /
/
/
/
1   2 1   2   H H ult 
 Fhy  Fhyo 1  u max ucap

 F  F  1  sign(du ) / 2  F  1  sign(du ) / 2
hy
hy
 hy
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b) Calculation of the average Fhy and uy and modification of Eq. (8) to calculate Hult is modified
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as:
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 F   F   F  / 2, u   u   u  / 2
y av
y
y
 hy av hyo hyo

 H ult   Fhy av . u y av / 2   Fhy av ucap  u y av

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c) Equation (6), used to calculate the stiffness degradation factor, is modified as:
219
 R  /    F + F /   / k u   F  /  ,

c
yo
o c
 k

yo


 Rk  Rk 1  sign(du ) / 2  Rk 1  sign(du ) / 2 
220


It may be observed that Equation sets (10) and (12) are formulated such that Fhyo and Fhy will be
221
used in the calculations when the load-displacement path approaches a positive yield line, while




















(10)
(11)

(12)
12
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

Fhyo
and Fhy will be used when it approaches a negative yield line. In addition, these formulations
223
offer the flexibility of choosing different degradation parameters,  , 1 and  2 , for positive and
224
negative loading. In Equation set (11), the average of the positive and negative yield forces and
225
positive and negative displacements are used to calculate Hult.
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Extended Smooth Hysteretic Model
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The current re-formulation of the smooth hysteretic model of Sivaselvan and Reinhorn (2001) is
228
extended herein by including the aforementioned features as follows.
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Smooth Hysteretic Model with Nonlinear Hardening: Hysteresis with nonlinear elastic
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behavior is modeled considering two springs acting in parallel. The first is the Hysteretic Spring
231
with yield force Fhy, yield displacement uy and elastic hysteretic stiffness kh. The additional
232
spring is the Nonlinear Elastic Spring, whose stiffness, k(u), depends upon the displacement u.
233
k(u) can be expressed as the summation of a constant linear elastic stiffness kle, and a nonlinear
234
elastic stiffness component kne.f<𝑢̅>, which is discretely applied at 𝑢̅. 𝑢̅ is a displacement
235
variable normalized by the a gap length, while kne is a dimensional stiffness coefficient; and
236
<x> is the Macaulay bracket defined as <x>=x for x>0, and =0 otherwise.
237
<x> is expressed computationally as (x+|x|)/2 and f is a function that defines the nonlinearity of
238
k(u) with respect to u.
In this formulation,
Hence, k(u) is given by:
239
k (u)  kle  kne f u
240
The displacement dependent part of k(u) is activated after the absolute deformation |u| is
241
beyond a specified “gap” length ug or another shifting displacement. This can be generalized by
242
expressing the normalized displacement variable 𝑢̅ as a ratio of |𝑢|and ug, and defining f<𝑢̅ > as:
(13)
13
243
244
f u  u  ug
 u u 
g
or
 u / ug  1
u /u
g
 1
(14)
The instantaneous tangent stiffness ktangent of the two springs taken together is given by:
0.5  0.5sign  F .du   k (u )
245
ktan gent  kh 1  Fh Fhy

246
Fh is the current force and Fhy is the yield force strength of the hysteretic spring. Note that in the
247
presence of the initial hysteretic stiffness kh, the linear elastic component kle of the overall initial
248
stiffness ko is given by:
249
kle  ko  kh
250
In addition, the total apparent yield force Fy of the overall behavior is given by:
N
(15)
h
(16)
uy
251
Fy  Fhy   k (u ).du
(17)
0
252
Embedding Strength Degradation in the Instantaneous Stiffness: SHM can accommodate
253
strength degradation in its hysteretic part. An instantaneous tangent-stiffness of a deforming
254
section in a structural element can be obtained by differentiating both sides of Equation (2) with
255
respect to u as:
256
dF du  a. Fy u y  (1  a) Fy  dZ du   (1  a)Z   dFy du 
ktan gent  a. Fy u y  (1  a)  Fy u y  1  Z

N
or
0.5sign(Z .du)  0.5  (1  a)Z  dFy
(18)
du
257
Note that if Fy changes, which can happen during strength degradation, the last term in Equation
258
(18) comprising dFy/du defines the rate of variation (shrinking or expanding) of the yield surface.
259
Stiffness Degradation of Linear Elastic Spring: Experimental results often suggest that in
260
addition to degradation of elastic stiffness of the hysteretic spring kh, the linear elastic stiffness
261
kle of the nonlinear elastic spring k(u) also degrades, depending upon the displacement ductility.
14
262
Hence, a one parameter degradation model, based on maximum displacement umax, is proposed
263
for kle:
264
 kl e deg raded   kle initial 1 1  umax
265
(kle)degraded and (kle)initial are the degraded and initial stiffness of the elastic spring, respectively. η1
266
is a positive factor that is zero for non-degrading kle, while ucap is the maximum displacement
267
capacity.
268
Modified Gaussian Pinching Model: In order to overcome the previously discussed numerical
269
problems, the exponent in Equation (9),  1/2  Fh / Fhy  Z 
270
term stays within the limitations of the computing device and it allows a smooth formulation.
271
The slip length factor Rs is varied linearly according to the maximum achieved displacement as:
272
Rs  Rso  Rs max  Rso 2  umax / ucap 
273
Rso is the initial value of Rs, Rsmax is the maximum value of Rs, η2 is a positive factor, umax is the
274
maximum displacement, and ucap is the displacement capacity. For Rs to be constant η2=0.
275
Alternate Pinching Model based on Transcendental Function: As an alternative to the
276
Gaussian Pinching Model, a pinching model based on a Tangent Function is formulated. This
277
automatically alleviates the numerical difficulties faced by the Gaussian formulation and
278
provides more control of the slip stiffness kslip, especially for degradations. In this model, the
279
forcing function of the slip-lock spring is defined by:
280
F
281
Fslip and uslip are the force and displacement of the slip-lock spring, respectively, as shown in
282
Figure 7. As the slip-lock spring and hysteretic springs are connected in series, they essentially

slip

/ ucap 
(19)

Zs 

2
, is “capped” such that the exponential

Fs   tan  uslip 2us 
(20)
(21)
15
283
share the same force and thus, Fhys=Fslip. Fs is the crack-closing force associated with the bond-
284
slip. It is expressed as a fraction of the maximum hysteretic force Fhy as:
285
Fs  s1Fhy
286
us is the slip (displacement) and is equal to the permanent deformation:
287
us  umax  u y
288
Due to uncertainties in the crack opening and in the permanent deformations, an adjustment
289
factor s2 is introduced to the right side of Equation (23). In this case, the slip us is defined as:
290
us  s2  umax  u y 
291
The tangent stiffness of the slip-lock spring is obtained from the derivative of Equation (21) with
292
respect to uslip :
0<s1  1.0
(22)
(23)
0<s 2  1.0
(24)
kslip  dFslip duslip   Fs us  / 2  sec 2  uslip 2us 


293
or k slip  kslip min 1   Fslip / Fs 
294
where kslipmin is the minimum stiffness of the slip-lock spring and is defined as:
295
kslip min   Fs us  / 2
296
The slip-lock spring is attached in series with the hysteretic spring and their combined stiffness
297
(kcomb ) is given by:
298
kcomb  khys .kslip
299
k
hys
 k slip 
2
(25)
(26)
(27)
16
300
Examples of Applications of Smooth Hysteretic Models
301
The above formulations were implemented in MATLABTM and further integrated in the
302
computational package IDARC2D by Reinhorn et al. (2009). Some examples that illustrate the
303
sensitivity of the model to the specified parameters are presented below.
304
Example 1:
305
experiencing large lateral deformations require the influence of geometric nonlinearity to be
306
incorporated.
307
The
308
250, kle=20.0,N=20,kne=100;𝑢𝑐𝑎𝑝 = 25( 𝑢𝑦+ + 𝑢𝑦− ), 𝑢𝑔 = 0.75
309
parameters are considered: mild, moderate, and severe (Sivaselvan and Reinhorn, 1999). No
310
degradation of kle is considered.
Stiffening at Large Displacements:
Typical models of bending elements
Assume first a simple structural element subjected to sinusoidal displacement.
properties
of
the
element
are:
kh
=
980,
Three
+
−
𝐹ℎ𝑦𝑜
= 500, 𝐹ℎ𝑦𝑜
=
sets
of
degradation
311
In the first set of analyses, slip-pinching at large deformation is obtained using
312
parameters Rso = 0.25, η2=0, Zs = 0.10, 𝑍̅ = 0.0, according to the Gaussian Model. In the second
313
analysis set (random displacement), all other primary variables except ug are kept the same as in
314
the first set of analyses. The “gap” displacement ug is assumed to be 0.5 in. for this analysis.
315
Also, the effect of pinching is discarded in the second set of analyses. The force-displacement
316
responses and the displacement history chosen for the above analyses are shown in Figure 8.
317
The sinusoidal displacement analysis demonstrates the effect of stiffening at large
318
displacement, coupled with pinching and degradation of strength and stiffness. Adherence of the
319
model to the kinematic hardening principle is also visible in this analysis. The random
320
displacement analysis shows the capability of the model to capture the effect of strength and
321
stiffness degradation without any instability.
17
322
Example 2: Softening at Large Displacement:
Many structural elements show nonlinear
323
hysteretic behavior with softening at larger deformations. The foregoing formulations are
324
implemented in MATLABTM to simulate a lateral force deflection diagram for a beam-column
325
moment connection (Specimen UCSD-1R from the SAC Joint Venture 1996, Sivaselvan and
326
Reinhorn 2001). The following parameters are used in the modeling: Fyho =120 kips; kho = 120
327
kip/inch; kle = 10 kip/inch; kne = -20 kip/inch; ug = 2.5 inch; N=2.0. The strength and stiffness
328
degradation are modeled using the following parameters: α = 5.0; β1 = 0.4; β2=0.2; η1=0.0; ucap =
329
20.0uy. A comparison between the experimental and analytical responses is shown in Figure 9.
330
Example 3 : Hysteretic Behavior with Geometric Nonlinearity: Laterally loaded elements, such
331
as flexible columns or partition wall structures, may exhibit large deformations when integrated
332
in ductile systems. At such large deformations, the pure bending and transverse shear
333
deformations are coupled with the axial deformations, as shown in Figure 10. However, the
334
large deformations may recover upon release of loading and the axial coupling behavior remains
335
elastic. Thus, the entire lateral force displacement behavior becomes nonlinear. The tangent
336
stiffness ktangent of this nonlinear behavior will have contributions from flexural stiffness kf and
337
shear stiffness ks (both of which are hysteretic), and also from the elastic axial stiffness kao.
338
Formulation of tangent stiffness:
339
stiffness, ktangent, depend upon their respective bilinear hysteresis and can be determined as
340
described for infill walls by Reinhorn et al. (2009). In this example, the lateral stiffness,
341
irrespective of the contributions from either or both kf and ks, is called klateral.
The nature and thus the value of kf and ks in the tangent
342
The laterally loaded member of length L shown in Figure 10 displaces from point A to
343
A’, by a distance u, through the action of lateral load F. The member is fixed at point O. At A’,
344
the slope of the member is  . The lateral force F can be resolved into component Fs, normal to
18
345
the deflected shape at A’, and parallel to the deflected shape at Fa. The components of total
346
displacement u are us along Fs, and ua along Fa. The transverse force Fs is resisted by the
347
stiffness klateral and the axial force Fa by kao. The displacement components are:
348
ua  u sin  ; us  u cos 
349
After resolving F into components, the total force is expressed as:
350
F  Fa sin   Fs cos   kaou sin 2   klateral u cos 2 
351
When u is small compared to L,   u / L, sin     u / L , cos  1.0 . Using these
352
approximations, Equation (29) is modified to:
353

(28)

F  kao u 3 / L2  klateral u
(29)
(30)
354
The derivative of both sides of Equation (30) produces the instantaneous (tangent) stiffness:
355
dF du  k tan gent  klateral  3kao  u / L 
356
Note that in Equation (31), klateral is hysteretic with finite post-elastic stiffness and the term
357
containing kao is nonlinear elastic. Assuming the hysteretic part of klateral to be khel and the post-
358
elastic lateral stiffness to be kpel, coupling of axial and lateral stiffness can be incorporated in the
359
present hysteretic model by defining the primary parameters of the model as follows:
360
kh  khel ,
361
f u  3 u / L
kel  k pel ,
2
ken  kao
(31)
(32)
2
(33)
362
If the physical situation demands that the coupling of axial stiffness occurs after the displacement
363
u exceeds a certain gap length ug, then Equation (33) can be modified as:
364

f u  3 u  ug
L

2
(34)
19
365
Implementations and Sample Results: A model of a partition wall element, subjected to
366
sinusoidal quasi-static displacements applied at the top of the wall, was tested in University at
367
Buffalo (SUNY) by Filiatrault et al. (2010). It is used here to illustrate the performance of the
368
hysteretic model with large deformations.
369
From the experimental Base-Shear vs. Top Displacement hysteresis shown in Figure
370
11(a), it is observed that the gap length ug is a varying parameter dependent on the maximum slip
371
umax reached before the wall collapsed. The variation can be expressed as:
372
ug  ugo 1  umax  ugo
373
ugo is the initial gap length, g1 is a factor (<1) and umax is the absolute maximum displacement
374
reached in the last cycle. The gap length was measured by Filiatrault et al. (2010) during the
375
experiments, when the tested partition wall slipped at both the top and bottom, depending on the
376
maximum displacements. It was also observed that while using the Alternative Pinching Model
377
based on the tangent function, the parameters s1 and s2 should also be varied because Fs and us
378
change nonlinearly with umax at large displacements. The variation of s1 and s2 can be expressed
379
as:
380
 s1  s1o 1  s11  umax u _ cap   s1min 


 s2  s2 o 1  s22  umax u _ cap   s2max 
381
s1o and s2o are the initial values and s1max and s2max are the maximum possible values of s1 and s2.
382
The properties of the partition wall estimated for the Gaussian Pinching Model are shown in
383
Table 1 (left). A comparison between the experimental and analytical responses for the Modified
384
Gaussian Pinching Model is shown in Figure 11(b).

u
max

 u go   g1umax umax  u go
u
max
 u go 
(35)
(36)
20
385
The parameters of the analysis performed using the new formulation based on the tangent
386
function for kslip and Equation (35) for gap length are shown in Table 1 (right). The
387
corresponding comparison between the experimental and analytical responses is shown in Figure
388
11(c).
389
directly from the experimental results. All the other input parameters were determined from
390
material data. Since these experiments were displacement driven, the maximum force levels of
391
the numerical simulations at different displacements are compared with those of the experimental
392
results. It is found that the analytical force magnitudes were within 10% of the experimental
393
values. If the nonlinear parameters are not rate dependent (such as velocity dependent damping),
394
the selected parameters produce reasonable results in dynamic analyses, within the ductility
395
range considered.
396
Reinhorn (2001), and the comparison of the simulation with the experimental results is shown in
397
Figure 12. It is observed that the previous SHM, which uses the Gaussian pinching model,
398
shows unstable behavior for sharper slip (pinching) and at large displacements. It also fails to
399
explicitly capture the degradation of nonlinear elastic stiffness and stiffening at large
400
deformations.
401
Remarks and Conclusions
402
The models presented in this paper account for the basic re-formulations of the SHM that are
403
time-independent, as opposed to the more classical Bouc–Wen model, which seems to be
404
dependent on force and displacement rates (velocities). Therefore, the model presented herein
405
can be used in nonlinear static analysis as well as in dynamic analyses, to consider degradations
406
and deteriorations. It is also observed that the Alternative Pinching Formulation based on a
Note that the initial stiffness and yield force in the last two examples were obtained
This example is also simulated using the previous SHM by Sivaselvan and
21
407
tangent function offers more flexibility and control for “pinched” modeling, especially when
408
compared to the more classical Gaussian Pinching Model.
409
The stiffness properties of nonlinear structural elements can be formulated using
410
information based on the material and geometry of sections derived from elementary nominal
411
formulations; however, for degradation and pinching, a database must be developed, as was done
412
in the experimental study developed by ASCE/SEI (2006) and Filiatrault et al. (2010). Kunnath
413
et al. (1997) also carried out system identification of degrading and pinched reinforced concrete
414
structures in order to calibrate the extended Bouc-Wen SHM (Kunnath et al. 1990).
415
The stability of the model is related to the continuous positive-definite stiffness matrix.
416
The strategy offered here is to use this positive stiffness, while loss of strength due to
417
degradation is coupled to the solution of the global system. There are, however, three possible
418
conditions that can lead to instability in the SHM: a) the hysteretic stiffness kho is much larger
419
(more than 100 times) than the linear elastic stiffness kle, b) the exponent of transition from the
420
elastic to the post-elastic state is very large (N larger than 100), and c) the "pinching" exponential
421
term is beyond the capacity of the computing device in the previous Gaussian pinching model
422
(Baber and Noori 1985). While the first two cases can be handled by proper selection of the
423
model parameters and by reduction of computational step size, the third case is addressed herein
424
by modifying the Gaussian model.
425
The enhancements to the proposed SHM are incremental to the previous version
426
developed by Sivaselvan and Reinhorn (2001). The use of the previous version without the
427
improvements remains valid when suitably fitted coefficients are used. The new model was
428
integrated in the computational platform IDARC2D (Reinhorn et al. 2009) and was found to be
22
429
backward compatible and able to successfully simulate all the examples included in the software
430
validation package simulated with PHM and the previous version of SHM.
431
The one-dimensional tangent stiffness calculated by the proposed models can be further
432
incorporated into the global stiffness matrix in order to analyze the structure as a whole. The
433
model with geometric nonlinearity presented herein may be further modified to incorporate the
434
effect of axial load on the element as discussed in Kikuchi et al. (2007). The model may also be
435
extended to bi-axial force-deformation (or moment-curvature) relationship required for three-
436
dimensional analysis as described in Sivaselvan and Reinhorn (2004). Finally, the one-
437
dimensional model developed herein can serve as a base for three-dimensional models developed
438
by Ray (2012).
439
23
440
References
441
442
06
ASCE/SEI (2006). "Seismic Rehabilitation of Existing Buildings (41-06) " ASCE/SEI 41-
443
444
Baber, T. T., and Noori, M. N. (1985). "Random Vibrations of Degrading Pinching
Systems." Journal of Engineering Mechanics, 111(8), 1010-1026.
445
446
Bouc, R. (1967). "Forced Vibration of Mechanical System with Hysteresis." Proc., 4-th
Conf. on Nonlinear Oscillations.
447
448
449
Casciati (1987). "Nonlinear Stochastic Dynamics of Large Structural Systems By
Equivalent Linearization." Proc., ICASP5, Fifth Internation Conference on Application of
Statistics and Probability in Soil and Structural Engineering.
450
451
Chen, Y. F., and Han, D. J. (2007). "Plasticity for Structural Engineers." J. Ross
Publishing, Inc., Fort Lauderdale, 14-16.
452
453
454
Clough, R. W., and Johnston, S. B. (1966). "Effects of Stiffness Degradation on
Earthquake Ductility Requirements." Proc., Second Japan Earthquake Engineering Symposium,
227-232.
455
456
Constantinou, M. C., and Adnane, M. A. (1987). "Dynamics of Soil-Base-IsolatedStructure Systems." National Science Foundation.
457
458
459
Erlicher, S., and Bursi, O. S. (2008). "Bouc-Wen-Type Models with Stiffness
Degradation: Thermodynamic Analysis and Applications." ASCE Journal of Engineering
Mechanics, 134(10), 843-855.
460
461
462
Filiatrault, Andre, Mosqueda, Gilberto, Retamales, Rodrigo, Davies, Ryan, Tian, Yuan,
and Fuchs, Jessica (2010). "Experimental Seismic Fragility of Steel Studded Gypsum Partition
Walls and Fire Sprinkler Piping Subsystems." ASCE, 237-237.
463
464
465
Ibarra, L. F., Medina, R. A., and Krawinkler, H. (2005). "Hysteretic models that
incorporate strength and stiffness deterioration." Earthquake Engineering and Structural
Dynamics, 34, 1489-1511.
466
467
Iwan, W. D. (1966). "A Distributed Element Model for Hysteresis and its Steady-State
Dynamic Response." Journal of Applied Mechanics, ASME, 33(4), 893-900.
468
469
470
Kikuchi, M., Yamamoto, S., and Aiken, I. D. (2007). "An Analytical Modeling of LeadRubber Bearings under Large Deformation." 8th Pacific Conference on Earthquake
EngineeringSingapore.
471
472
Kunnath, S., Reinhorn, A., and Park, Y. (1990). "Analytical Modeling of Inelastic
Seismic Response of R/C Structures." Journal of Structural Engineering, 116(4), 996-1017.
24
473
474
475
Kunnath, Sashi K., Mander, John B., and Fang, Lee (1997). "Parameter identification for
degrading and pinched hysteretic structural concrete systems." Engineering Structures, 19(3),
224-232.
476
477
478
Kunnath, S. K., and Reinhorn, A. M. (1994). "Efficient Modeling Scheme for Transient
Analysis of Inelastic RC Structures." Microcomputers in Civil Engineering, International
Journal of Computer-Aided Civil and Infrastructure Engineering, 10, 97-100.
479
480
481
Nagarajaiah, Satish, Reinhorn, Andrei M., and Constantinou, M. C. (1991a). "Nonlinear
Dynamic Analysis of 3-D-Base-Isolated Structures." ASCE/Journal of Structural Engineering,
117(7), 2035-2054.
482
483
484
Nagarajaiah, Satish, Reinhorn, Andrei M., and Constantinou, M. C. (1991b). "3D-BASIS:
Nonlinear dynamic analysis of three-dimensional base isolated structures - Part II." NCEER-910005, National Center for Earthquake Engineering Research, SUNY, Buffalo, New York
485
486
487
Park, Y. J., Reinhorn, A. M., and Kunnath, S. K. (1987). "IDARC:Inelastic Damage
Analysis of Reinforced Concrete Frame - Shear-Wall Structures." Technical Report NCEER,
University at Buffalo, SUNY.
488
489
490
Reinhorn, A. M., Madan, A., Valles, R. E., Reichmann, Y., and Mander, J. B. (1995).
"Modeling of Masonary Infill Panels for Structural Analysis." Technical Report NCEER,
University at Buffalo, SUNY.
491
492
493
Reinhorn, A. M., Roh, H. S., Sivaselvan, M. V., Kunnath, S. K., Valles, R. E., Madan,
A., Li, C., Lobo, R., and Park, Y. J. (2009). "IDARC2D Version 7.0: A Program for the Inelastic
Damage Analysis of Structures." Technical Report MCEER, University at Buffalo, SUNY.
494
495
Rosenbrock, H. H. (1963). "Some general implicit processes for the numerical solution of
differential equations." The Computer Journal, 329-330.
496
497
Sivaselvan, M., and Reinhorn, A. M. (2001). "Hysteretic Models for Deteriorating
Inelastic Structures." ASCE/Journal of Engineering Mechanics, 126(6), 633-640.
498
499
500
Sivaselvan, M. V., and Reinhorn, A. M. (2004). "Nonlinear Structural Analysis Towards
Collapse Simulation : A Dynamical Systems Approach " MCEER-04-0005, University at
Buffalo, SUNY, Buffalo.
501
502
Sivaselvan, M.V., and Reinhorn, A.M. (1999). "Hysteretic Models for Cyclic Behavior of
Deteriorating Inelastic Structures." Technical Report MCEER, University at Buffalo, SUNY.
503
504
505
Takeda, T., Sozen, M. A., and Neilsen, N. N. (1970). "Reinforced Concrete Response to
Simulated Earthquake." Journal of Structural Division, Proceedings of the American Society of
Civil Engineers, 96(12), 2557-2573.
506
507
Thyagarajan, R. S. (1989). "Modeling and Analysis of Hysteretic Structural Behavior."
PhD, California Institute of Technology, Pasadena.
25
508
509
Wen, Y. K. (1976). "Methos for Random Vibration of Hysteretic Systems." J. Engrg.
Mech. Div., ASCE, 102(2), 249-263.
510
511
Ray, T. (2012). "Modeling of Multidimensional Nonlinear Elastic and Inelastic Structural
Systems."PhD, University at Buffalo, SUNY.
512
513
514
515
Rodgers, Geoffrey W., Mander, John B., and Geoffrey Chase, J. (2012). "Modeling
cyclic loading behavior of jointed precast concrete connections including effects of friction,
tendon yielding and dampers." Earthquake Engineering & Structural Dynamics, 41(15), 22152233.
516
517