A novel family of multiple springs models suitable for biaxial rate-independent hysteretic behavior

Highlights

• A novel family of biaxial rate-independent hysteretic models is presented.

• The family is characterized by n springs evenly spaced in a circular configuration.

• Coupled anisotropic or isotropic biaxial behavior can be simulated.

• Multiple Springs Bilinear and Exponential Models are developed.

• The MSEM accuracy and computational efficiency are assessed.

Abstract

This paper presents a novel family of multiple springs models capable of reproducing the nonlinear response typical of mechanical systems and materials having a biaxial kinematic rate-independent hysteretic behavior. In such a formulation, the generalized force vector, representing the output variable, is computed by summing the contribution of $n$ springs, respectively made up of a nonlinear elastic spring in parallel with a rate-independent hysteretic one. In particular, the generalized force of each spring is computed as a function of the related generalized displacement and history variable. Two isotropic biaxial hysteretic models are derived from the proposed general formulation: the Multiple Springs Bilinear Model and the Multiple Springs Exponential Model. The former is an algebraic model that is illustrated to clearly explain the meaning of the parameters and variables adopted in the formulation. Conversely, the latter is a transcendental model that is presented not only to demonstrate the potentiality of the family in terms of accuracy and computational efficiency, but also to show the possibility of developing models that can reproduce different types of biaxial hysteretic behavior with few parameters having a clear mechanical significance. Such a sophisticated model is validated through numerical and experimental tests.

Introduction

Hysteresis represents a complex phenomenon typically characterizing the nonlinear behavior of mechanical systems and materials employed in numerous fields of engineering, such as aerospace, civil, and mechanical engineering [6], [37], [22], [25], [39].

Mechanical hysteresis phenomena are defined rate-independent when the output variable depends only on the input one and not also on its first time derivative [31]. The input and output variables may be represented, respectively, by a generalized displacement and a generalized force, or vice versa.

According to the nature of their variables, we may distinguish between uniaxial and biaxial rate-independent mechanical hysteresis phenomena: in the former (latter), both the input and output variables are scalar (vector) quantities. Specifically, in biaxial hysteresis phenomena, these variables are represented by two-dimensional vectors [11]. Some examples of biaxial hysteresis phenomena are illustrated in [44], [4], [20], [27], [24].

Biaxial rate-independent hysteretic mechanical systems and materials display a coupled behavior when the two components of the output variable, along two generic orthogonal directions, exhibit bidirectional interaction and when the shape of the output variable path, obtained by applying a circular input variable orbit, is not a square. In particular, such a coupled behavior is referred to as isotropic (anisotropic) when the interaction curve of the output variable components is (not) circular.

A coupled biaxial hysteretic behavior is characterized by kinematic hardening (softening) when the typical hysteresis loop, displayed along a generic direction, is limited by two parallel straight lines or curves whose distance remains constant and the output variable increases (decreases) for increasing values of the input one.

There exist other complex phenomena that may affect the behavior of biaxial rate-independent hysteretic mechanical systems and materials, such as cyclic hardening and softening, asymmetry, as well as pinching [44], [27]; such phenomena are not taken into account in this paper.

Several phenomenological models have been formulated by researchers to reproduce the typical complex response that characterizes mechanical systems and materials with biaxial kinematic rate-independent hysteretic behavior. According to the type of equations that mathematically define the vector expression of the output variable, existing models may be classified as follows:

• biaxial algebraic models: those requiring the solution of an algebraic vector equation, such as the one developed by Wada and Hirose [43], subsequently used by Furukawa et al. [13] and Hoso et al. [16],

• biaxial transcendental models: in which a transcendental vector equation needs to be solved, as it happens in the model introduced by Kikuchi et al. [19] and employed by Ishii and Kikuchi [17],

• biaxial differential models: those that are characterized by a differential vector equation, such as the one formulated by Park et al. [30], employed by Nagarajaiah et al. [26], and then improved by Harvey and Gavin [15], or the one developed by Abe et al. [2].

Each existing biaxial model has its own advantages and disadvantages in terms of accuracy, computational efficiency, number and physical meaning of adopted parameters, and implementation properties. For instance, the biaxial algebraic model proposed by Wada and Hirose [43], employed to simulate the behavior of columns in building frames, requires a set of only three parameters but it is not accurate since the typical hysteresis loop, that can be simulated along a generic direction, is described by a piecewise linear function. On the other hand, the biaxial transcendent model developed by Kikuchi et al. [19], adopted to reproduce the behavior of elastomeric bearings, is much more accurate than the previous one but requires a set of twenty-three parameters having no clear mechanical significance.

The biaxial differential models represent the most used ones because of their accuracy and the relatively small number of employed parameters. For instance, the models formulated by Park et al. [30] and Abe et al. [2] allow one to accurately reproduce complex hysteresis phenomena by means of a set of only 5 and 13 parameters, respectively. Unfortunately, biaxial differential models require parameters with no clear physical significance and are not computationally efficient since they need the numerical solution of a differential vector equation to compute the output variable at each time step of a nonlinear time history analysis.

In this work, we develop a novel family of biaxial kinematic rate-independent hysteretic models that is formulated by combining the multiple shear springs model, introduced by Wada and Kinoshita [42], with the class of uniaxial phenomenological models recently introduced by Vaiana et al. [38].

The main idea behind the proposed family of multiple springs models is to permit the researchers to create their own models according to the specific problem to be simulated. Thus, by particularizing the general expressions of the proposed formulation, they may have the possibility to develop simple biaxial models, such as the bilinear one, or more accurate models capable of reproducing, with only one set of parameters, different types of biaxial hysteretic behavior, namely coupled biaxial kinematic hardening or softening behavior, with or without stiffening.

In particular, the proposed formulation may allow for the development of novel biaxial models having important advantages over hysteretic models already available in the literature. As an example, it is possible to develop models which are as accurate as the differential ones but that, at the same time, not only need a smaller set of parameters with clear physical meaning, but also allow for a considerable reduction of the computational effort, being the numerical solution of a differential vector equation, typically performed by means of multi-steps [33] or Runge-Kutta methods [34], not required.

Such novel biaxial models have a broad range of structural applications. As an example, they may be adopted to simulate the coupled biaxial hysteretic response characterizing reinforced concrete or steel cross sections [44], [27] when complex phenomena, such as cyclic hardening or softening, can be neglected. In addition, they may be implemented to reproduce the complex response of building frames [43], sheathing-to-framing connections employed in platform framing buildings [7], [3], [10], as well as the transverse behavior of seismic isolation devices [1].

The paper is organized into three parts. In the first one (Section 2), the formulation of the proposed family of multiple springs models is presented. In particular, we first illustrate the expressions for evaluating the two components of the output vector variable, namely the generalized force vector, that is obtained by summing the contributions of n springs; then, we describe the expressions defining the angle as well as the generalized displacement and velocity of the i-th spring. Subsequently, we introduce the general expressions of the generalized tangent stiffness, generalized force, and generalized history variable of the i-th spring, and the expressions relating the i-th spring parameters.

In the second part (Section 3), we particularize the general expressions illustrated in Section 2 on the basis of appropriate functions selected for the generalized tangent stiffness of the i-th spring, thus developing two specific models denominated Multiple Springs Bilinear Model (MSBM) and Multiple Springs Exponential Model (MSEM), respectively. The former, representing one of the simplest models that can be derived from the proposed formulation, is illustrated to allow the reader to easily understand the meaning of the adopted parameters and variables. On the contrary, the latter, representing a more complex model, is proposed to show the reader the potentiality of the family in terms of accuracy and computational efficiency. In addition, we also illustrate the flowcharts of the above-described biaxial models to help the reader with the implementation procedure and to show that such models can be easily implemented in a computer program.

Finally, in the third part (Sections 4 and 5), we validate the MSEM by means of numerical simulations and experimental tests. Specifically, in Section 4, we first check the numerical accuracy and the computational efficiency of the MSEM by performing some nonlinear time history analyses on a single degree of freedom mechanical system and by comparing the results with those obtained by employing one of the most adopted biaxial differential models available in the literature, namely the Biaxial Bouc-Wen Model (BBWM), reformulated by Harvey and Gavin [15]. Then, in Section 5, we carry out a nonlinear time history analysis on a seismically base-isolated structure and we compare the numerical results with those obtained during some experimental shaking table tests performed by Losanno et al. [23], thus verifying the accuracy of the MSEM in simulating the coupled biaxial hysteretic behavior displayed by the adopted seismic isolation devices.