1 Introduction

It is commonly recognized that the damage caused by earthquake induced ground motion to civil structure is adverse depending on its intensity, especially in those earthquake-prone region. Traditional approach addressing this impact is to either increase structure’s strength or energy absorption capacity, while the latter one is usually achieved by energy dissipation contributed by plastic hysteresis at element level (well-known as ‘plastic hinge zone’), which is demonstrated and observed as effective but with inevitable compensation as leaving localized structural damage [1].

According to the statistics, up to 2011 around 20,000 structures spreading in more than 30 countries have been equipped by either seismic isolation (SI) or supplemental energy dissipating (ED) techniques [2]. Drawing focus into ED system that this research relates to, which functions by adding additional dam** capacity to the original structure, therefore reduces the demand of inelastic hysteretic energy dissipation by structural frame [3, 4].

The prevalent and commercially available ED systems can be classified by controlling mechanisms as passively controlled, active controlled, and semi-active controlled [5]. Although with comparison to passive ED system, active ED system can be more event-adaptive, intelligent and can tackle wider band width of external excitation (mainly of earthquake, wind), its effectivity strongly relies on large external power supply which is probable to be inoperative during extreme earthquake event or energy-wasting if building is exposed to wind-frequent environment [6, 7]. Semi-active controlled ED system has been researched by many scholars [8, 9], which can be categorized as an intermediate portfolio between active and passive controlled ED systems. Differing to active ED, semi-active ED requires much less power supply in aspects of both life-time consumption and instantaneous power peak [10], which is only used for shifting specific mechanical parameters as an automated dynamic real-time optimization to improve efficiency of ED system [5, 11]. Despite that high efficiency of energy dissipation plus considerable economy that semi-active ED solution can achieve [12, 13], hindrances still exist, such as complexity in control system design and manufacturing [5], relatively low dam** capacity and limited reliability due to the complexity of its constituent mechanics [10, 14]. The last part forming the ground of ED systems that is right the one dominantly adopted in earthquake engineering by this far across worldwide, belongs to passive ED systems. Regarding energy-dissipating mechanisms, they can be categorized as displacement-dependent (metallic yield damper, friction device, buckle-restrained brace, etc.), velocity-dependent (viscous fluid damper, etc.) and mass tuning (tuned mass damper, tuned liquid damper, etc.) and hybrid type (viscoelastic damper, etc.) [1, 15].

This research focuses on a newly developed branch of friction ED systems, also called friction damper (FD), which follows or quasi-follows Coulomb’s friction law. By this far many prototypes of FD have been researched and developed [3]. Owing to its good economy, mechanical simplicity, easy applicability and large hysteretic energy dissipation capacity, FD has gained very wide practical application hence big market. However, without a high initial static force that builds up sufficiently large hysteretic loop at which FD starts sliding to dissipate energy, there will not be such large dam** capacity it can supply to structure as an impactful advantage of FD, whereas large initial static force does stiffen global structure and consequently induces large floor acceleration [16]. Namely, under frequent dynamic loading case with low or moderate intensity, it rarely plays the role of dissipating energy, but inversely induces larger floor acceleration instead [17].

To overcome the shortcoming of FD outlined above, a new branch of FD portfolio has been developed in various patterns, called passive variable friction damper (PVFD). It is worth of clarification herein that, there is no configuration of response sensor nor real-time optimizing computing chip configured to vary the factors (mostly to shift normal force) that determines the friction force, which shall be categorized as semi-active damper. Instead, PVFD employs pure mechanical adaptivity to adjust friction force via varying normal force, as a response to either displacement or rotation [1) are 3 typical patterns of hysteretic loop of PVFD devices, according to illustrations originally delivered by Wang et al. [3, along with experiment configuration (Fig. 3). External load is exerted periodically in displacement pattern (Fig. 4).

Fig. 2
figure 2

Schematic of passive variable friction damper

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Fig. 3
figure 3

a 3D model of specimen; b experiment configuration

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Fig. 4
figure 4

Step history of displacement loading

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Hysteresis of developed and tested specimen is in pattern of butterfly-hysteresis one (Fig. 1a). Yielded hysteretic loop is smoothed into piecewise form and fitted regarding theoretical derivations (Fig. 5) given in Eqs. 1–5. Reaction force represented by each colored piece on loop is derived based on Coulomb’s friction theory when in sliding, and linear elastic assumption while deforming without sliding.

$$F_{1} = k_{i} \cdot X$$
(1)
$$F_{2} { = }\mu \cdot f_{c}$$
(2)
$$\begin{gathered} F_{3} = f_{{x1}} + f_{{x2}} \hfill \\ \quad = \mu \cdot (f_{c} + k_{s} \cdot d_{s} ) \cdot \cos ^{2} \theta + f_{c} \cdot \tan \theta \hfill \\ \quad = \frac{1}{2} \cdot \mu \cdot [f_{c} + k_{s} \cdot (X - x_{2} )] \cdot \sin 2\theta + f_{c} \cdot \tan \theta \hfill \\ \end{gathered}$$
(3)
$$F_{4} = \mu \cdot f_{c} + \frac{1}{2} \cdot \mu \cdot [f_{c} + k_{s} \cdot (x_{3} - x_{2} )] \cdot \sin 2\theta + k_{i} \cdot (X - x_{3} )$$
(4)
$$\begin{gathered} F_{5} = f_{{x1}} - f_{{x2}} \hfill \\ \quad = \frac{1}{2} \cdot \mu \cdot [f_{c} + k_{s} \cdot (X - x_{2} )] \cdot \sin 2\theta - f_{c} \cdot \tan \theta \hfill \\ \end{gathered}$$
(5)
Fig. 5
figure 5

Theoretically fitted hysteresis against experimental one

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In Eqs. 1 ~ 5, F1–F5 denote reaction forces represented by colored segments on hysteretic loop (Fig. 5), respectively. Grayed segments denoting the central symmetric half can be obtained via rotating the colored pieces altogether around origin by 180 degrees in F-X plane. Symbols x1–x4 denote the deformations of 4 joint points at which the reaction force changes. Symbols ki, fc, µ, ks, θ represent 5 constants of, respectively, stiffness of elastic-slip, total precompression force, friction coefficient of slip** faces, aggregated stiffness by all spring fasteners, angle of end sloped faces measured to the flattened middle face. Regarding Fig. 2, symbols fx1, fx2, ds denote 3 intermediate variables, which are respectively, projection of friction force to the spatial plane coplanar with the flattened middle face where initial friction F2 acts on, projection of normal compression force to the same spatial plane, compressive deformation of spring fasteners. Particularly, regarding Fig. 5 and referring to Eqs. 3 and 5, F3 has relatively larger magnitude than F5 so that the slope of orange segment is steeper than that of purple, because fx2 plays a resistive role during off-centering motion, whereas plays a motivating role in centering motion.

3 Equivalent modelling approach

By this far, there is very few built-in link elements available for modelling the butterfly-hysteresis PFVD in structural FEA software, especially of those prevalent and commercially available ones, while such innovative and evidenced as efficient ED device calls for more promotion and wider application in real engineering practice. In this research, an equivalent modelling approach has been proposed, via assembling 3 built-in link elements that are dual-flag friction spring (DFFS), negative-stiffness multi-linear elastic (NSMLE), and plastic Wen (PW) dampers, into parallel as a ‘super-link’ (Fig. 6). 4 shoulders in stiffening stage of the super-link are not as one single slope but linked two, which can be reasoned from the perspective of geometry to detailed view where the hysteresis of DFFS begins to be cut through by NSMLE. If the slopes of these 4 shoulders in fitted hysteresis (Fig. 5) can well match the secant slopes of the numerically assembled one (Fig. 6), such distortion in shape can be deemed as acceptable.

Fig. 6
figure 6

Illustration of assembly of super-link element

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The super-link equivalently modelling butterfly-hysteresis PFVD, has been numerically tested in SAP2000 environment using single-degree-of-freedom (SDoF) subjected to harmonic displacement loading with amplitude valued by maximum deformation capacity of tested PVFD device, yielded hysteresis of which is given in Fig. 7.

Fig. 7
figure 7

Hysteresis yielded from SAP2000 analysis of SDoF model connected by super-link

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4 Optimization of modelling parameters

4.1 Decomposition of hysteretic loop

Unlike those built-in link elements, parameters of assembled super-link used to equivalently model the peculiar hysteretic pattern cannot be determined from the real mechanical property of PVFD device, neither from experiment nor standardized database of product portfolio. On the other hand, because of the geometric shift at 4 shoulders on the hysteresis of super-link comparing to the experimental one, there is not exact analytical solution of modelling parameters from given real mechanical parameters of PVFD that determines its unique hysteretic loop. Therefore, an optimization methodology better with automated workflow for practical ease, is called to be invented for conducting modelling parameters to fit various PVFD device as representable as possible.

Figure 8a–c symbolized parameters determining complete nonlinear behavior in SAP2000 environment for link components DFFS, NSMLS and PW, respectively, which are all pending to be yielded as final output from optimization work. According to the complete coupling of deformation of 3 components assembled in parallel, analytical piecewise F-X function of the super-link can be derived via making arithmetic summation of 3 components that geometrically illustrated in Fig. 6.

Fig. 8
figure 8

Parameters defining adopted 3 link components in SAP2000

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The decomposition diagram given in (Fig. 9) directs the focus from the total hysteresis of super-link into the isolated shoulder region, because the contribution of PW is relatively the most simplistic one out of all 3 link components hence the most challenging geometric complexity of this synthetic hysteresis drops into the 4-shoulder region, which is completely dependent on contribution of DFFS and NSMLE components. Nonetheless PW’s contribution will be accounted for optimization in terms of energy dissipation within one loading circle and will be addressed in following section.

Fig. 9
figure 9

Decomposition of shoulder region of super-link’s hysteretic loop

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4.2 Numerical optimization approach

The isolated region incorporating 2 shoulders in 1st and 4th quadrants that red-outlined in Fig. 9, has been symbolized into Fig. 10, symbols assigned to coordinates of inflection points are collected in Table 1, each of which can be mathematically determined based on input 7 parameters that extracted from the real hysteretic loop of targeted PVFD device. Derivation of tabularized coordinates in Table 1 are given by Eqs. 6–17.

$$X_{a} = \frac{{\left( {d_{0} + \left( {\frac{{\left( {d_{0} \cdot k_{f0} - d_{0} \cdot \left( {k_{f1} + k_{n2} } \right)} \right)}}{{\left( {k_{f1} + k_{n2} } \right)}}} \right) \cdot \frac{{\left( {k_{f2} + k_{n2} } \right)}}{{\left( {k_{f0} - k_{f2} - k_{n2} } \right)}}} \right)}}{2}$$
(6)
$$F_{a} = 0$$
(7)
$$X_{b} = d_{0}$$
(8)
$$F_{b} = d_{0} \cdot k_{f0} - k_{n2} \cdot d_{0} - \left( {\frac{{\left( {d_{0} \cdot k_{f0} + k_{0} \cdot \left( {\frac{{\left( {d_{0} \cdot k_{f0} - d_{0} \cdot \left( {k_{f1} + k_{n2} } \right)} \right)}}{{\left( {k_{f1} + k_{n2} } \right)}}} \right) \cdot \frac{{\left( {k_{f2} + k_{n2} } \right)}}{{\left( {k_{f0} - k_{f2} - k_{n2} } \right)}}} \right)}}{2} - \frac{{\left( {d_{0} \cdot k_{n2} + k_{n2} \cdot \left( {\frac{{\left( {d_{0} \cdot k_{f0} - d_{0} \cdot \left( {k_{f1} + k_{n2} } \right)} \right)}}{{\left( {k_{f1} + k_{n2} } \right)}}} \right) \cdot \frac{{\left( {k_{f2} + k_{n2} } \right)}}{{\left( {k_{f0} - k_{f2} - k_{n2} } \right)}}} \right)}}{2}} \right)$$
(9)
$$X_{c} = d_{0}$$
(10)
$$F_{c}=(k_{f1}+k_{n2})\cdot d_{u}+d_{0}\cdot k_{f0}-d_{0}\cdot (k_{f1}\ +k_{n2})-d_{u}\cdot k_{n2}-\frac{\left({d}_{0}\cdot {k}_{f0}\cdot +{k}_{f0}\cdot \left(\frac{\left({d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)\right)}{\left({k}_{f1}+{k}_{n2}\right)}\right)\cdot \frac{\left({k}_{f2}+{k}_{n2}\right)}{\left({k}_{f0}-{k}_{f2}-{k}_{n2}\right)}\right)}{2}+\frac{\left({d}_{0}\cdot {k}_{n2}\cdot +{k}_{n2}\cdot \left(\frac{\left({d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)\right)}{\left({k}_{f1}+{k}_{n2}\right)}\right)\cdot \frac{\left({k}_{f2}+{k}_{n2}\right)}{\left({k}_{f0}-{k}_{f2}-{k}_{n2}\right)}\right)}{2}$$
(11)
$${X}_{d}=\frac{\left(\left(\frac{\left({d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)\right)}{\left({k}_{f1}+{k}_{n2}\right)}\cdot \left({k}_{f2}+{k}_{n2}\right)\right)-\left(\left({k}_{f1}+{k}_{n2}\right)\cdot {d}_{u}+{d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)-{d}_{u}\cdot {k}_{f0}\right)\right)}{\left({k}_{f0}-{k}_{f2}-{k}_{n2}\right)}$$
(12)
$${F}_{d}=\left({k}_{f2}+{k}_{n2}\right)\cdot \left(\left(\left(\left({d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)\right)/\left({k}_{f1}+{k}_{n2}\right)\cdot \left({k}_{f2}+{k}_{n2}\right)\right)-\left(\left({k}_{f1}+{k}_{n2}\right)\cdot {d}_{u}+{d}_{0}\cdot {k}_{0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)-{d}_{u}\cdot {k}_{f0}\right)\right)/\left({k}_{f0}-{k}_{f2}-{k}_{n2}\right)\right)+\left({d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)\right)/\left({k}_{f1}+{k}_{n2}\right)\cdot \left({k}_{f2}+{k}_{n2}\right)-{k}_{n2}\cdot \left(\left(\left(\left({d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)\right)/\left({k}_{f1}+{k}_{n2}\right)\cdot \left({k}_{f2}+{k}_{n2}\right)\right)-\left(\left({k}_{f1}+{k}_{n2}\right)\cdot {d}_{u}+{d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)-{d}_{u}\cdot {k}_{f0}\right)\right)/\left({k}_{f0}-{k}_{f2}-{k}_{n2}\right)\right)-\left(\left({d}_{0}\cdot {k}_{f0}+{k}_{f0}\cdot \left(\left({d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)\right)/\left({k}_{f1}+{k}_{n2}\right)\right)\cdot \left({k}_{f2}+{k}_{n2}\right)/\left({k}_{f0}-{k}_{f2}-{k}_{n2}\right)\right)/2-\left({d}_{0}\cdot {k}_{n2}+{k}_{n2}\cdot \left(\left({d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)\right)/\left({k}_{f1}+{k}_{n2}\right)\right)\cdot \left({k}_{f2}+{k}_{n2}\right)/\left({k}_{f0}-{k}_{f2}-{k}_{n2}\right)\right)/2\right)$$
(13)
$${X}_{e}=\left({d}_{0}+\left(\left({d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)\right)/\left({k}_{f1}+{k}_{n2}\right)\right)\cdot \left({k}_{f2}+{k}_{n2}\right)/\left({k}_{f0}-{k}_{f2}-{k}_{n2}\right)\right)/2$$
(14)
$${F}_{e}=\left({k}_{f2}+{k}_{n2}\right)\cdot \left(\left({d}_{0}+\left(\left({d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)\right)/\left({k}_{f1}+{k}_{n2}\right)\right)\cdot \left({k}_{f2}+{k}_{n2}\right)/\left({k}_{f0}-{k}_{f2}-{k}_{n2}\right)\right)/2\right)+\left({d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)\right)/\left({k}_{f1}+{k}_{n2}\right)\cdot \left({k}_{f2}+{k}_{n2}\right)-\left(\left({d}_{0}\cdot {k}_{f0}+{k}_{f0}\cdot \left(\left({d}_{0}\cdot {k}_{f0}-{d}_{0}\cdot \left({k}_{f1}+{k}_{n2}\right)\right)/\left({k}_{f1}+{k}_{n2}\right)\right)\cdot \left({k}_{f2}+{k}_{n2}\right)/\left({k}_{f0}-{k}_{f2}-{k}_{n2}\right)\right)/2\right)$$
(15)
$$X_{f} = \left( {\left( {d_{0} \cdot k_{f0} - d_{0} \cdot \left( {k_{f1} + k_{n2} } \right)} \right)/\left( {k_{f1} + k_{n2} } \right)} \right) \cdot \left( {k_{f2} + k_{n2} } \right)/\left( {k_{f0} - k_{f2} - k_{n2} } \right)$$
(16)
$$F_{f} = 0$$
(17)
Fig. 10
figure 10

Isolated 2 shoulders in 1st and 4th quadrants

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Table 1 Symbols assigned to coordinates of inflection points
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In Eqs. 6–17, d0, du, respectively denote the positive displacement value at which PVFD initiates its friction increasing mechanism during off-centering motion, and absolute value of its maximum deformation capacity, hence identical to x2, x3 in Fig. 5. Kf0, kn2, kf1, kf2 are variables pending to be optimized into one optimal combination within predefined searching range (called optimization variables), kn1 is forced to equate kf0 for purpose of cancelling each other in inner constant friction platform. Shifts of hysteretic geometry of isolated part if arbitrarily varying these 4 optimization variables are exemplified in Fig. 11, each set is obtained via varying only one variable while kee** the rest 3 as constant. 5 indices quantifying fitting quality (Fig. 12) of optimization output, are set as yielding force (Fy), maximum positive reaction forces (Fmax), minimum negative force (Fmin), displacement where Fmin is reached (x4), and maximum energy dissipation in one hysteretic cycle (Emax). Fy, Fmax, and Emax are commonly recognized as significant mechanical parameters of displacement-dependent ED device in terms of anti-earthquake performance, while Fmin and x4 make significant impact the geometric fidelity in 1st and 4th quadrants. Regarding illustration delivered by Fig. 12, Fy is identical to Fpy (Fig. 8(c)), Fmax can be obtained by summation of Fc (Table 1) and Fpy,. Fmin is represented by Fd (Table 1), x4 is represented by Xd (Table 1). In following content, superscript dot notated over these 5 indices is to distinguish the value of super-link with corresponding one from experimental PVFD hysteretic loop.

Fig. 11
figure 11

Geometric shift that one optimization variable monotonically changes

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Fig. 12
figure 12

5 indices quantifying fitting quality

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The area of hysteresis, denoted as Emax, reflecting energy that the super-link can dissipate in one hysteretic cycle, can be calculated via making summation of its 3 decomposed regions given in Fig. 13. Analytical solution of E’max is derived and presented by Eq. 18. Particularly, real values Fy, Fmax, Fmin, x4 can be directly read from hysteretic loop of tested PVFD, Emax should instead be calculated via integrating ‘(Fn+1 + Fn)·dx/2’ over recorded data points series within one complete loading cycle.

$$\begin{aligned} E_{{\max }}^{\prime } = & A_{1} + 2 \cdot (A_{2} + A_{3} + A_{4} + A_{5} ) \\ & = 4 \cdot F_{{pu}} \cdot d_{{pu}} + \left| {\det \left[ {\begin{array}{*{20}c} 1 & {X_{a} } & {F_{a} } \\ 1 & {X_{b} } & {F_{b} } \\ 1 & {X_{c} } & {F_{c} } \\ \end{array} } \right]} \right| + \left| {\det \left[ {\begin{array}{*{20}c} 1 & {X_{a} } & {F_{a} } \\ 1 & {X_{c} } & {F_{c} } \\ 1 & {X_{d} } & {F_{d} } \\ \end{array} } \right]} \right| + \left| {\det \left[ {\begin{array}{*{20}c} 1 & {X_{a} } & {F_{a} } \\ 1 & {X_{d} } & {F_{d} } \\ 1 & {X_{e} } & {F_{e} } \\ \end{array} } \right]} \right| + \left| {\det \left[ {\begin{array}{*{20}c} 1 & {X_{a} } & {F_{a} } \\ 1 & {X_{e} } & {F_{e} } \\ 1 & {X_{f} } & {F_{f} } \\ \end{array} } \right]} \right| \\ \end{aligned}$$
(18)
Fig. 13
figure 13

Geometric partitions of modelled hysteretic loop for area calculation

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5 Summary of optimization workflow

Figure 14 presents the interactive workflow for engineers to generate optimized modelling parameters via simply inputting the mechanical parameters regarding targeted PVFD device that is to be applied into engineering practice.

Fig. 14
figure 14

Interactive workflow for generating optimized modelling parameters

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In particular, the second step requires the searched ranges of 4 variables to be defined, according to either analytical derivations employed for optimization, or visual patterns of hysteresis of equivalent super-link to experimental one, kf1 and kf2, of DFFS component, must be less than largest secant value from 4 shoulder regions of experimental hysteresis, kn2 must be less than kf0. Kfo on pure analytical ground, can be arbitrary adopted up to infinite, because it will be cancelled out by kn1 (identical to kf0) of NSMLS component. However, larger searching range defined for kf0 will not only consume more computational efforts, but also cause convergence difficulty if being modelled in SAP2000. Based on above considerations, ground of all 4 searching ranges can be set as 0, upper bound of searching range of kf0, is suggested to be set no more than twice of ke from experimental hysteresis; upper bound of kn2 can be set identical to kf0,; upper bound of kf1 can be set as secant value of 1st quadrant shoulder region from experimental hysteresis, and lower bound of kf2 can be set as negative of value set to kf1.

6 Example of optimization work and numerical accuracy assessment

6.1 Optimization for modelling parameters of tested specimen

To demonstrate an application of using proposed optimization method for fast determining the optimal parameters of components assembled to equivalently simulate nonlinear behavior of butterfly-hysteresis PVFD, numerical modelling and analysis of such super-link connecting a SDoF, within SAP2000 environment, has been conducted. Property definition of 3 link components assembling into the super-link, are for instance displayed in Fig. 15. Displacement load imposed to SDoF model is set in harmonic pattern with amplitude of 0.03 m identical to the maximum displacement capacity of tested PVFD. Nonlinear direct integration method is employed for computation with no additional dam** defined, time step for which is set as sufficiently fine for catching desired nonlinear behavior of super-link. Parameters from hysteresis of tested PVFD specimen are referred as optimization input (as constants) (Table 2). For illustration purpose, in comparison to experimental one, 16 hysteresis graphically presented in "Appendix A", are yielded from SDoF assigned super-link in various parameter combinations, each model in collection differs from the others by shifted error ratio predefinitions hence completely different output modelling parameters. Table 3 gives the error ratio predefinition and output optimization variables for all trials that correspond to yielded hysteresis given in "Appendix A" by same order. Particularly, error ratio ‘Err_Fmax’ is employed to judge how well F`max can match Fmax, which is not predefined but output by each turn of running code instead.

Fig. 15
figure 15

User interface of defining property for 3 link components in SAP2000

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Table 2 Optimization input (as constants) regarding experimentally output hysteretic loop of tested specimen
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Table 3 Optimization outputs upon configured multiple combinations of error ratio predefinitions
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Because of the geometric difference between the real hysteresis of PVFD and equivalently modelled one, there will yield no solution if input 4 error ratio predefinitions (Err_Fy, Fmin, xm, Emax) and 1 output judging ratio (Err_Fmax) were all to be 0. In another words, the optimization process involves personal engineering judgement, any one larger of 4 input error ratio predefinitions will leads to smaller output of F`max hence larger absolute value of Err_Fmax, and vice versa. Therefore, engineering judgement should be employed to determine those relatively more interested variables according to the physical meanings they represent, then to diminish allowable error ratios for interested those, error ratios of the rest can hence be allowed to be adjusted incrementally until yielding a solution.

Subject to judgement from engineering perspective in this case, indices Emax, Fy, and the post-evaluated Fmax, are key mechanical parameters as for added-structure-dam** use, in terms of its energy dissipation capacity and dam** force magnitude. X4 is relatively less significant but will impact small-displacement behavior if deviate too much from experimental value. Comparing to mentioned 3 indices, Fmin is the least impactful one out of all 5 ones. Therefore, the optimized parameter set for modelling super-link equivalent to tested PVFD product, is determined to be the one having relatively smaller error ratios in Fy, x4, Emax, while adjusting error ratio of Fmin to run code until yielded F`max satisfies to have small enough error ratio Err_Fmax. Error ratio predefinitions finally justified as optimal for tested PVFD, are given in Table 4, while output modelling parameters are given in Table 5. Hysteretic loop yielded by the super-link modelled in SAP2000 configured with output parameters (Table 5), is presented in Fig. 16 against experimental one.

Table 4 4 error ratio predefinitions under acceptance judgement
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Table 5 Modelling parameters for 3 link components output by running coded algorithm
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Fig. 16
figure 16

Hysteresis of super-link defined by product-wise optimized parameters

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As Fig. 16 depicts, the hysteretic loop of modelled SDoF system assigned with super-link assembly defined by output optimized modelling parameters, gives close match to the one yielded from experiment. Drawback can however be observed at centering motion stage (negative stiffness shoulders), the secant value of numerical one is noticeably smaller than experimental one, which is as expected according to calculated Err_Fmin of 15% (Table 4). F´max reaches 5.26 × 105 N, which is well below 5% error ratio to experimental one (Fmax = 5.45 × 105 N) and can be deemed as acceptable.

6.2 Configuration of solid FEA model

The equivalent modelling approach for butterfly-hysteresis PVFD is digitally tested under SAP2000 environment, yielded F-X relationship meets expectation and sits strictly within error-ratio-confined numeral domain. However, whether the numerical accuracy of modelling PFVD from proposed super-link assembly drops in commonly acceptable level while conducting practical structural simulation using structural FEA (in this research, SAP2000), remains uncertain. To verify whether the equivalent modelling approach achieved in SAP2000 can give desired accuracy, another digital verification work has been conducted via setting a comprehensive solid FEA model simulating tested PVFD, utilizing Abaqus CAE software.

As shown in Fig. 17, according to the geometry and physical property of tested PVFD product, a solid FEA model has been built in Abaqus CAE. Q355 steel is assigned to clamp board and middle slider while material property of a patented highly compacted layered polymer is assigned to friction panel part. Tie constraint is assigned onto the interface of steel clamp board to friction panel. Because the nonlinearity of PVFD is purely induced by friction mechanism, materials of both steel and polymer are defined with isotropic elasticity only. According to product specification, Young’s modules are set as 2 × 1011 Pa and 4 × 1010 Pa for steel and polymer material, respectively. Poisson ratio for steel is set as 0.3 and 0.28 for polymer material. Mass density for both is assigned unity because the analysis is to be quasi-static and not accounting for gravity, so that mass induced inertia force and gravity can be excluded. Analytical section for all parts is defined as solid homogenous. In terms of the two-sided friction face, hard contact is defined for its normal behavior, penalty friction formulation is employed to catch its tangential sliding behavior with friction coefficient 0.62 and absolute elastic slip** distance 1 mm defined, while the elastic slip** distance cannot be determined directly from neither test output nor product specification hence is adjusted to match the tested non-slip** force–deformation relationship. According to specimen test results, the slip** behavior does not manifest rate-dependent manner within tested cycles, so the friction force is not assigned rate-dependency definition. To accurately catch the process of stiffness varying mechanism, structured mesh is employed and size of which is capped at 0.007 m (less than 0.01 m where the modelled PVFD varies stiffness). Screw fastener is equivalently modelled as linear spring connector with only axial degree of freedom enabled and constrained each end to the outermost surface edge of circular hole that preserved for fastener. Stiffness value for each connector section is defined identical to the one of pre-compressed stacked-spring disk.

Fig. 17
figure 17

Illustration of solid FEA model of tested specimen

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Load is assigned in multi-stepped displacement form on to a reference point that being constrained to lead the motion of middle slider, in correspondence to SDoF simulated in SAP2000. Fix-end boundary condition is then assigned to clamp board’s end on the side opposite to loading with U3 degree of freedom released. Pre-compression force acting to fastener is defined as connector force exerted in multi-stepped ramp pattern prior to and propagated in displacement loading process.

Like other categories of structural ED system, PVFD product aims to dissipate earthquake energy input to structure, so the uncertainty of ground motion should be considered while assessing performance of ED system. Inspired by approach for such digital testing24, to account such uncertainty, ground motions of 4 well-known big earthquake events occurring in 3 different regions, have been selected from PEER ground motion record database [24], as direct displacement loading function (integrated twice from original acceleration into displacement form) to compel PVFD modelled in both SAP2000 and Abaqus, instead of using simplistic harmonic function. Figure 18 displays these 4 waves, maximum amplitudes of which are all normalized and scaled to match the maximum displacement capacity of tested PVFD.

Fig. 18
figure 18

Displacement motion input (scaled): a Chi-Chi earthquake; b Kobe earthquake; c Loma Prieta earthquake; d Northridge earthquake

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6.3 Result and discussion

Response time history extracted from analysis result is given (Fig. 19) in comparative pattern of SAP2000’s SDoF against Abaqus’s solid FEA.

Fig. 19
figure 19

Comparison of Response time history yielded from SAP2000 SDoF and Abaqus solid FEA model

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The comparisons above manifest very close match in each pair. Therefore, the feasibility and accuracy of applying such equivalent modelling approach to simulate the peculiar nonlinear behavior of butterfly-hysteresis PVFD in engineering practice, is hereby validated in digital way.

7 Conclusions

PVFD has been developed by many researchers so far, which is demonstrated as possessing improved anti-earthquake performance comparing to traditional FD. To simulate its peculiar nonlinear behavior including negative stiffness, several methods have been proposed by researchers, the numerical accuracy and applicability however vary, nonetheless the threshold of applying of PVFD in real engineering practice remains higher than the other general link elements that are widely built in popular structural modelling and analysis software. In this research, an innovative equivalent modelling approach addressing such difficulty has been developed comprehensively. In advance, a product of PVFD has been developed, manufactured, and tested, which features butterfly-pattern hysteretic behavior.

An equivalent approach for modelling the peculiar hysteretic behavior of butterfly-hysteresis PVFD that assembles 3 general link elements in parallel into a super-link element, has been proposed and numerically tested in SAP2000. Hysteresis yielded from this super-link under displacement loading gives close match to the tested one.

An optimization method has been developed, via which to yield modelling parameters best fitting targeted butterfly-hysteresis PVFD product within predefined searching ranges and error ratios. To enable most engineers to apply this method in engineering practice with ease, a coded algorithm has been written, which can automate entire workflow producing product-wise modelling parameters for defining ‘super-link’ element, while being optimized to meet customizable criteria that judges fitting rationality and fidelity. An example of applying such workflow for tested butterfly-hysteresis PFVD specimen has been conducted, yielded super-link element meets predefined optimization criteria as expected. Only limited human intervention is required throughout whole process.

Numerical accuracy of using assembled super-link element has been verified by means of digital comparative work. In the work, generic 3D solid FEA model has been built in Abaqus that in real physical detail simulates tested PVFD device. According to comparison made to hysteresis yielded by super-link in SAP2000 and Abaqus model, under several displacement loadings accounting for random feature of natural ground motions, a very close match is achieved of using these 2 modelling approaches. Therefore, the numerical accuracy of modelling hysteretic behavior of butterfly-hysteresis PVFD using equivalent super-link has been verified.

Although the equivalent approach modelling the peculiar nonlinear behavior of PVFD is straightforward for both understanding and practicing, with acceptable error rate in terms of key parameters and digitally verified good accuracy, the work of validation and verification still need to be undertaken in other ways. In future research, it is recommended to use comprehensive experimental tests to sufficiently verify the accuracy and effectiveness of proposed equivalent modelling approach. As for more detailed comparison and validation purpose, this approach is suggested to be digitally tested within different structural analysis software with 3 basic link elements built-in, or conducting same work by means of investigating the direct formula of the hysteretic model then programing which into analysis software that supports post development functionality in terms of defining new hysteretic rule.