1 Introduction

Seismic isolation is nowadays widely used across the world as one of the most effective techniques for the protection of buildings and bridges from earthquakes. By introducing a layer of low lateral stiffness devices between the structure and the foundation, base isolation has the effect of reducing the earthquake-induced forces in the superstructure. The seismic devices that are most commonly adopted are basically of two types, (1) rubber bearings and (2) sliding isolators. Extensive experimental and analytical work has been carried out to investigate and simulate the behavior of these devices. For a comprehensive review of the literature on modelling and analysis of rubber bearings and sliding isolators, the reader is referred to recent work by Markou et al. [1] and Calvi and Calvi [2]. With a few exceptions [3,4,5,6], existing models are generally unidirectional [7,8,9,10,11] and, in most cases, difficult to extend to general bidirectional loading. However, experiments on different types of rubber isolators [3, 12, 13] have shown that coupling effects between orthogonal components of response should be considered when the bearings are subjected to bidirectional loading. The aim of the present paper is to develop a simple, accurate and robust formulation for the 2D nonlinear dynamic analysis of base isolation systems, and to provide a basis for the dynamic identification of these systems under any kind of environmental and engineered excitation, including earthquake ground motions and release tests. To this end, a biaxial bilinear model is used in this work to characterize the behavior of the isolators. Nonlinear structural mechanics problems in general, as those related to dynamic analysis of base-isolated structures, are usually analyzed using displacement-based strategies where time integration methods, generally coupled with an iterative procedure of the Newton type, are used to solve the nonlinear equilibrium equations [14,15,16]. A different approach is taken in this paper where the computation in each time increment is cast as a mathematical program. Classic examples of mathematical programming include convex optimization, linear complementarity problems, and conic optimization (or complementarity over cones) [17,18,19]. By formulating the incremental state update problem as a mathematical program, the following attractive features may be exploited: (1) algorithms and solvers exist that can be invoked for the solution of a particular mathematical program, (2) these algorithms have excellent convergence characteristics, (3) questions about whether the problem is well-posed, and about the existence and uniqueness of solutions may be explored and answered. The use of mathematical programs for nonlinear structural analysis was first proposed in the seventies by Maier et al. [20, 21]. More recently, Sivaselvan et al. [22] developed an optimization-based algorithm for collapse simulation of large-scale structural systems. Sivaselvan [23], later developed a formulation for the nonlinear dynamic analysis of framed structures with softening plastic hinges based on casting the incremental state update as a complementarity problem. Oliveto and Sivaselvan [24] then extended such formulation to the dynamic analysis of tensegrity structures. In this work, the incremental state update problem is formulated as a convex minimization problem. The constraints of the optimization problem are yield functions representing interaction between the two horizontal components of isolator response. The organization of the paper is as follows. The equations of motion of a base-isolated building and the constitutive behavior used to model the isolators are illustrated in Sect. 2. Next, in Sect. 3, the governing equations of the spatially discretized base isolation system are discretized in time and their solution is set up as a convex optimization problem. Finally, in Sect. 4, a real-world base isolation system is introduced and static and dynamic numerical simulations are carried out to illustrate the proposed approach.

2 Modeling of base isolation systems

The equations of motion of a base isolation system may be written as:

$${\mathbf{M\dot{v}}}_{0} + {\mathbf{Cv}}_{{\mathbf{0}}} + {\mathbf{H}}^{T} {\mathbf{F}} = {\mathbf{P}}$$
(1)

where F ∈ \({\mathbb{R}}^{2N}\) is the vector of shear forces in the isolators, N being the total number of isolators in the system, v0 ∈ \({\mathbb{R}}^{{N_{DOF} }}\) is the vector of velocities at the NDOF degrees of freedom of the base isolation system, H∈ \({\mathbb{R}}^{{2N \times N_{DOF} }}\) is the transformation matrix that relates displacements, velocities, and accelerations (u, v, and a) of the isolators, to u0, v0, and a0, the global NDOF degrees of freedom of the system, M ∈ \({\mathbb{R}}^{{N_{DOF} \times N_{DOF} }}\) is the mass matrix, C ∈ \({\mathbb{R}}^{{N_{DOF} \times N_{DOF} }}\) is the damping matrix, and P ∈ \({\mathbb{R}}^{{N_{DOF} }}\) is the vector of external forces.

The constitutive behavior of the isolators is herein formulated using the Generalized Standard Material Framework [25, 26]. Representations of this kind apply to any dynamic model of a structural system, or component, whose constitutive behavior can be derived from stored energy and dissipation functions [22]. In this work, the constitutive equations are described in terms of biaxial shear forces in the isolators. For a generic isolator, i, we have Fi = (Fi1, Fi2)T. In the following subsections, we characterize the constitutive behavior of high damping rubber bearings (HDRBs) and low friction sliding bearings (LFSBs). A hybrid base isolation system (HBIS) comprising these two types of isolators will be the subject of a series of numerical simulations presented in a later section.

2.1 High damping rubber bearings

The shear behavior of the HDRBs is herein characterized by extending to two dimensions the bilinear model with kinematic hardening shown in Fig. 1. This kind of a model, tuned according to the maximum strain amplitude of interest, is widely adopted in practice for the analysis and design of HDRBs [27]. Using the generalized material formalism, we define the following form of complementary stored energy:

Fig. 1
figure 1

1D constitutive model for HDRBs

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$$\psi^{c} \left( {{\mathbf{F}}_{i} ,{\varvec{\upzeta}}_{i} } \right) = \frac{1}{2}{\mathbf{F}}_{i}^{T} {\mathbf{A}}_{Ri} {\mathbf{F}}_{i} + \frac{1}{2}{\varvec{\upzeta}}_{i}^{T} {\mathbf{A}}_{hi} {\varvec{\upzeta}}_{i}$$
(2)

where ARi = diag(1/k0,1/k0) is the matrix of elastic compliances, Ahi = diag(1/kh,1/kh) is the matrix of compliances related to hardening, and ζi denotes the internal variables for plasticity. Moreover, the following complementary dissipation function is used:

$$\phi^{c} \left( {{\mathbf{F}}_{i} ,{\varvec{\upzeta}}_{i} } \right) = \chi_{E} \left( {{\mathbf{F}}_{i} ,{\varvec{\upzeta}}_{i} } \right)$$
(3)

where χE is the indicator function of the elastic region given by the convex set

$$E = \left\{ {\left. {\left( {{\mathbf{F}}_{i} ,{\varvec{\upzeta}}_{i} } \right)} \right|\varphi \left( {{\mathbf{F}}_{i} ,{\varvec{\upzeta}}_{i} } \right) < 0} \right\}$$
(4)

and φ is a yield function. The indicator [28, 29] function, χE, is defined as

$$\chi_{E} \left( {\mathbf{x}} \right) = \left\{ {\begin{array}{*{20}c} 0 & {{\text{if}}\,\quad{\mathbf{x}} \in E} \\ \infty & \quad {\text{otherwise}} \\ \end{array} } \right.$$
(5)

In the case of one-dimensional kinematic hardening, as shown in Fig. 1, complementary stored energy, ψc, and yield function, φ, are given by

$$\psi^{c} \left( {F,\zeta } \right) = \frac{1}{2}\frac{{F^{2} }}{{k_{0} }} + \frac{1}{2}\frac{{\zeta^{2} }}{{k_{h} }}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\varphi \left( {F,\zeta } \right) = \left| {F - \zeta } \right| - F_{y}$$
(6)

We note that the following important relationship holds:

$$\frac{1}{{k_{0} }} + \frac{1}{{k_{h} }} = \frac{1}{{k_{1} }}$$
(7)

Yield functions accounting for biaxial shear interaction are presented in a later section.

2.2 Low friction sliding bearings

The behavior of the LFSBs is defined by extending to two dimensions the Coulomb friction model shown in Fig. 2. We note that a finite, albeit large, initial stiffness, k0c, is introduced in the friction model. Such a stiffness is large enough so as not to change the physical behavior of the sliding isolators and it guarantees strict convexity of the optimization problem stated in a following section. For the sliding isolators, the following form of complementary stored energy and dissipation functions are used:

Fig. 2
figure 2

Constitutive model for LFSBs

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$$\psi^{c} \left( {{\mathbf{F}}_{i} } \right) = \frac{1}{2}{\mathbf{F}}_{i}^{T} {\mathbf{A}}_{Si} {\mathbf{F}}_{i} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\phi^{c} \left( {{\mathbf{F}}_{i} } \right) = \chi_{E} \left( {{\mathbf{F}}_{i} } \right)$$
(8)

where ASi = diag(1/k0c,1/k0c), and χE is in this case the indicator function of the convex set:

$$E = \left\{ {\left. {\left( {{\mathbf{F}}_{i} } \right)} \right|\varphi \left( {{\mathbf{F}}_{i} } \right) < 0} \right\}$$
(9)

When one-dimensional ideal plastic behavior is considered, as shown in Fig. 2, we have:

$$\psi^{c} \left( F \right) = \frac{1}{2}\frac{{F^{2} }}{{k_{0c} }}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\varphi \left( F \right) = \left| F \right| - F_{c}$$
(10)

3 Proposed convex optimization procedure

In this section, the equations governing the behavior of a hybrid base isolation system composed of HDRBs and LFSBs are presented. Time discretization and manipulation of these equations is shown to lead to the formulation of a convex optimization problem for the incremental state update.

3.1 Governing equations

The governing equations of a spatially discretized HBIS model can be obtained as the Euler–Lagrange equations of a generalized Hamilton’s principle [30] and written as

$$\begin{aligned} &{\mathbf{M\dot{v}}}_{0} + {\mathbf{Cv}}_{{\mathbf{0}}} + {\mathbf{H}}^{T} {\mathbf{F}} = {\mathbf{P}} \hfill \\ &\frac{d}{dt}\left( {\partial_{{\mathbf{F}}} \psi^{c} \left( {{\mathbf{F}},{\varvec{\upzeta}}} \right)} \right) + \partial_{{\mathbf{F}}} \phi^{c} \left( {{\mathbf{F}},{\varvec{\upzeta}}} \right) - {\mathbf{Hv}}_{0} = {\mathbf{0}} \hfill \\ &\frac{d}{dt}\left( {\partial_{{\varvec{\upzeta}}} \psi^{c} \left( {{\mathbf{F}},{\varvec{\upzeta}}} \right)} \right) + \partial_{{\varvec{\upzeta}}} \phi^{c} \left( {{\mathbf{F}},{\varvec{\upzeta}}} \right) = {\mathbf{0}} \hfill \\ \end{aligned}$$
(11)

The first of Eq. (11), already presented in Sect. 2, is the classic equation of motion expressing momentum conservation, the second expresses compatibility of deformation in the isolators and the third represents the evolution of the constitutive internal variables. ζ∈ \({\mathbb{R}}^{{N_{r} }}\) is the vector of conjugates of the plastic strains in the HDRBs, Nr being the number of HDRBs in the system. When specialized to the complementary stored energy, ψc, given in Eqs. (2) and (8)1, (11) may be written as

$$\begin{aligned} &{\mathbf{M\dot{v}}}_{0} + {\mathbf{Cv}}_{{\mathbf{0}}} + {\mathbf{H}}^{T} {\mathbf{F}} = {\mathbf{P}} \hfill \\ &{\mathbf{A\dot{F}}} + \partial_{{\mathbf{F}}} \phi^{c} \left( {{\mathbf{F}},{\varvec{\upzeta}}} \right) - {\mathbf{Hv}}_{0} = {\mathbf{0}} \hfill \\ &{\mathbf{A}}_{h} {\dot{\mathbf{\zeta }}} + \partial_{{\varvec{\upzeta}}} \phi^{c} \left( {{\mathbf{F}},{\varvec{\upzeta}}} \right) = {\mathbf{0}} \hfill \\ \end{aligned}$$
(12)

where A∈ \({\mathbb{R}}^{2N \times 2N}\) is the diagonal matrix of elastic compliances in all the isolators, and Ah∈ \({\mathbb{R}}^{{2N_{r} \times 2N_{r} }}\) is the diagonal matrix of compliances related to hardening in the HDRBs. In the next section, the governing equations are discretized in time leading to the formulation of a convex minimization problem.

3.2 Time discretization and optimization problem

Following [22], Eq. (12) are discretized in time using central differences and the midpoint rule. We note that in the case of linear elasticity this choice of discretization leads to Newmark’s integration method with constant average acceleration [30]. Discretizing the first of Eq. (12) gives:

$${\mathbf{M}}\left( {\frac{{{\mathbf{v}}_{0}^{n + 1} - {\mathbf{v}}_{0}^{n} }}{\Delta t}} \right) + {\mathbf{C}}\left( {\frac{{{\mathbf{v}}_{0}^{n + 1} + {\mathbf{v}}_{0}^{n} }}{2}} \right) + {\mathbf{H}}^{T} \left( {\frac{{{\mathbf{F}}^{n + 1} + {\mathbf{F}}^{n} }}{2}} \right) = \frac{{{\mathbf{P}}^{n + 1} + {\mathbf{P}}^{n} }}{2}$$
(13)

where Δt is the time increment and superscripts denote discrete times. Equation (13) may then be rearranged leading to:

$${\mathbf{v}}_{0}^{n + 1} = {\bar{\mathbf{M}}}^{ - 1} \left( {{\mathbf{b}}_{1} - \frac{\Delta t}{2}{\mathbf{H}}^{T} {\mathbf{F}}^{n + 1} } \right)$$
(14)

where \({\bar{\mathbf{M}}} = {\mathbf{M}} + \frac{\Delta t}{2}{\mathbf{C}}\) and:

$${\mathbf{b}}_{1} = \left( {{\mathbf{M}} - \frac{\Delta t}{2}{\mathbf{C}}} \right){\mathbf{v}}_{0}^{n} + \frac{\Delta t}{2}\left( {{\mathbf{P}}^{n + 1} + {\mathbf{P}}^{n} } \right) - \frac{\Delta t}{2}{\mathbf{H}}^{T} {\mathbf{F}}^{n}$$
(15)

We now discretize the second of Eqs. (12), which after rearranging and substituting Eq. (14) for \({\mathbf{v}}_{0}^{n + 1}\) gives:

$$\left( {{\mathbf{A}} + \frac{{\Delta t^{2} }}{4}{\mathbf{H\bar{M}}}^{ - 1} {\mathbf{H}}^{T} } \right){\mathbf{F}}^{n + 1} + \frac{\Delta t}{2}\partial_{{\mathbf{F}}} \phi^{c} \left( {{\mathbf{F}}^{n + 1} ,{\varvec{\upzeta}}^{n + 1} } \right) = {\mathbf{b}}_{2}$$
(16)

where

$${\mathbf{b}}_{2} = {\mathbf{AF}}^{n} + \frac{\Delta t}{2}{\mathbf{H\bar{M}}}^{ - 1} {\mathbf{b}}_{1} + \frac{\Delta t}{2}{\mathbf{Hv}}_{0}^{n} - \frac{\Delta t}{2}\partial_{{\mathbf{F}}} \phi^{c} \left( {{\mathbf{F}}^{n} ,{\varvec{\upzeta}}^{n} } \right)$$
(17)

Finally, by discretizing the third of Eqs. (12) we get:

$${\mathbf{A}}_{h} {\varvec{\upzeta}}^{n + 1} + \frac{\Delta t}{2}\partial_{{\varvec{\upzeta}}} \phi^{c} \left( {{\mathbf{F}}^{n + 1} ,{\varvec{\upzeta}}^{n + 1} } \right) = {\mathbf{b}}_{3}$$
(18)

where

$${\mathbf{b}}_{3} = {\mathbf{A}}_{h} {\varvec{\upzeta}}^{n} - \frac{\Delta t}{2}\partial_{{\varvec{\upzeta}}} \phi^{c} \left( {{\mathbf{F}}^{n} ,{\varvec{\upzeta}}^{n} } \right)$$
(19)

It is easy to recognize nonlinear Eqs. (16) and (18) as the first-order necessary optimality conditions of the following minimization problem:

$$\begin{aligned} &\mathop {\hbox{min} }\limits_{{{\mathbf{F}},\,{\varvec{\upzeta}}}} \,\,\frac{1}{2}{\mathbf{F}}^{T} \left( {{\mathbf{A}} + \frac{{\Delta t^{2} }}{4}{\mathbf{H\bar{M}}}^{ - 1} {\mathbf{H}}^{T} } \right){\mathbf{F}} \\ & \quad + \frac{1}{2}{\varvec{\upzeta}}^{T} {\mathbf{A}}_{h} {\varvec{\upzeta}} + \frac{\Delta t}{2}\phi^{c} \left( {{\mathbf{F}},{\varvec{\upzeta}}} \right) - {\mathbf{b}}_{2}^{T} {\mathbf{F}} - {\mathbf{b}}_{3}^{T} {\varvec{\upzeta}} \end{aligned}$$
(20)

Given that the complementary stored energy and dissipation functions are convex, and that matrix \({\bar{\mathbf{M}}}\) is positive definite, (20) represents a convex minimization problem and, as such, the first-order optimality conditions are also sufficient for the solution to be a global minimum. Since the dissipation potential, ϕc, is represented by indicator functions of the elastic region of the isolators, minimization problem (20) may be restated as:

$$\begin{aligned} & \mathop {\hbox{min} }\limits_{{{\mathbf{F}},\,{\varvec{\upzeta}}}} \,\,\frac{1}{2}{\mathbf{F}}^{T} \left( {{\mathbf{A}} + \frac{{\Delta t^{2} }}{4}{\mathbf{H\bar{M}}}^{ - 1} {\mathbf{H}}^{T} } \right){\mathbf{F}} + \frac{1}{2}{\varvec{\upzeta}}^{T} {\mathbf{A}}_{h} {\varvec{\upzeta}} - {\mathbf{b}}_{2}^{T} {\mathbf{F}} - {\mathbf{b}}_{3}^{T} {\varvec{\upzeta}} \hfill \\ & {\text{subject}}\,{\text{to}}\,\,\,\,\,\,\varphi \left( {{\mathbf{F}},{\varvec{\upzeta}}} \right) \le 0 \hfill \\ \end{aligned}$$
(21)

where the yield functions, φ, of the isolators appear as inequality constraints and the corresponding Lagrange multipliers are the incremental plastic strains. The constraints represent the convex elastic region and, therefore, (21) is an inequality constrained convex minimization problem. Several strategies and algorithms exist for the solution of this kind of convex quadratic programming problems [17, 18]. Particularly appealing are interior-point methods as they are relatively easy to implement and quite efficient on certain types of problems. They were first developed for large linear programming problems in the 1980s and quickly became strong competitors of the simplex method. Another important class of methods for constrained minimization attempt to solve the original constrained problem through a sequence of unconstrained subproblems. These are the penalty, barrier and augmented Lagrangian methods. A minimization algorithm based on an augmented Lagrangian approach [17] and Newton’s method was developed for collapse simulations of large-scale structures by Sivaselvan et al. [22].

Conventional displacement-based approaches such as Newton’s method based on the tangent stiffness matrix require some sort of iteration procedure that is not always guaranteed to converge. On the other hand, one advantage of relying on convex optimization procedures combined with interior point algorithms is that convergence is theoretically guaranteed [19]. Moreover, Sivaselvan et al. [22] have shown that the computational effort required for convergence of the minimization procedure does not increase significantly, even with a considerable increase of the time increment. As a first step, in this paper we discuss the use of readily available MATLAB [31] minimization functions quadprog and fmincon for the solution of (21). The former solves convex quadratic programming problems with linear inequality constraints using a predictor–corrector interior-point algorithm proposed by Mehrotra [32]. The latter minimizes nonlinear functions subjected to nonlinear inequality constraints using an interior-point algorithm based on the logarithmic barrier method [33,34,35].

3.3 Yield functions and constraints

The purpose of this subsection is to show how different interaction diagrams can be considered using the proposed formulation and how these can be implemented using Matlab functions quadprog and fmincon [31]. Minimization function quadprog can be used when the interaction diagrams may be approximated by convex polygons, while function fmincon must be used when considering convex nonlinear interaction domains. When using functions quadprog and fmincon, we set the optimization problem (21) in the following form:

$$\begin{aligned} & \mathop {\hbox{min} }\limits_{{\mathbf{x}}} \,\,\frac{1}{2}{\mathbf{x}}^{T} {\mathbf{Qx}} + {\mathbf{f}}^{T} {\mathbf{x}} \hfill \\ & {\text{subject}}\,{\text{to}}\,\,\,\,\,\,\left\{ {\begin{array}{*{20}c} {{\mathbf{A}}_{in} {\mathbf{x}} \le {\mathbf{b}}} & {{\text{linear}}\,{\text{constraints}}\, ( {\text{quadprog)}}} \\ {{\mathbf{c}}\left( {\mathbf{x}} \right) \le {\mathbf{0}}} & {{\text{nonlinear}}\,{\text{constraints}}\, ( {\text{fmincon)}}} \\ \end{array} } \right. \hfill \\ \end{aligned}$$
(22)

where

$${\mathbf{x}} = \left[ {\begin{array}{*{20}c} {\mathbf{F}} \\ {\varvec{\upzeta}} \\ \end{array} } \right]\,\,\,\,\,\,\,\,{\mathbf{Q}} = \left[ {\begin{array}{*{20}c} {{\mathbf{A}} + \frac{{\Delta t^{2} }}{4}{\mathbf{H\bar{M}}}^{ - 1} {\mathbf{H}}^{T} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{A}}_{h} } \\ \end{array} } \right]\,\,\,\,\,\,\,\,{\mathbf{f}} = \left[ {\begin{array}{*{20}c} { - {\mathbf{b}}_{1} } \\ { - {\mathbf{b}}_{2} } \\ \end{array} } \right]$$
(23)

The linear and nonlinear constraints in (22) represent the yield functions for the isolators. Different forms of biaxial shear interaction may be considered such as those plotted in Fig. 3. The square diagram (a) assumes no interaction whereas a more realistic shear interaction diagram for circular seismic isolation bearings is given by (d). Diagrams (b) and (c) are represented to illustrate how by increasing its number of sides, a polygon interaction diagram may be used to approximate the actual circular one. For large scale problems, it could be convenient to select the algorithm to be used based on a tradeoff between accuracy and computational cost.

Fig. 3
figure 3

Bidirectional shear force interaction diagrams

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The constraint inequalities related to biaxial shearing of a single rubber isolator are given below. For the sake of illustration, biaxial shear interaction diagrams (b) and (d) of Fig. 3 are considered. In the first case, the constraints are linear and given by:

$$\begin{aligned} \,\,\,F_{1} - \zeta_{1} + F_{2} - \zeta_{2} \le F_{y} \,\,\,\,\,\,\,\,\,\,\,\,\,\, - F_{1} + \zeta_{1} + F_{2} - \zeta_{2} \le F_{y} \hfill \\ - F_{1} + \zeta_{1} - F_{2} + \zeta_{2} \le F_{y} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,F_{1} - \zeta_{1} - F_{2} + \zeta_{2} \le F_{y} \hfill \\ \end{aligned}$$
(24)

These constraints can be written in the form (22) by setting:

$${\mathbf{A}}_{in} = \left[ {\begin{array}{*{20}c} 1 & 1 & { - 1} & { - 1} \\ { - 1} & 1 & 1 & { - 1} \\ { - 1} & { - 1} & 1 & 1 \\ 1 & { - 1} & { - 1} & 1 \\ \end{array} } \right]\,\,\,\,\,\,\,\,{\mathbf{x}} = \left[ {\begin{array}{*{20}c} {F_{1} } \\ {F_{2} } \\ {\zeta_{1} } \\ {\zeta_{2} } \\ \end{array} } \right]\,\,\,\,\,\,\,\,{\mathbf{b}} = F_{y} \left[ {\begin{array}{*{20}c} 1 \\ 1 \\ 1 \\ 1 \\ \end{array} } \right]$$
(25)

In the second case, the one quadratic constraint is given by:

$$c\left( {F_{1} ,F_{2} ,\zeta_{1} ,\zeta_{2} } \right) = \left( {F_{1} - \zeta_{1} } \right)^{2} + \left( {F_{2} - \zeta_{2} } \right)^{2} - F_{y}^{2} \le 0$$
(26)

3.4 Proposed algorithm

The algorithm used for implementation of the proposed procedure is summarized in Procedure 1. In Sect. 4, numerical examples are presented, to illustrate the working of the numerical algorithm.

3.5 Static displacement-controlled simulations

In the case of static displacement-controlled simulations as those presented in Sect. 4.2, the system of governing Eq. (12) may be reduced to:

$$\begin{aligned} & {\mathbf{A\dot{F}}} + \partial_{{\mathbf{F}}} \phi^{c} \left( {{\mathbf{F}},{\varvec{\upzeta}}} \right) - {\dot{\mathbf{u}}} = {\mathbf{0}} \hfill \\ & {\mathbf{A}}_{h} {\dot{\mathbf{\zeta }}} + \partial_{{\varvec{\upzeta}}} \phi^{c} \left( {{\mathbf{F}},{\varvec{\upzeta}}} \right) = {\mathbf{0}} \hfill \\ \end{aligned}$$
(27)

where F = (F1, F2)T, u = (u1, u2)T, A = diag(1/k0, 1/k0), Ah= diag(1/kh, 1/kh). Discretizing Eqs. (27) in time using Backward–Euler gives:

$$\begin{aligned} & {\mathbf{A}}\frac{{{\mathbf{F}}^{n + 1} - {\mathbf{F}}^{n} }}{\Delta t} + \partial_{{\mathbf{F}}} \phi^{c} \left( {{\mathbf{F}}^{n + 1} ,{\varvec{\upzeta}}^{n + 1} } \right) - \frac{{\left( {{\mathbf{u}}^{n + 1} - {\mathbf{u}}^{n} } \right)}}{\Delta t} = {\mathbf{0}} \hfill \\ & {\mathbf{A}}_{h} \frac{{{\varvec{\upzeta}}^{n + 1} - {\varvec{\upzeta}}^{n} }}{\Delta t} + \partial_{{\varvec{\upzeta}}} \phi^{c} \left( {{\mathbf{F}}^{n + 1} ,{\varvec{\upzeta}}^{n + 1} } \right) = {\mathbf{0}} \hfill \\ \end{aligned}$$
(28)

Equations (28) can be rearranged and written as:

figure a

Procedure 1 Time increment algorithm

$$\begin{aligned} {\mathbf{AF}}^{n + 1} + \Delta t\partial_{{\mathbf{F}}} \phi^{c} \left( {{\mathbf{F}}^{n + 1} ,{\varvec{\upzeta}}^{n + 1} } \right) = {\mathbf{b}}_{2} \hfill \\ {\mathbf{A}}_{h} {\varvec{\upzeta}}^{n + 1} + \Delta t\partial_{{\varvec{\upzeta}}} \phi^{c} \left( {{\mathbf{F}}^{n + 1} ,{\varvec{\upzeta}}^{n + 1} } \right) = {\mathbf{b}}_{3} \hfill \\ \end{aligned}$$
(29)

where b2 = AFn+ (un+1-un) = AFn+ Δun and b3 = AhFn. It is easy to recognize nonlinear Eqs. (29) as the first-order necessary optimality conditions of the following minimization problem:

$$\mathop {\hbox{min} }\limits_{{{\mathbf{F}},\,{\varvec{\upzeta}}}} \,\,\frac{1}{2}{\mathbf{F}}^{T} {\mathbf{AF}} + \frac{1}{2}{\varvec{\upzeta}}^{T} {\mathbf{A}}_{h} {\varvec{\upzeta}} + \frac{\Delta t}{2}\phi^{c} \left( {{\mathbf{F}},{\varvec{\upzeta}}} \right) - {\mathbf{b}}_{2}^{T} {\mathbf{F}} - {\mathbf{b}}_{3}^{T} {\varvec{\upzeta}}$$
(30)

which can be restated as:

$$\begin{aligned} & \mathop {\hbox{min} }\limits_{{{\mathbf{F}},\,{\varvec{\upzeta}}}} \,\,\frac{1}{2}{\mathbf{F}}^{T} {\mathbf{AF}} + \frac{1}{2}{\varvec{\upzeta}}^{T} {\mathbf{A}}_{h} {\varvec{\upzeta}} - {\mathbf{b}}_{2}^{T} {\mathbf{F}} - {\mathbf{b}}_{3}^{T} {\varvec{\upzeta}} \hfill \\ & {\text{subject}}\,{\text{to}}\,\,\,\,\,\,\varphi \left( {{\mathbf{F}},{\varvec{\upzeta}}} \right) \le 0 \hfill \\ \end{aligned}$$
(31)

4 Numerical simulations

In this section, use of the proposed convex optimization procedure is illustrated through a series of numerical examples. The first example consists of subjecting one of the high damping rubber bearings of a real-world HBIS to bidirectional shearing under imposed displacement orbits; a second example concerns 2D dynamic analyses of the entire base isolation system subjected to earthquake motion.

4.1 Solarino base isolation system

At the turn of the century, a hybrid base isolation system composed of HDRBs and LFSBs was used for the seismic retrofit of two four-story reinforced concrete buildings in the town of Solarino in Sicily [36]. After the completion of the retrofitting works, one of the buildings was subjected to a set of free vibration tests by sudden release of a statically imposed initial displacement. Extensive structural identification studies [37,38,39,40] and earthquake response simulations [41, 42] were then conducted on the building using one-dimensional analytical and numerical models. In this work, these analyses are extended to the dynamic response of the Solarino base isolation system under two-dimensional excitation.

As shown in Fig. 4, the base isolation system of the Solarino buildings is composed of 12 HDRBs and 13 LFSBs. Figure 5 shows one isolator of each kind as they are mounted in the base isolation system.

Fig. 4
figure 4

Solarino hybrid base isolation system

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Fig. 5
figure 5

Bearings mounted in Solarino base isolation system: a HDRB, b LFSB

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We define a coordinate system with origin in the center of rigidity of the elastomeric bearings. The coordinates, x and y, of each isolator are given in Tables 1 and 2. As illustrated in Fig. 6, let u0 = (u01, u02, ϑ0)T be the vector of global degrees of freedom of the system. Assuming small rotations, the horizontal displacements, ui = (ui1, ui2)T, of generic isolator i may then be expressed as:

Table 1 Coordinates of HDRBs in Solarino base isolation system
Full size table
Table 2 Coordinates of LFSBs in Solarino base isolation system
Full size table
Fig. 6
figure 6

Displacements and forces on generic isolator and at the global degrees of freedom

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$${\mathbf{u}}_{i} = {\mathbf{H}}_{i} {\mathbf{u}}_{0}$$
(32)

where

$${\mathbf{H}}_{i} = \left[ {\begin{array}{*{20}c} 1 & 0 & { - y_{i} } \\ 0 & 1 & {x_{i} } \\ \end{array} } \right]$$
(33)

Owing to the static-kinematic duality established via the principle of virtual work, a similar expression relates the horizontal forces in isolator i, Fi = (Fi1, Fi2)T, to forces in the global degrees of freedom, F0 = (F01, F02, M0)T, that is

$${\mathbf{H}}_{i}^{T} {\mathbf{F}}_{i} = {\mathbf{F}}_{0}$$
(34)

4.1.1 Modeling rubber isolator properties

The mechanical properties of the HDRBs were obtained by fitting a bilinear model to unidirectional shear force–deformation data obtained from displacement controlled harmonic tests performed at the University of Basilicata as part of the DPC-ReLUIS 2014 project [43]. The isolator tested has the following characteristics: total height 169 mm, diameter 500 mm, rubber height (tr) 8 × 12 mm = 96 mm, steel height 3 × 11 mm = 33 mm, end plates 2 × 20 mm = 40 mm. Several tests were performed with a frequency of 0.5 Hz and maximum shear deformation, γmax= umax/tr, varying between 5 and 200%. As shown in Fig. 7, the parameters of the bilinear model used for the simulations were calibrated using the measured cyclic force–deformation curve at γmax= 160%. The identified parameters are: initial elastic stiffness, k0 = 3674.10 kN/m; post-elastic stiffness, k1 = 739.68 kN/m; yield strength Fy = 27.87 kN.

Fig. 7
figure 7

Experimental and fitted force–deformation response of HDRB (γmax = 160%)

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4.1.2 Modeling sliding isolator properties

The dynamic coefficient of friction, μ, for the LFSBs, determined from previous identification studies [2] of the Solarino base isolation system, was taken equal to 0.75%. The Coulomb friction force, Fc = P, on each isolator was then calculated based on its axial load, P, under the seismic mass of the building. These axial forces were computed via static analysis in SAP2000 [44] of a detailed finite element model of the building. For details of the building, the reader is referred to [45]. Axial loads and estimated friction forces on the sliding isolators are given in Table 3. The value used as the artificial initial stiffness for the Coulomb friction model is k0c= 1e6 kN/m.

Table 3 Axial loads, P, and friction forces, Fc, on LFSBs
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4.1.3 Mass matrix and effective earthquake forces

Due to mass eccentricity and small asymmetries in the distribution of the sliding bearings, the Solarino isolation system shown in Fig. 4 is not perfectly symmetric. The mass matrix is given by:

$${\mathbf{M}} = \left[ {\begin{array}{*{20}c} m & 0 & { - S_{x} } \\ 0 & m & {S_{y} } \\ { - S_{x} } & {S_{y} } & {I_{0} } \\ \end{array} } \right]$$
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where m is the total mass of the superstructure, Sx= mey and Sy= mex are the first moments of mass with respect to X and Y respectively, and I0 = Icm+ m(e2x + e2y) is the mass moment of inertia about the vertical axis through the center of rigidity (subscript cm stands for center of mass). The position of the center of mass, along with the values of the total mass, m, and mass moment of inertia about the vertical axis through the center of mass, Icm, were determined through a detailed finite element model of the building developed in SAP2000 and are given in Table 4. The effective earthquake forces are:

Table 4 Mass properties and eccentricity
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$${\mathbf{P}} = - {\mathbf{M}} {{\varvec{i}}_{x}}{ {\ddot{u}}_{gx} } - {{\mathbf{M}}{{\varvec{i}}_{y}} {{\ddot{u}}_{gy}}}$$
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where ügx and ügy are the two orthogonal components of ground motion, EW and NS respectively, and the influence vectors, ix= [1, 0, 0]T and iy= [0, 1, 0]T, represent the displacements of mass m resulting from static application of unit ground displacements, ugx and ugy.

4.2 Example 1: Bidirectional shearing of high damping rubber bearing

In this first example, one single rubber bearing of the Solarino base isolation system was subjected to the bidirectional displacement orbits shown in Fig. 8. Several simulations were conducted using the bidirectional shear interaction diagram of Fig. 3d.

Fig. 8
figure 8

Imposed displacement orbits

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The results in terms of biaxial resisting forces in the bearing are shown in Figs. 9 and 10 for different levels of maximum shear deformation, γmax= 50, 100 and 150%.

Fig. 9
figure 9

Biaxial shear response of HDRB to box orbits

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Fig. 10
figure 10

Biaxial shear response of HDRB to figure eight orbits

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Coupling between components of motion is clear. The numerical simulations show that increasing the displacement in one direction while keeping the displacement in the orthogonal direction fixed (box orbit) affects the shear force in both directions. Moreover, due to interaction between components of motion, the shape of the loops at maximum strain is considerably changed (figure eight orbit) as compared to those obtained under unidirectional loading (Fig. 7). The results of the simulations compare favorably, in a qualitative sense, to those obtained from bidirectional tests on similar bearings performed as part of the Caltrans Protective Systems Project at the University of California, Berkeley [3, 46,47,48]. Similar trends can also be seen in experiments by Abe et al. [12] and Yamamoto et al. [13].

4.3 Example 2: earthquake simulations

In a second numerical example, the Solarino base isolation system was subjected to bidirectional earthquake records scaled so that the 5% damped spectral acceleration of the EW component at the fundamental period of the base-isolated building (T = 2.35 s) be equal to that of the design spectrum provided by the seismic regulations at the time of the retrofit [49]. Two Italian acceleration records were considered, one from the L’Aquila 2009 event and another from the Irpinia 1980 earthquake. Characteristics and scaling factors of the ground motions are provided in Table 5. Design spectrum and response spectra of the scaled records are given in Fig. 11.

Table 5 Characteristics and scaling factors of earthquakes selected for analysis (R is the minimum distance from the fault)
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Fig. 11
figure 11

Design spectrum and response spectra of bidirectional ground motions considered: a EW components, b NS components

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4.3.1 Response of the Solarino base isolation system

The simulations were carried out using the interaction diagram of Fig. 3d. The maximum displacements of the bearings were 19.64 cm under the AQK earthquake and 6.61 cm under the STR earthquake. These, both smaller than the maximum permissible displacement of 20 cm specified by the manufacturer, were exhibited by corner bearings, R1 and R12, respectively. We emphasize that the purpose of the simulations was not to assess the performance of the Solarino base isolation system but to illustrate the ability of the proposed formulation to compute the response of the base isolation system under bidirectional earthquake excitation. The bidirectional earthquake response of bearings R1 and R12 is shown in Figs. 12 and 13. Satisfaction of the plastic constraints during the simulations is illustrated in Fig. 14.

Fig. 12
figure 12

Response of HDRB R1 to bidirectional AQK earthquake record

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Fig. 13
figure 13

Response of HDRB R12 to bidirectional STR earthquake record

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Fig. 14
figure 14

Shear interaction diagram and plastic constraint satisfaction: a response of HDRB R1 to bidirectional AQK earthquake record, b response of HDRB R12 to bidirectional STR earthquake record

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5 Concluding remarks

A formulation has been derived for the dynamic analysis of base isolation systems under bidirectional excitation. The force-based approach consists of casting the computation at each time step as a convex minimization problem. Coupling between the two horizontal components of response in the isolators is considered through bidirectional shear force interaction diagrams representing the constraints of the optimization problem. Numerical examples have been carried out to illustrate the approach and to simulate the response of the Solarino hybrid base isolation system to bidirectional ground motions. The proposed formulation can be easily applied to any base isolation system made with isolation devices whose unidirectional shear behavior can be approximated using bilinear models. Inclusion in the current framework of more complex hysteretic models [9, 11], as well as the effects of vertical loading on the bearings and coupling of vertical and horizontal motion, are the subject of ongoing work.