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date: 22 April 2019

# Assessment of Earthquake Performance of Structures by Hybrid Simulation

## Summary and Keywords

With current rapid growth of cities and the move toward the development of both sustainable and resilient infrastructure systems, it is vital for the structural engineering community to continue to improve their knowledge in earthquake engineering to limit infrastructure damage and the associated social and economic impacts. Historically, the development of such knowledge has been accomplished through the deployment of analytical simulations and experimental testing. Experimental testing is considered the most accurate tool by which local behavior of components or global response of systems can be assessed, assuming the test setup is realistically configured and the experiment is effectively executed. However, issues of scale, equipment capacity, and availability of research funding continue to hinder full-scale testing of complete structures. On the other hand, analytical simulation software is limited to solving specific type of problems and in many cases fail to capture complex behaviors, failure modes, and collapse of structural systems. Hybrid simulation has emerged as a potentially accurate and efficient tool for the evaluation of the response of large and complex structures under earthquake loading. In hybrid (experiment-analysis) simulation, part of a structural system is experimentally represented while the rest of the structure is numerically modeled. Typically, the most critical component is physically represented. By combining a physical specimen and a numerical model, the system-level behavior can be better quantified than modeling the entire system purely analytically or testing only a component. This article discusses the use of hybrid simulation as an effective tool for the seismic evaluation of structures. First, a chronicled development of hybrid simulation is presented with an overview of some of the previously conducted studies. Second, an overview of a hybrid simulation environment is provided. Finally, a hybrid simulation application example on the response of steel frames with semi-rigid connections under earthquake excitations is presented. The simulations included a full-scale physical specimen for the experimental module of a connection, and a 2D finite element model for the analytical module. It is demonstrated that hybrid simulation is a powerful tool for advanced assessment when used with appropriate analytical and experimental realizations of the components and that semi-rigid frames are a viable option in earthquake engineering applications.

# Introduction

Recent earthquake events around the world have demonstrated the substantial infrastructure damage that can result from such events and the potential subsequent devastating life and economic consequences. Examples of these ample earthquakes include the 2004 Sumatra earthquake (Mw 9.3) in Indonesia, the 2010 Chile earthquake (Mw 8.8), the 2010 Haiti earthquake (Mw 7.0), the 2011 Tōhoku earthquake in Japan (Mw 9.0), and the 2011 Christchurch earthquake in New Zealand (Mw 6.3). The losses from these earthquakes were on the order of hundreds of billions of dollars with tens or hundreds of thousands of people dead or injured and in some cases over a million displaced as in the Haiti earthquake. In some cases, secondary events resulting from the earthquakes, such as tsunamis, fires, and landslides, can magnify the resulting damage even further. The devastating effects of the earthquakes and their consequences have been highlighted in a recent report published by the Center for Disaster Management and Risk Reduction (CEDIM) in Germany (Daniell & Vervaeck, 2012). The report indicated that in 2011 alone, earthquakes and their consequences, including tsunamis and landslides, caused damages of approximately \$365 billion U.S. dollars with more than half of the damage resulting from the Tōhoku earthquake and the subsequent tsunami. These seismic events and their corresponding losses not only hamper the economic growth of the affected countries, setting their societal development many years back, but also affect other countries that might depend at large on them. For example, the 2011 Tōhoku earthquake in Japan led to a 20% drop in vehicle production in Thailand (World Bank, 2014). Therefore, proper assessment of seismic risk is critical for the development of effective preventive strategies that can be implemented to reduce the potential for earthquake losses and the resulting economic impact, improving the resiliency of both the infrastructure and the community.

Components of seismic risk analysis include hazard quantification, evaluation of exposure in relation to available inventory, and the development of fragilities for the assessment of infrastructure vulnerability. Undoubtedly, infrastructure vulnerability, measured by physical damage, is essential for proper assessment of the earthquake impact as it directly influences life and economical losses.

Traditionally, vulnerability of structural systems to earthquake or any other extreme demands, has been evaluated through either experimental testing of the various components making up the structure or through analytical assessment of structures on both the component and/or system levels. Analytical models are still limited in their applications despite recent advances in computational capabilities. This is because of their inability in some cases to properly capture complex failure modes including for example combined shear and tension failures (Wen & Mahmoud, 2015), local buckling, formations of cracks, and collapse of systems. Other issues include lack of convergence due to the presence of large material inelasticity and material nonlinearity, which are often needed for accurate simulations. On the other hand, various limitations impede the complete use of experimental facilities for testing full structural systems including issues of scale, funding, and availability of proper equipment. However, it is well agreed upon that experimental testing is the most reliable means by which the seismic assessment of structures can be conducted if the above-mentioned limitations can be overcome. There are currently three methods for conducting experimental tests under seismic excitations; namely shake table testing, quasi-static testing, and hybrid simulations. Hybrid simulations can be classified as quasi static, real time or pseudo-real time, with the first two types being more common, and can either be distributed among different facilities or performed at the same site.

Shake table tests can produce more realistic testing environment than quasi-static testing and hybrid simulations. The shaking tests involve fixing structures at their bases on a table that is dynamically operated with hydraulic actuators using a desired input motion. Most shake tables are configured with small size tables that only allow for dynamic excitation in one direction (i.e. uniaxial). Newly developed experimental facilities, however, are equipped with tables that are large in size and can mimic motion in multiple degrees-of-freedom (DOFs) including the 2DOFs shake table at the University of California San Diego in San Diego, California, and the 6DOFs shake table at the Hyogo Earthquake Engineering Research Center (E-Defense) in Hyogo, Japan. The input motions used to perform the tests can be as simple as a sinusoidal function or an input resembling an actual ground motion. The benefit of shake table tests is the potential for gaining significant insight on global system response including failure modes and the associated collapse mechanisms. Undoubtedly, dynamic testing using a full-scale shake table is viewed as the most realistic method for the seismic evaluation of structural models. However, this requires full-sized tables, which are not readily available in most structural laboratories due to the large space they typically occupy, and the cost associated with building them.

Hybrid simulation has emerged as a potentially accurate and efficient tool for the evaluation of the response of large and complex structures under earthquake loading. The concept of hybrid simulation hinges on combining experimental and analytical models in a single simulation while taking advantage of what each tool has to offer. In an experimental-analytical hybrid simulation, part of a structural system is physically represented in the laboratory while the remainder of the structure is numerically modeled. Typically, the most critical component, or the component whose behavior cannot be well presented with a numerical model, is physically simulated. By combining a physical specimen and a numerical model, the system-level behavior can be better quantified than modeling the entire system purely analytically or testing only a component. The accuracy of the simulation is governed by how well the physical and numerical models are representative of reality as well as the number of DOFs controlled in the simulation. In most cases, not all DOFs need to be considered in the simulation. Such decision, however, has to be well argued prior to testing based on prior understanding of the expected behavior, preferably by conducting numerical assessment, or at the very least by relying on one’s experience and intuition. The decision to exclude specific DOFs from the simulation can be justified for example if planar behavior is governing the response where no or minimal out-of-plane deformations or buckling is expected. In this case, including the out-of-plane DOFs in the simulation will not pay significant dividends and will only add to the complexity of the test.

As noted earlier, hybrid simulation can generally be classified as real time or pseudo static. Real-time hybrid simulation (RTHS) has been relied upon as a testing framework when the specimen dynamics are of particular interest. In RTHS, the specimen is excited in real-time through accurate timing of the action and reaction loop between experimental and numerical substructures. The real-time constraint necessitates high-speed actuator control, specialized digital signal processing hardware, and algorithms with low computational cost. One of the main challenges in RTHS is accurate tracking of the desired specimen trajectories using hydraulic actuators. The actuators themselves introduce unwanted dynamics into the RTHS loop, most significantly in the form of a time lag. The actuator control challenge is compounded by control-structure interaction (CSI), observed by Dyke et al. (1995). Through the intrinsic coupling of the actuator to the specimen, changes in specimen behavior will lead to changes in actuator behavior. CSI further complicates the control problem when multiple actuators are connected to the same specimen. With multiple actuators, the dynamics of the actuators become coupled to each other through the specimen. To account for actuator dynamics including CSI, model-based control approaches have been proposed by the PI Phillips, providing high-fidelity control over a broad frequency range (Phillips & Spencer, 2011).

To date, RTHS has proven a valuable tool for earthquake engineering studies and in particular the evaluation of dampers for structural control (Nakashima et al., 1992; Nakashima & Masaoka, 1999; Carrion & Spencer, 2007; Christenson et al., 2008; Phillips et al., 2010; Jiang et al., 2013). Alongside advances in actuator control, researchers have developed numerical integration strategies (Chen & Ricles, 2008) and outlined complete RTHS frameworks (Nakata & Stehman, 2012). A comprehensive overview of the development and application of real-time hybrid simulations can be found in Friedman et al. (2015).

Due to its mature development, quasi-static testing protocols have been the approach of choice for conducting hybrid simulations, unless otherwise needed as indicated above, and is the focus in this article. The quasi-static approach comprises a number of steps resulting from marching through the input excitation using a desired integration scheme. At a given step in the simulation, the equation of motion is solved and the calculated deformation commands are sent to both the analytical and experimental components. Once the commands are executed, the restoring forces are obtained and used for the numerical time-stepping through the input record. A depiction of the hybrid simulation approach is shown in Figure 1 for the evaluation of a two-story two-bay frame. In this figure, the experimental substructure is chosen as the far-right column in the first story while the analytical substructure is the remainder of the frame. The numerical integration is performed on a computer where for a given step, target displacements, x, are sent to both of the substructured models. Once executed, the models return back the resulting restoring forces, R(x), and a new displacement command is calculated for the next step and sent to the models. It is worth noting that in the schematic representation shown in Figure 1, one actuator is being used to control the lateral displacement at the top of the column. In other applications, the target deformational commands might comprise both displacements and rotations, depending the DOFs being controlled.

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Figure 1. Overview of the hybrid simulation approach.

# Chronicled Development and Recent Examples of Hybrid Simulations

Pseudo-dynamic (PSD) testing is another testing technique that has been widely used by many researchers (Hakuno et al., 1969; Mahin & Shing, 1985; Nakashima & Kato, 1987; Elnashai et al., 1990; Negro et al., 2004; Jeong & Elnashai, 2005). In this testing method, the use of a shake table is substituted by hydraulic actuators connected to the structure. The problem associated with over estimating the loading during cyclic quasi-static testing is overcome by imposing realistic loading on the structure through numerical integration of the dynamic equation of motion while using an actual earthquake. The major shortcoming of using PSD testing is that it requires testing of the whole structure, which is not feasible in some cases due to laboratory space and equipment capacity limitations.

A more attractive approach is to use the concept sub-structuring PSD (SPSD) testing which is nothing but a derivative of PSD. In this method, the structure can be portioned into various components comprising of experimental or analytical modules or a combination of both. Combining analytical and experimental modules in a single simulation is known as hybrid simulation. The concept of hybrid simulation was first developed for the analysis of a single DOF system under earthquake loading (Hakuno, Shidawara, & Hara,1969). The simulation setup comprised an analog computer for solving the equations of motion and an electromagnetic actuator to load the structure. Since then, simulation techniques have significantly evolved to include sub-structuring techniques with hybrid simulation, making it possible to consider distributed hybrid simulation and real-time hybrid simulation (Nakashima et al., 2008).

Hybrid simulation has been used by many researchers for the seismic evaluation of structures and has proven to be very valuable in overcoming the limitations of using conventional PSD (Watanabe et al., 2001; Spencer et al., 2004; Kim et al., 2006; Stojadinovic et al., 2006; Mahmoud et al., 2013). The attractiveness of this option lies in the fact that it captures the complex interaction between the various modules while providing information on the global system behavior. That being said, attention must be given to the experimental and analytical modules used in the simulation since oversimplified representation of either or both can lead to unreliable results. On the other hand, complex representation of the experimental or analytical modules can result in undesired consequences including lengthy simulations, potential for network communication losses, and in some cases divergence of the entire simulation.

Notable advances in hybrid simulations have been made in the past few decades with substantial developments and full applications realized in the last decade alone. The spike in advances and applications was driven by the initiative put forward by the National Science Foundation (NSF), who established the George E. Brown, Jr. Network for Earthquake Engineering Simulation (NEES) to provide researchers the tools to learn how earthquakes impact buildings, bridges, utility systems and other critical components of today’s society. The network includes 15 large-scale, experimental sites that comprised of various advanced tools and were bridged together by the high-speed Internet and linked to a centralized data pool, real-time data viewers, and various tele-presence technologies, allowing both on-site and off-site interactions among researchers.

It is worth noting that at its inception, hybrid simulation relied upon simple linear integration schemes. However, with the move towards more advanced simulations where system collapse is of an interest, it is critical to employ proper integration methods for solving nonlinear equations of motions. Various studies on the development of such have been conducted and include the work by Nakashima, Kaminoso, Ishida, and Kazuhiro (1990) and Shing, Vannan, and Cater (1991), Ahmadizadeh and Mosqueda (2008), Chen and Ricles (2008), and Schellenberg, Mahin, and Fenves (2009).

# Overview of Hybrid Simulation Environments

The general components of a hybrid simulation comprise of a numerical integration scheme, simulation components (either physical, analytical, or numerical, or a combination of such), object classes, and communication protocols. These main components are discussed blow in more details.

## Numerical Integration Schemes

Integration methods utilized in hybrid simulations can be considered its backbone, as it has a direct effect on the accuracy and stability of the whole simulation. Since a hybrid simulation is an aggregation of numerical and experimental components, several terms that form the equations of motion are assembled from both the analytical and the experimental modules. Because experimental modules represent physical specimens in a laboratory there are limitations on the command they can execute. This is because actuators function in either force, displacement, or a mixed force-displacement control. Analytical modules on the other hand can be configured to receive any type of command (i.e., force, displacement, acceleration, among others). This fact leads to a range of special requirements that need to be provided by the integration schemes. When the system being analyzed is expected to behave mostly in a linear fashion (i.e., small material inelasticity and geometrical nonlinearity), the resulting equations of motion are expected to be linear where the mass, damping, and stiffness matrices remain constant throughout the simulation. In such case, numerical integration methods that do not account for nonlinearities can be utilized. However, in cases where large system deformation is expected, which will require nonlinear equations of motion, a numerical integration method suited for such deformation needs to be used. Several integration methods addressing nonlinear equations of motion have been proposed over the years. In this section some of the most commonly used integration methods will be discussed, along with their limitations and advantages. While integration schemes have been developed to accommodate displacement, force, or mixed-mode testing protocols, conducting tests under force control has proven challenging and on the contrary, testing using displacement commands is rather the norm.

The basic principle of hybrid simulation revolves around solving the equation of motion shown in Equation 1. The equation is discretized at the time instant $ti+1$ and encompasses both the physical and numerical model. The mass matrix, M, damping matrix, C, resisting force vector, $Pr$, and the external force vector, $Pi+1$, are the known terms in the equation at a given time step. The coefficients for the M, C, and Pr matrices could either remain unchanged in the case of a linear equation of motion or be updated at a given time step in a nonlinear case. The equation is solved for the displacement $(Ui+1)$, velocity $(U˙i+1)$, and acceleration $(U¨i+1)$ vectors using time-stepping integration algorithms.

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Both explicit and implicit integration algorithms can be used for solving the equations of motion. The explicit methods uses the state of the structure at the previous time step $(U¨i, Ui, Ui)$ to compute the response at the end of the current time step $(U¨i+1, U˙i+1, Ui+1)$. The appeal of the explicit method is that it does not require knowledge of properties at the target command for the determination of the required command to be executed at that step. In addition, the method does not require iterations to be performed and neither does it require knowledge of the initial and tangential stiffness. The method is also easy to implement and computationally efficient. Despite its attractiveness, the conditional stability of this method and its inadequacy to be applied when singular mass or stiffness matrices or massless DOFs are present, limit its application in some cases (Schellenberg et al., 2009). This could be an issue when large material inelasticity and/or geometrical nonlinearities are expected. The method is also not suitable for stiff systems with small periods. Unlike explicit methods, implicit methods require knowledge of the properties at the target command in order to compute the response while satisfying the kinematic and equilibrium conditions at the end of the time step using iterations. One of the main limitations of the implicit method is its inability to accommodate nonuniform command increments that could be either specified or produced during equilibrium iteration. The nonuniform increments are characteristics of the Newton-Raphson algorithm, typically used in the implicit method, where the increments are expected to decrease rapidly during iteration (Mosqueda & Ahmadizadeh, 2011). From a safety perspective, the iterative nature of the implicit method could result in command overshoot that could essentially damage the physical specimen or at the very least cause it to experience significant deformation. The method requires knowledge of both the initial and tangential stiffness matrices. It is, however, difficult to obtain tangential stiffness matrices for experimental substructures, making it computationally demanding and cumbersome to implement. The implicit method is generally unconditionally stable, suitable for stiff systems with small periods, and applicable to systems with singular mass matrices, such as structures with rotational DOFs. A general overview of the time-stepping approach utilized in hybrid simulation is shown is in Figure 2.

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Figure 2. Typical time-stepping approach in hybrid simulation.

In the context of hybrid simulations, errors could arise due to various reasons including errors due to the experimental setup, numerical integration, and errors directly associated with splitting the system as a hybrid one (Mahin & Shing, 1985). For the numerical integration errors, as noted earlier, some complications could arise during the execution of hybrid simulations, which will need to be considered when selecting an appropriate integration method. The integration method should be equal to or higher than second order so that the numerical approximation errors may be kept to a minimum. This is essential to reduce the potential for experimental errors during the simulations. The resisting force vector in the equation of motion is populated from the numerical as well as the experimental components. Thus, in each iteration of the integration method certain calls would be made to acquire the required values from the physical substructure, which could potentially also increase the chances of experimental errors. This is because at the end of each step some oscillations in the recorded actions are typically present before a steady-state condition is reached, which is the nature of experimental testing. Therefore, the number of calls to the physical substructure should be kept at a minimum to lessen the potential error. It is worth noting that the propagation of error in subsequent steps not only invalidates the results but also could lead to divergence of the whole simulation, which can cause dynamic instability, posing serious safety concerns. Therefore, an important characteristic of the integration schemes used is to possess the ability to provide adjustable damping to reduce higher mode effects and reduce the potential for a high rate of error accumulation. The deformation increments produced by the integration method should be uniform, whenever possible. Nonuniform increments can be challenging to the physical substructure since actuators may have trouble following nonuniform increments. These considerations are quite vital when considering an integration method for hybrid simulations, as they are quite sensitive to instability. Based on these recommendations some of the most prominent and widely used methods are discussed below including the Implicit Newmark Method (INM), the Modified Implicit Newmark Method for Hybrid Simulation (INM-HS), the Implicit Generalized-Alpha Method for Hybrid Simulation (IGα‎-HS), the Generalized Alpha Operator Splitting Method (Gα‎-OS1), and the Modified Generalized Alpha-OS Method (Gα‎-OS2).

### Implicit Newmark Method (INM)

Although not used in hybrid simulations, a discussion of the INM is required prior to introducing the INM-HS in the following section. The INM was originally developed for purely numerical simulations, the implicit form of the Newmark method (Newmark, 1959) is the most widely used integration method in structural dynamics. Although an efficient tool, the implicit method has certain limitations regarding its application to hybrid simulation, which requires adjustment for proper interaction of the physical substructures. The residual form of the equation of motion can be written as shown in Equation 2, and the system of nonlinear equations can be solved using the iterative Newton-Raphson method as shown in Equation 3.

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where, $J(Ui+1k)$ is the iteration matrix or the Jacobian matrix, $ΔUk$ is the increment deformation vector, and $−FUi+1k$ is the negative residual. The Jacobian matrix and the negative residual vector are known as the effective stiffness matrix and the effective (unbalanced) force vector, respectively, and are related using Equation 4 below.

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Since the equations are presented where the solution is derived by solving for deformation increments, the Newmark equations are reformulated to obtain expressions for velocity and acceleration in terms of displacements as shown in equations 7 and 8, where $γ$ and $β$ define the variation of acceleration over a time step and determine the stability and accuracy.

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The Newton-Raphson algorithm requires a good initial approximation (i.e., the initial assumption has to be close to the actual solution). Therefore equations 911 are used to calculate the initial values of displacement, velocity, and acceleration vectors to start the Newton-Raphson equilibrium iterations.

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After solving for the incremental displacement vector using equation 4, the initial displacement, velocity, and acceleration vectors are updated using equations 1214 and the process is repeated until convergence is achieved. Convergence can be measured using several criteria, which have been comprehensibly listed in Schellenberg et al. (2009). Typically, displacement increment is the most commonly used as a tolerance criterion; however other variables such as the norm of unbalanced force vector or an energy measure can also be considered a viable option.

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### Modified Implicit Newmark Method for Hybrid Simulation (INM-HS)

The Implicit Newmark method, although applicable to hybrid simulations, results in nonuniform displacement increments, which tends to create experimental errors. To eliminate such issue, Schellenberg et al. (2009) modified the implicit Newmark method to improve its accuracy and make it compatible with the framework of hybrid simulations. This approach was first introduced by Dorka and Heiland (1991) and Shing et al. (1991), with later refinements by Zhong (2005). This modified version overcomes the problems of the implicit Newmark method by performing a constant number of iterations per integration step. The iterative process is terminated after reaching a specified number of iterations defined by the user, instead of utilizing a convergence criterion as discussed in the previous section. Figure 3 describes the procedure of the modified Newmark method for hybrid simulation. As shown in the figure, this modified method utilizes an interpolation function (actual displacement) to update the model or generate the displacement command, based on the trial displacements ($U¯i+12,U¯i+13...U¯i+1k+1$) that are produced at each increment. In other words, the trial displacements are not directly used to update the model. The interpolation is performed between converged solutions and the current trial displacement using Lagrange polynomials.

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Figure 3. Illustration of the modified implicit Newmark method (after: Zhong, 2005).

In order to account for this modification, the corrector equations are reformulated as outlined in equations 1517, where the scaled displacement values are evaluated using equation 18 given by Schellenberg et al. (2009). In these equations, $ΔUi+1k−1$ are the trial displacements at previous iterations (k-1), $Uj$ are the converged displacements at previous time step, which are also the trial displacements of current iteration, and $Ln,j(x)$ are the Lagrange functions of order n and x is the location of interpolation given by the ratio of iterations to the maximum number of iterations performed at each step of the iteration, i.e. k/kmax.

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The proposed modifications outlined above result in an integration method with uniform displacements increments; thereby significantly improving the convergence characteristics. More details of this integration method can be found in the original document by Schellenberg et al. (2009).

### Implicit Generalized-Alpha Method for Hybrid Simulation (IGα‎-HS)

In hybrid simulations experimental errors can excite higher modes thereby influencing the results significantly. To alleviate this problem, the alpha-methods of numerical integrations can be relied upon, since these methods are characterized by high-frequency numerical dissipation, which tends to remove the high-frequency response modes that could result in significant overshoot. The overshoot phenomena refer to the over-estimation of the target displacement by the actuators. Early work on developing an implicit generalized-alpha method was conducted by Chung and Hulbert (1993) who introduced numerical dissipation to improve the above-mentioned shortcoming of the implicit Newmark method. Chung and Hulbert reported that the high-frequency numerical dissipation improves the convergence of iterative integration methods and can eliminate the overshoot issue. To introduce algorithmic damping, weighted equations of motion are formulated between time steps $ti$ and $ti+1$, instead of at the exact time steps, using the weighing parameters $αm$ and $αf$ (Equation 19).

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The internal resisting force, external force, acceleration, and velocity vectors are evaluated between time steps using the generalized trapezoidal rule. As shown in Equations 2023, the terms are represented as the weighted sum between the values at time i and i+1. The final modified equations of motion are given by substituting the weighted terms in the original equation of motion (Equation 19) and given by Equation 24.

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Similar to the methods discussed previously, Newton-Raphson procedure is used to solve the system of nonlinear equations given by Equation 24. The procedure followed is exactly similar to that of modified Newmark method. The difference is in the fact that the initial values of displacement, velocity and acceleration are updated by running a specific number of iterations, assigned by the user, instead of using a convergence criterion. The amount of algorithmic damping is controlled by the damping parameter $(ρ)$. Chung and Hulbert (1993) derived the $α$ and $β$ terms shown in Equation 25, along with their respective feasible domains, to maximize the high-frequency dissipation while maintaining second-order accuracy.

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### Generalized Alpha Operator Splitting Method (G$α$-OS1)

The generalized alpha OS integration method is similar to the alpha-OS method presented by Nakashima et al. (1988, 1990). It is a combination of the operator splitting algorithm originally developed by Hughes et al. (1979) and the generalized alpha method developed by Chung and Hulbert (1993). The former component works as a corrector and the latter as a predictor. The advantage of this algorithm lies in the fact that it does not require iterative solution algorithms and it inherits the dissipation property of the alpha family, as previously mentioned. Therefore, it could achieve stability without incurring significant processing time. The algorithm starts off on a similar note to the generalized alpha method as it utilizes the same equation of motion (Equation 19). The displacement, velocity, and acceleration terms are written in a predictor-corrector form as shown below in Equations 26 and 27. The nonlinear internal resisting forces are defined using the operator-splitting technique as the difference between elastic and nonlinear forces at the predictor displacements, as shown in Figure 4 and given by Equation 28.

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Figure 4. Approximation of nonlinear resisting force using $Ki$ (after: Schellenberg et al., 2009).

Equation 29 is derived by substituting Equation 28 in the respective equation of motion (Equation 19). Similar to the implicit generalized-alpha method, the respective equations of displacement, velocity, acceleration, internal resisting force, and external force vectors between time steps i and i+1 are obtained by the weighted sum of these time steps. Equations 30 and 31 show the velocity and acceleration in terms of the predictor displacement, which are substituted in Equation 29 to obtain system of linear equations that are solved for displacement increments between the predicted and trial displacements. Finally, the displacement, velocity, and acceleration terms are updated using Equations 3234.

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The operator-splitting integration method presents certain advantages for hybrid simulations. Given that the method is a predictor-one-corrector, only one force acquisition from the experimental substructure is necessary per integration step so the chances of incurring experimental errors are substantially reduced. The OS method requires only the initial stiffness matrix for the simulation, instead of the tangent stiffness matrix, which can be difficult to extract for physical substructures. The OS method, although computationally efficient, has a limitation as well since it does not utilize iterative algorithms and therefore cannot be implemented for structures with high geometric nonlinearity.

### Modified Generalized Alpha-OS Method (Gα‎-OS2)

As discussed in the previous section, the generalized alpha-OS method cannot be implemented for cases with high geometric nonlinearities. To improve the method’s stability for such cases a mixed stiffness matrix replaces the initial stiffness matrix for estimating the restoring forces. The mixed stiffness matrix is formed from the tangent stiffness matrices of the analytical and experimental components. In cases where the tangent stiffness matrices are not readily available for the experimental components (due to difficulty in obtaining them) then the initial experimental stiffness matrices can be used. The nonlinear internal resisting force vector of the equation is now approximated using Equation 35.

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The mixed stiffness matrix is calculated at the predictor displacements $U˜i+1$ or $U˜i+αf$. Therefore, the Jacobian has to be evaluated at least once per iteration since it is not uniform anymore, unlike the generalized-alpha OS method. This tends to negatively affect the computational cost of the algorithm. The internal resisting force vector is substituted back into the equation of motion to obtain the approximate equation of motion given by Equation 36.

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The rest of the procedure for finding the solution is the same, as is that of generalized alpha OS method; hence, it is not explained here. The modified method has demonstrated stable characteristics compared to its counterpart, due to the use of mixed global stiffness.

## Simulation Components, Object Classes, and Communication Protocols

The computational components in hybrid simulation include the analytical model of the numerical substructure, the time-stepping integration algorithms, and communications between the numerical and experiment substructures. Orchestration of the entire simulation is typically conducted by one component who serves as the so-called coordinator. Typically, there are two different types of coordinator configuration used in hybrid simulation. In the first configuration, the time-stepping integration as well as communications are handled by the coordinator, and the numerical and experimental substructures are represented by in a finite element software and a physical laboratory, respectively. The coordinator sends the target commands to the substructured finite element and experimental modules and receives the restoring force back. An example of a coordinator utilizing this simulation arrangement is UI-SIMCOR, which is developed and maintained by researchers at the University of Illinois at Urbana-Champaign. In the second configuration there is no designated coordinator; instead finite element software serves as the master simulation that handles communication with the physical laboratory where experimental substructures are treated as outsourced elements. Both the time-stepping integration and the numerical substructures are processed in the master simulation. An example of this arrangement is OpenSees, which is developed and maintained by researchers at the University of California at Berkeley. Specifically, in the OpenSees platform, the substructure modules are treated as element classes. OpenSees is a finite element object-oriented software developed using the C++ programming language, which has been modified to act as a simulation coordinator through interfacing with the experimental class through a middleware software, OpenFresco. Two scripts are utilized in OpenSees; namely a master and slave script. The master script includes the analytical module while the slave is the experimental module. In addition, OpenSees is also responsible for solving the equations of motion of the whole systems and for sending a vector of deformations to the slave script through Open Fresco.

Physical components in a hybrid simulation receive the target deformations from the computational components (either through coordinator or master simulation), impose the target commands to the experimental substructure, extract the corresponding measured actions, and finally send the restoring forces back to the computational components. These deformations could be in the form of either translational or rotational commands. The experimental substructure should therefore be properly configured to allow for the transformation of these Cartesian deformation commands to the proper actuator translation displacement. This often requires the use of multiple actuators to produce the desired rotation in cases where rotational DOFs are being controlled.

Simulation coordinators, such as UI-SIMCOR are typically equipped with two object classes: namely a restoring force module and an auxiliary module. The objects of the restoring force module represent structural components at remote sites irrespective of whether the components are experimental specimens or analytical models. The restoring force module also serves as a communication enabler between the simulation coordinator and the remote site. Specifically, when deformations are imposed on a structural component represented by an object in the restoring force module, the object reformats data for a particular protocol, establishes the connection to the remote sites, and sends the reformatted data. Different data formats are typically utilized for such communication including ASCII and/or binary formats. Other functionalities are employed in the retorting force modules including examining the developed actions and deformations at every numerical iteration and imposing set limits to reduce the potential for large instability to ensure a safe testing environment. The auxiliary module is responsible for controlling commands that are not related to deformations (software and/or hardware). The function of the object of this class is to send out pre-specified commands to remote sites to control for example data acquisition systems, still cameras, and video cameras.

Network Telepresence Control Protocol (NTCP) provides a secure communication language for both computational and experimental modules. Specifically, the deformational commands sent from the simulation coordinators are first encrypted by being converted to a protocol language prior to being sent over a TCP/IP connection. Communication portal that has been opened using the Network Telepresence Control Protocol is referred to as an NTCP port. The port serves as a facilitator to allow communication between the simulation coordinator and the computational tools. Different plug-ins are typically required depending on the finite element software being used and are typically either MATLAB-based or LabVIEW-based plug-ins. An example of the various module classes and their interface with UI-SIMCOR is shown in Figure 5.

Click to view larger

Figure 5. Architecture of UI-SIMCOR hybrid simulation framework (used with permission: Kwon et al., 2008).

# Application Example on Semi-Rigid Steel Frames

In this section, a hybrid simulation application example is provided for the assessment of semi-rigid steel frames. The main components describing the case study application include connection topology and frame configuration, selection and scaling of the ground motion, the analytical module, the experimental module, the integration scheme and finally the main observations and results of the simulation.

## Connection Topology and Frame Configuration

There exist many benefits for using semi-rigid connections for the design and construction of steel frames under seismic loads. The connections have demonstrated stable hysteretic behavior and sufficient ductility and have been successfully used in low- and mid-rise steel frames subjected to low-to-moderate seismic actions. However, existing research on the seismic performance of these types of connections is from testing of beam-to-column subassemblies under idealized load and boundary conditions, from formulating analytical/mathematical representation of the moment-rotation curve, and from developing numerical 2D and 3D nonlinear finite element models. The system-level experimental behavior of semi-rigidly connected frames using real earthquake motions conditions is generally lacking. To conclusively investigate the full potential of semi-rigidity (implying also partial-strength) in earthquake resistance, assessment of its local behavior and its effect on the global behavior of the frame is needed. This example provides high-level overview of a recently conducted study to develop a system-level hybrid simulation framework for exploring the effect of top, seat, and double web angle connection stiffness; capacity; and ductility on the seismic performance of steel frames. The experimental component of the simulation comprised a full-scale beam-column subassembly. The simulation was conducted using the state-of-the-art instrumentation and control equipment at the MUST-SIM facility at the University of Illinois, part of the Network for Earthquake Engineering Simulation (NEES).

In this application example, a two-story, four-bay (longitudinal) and two-bay (transverse) steel frame building is considered and is assumed to be located in Los Angeles, California, on soft soil. The height of the first and second story is 4.57 m and 4.11 m, respectively, and the bay width is 9.14 m. The perimeter frames are special moment resisting frames (SMRF) designed according to the Structural Seismic Design Manual, Volume 3 of the International Building Code (IBC, 2006). The inner frames are the gravity loading frames. For the design of the SMRF, the importance factor (I) is 1, the force modification factor (R) is 8, and the overstrength factor (Ω‎) is 3. The beam and column sizes are W18x40 and W14x159, respectively. Sizing of the beams and columns was based on the strong-column weak-beam design methodology with an elastic panel zone. A plan view of the entire structure and an elevation of a typical SMRF are shown in Figure 6. Detail 1 on the elevation view marks the physical substructure component of the hybrid simulations. The assumed rigid connections in the frame were configured to reflect partial strength and semi-rigidity. Specifically, three different frames were considered with connections designed according to the Eurocode 3 (ENV, 1998). The sizes of the angles and the bolts were selected to result in three different frames with three distinct connection capacities—30%, 50%, and 70% of the plastic moment capacity of the beam. The standardized parameters typically used for describing the geometry of these types of connections are shown in Figure 6. The geometrical parameters for all three connections are also listed in the figure.

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Figure 6. Geometric layout of the sample frames with detail 1 on the elevation view marking the physical component of the simulations.

## Selection and Scaling of Ground Motion

As previously mentioned, the building is designed assuming it is to be located in Los Angeles, California. Therefore, records collected during the 1994 Northridge and 1989 Loma Prieta earthquakes were considered as they fit the location criteria. A total of 40 records were selected with approximately 30 to 40 seconds of motion duration and 0.005 to 0.02 seconds of varied time steps. The number of records was further reduced to 20 based on epicentral distances of 5 to 10 km and 15 to 20 km such that both short-period and long-period structures would be excited. The records were further narrowed down to eight time histories based on the spectral acceleration to ensure that structures with periods between 0.5 sec and 1.2 sec would be stimulated. This period range provides a reasonable lower and upper bound of the frames’ fundamental periods. In order to select one of the eight records to be used in the hybrid simulations, an eigenvalue analysis was conducted to determine the natural period of the structures and ensure high demand on the frame in its elastic and inelastic ranges. The selected record for the hybrid simulation was the 1989 Loma Prieta earthquake, recorded at station USGS 1662 Emeryville, which is 77 km from the epicenter of the earthquake on soft soil (Vs­ = 199 m/s) with peak ground acceleration of 0.26 g as shown in Figure 7.

Figure 7. Loma Prieta acceleration time-history record with time step of 0.005 seconds.

The record acceleration was scaled to ensure constant capacity-to-demand ratio for all three frames where the demand is chosen to be 5% higher than the capacity. In this example, the capacity is defined as the base shear value at which a plateau is observed in the base shear versus displacement curve. A conventional pushover analysis of the structures, using an analytical model, was conducted to obtain load-displacement curve of the frames. The resulting equation used for calculating the factor used to scale the record is given by Equation 37.

$Display mathematics$
(37)

where; n is the scaling factor, $Vcapacity$ is the capacity of the structure (defined from pushover analysis); W is the weight of the structure, and Sa is the spectral acceleration at the fundamental period of the structure. The scaling factor used in the simulations was calculated to be 0.830, 0.810, and 0.763 for the 70% Mpbeam, 50% Mpbeam, and 30% Mpbeam frames, respectively. Table 1 shows the characteristics of the ground motion used in the hybrid simulations and the corresponding scaling factors for each frame.

Table 1. Ground Motion Characteristics and Scale Factors

Earthquake

Mw

Station

Fault Distance

Hor. PGA (g)

Scale Factor

Epicentral (km)

Hypocentral (km)

30%

50%

70%

Loma Prieta (17/01/09)

7.1

Emeryville/Pacific Park Plaza Building

96

17.48

0.245

0.763

0.810

0.83

## Analytical Module

Mostly, analytical models of frames utilize line elements connected with springs representing the load deformation characteristics connections. This modelling approach has been viewed as the best alternative for hybrid simulation since the number of elements in this case is small and significant time is not required to complete a simulation step. However, adopting such approach can result in models that represent idealized behavior and, in many cases, cannot capture the true local response at critical details. The lack of proper representation of the localized behavior will consequently have an effect on the spread of yielding in the beam. In the case of top- and seat-angle with double web-angle connections, using line element models will not allow for the prying actions of the angles to be captured since the top and seat angles are neither physically modeled nor accounted for. In light of these arguments, an inelastic finite element model was employed in the presented example. The model comprised 2D plane strain elements for the beam-to-column connections and 1D beam elements between subsequent connections. The model was developed using the commercial finite element software, ABAQUS, which is a general purpose commercial package (Simulia, 2007). Various behavioral features were included in the model such as (1) bolt preload, (2) friction between faying surfaces, (3) connection slip, (4) the effect of bolt-hole ovalization, and (5) hot-rolling residual stresses in the angles. In addition, the effect of the inner gravity frames on the stability of the moment resisting frame (i.e., large P-Δ‎ effect) was included through the use of a leaner column modeled as two truss elements pinned at the base and at the first-floor level. In addition, tie multi-point constraints were used to provide rigid links between the semi-rigid frames and the leaner column. The described model with a zoom-in view of the deformed shape of the middle connection in the first floor is shown in Figure 8.

Click to view larger

Figure 8. Analytical frame model used in the hybrid simulations with a zoomed view of the deformed shape of an interior connection (Mahmoud et al., 2013).

## Experimental Module

The experimental component of the simulation utilized the Multi-Axial Full-Scale Substructured Testing and Simulation Facility (MUST-SIM), which is part of the 15 sites of the Network of Earthquake Engineering Simulations (NEES). Details on the advanced capabilities of the MUST-SIM facility can be found in Elnashai et al. (2004). The main components of the facility include three Load and Boundary Condition Boxes (LBCBs) and the L-shaped strong wall. The main loading units (i.e., the LBCBs) are capable of providing displacement and forces in all six DOFs at different contact points.

The experimental component represented a full-scale bolted beam-column subassembly. The beam included a portion of first-story beam in the first bay while the column included portion of the first- and second-story columns in the same bay for a total number of three points to be controlled during the simulation. At each of the three control points, an LBCB should be used to provide the required deformation commands during the simulation. However, the base of the column was fixed to the lab floor, and only two LBCBs were used as shown in Figure 9. The fixity of the column base was accounted for during the simulation through the use of relative deformation between all three control points. In this example, all three planar DOFs were controlled in the simulations at each of the control points.

The instrumentation plan was developed and installed to capture the local response of the connection and the global response of the beam-column subassembly. In each test, a total of 175 channels were installed on the specimen and recorded using a National Instrument Data Acquisition (NI-DAQ) system. In addition, each LBCB houses six load cells and six LVDTs for displacement and load measurements, respectively, for each actuator. Figure 9 shows the overall experimental setup and an example of linear pot arrangements used for measuring connection rotation.

Click to view larger

Figure 9. Experimental setup (left) and detailed instrumentation of the connection (right).

## Integration Scheme for PSD Testing

The integration of the experimental and analytical modules was realized through the use of UI-SIMCOR (Kwon, Nakata, Elnashai, & Spencer, 2005). The simulation started with the stiffness evaluation step, followed by gravity loading, and finally the earthquake/dynamic simulation steps. In the earthquake application stage, time stepping was performed using the α‎-Operator Splitting method. As previously indicated, in this approach the earthquake force is calculated numerically using time-step integration of the equation of motion. The corresponding deformations are then applied simultaneously to the experimental and analytical substructures. Following the execution of the deformation commands, the resulting restoring forces are measured for each substructure and used in a feedback loop for the calculation of the deformation command corresponding to the next step. The numerical integration in UI-SIMCOR uses the OS method with a modified α‎- parameter through the Newmark integration scheme (α‎-OS method), which applies numerical damping to the undesired oscillations as previously discussed in detail. A full description of UI-SIMCOR and its components can be found in (Kwon et al., 2005). A schematic of the hybrid simulation application is shown in Figure 10.

Click to view larger

Figure 10. Hybrid simulation framework for the investigated frames.

## Simulations Results and Observations

The section below provides a brief summary of some key performance measures obtained from the simulations including interstory drift, bases shear, and moment-rotation relationships.

### Interstory Drift and Base Shear

A comparison of the second-story interstory drift ratio (IDR) and the base shear normalized by the total weight for all three frames during the hybrid simulations is shown in Figure 11. It is noted from the figure that the IDR response of the frames are similar up to time 5 sec. In the time range of 5 sec to 6.42 sec, larger period elongation is observed in the 30% Mpbeam frame when compared to the other frames. This is likely due to the nonlinearity experienced by the 30% Mpbeam frame at lower lateral deformation. Noteworthy that for the 30%Mpbeam, the simulation was not completed and stopped after time 6.42 sec of the earthquake motion due to conversion problems associated with the contact formulation in the analytical model.

Click to view larger

Figure 11. Comparison of the interstory drift ratio and base shear normalized by the total weight of the structure.

In addition to calculating the maximum IDR, the frames are assessed using two different performance levels, namely Design Basis Earthquake (DBE) and Maximum Considered Earthquakes (MCE). The acceptance criteria used in this study was to limit the interstory drift ratio to 2.5% and 5% for DBE and MCE, respectively, as defined by ASCE 41-06 (ASCE/SEI 41-06, 2007). The calculated IDR for the first and second story and their normalized values with respect to ASCE 41-06 requirements are listed in Table 2. As shown in the table, the second-story IDR normalized to that of MCE approaches 1 for the 70% Mpbeam and 50% Mpbeam frames and slightly exceeds 1 for the 30% Mpbeam frame while the value is well below 1 for the DBE. The first-story IDR normalized to that of MCE and DBE is always well below 1. These results clearly show the potential use of these frames in high seismic regions since the limit specified by ASCE 41-06 was exceeded in only one case by only 8%.

Table 2. Maximum and Normalized IDR

Mpbeam

$IDRMax1st(%)$

$IDRMax1stIDRASCE41DBE$

$| IDRMax1st |IDRASCE41MCE$

$IDRMax2nd(%)$

$| IDRMax2nd |IDRASCE41DBE$

$| IDRMax2nd |IDRASCE41MCE$

70%

1.61

0.644

0.322

2.32

0. 928

0.464

50%

1.86

0.744

0.372

2.42

0. 968

0.484

30%

1.58

0.632

0.316

2.70

1.080

0.540

### Moment-Rotation Relationship and the Corresponding Joint Deformation

Figure 12 shows a depiction of connection deformation and comparison of the derived moment-rotation relationships for all three connections. As shown in the figure, large pinching and hardening was observed in the response of the 70% Mpbeam connection when compared to the other two connections. The highest calculated stiffness degradation was observed in the 50% Mpbeam specimen, which is 46.05% of the original stiffness, followed by the 30% Mpbeam specimen, which experienced degradation in its stiffness of 33.59% and finally the 70% Mpbeam specimen with stiffness degradation of 23.47%. The characteristics of the connections during the simulations are summarized in Table 3. The table includes values for the initial stiffness (ki), the unloading stiffness (ku), the percent of stiffness degradation (kdeg), the maximum moment and rotations experienced by the connections, and the energy dissipated by each connection during the simulations. The maximum moment sustained by the 70% Mpbeam, 50% Mpbeam, and 30% Mpbeam connections is equal to 36,404 kN.mm, 28,879 kN.mm, and 19,298 kN.mm, with corresponding rotations of 0.0196 rad, 0.0271 rad, and 0.3400 rad, respectively. The sustained rotations are below the 0.04 rad required by AISC since the demand did not reach such limit. At the maximum level of rotation, no failures were observed in any of the subassembly components.

Click to view larger

Figure 12. Comparison between the moment-rotation relationships.

Table 3. Characteristics of the Connections During the Simulations

Mpbeam

kdeg (%)

|M|Max (kN.mm)

%Mpbeam

70%

5,769,941

4,415,751

23.4

36,404

82.0

0.0196

2,205

50%

5,584,996

3,013,508

46.0

28,879

65.2

0.0271

2,005

30%

3,463,221

2,299,975

33.6

19,298

43.6

0.034

1,238

# Conclusions

In summary, the chronicled development of hybrid simulation is presented and a detailed overview of its main components is discussed including the most widely used implicit and explicit numerical integration schemes, the simulation components, object classes, and communication protocols. In addition, a new substructured PSD hybrid simulation approach is developed for system-level evaluation of semi-rigid partial-strength steel frames. In this sample example, an analytical and experimental components of a two-bay two-story semi-rigid partial-strength frame are used in the simulations for complete analysis of the entire frames. Specifically, three connections with capacities equal to 30%, 50%, and 70% of the plastic moment capacity of the beam are employed in three different frames to investigate the response of the frames under seismic events under the 1989 Loma Prieta earthquake record. The following conclusions can be made on the discussed hybrid simulations framework and its application to the subject frames.

• Hybrid simulations allow for collaboration of various analytical platforms and experimental sites into a simulation of large, complex, and interacting system.

• Early work on utilizing hybrid simulations started back in the 1960s with a substantial increase in development and implementations in the past two decades.

• Both implicit and explicit integration schemes can be used for time-stepping through the time-history record.

• The stability of the chosen method is critical for successful completion of the simulations

• Accurate simulations depend heavily on accuracy of the experimental and analytical modules.

• The application of hybrid simulations for the seismic assessment of semi-rigid steel frames highlights the feasibility of using these types of frames in high seismic regions.

• The maximum base shear was developed in the 70% Mpbeam frame followed by the 50% Mpbeam frame then the 30% Mpbeam frame.

• The maximum IDR is equal to 2.7% in the second story of the 30% Mpbeam frame, which is slightly above the 2.5% limit specified by ASCE 41-06.

• The moment sustained by the connections did not reach a plateau and the behavior of all three connections was highlighted by stable hysteretic behavior and high energy dissipation.

• The sustained rotations are below the 0.04 rad required by AISC since the demand did not reach such limit.

• At the maximum level of rotation, no failures were observed in any of the subassembly components.

• Large pinching and hardening observed in the 70% Mpbeam connection.

# Acknowledgments

The authors wish to acknowledge the valuable support received from researchers and staff at the Newmark Laboratory and the Civil and Environmental Engineering Department at the University of Illinois at Urbana-Champaign. Special thanks are owed to Prof. Billie Spencer and Prof. Dan Kuchma (currently at Tufts University) for their leadership and vision regarding the development of the MUST SIM facility. Thanks, are also due to Prof. Oh-Sung Kwon (currently at the University of Toronto) and Prof. Narutoshi Nakata (currently at Tokushima University) for their tireless effort in developing the simulation capabilities including UI-SIMCOR and multi-DOFs multi-mode control of the loading units.

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