IntroductionVulnerability assessments of buildings on shallow foundations in liquefiable soils have been mainly focused on soil deformation and the consequent damage to foundations. Since soil softening associated with liquefaction acts as a natural isolation barrier (Karatzia et al. 2019), the damage associated with strong ground shaking, in conjunction with ground deformation, is often neglected. However, several studies (Dashti et al. 2010; Kramer et al. 2011) have demonstrated that when liquefaction occurs later in the strong shaking, buildings can experience both significant settlement and strong shaking. In fact, partial liquefaction under a building leads to soil deformation and consequent changes in its dynamic properties, which can then potentially amplify its dynamic response. To better understand strong shaking-induced damage when liquefaction occurs, the interactions between the soil, foundation, and structure must be well defined, and a framework should be established to account for the combined damage and loss induced by both soil-foundation deformation and shaking (Millen et al. 2018).While there are efforts to quantify the vulnerability of buildings to liquefaction using empirical field data sets (Paolella et al. 2020), these approaches are limited by the availability and quality of the data. Three alternative analytical approaches have been identified and adopted to consider soil-liquefaction-foundation-structure interaction (SLFSI) in building vulnerability (Fig. 1). The building-soil system can be assessed directly by modelling both soil and building in a single numerical or experimental model (a full model approach), such as in Dashti and Bray (2013), and Ramirez et al. (2018). Alternatively, the building response and soil response can be completely decoupled. In this case, shaking and liquefaction damage are assessed independently and then combined through an interaction function (separate hazards approach), such as in the HAZUS methodology (FEMA 2003), the proposal by Bird et al. (2006), or imposing ground deformations directly or indirectly to a building (Fotopoulou et al. 2018; Gómez-Martinez et al. 2020).The full model approach, where both soil and structure are modelled together with adequate constitutive laws in a single simulation, is advantageous because the interactions between all mechanisms are accounted for implicitly. However, it is a demanding task to develop an adequate full model that accurately captures the complex effective stress-controlled soil behavior and the various nonlinear mechanisms in a building. Notably, none of the widely used commercial software currently contain both a three-dimensional state-compatible constitutive model for the soil, and elements for modelling existing structures (e.g., reinforced concrete buildings) with a suitable array of nonlinear material constitutive models. Therefore, a trade-off must be made by reducing the accuracy for the soil or the structural modelling. In addition, the full model approach is computationally demanding as nonlinear structural models with degrading behavior often require very small time steps to achieve convergence, but in a large soil domain medium, there are many degrees of freedom that must be assessed at each time step.The separate hazards approach is numerically efficient and can make use of existing results for ground shaking damage (e.g., fragility functions), but it suffers from some significant drawbacks. The use of an interaction function to combine shaking and liquefaction damage is non-trivial as will be shown in the next section. Essentially, liquefaction modifies the shaking demand, and differential settlements affect the lateral resistance capacity of the building, which influences the shaking damage. Meanwhile, liquefaction-induced settlement and tilt are dependent on the inertial load (shaking) of the building.In this article, an innovative modular approach is proposed for reinforced concrete-framed structures on shallow foundations in liquefiable soil deposits, where different macro-mechanisms are first quantified and then connected based on their interactions. The idea is to capture the full system behavior through a combination of sub-models (e.g., a pore pressure model, a settlement model) that focus on a particular mechanism or system behavior, rather than at the constitutive level. For this reason, it has been termed the macro-mechanism approach. The procedure presented here has been applied in an extensive parametric study described in Viana da Fonseca et al. (2018) for the development of the fragility functions for the loss assessment LIQUEFACT software (Meslem et al. 2019), and to assess liquefaction-induced loss at critical infrastructure as described by Meslem et al. (2021).In this work, the approach is described in detail and then a validation study is performed focusing on the behavior of the Public Education Center (PEC) building in Adapazari, Turkey, during the 1999 Kocaeli earthquake. As part of the validation study, the PEC building is modelled using the macro-mechanism approach in OpenSees, as well as using two full numerical models in FLAC (version 8.0) (Itasca 2016) and PLAXIS (Bentley 2020) that were developed by separate research teams at the University of Porto and Istanbul University-Cerrahpasa, respectively, to allow for comparisons between the performances of the different models.The Macro-Mechanism ApproachMethodology DescriptionA nonlinear time history analysis procedure that models the macro-mechanisms of the soil, foundation, and structure during a shaking event has been developed. This procedure is a sub-structuring approach and has been developed to provide an efficient method to consider the impact of liquefaction on the performance of buildings. The problem domain of the proposed procedure is reinforced concrete buildings founded on shallow foundations on flat ground, and subjected to a ground motion applied only in one principal direction of the building. The so-called macro-mechanism approach addresses the three liquefaction-induced macro-mechanisms of changes in ground shaking, changes in foundation impedances, and settlement, through a series of time-dependent sub-models that are subsequently integrated into a time history analysis of the building using springs, dashpots, and imposed displacements. The development of the model requires four steps that can be performed separately or in combination: 1.Quantify the liquefaction potential of the soil profile in terms of depth and thickness of the liquefiable layer(s) and the resistance to liquefaction. The free-field pore pressure ratio is then quantified throughout the duration of shaking.2.Estimate the near-field ground shaking time series accounting for liquefaction for use as acceleration input for the building time history analysis.3.Estimate the soil-foundation stiffness using springs and dashpots and account for the change in soil characteristics due to liquefaction and nonlinear shear deformation throughout the duration of shaking.4.Estimate the expected load-settlement behavior of each footing accounting for the expected level of pore pressure build up throughout the duration of shaking.The key aspects of this approach can be seen in Fig. 2, where the outputs of the four steps are used as inputs into a nonlinear model of the building. The input motion is the expected near-field motion from Step 2 and the expected differential settlement behavior is captured through a combination of imposed settlement (Step 4) and changes in the stiffness of the soil springs (Step 3). In this paper, a procedure for completing each sequential step is presented. It is recognized that this decoupling introduces inaccuracies, and in the case study, an additional analysis is performed where ground shaking, pore pressure, and settlement are obtained from a fully-coupled 2D FLAC analysis to demonstrate the flexibility of the macro-mechanism approach.Estimation of Excess Pore PressureThe build up of excess pore pressure in free-field conditions can be either estimated through a 1D effective stress analysis, or through simplified pore pressure models. There are several simplified pore pressure models to estimate liquefaction that are often separated into stress-based, strain-based, or energy-based. Some of them only provide the factor of safety against liquefaction triggering, but others provide the whole pore pressure evolution during the earthquake duration including the time of liquefaction triggering. In this work, the strain energy-based model proposed by Millen et al. (2020) was used, but other simplified models that provide the pore pressure time series could be used instead (see Seed et al. 1975; Green et al. 2000; Kokusho 2013; Rios et al. 2019). In the method developed by Millen et al. (2020), liquefaction resistance is measured in terms of normalized cumulative absolute strain energy (NCASE), which was shown to be insensitive to loading amplitude but sensitive to soil properties. NCASE is defined as the cumulative change in absolute elastic strain energy divided by the initial vertical effective stress, σvo′, [Eqs. (1) and (2) and Fig. 3] which can be determined directly from laboratory testing, or a semi-analytical correlation (see Millen et al. 2020), where τ is the shear stress and γ is the shear strain (1) NCASE=1σv0′∑j=0npeaks|τav.,j|·|γj+1−γj|(2) τav.,j={|τj+1+τj|2τj+1·τj≥0|τj+12+τj2|2·|τj+1−τj|τj+1·τj<0}The estimated NCASE demand can be obtained using the nodal surface energy spectrum (NSES), which provides an exact solution for the NCASE at any depth in a homogeneous purely linear elastic soil deposit, and additional correction factors have been proposed by Millen et al. (2020) to account for soil nonlinearity and heterogeneous soil profiles. The NSES method takes the upward motion and combines it with an approximate downward wave (itself but with a depth-dependent time shift that corresponds to the time taken for the seismic wave to travel from the depth of interest to the surface and back). The upward and downward waves are subtracted and the cumulative absolute change in kinetic energy of this combined motion is computed, which corresponds to the NCASE at the depth of interest. This procedure has been implemented into the open-source Python package Liquepy (Millen and Quintero 2022) (see Millen et al. 2020; Rios et al. 2022, for details).Estimation of Surface Ground MotionIn terms of the dynamic response of soil deposits, the soil profile acts as a filter and converts the upward propagating wave to a surface motion. This step can be achieved through effective stress modelling, either of a one-dimensional soil column or a multi-dimensional half space, or through simplified models that account for the change in dynamic behavior due to liquefaction. In this work, the Equivalent Linear Stockwell Analysis (ELSA) method, initially proposed in Viana da Fonseca et al. (2018) and extended by Millen et al. (2021), is used for this purpose. The method makes use of the Stockwell transform to apply time-dependent equivalent linear transfer functions to the input motion to simulate the site response of a liquefying deposit. In the ELSA method, the pore pressure in the free-field is used as there is currently no suitable method to directly estimate excess pore pressure under a building.The method from Millen et al. (2021) consists of the eight steps below, as implemented in the Liquepy Python package (Millen and Quintero 2022) 1.Define modulus reduction and damping curves for each layer of soil (Darendeli 2001).2.Perform an equivalent linear analysis and obtain the strain time series in each layer. In this study, the analysis was performed using the open-source Python package PySRA v0.3.0 (Kottke 2018).3.Adjust the modulus reduction and damping values based on the time series for each layer, where each strain time series is split into one-second intervals and the peak shear strain is taken as the maximum value from a three-second window centered around each one-second interval. The effective strain for each segment is then taken as 90% of the peak strain.4.Estimate the excess pore pressure ratio time series in each liquefiable layer, using available methods [the method from Millen et al. (2020) outlined in the section above was used in this study].5.Using the pore pressure (Step 4) and the strain (Step 3), calculate a pore pressure-adjusted shear modulus and damping in each layer for each time interval using equations from Millen et al. (2021).6.Develop transfer functions for each time interval that convert the input motion to the surface acceleration.7.Compute the Stockwell transform of the input motion.8.Apply each transfer function to the input Stockwell transform at the appropriate time interval to produce the surface Stockwell transform.9.Invert the surface Stockwell transform to obtain the surface acceleration time series.Estimation of Soil-Foundation Interface StiffnessSoil-foundation impedance was modelled using the procedure outlined in Millen et al. (2019) where a single spring and dashpot were used for each vertical, horizontal, and rotational degree of freedom of each footing. The initial spring and dashpot properties were determined using the formulations proposed in Gazetas (1991). The horizontal stiffness was modelled as constant linear elastic, while the vertical and rotational springs were degraded based on the change in excess pore pressure ratio, ru (where ru is defined as the excess pore water pressure divided by the initial vertical effective stress).The vertical stiffness was modelled as a linear no-tension spring. However, the spring stiffness decreased linearly with an increase in ru from an initial value Kv;I, corresponding to ru=0, to a residual value Kv;res at ru=1, where the residual value was calculated based on Karatzia et al. (2017) and a shear wave velocity for the liquefied soil of 30  m/s [15% of the initial value consistent with the range of 10%–30% provided by Karatzia et al. (2017)]. The stiffness was calculated at each step by reading the ru time series from the center of the liquefiable layer in the free-field. Original research by Karatzia et al. (2017) assumed that the liquefiable layer had the same properties under the building as in the free-field, and this assumption is also made here and may under- or over-estimate the soil stiffness, as the presence of the building modifies pore pressure build up and confining stress. The change in the vertical stiffness was not intended to capture the liquefaction-induced settlement (which was modelled through vertical displacements at the spring ends and is typically driven by several mechanisms as well as vertical loading) as the spring deformation in the case study below was less than 2 mm. However, the vertical resistance of the soil can be a crucial parameter in the redistribution of loads and in the change of the dynamic response period of the soil-foundation-structure system. The macro-mechanism structural model was developed in OpenSees and the vertical spring was modelled using the elastic no-tension material (ENT material), which allowed for regular updating of the stiffness.The rotational spring was both deformation- and pore pressure-dependent and was modelled in OpenSees using the uniaxial p-y material PyLiq1 (Boulanger et al. 1999) that allows the stiffness and strength to decrease due to a pore pressure time series and can represent the rocking behavior of a foundation. Herein, the ultimate moment capacity was the input for the non-liquefied soil calculated using (3) where B is the footing width, N is the static vertical load, and Ncap is the foundation bearing capacity in static conditions. To estimate the residual moment capacity, required for the PyLiq1 material, the non-liquefied moment capacity was reduced down by a factor equal to the ratio of liquefied to non-liquefied rotational stiffness from Karatzia et al. (2017). The free-field pore pressure time series was inputted directly into the PyLiq1 material to reduce both the stiffness and capacity. The rotational damping was modelled within the PyLiq1 element, whereas the damping for the other modes was modelled with dashpot elements in parallel with the springs.Estimation of SettlementsThere are several methods available for the simplified assessment of foundation settlement on liquefied soil (Bray and Macedo 2017; Bullock et al. 2019b; Karamitros et al. 2013). Due to the modular nature of the macro-mechanism approach, all procedures can be applied to quantify epistemic uncertainty. The validation study presented in the next section uses the following equation proposed by Bray and Macedo (2017), which was developed from a best-fit regression using the results of a numerical two-dimensional parametric analysis (4) ln(DS)=c1+4.59·ln(Q)−0.42·ln(Q)2+c2·LBS+0.58·ln(tanh(HL))−0.02·B+0.84·ln(CAVdp)+0.41·ln(Sa)+εwhere DS is the foundation settlement, constants c1 and c2 depend on the liquefaction-induced building settlement index (LBS) defined by Eq. (5 ) and have values of −8.35 and 0.072, respectively, for LBS<16, and −7.48 and 0.014 otherwise. Q is the foundation contact pressure, HL is the liquefiable layer thickness, B is the building width, Sa is the spectral acceleration at a period equal to 1 s, and ε is a normal random variable with zero mean and a standard deviation of 0.50.The LBS index, as defined in Eq. (5), is a weighted liquefaction-induced shear strain in the free-field. The value of the free-field shear strain, εshear, is calculated using Zhang et al. (2004), based on the estimated relative density (Dr) of the liquefied soil layer and the calculated safety factor against liquefaction triggering (FSL). Parameter z (m) in Eq. (5) is the depth measured from the ground surface (>0) and W is a foundation-weighting factor wherein W is 0.0 for z values less than Df, which is the embedment depth of the foundation, and 1.0 otherwise.To determine the standardized Cumulate Absolute Velocity (CAVdp) as defined in Campbell and Bozorgnia (2012), Eq. (6) was used, where N is the number of discrete one-second time intervals and H(x) is 0 if x<0 or 1 otherwise. x is (PGAi−0.025) where PGAi is the value of the peak ground acceleration (in g) in each one-second time interval i, inclusive of the first and last values (5) (6) CAVdp=∑i=1N(H(x)∫i−1i|a(t)|dt)To convert the settlement value from Eq. (4) to a time series, either CAVdp can be used as a time series, with all other values as scalars, or additionally the LBS can be considered a time series where the εshear parameter is computed for a time-dependent factor of safety that can be determined using the acceleration time series and a cycle counting procedure or based on the free-field pore pressure time series. The latter option results in less settlement in the initial portion of shaking but ultimately obtains a similar value if the final εshear is similar to the scalar value. For the validation study in Adapazari, Eq. (4) was used with only CAVdp as a time series to estimate the settlements for the five footings of the building using each footing load and width. Then, an average settlement time series of all five footings was calculated to be used in the central footing, with other footings adjusted by the expected tilt.The settlement of each footing (except the central footing) was estimated by multiplying the average settlement pre-calculated using the Bray and Macedo (2017) method (S) by a constant coefficient based on the expected level of tilt (θ). The estimation of this coefficient is based on the simplified empirical model (Bullock et al. 2019a), which results in Eq. (7) below (7) where xfoot is the distance from each footing axis to the middle of the structure and θ is the global tilt.The Simplified Empirical Model for Residual Tilt (Bullock et al. 2019a) can be used to estimate the global tilt. Since this model is based entirely on case histories, it does not suffer from a simulation introduced bias. However, Bullock et al. (2019a) identified that direct numerical simulations resulted in general underprediction in tilt, and stated as potential causes, that the numerical models did not fully incorporate the 3D heterogeneity and ejecta observed in the field. The adopted Bullock et al. (2019a) model depends only on the width of the mat foundation (B, being in this case the total width of the structure), the thickness of the non-liquefiable crust (DS,T), and the average settlement experienced by the foundation (S), as shown in Eq. (8) (8) ln(θ)r=a1·ln(S)+a2·ln(B)+a3·DS,T+εrewhere a1=0.509, a2=−0.936 and a3=−0.102 are fitted parameters and εre is an error term that follows a normal distribution with zero mean and a standard deviation of 0.29.The global tilt (θ) used in Eq. (7) is a probabilistic value calculated using (9) where σre* represents the total uncertainty (Bullock et al. 2019a) expressed in Eq. (10) (10) σre=*{σre,1B≤B1*σre,1−(σre,1−σre,2)(B−B1*)(B2*−B1*)B1*

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