Analysis of the Out-Of-Plane Seismic Behavior of Unreinforced Masonry: A Literature Review

Although the issue of the out-of-plane response of unreinforced masonry structures under earthquake excitation is well known with consensus among the research community, this issue is simultaneously one of the more complex and most neglected areas on the seismic assessment of existing buildings. Nonetheless, its characterization should be found on the solid knowledge of the phenomenon and on the complete understanding of methodologies currently used to describe it. Based on this assumption, this article presents a general framework on the issue of the out-of-plane performance of unreinforced masonry structures, beginning with a brief introduction to the topic, followed by a compact state of art in which the principal methodologies proposed to assess the out-of-plane behavior of unreinforced masonry structures are presented. Different analytical approaches are presented, namely force and displacement-based, complemented with the presentation of existing numerical tools for the purpose presented above. Moreover, the most relevant experimental campaigns carried out in order to reproduce the phenomenon are reviewed and briefly discussed.

Numerical Assessment of the out-of-plane response of a brick masonry structure without box behaviour

This paper presents the assessment of the out-of-plane response due to seismic loading of a masonry structure without rigid diaphragm. This structure corresponds to real scale brick masonry specimen with a main façade connected to two return walls. Two modelling approaches were defined for this evaluation. The first one consisted on macro modelling, whereas the second one on simplified micro modelling. As a first step of this study, static nonlinear analyses were conducted to the macro model aiming at evaluating the out-of-plane response and failure mechanism of the masonry structure. A sensibility analyses was performed in order to assess the mesh size and material model dependency. In addition, the macro models were subjected to dynamic nonlinear analyses with time integration in order to assess the collapse mechanism. Finally, these analyses were also applied to a simplified micro model of the masonry structure. Furthermore, these results were compared to experimental response from shaking table tests. It was observed that these numerical techniques simulate correctly the in-plane behaviour of masonry structures. However, the

Effect of environmental ageing on the numerical response of FRP-strengthened masonry walls

Recent durability studies have shown the susceptibility of bond in fiber-reinforced polymer (FRP) strengthened masonry components to hygrothermal exposures. However, it is not clear how this local material degradation affects the global behavior of FRP-strengthened masonry structures. This study addresses this issue by numerically investigating the nonlinear behavior of FRP-masonry walls after aging in two different environmental conditions. A numerical modeling strategy is adopted and validated with existing experimental tests on FRP-strengthened masonry panels. The model, once validated, is used for modeling of four hypothetical FRP-strengthened masonry walls with different boundary conditions, strengthening schemes, and reinforcement ratios. The nonlinear behavior of the walls is then simulated before and after aging in two different environmental conditions. The degradation data are taken from previous accelerated aging tests. The changes in the failure mode and nonlinear response of the walls after aging are presented and discussed.

Seismic vulnerability of existing masonry buildings: Nonlinear parametric analysis

Existing masonry structures are usually associated to a high seismic vulnerability, mainly due to the properties of the materials, weak connections between floors and load-bearing walls, high mass of the masonry walls and flexibility of the floors. For these reasons, the seismic performance of existing masonry structures has received much attention in the last decades. This study presents the parametric analysis taking into account the deviations on features of the gaioleiro buildings - Portuguese building typology. The main objective of the parametric analysis is to compare the seismic performance of the structure as a function of the variations of its properties with respect to the response of a reference model. The parametric analysis was carried out for two types of structural analysis, namely for the non-linear dynamic analysis with time integration and for the pushover analysis with distribution of forces proportional to the inertial forces of the structure. The Young's modulus of the masonry walls, Young's modulus of the timber floors, the compressive and tensile non-linear properties (strength and fracture energy) were the properties considered in both type of analysis. Additionally, in the dynamic analysis, the influences of the vis-cous damping and of the vertical component of the earthquake were evaluated. A pushover analysis proportional to the modal displacement of the first mode in each direction was also carried out. The results shows that the Young's modulus of the masonry walls, the Young's modulus of the timber floors and the compressive non-linear properties are the pa-rameters that most influence the seismic performance of this type of tall and weak existing masonry structures. Furthermore, it is concluded that that the stiffness of the floors influences significantly the strength capacity and the collapse mecha-nism of the numerical model. Thus, a study on the strengthening of the floors was also carried out. The increase of the thickness of the timber floors was the strengthening technique that presented the best seismic performance, in which the reduction of the out-of-plane displacements of the masonry walls is highlighted.

Seismic evaluation of environmentally degraded FRP-strengthened masonry walls

Fiber Reinforced Polymers (FRPs) have been extensively used for externally bonded reinforcement of masonry structures during the last years. Available information shows that FRPs can significantly improve the seismic performance of masonry elements without altering their structural mass. However, the durability and long-term performance of the strengthened elements are not clearly known yet. Recent experimental results show that environmental conditions can lead to degradation of the bond between FRP and masonry and FRP delaminations. But the effect of these local degradation mechanisms on the global structural response is not studied yet. This paper is therefore aimed at numerically investigating the effect of environmental degradation on the global performance of strengthened masonry walls. The nonlinear behavior of masonry walls strengthened with FRP composites is initially simulated with the aim of a FE package. The adopted numerical modeling strategy is verified by comparison of numerical and experimental results. The model, once validated, is used for investigating the effect of materials and bond degradation on the global behavior and failure modes of strengthened walls. The effect of strengthening scheme on the long-term performance of strengthened walls is also investigated. The degradation data are taken from experimental tests previously performed by the authors. The numerical results show that the effect of local material degradation on the global response of strengthened walls depends on the strengthening schemes, and severity of the environmental conditions. Moreover, environmental induced degradations and FRP delaminations can lead to change of expected failure modes in the strengthened elements. These observations, that are usually neglected at the design stage, can be critical in the long-term performance of strengthened structures.

Seismic retrofit of masonry-to-timber connections in historical constructions

Tese de Doutoramento em Engenharia Civil.

Measurements of seismic attenuation and transient fluid pressure in partially saturated Berea sandstone: Evidence of fluid flow on the mesoscopic scale

A novel laboratory technique is proposed to investigate wave-induced fluid flow on the mesoscopic scale as a mechanism for seismic attenuation in partially saturated rocks. This technique combines measurements of seismic attenuation in the frequency range from 1 to 100?Hz with measurements of transient fluid pressure as a response of a step stress applied on top of the sample. We used a Berea sandstone sample partially saturated with water. The laboratory results suggest that wave-induced fluid flow on the mesoscopic scale is dominant in partially saturated samples. A 3-D numerical model representing the sample was used to verify the experimental results. Biot's equations of consolidation were solved with the finite-element method. Wave-induced fluid flow on the mesoscopic scale was the only attenuation mechanism accounted for in the numerical solution. The numerically calculated transient fluid pressure reproduced the laboratory data. Moreover, the numerically calculated attenuation, superposed to the frequency-independent matrix anelasticity, reproduced the attenuation measured in the laboratory in the partially saturated sample. This experimental?numerical fit demonstrates that wave-induced fluid flow on the mesoscopic scale and matrix anelasticity are the dominant mechanisms for seismic attenuation in partially saturated Berea sandstone.

Seismic attenuation due to wave-induced fluid flow at microscopic and macroscopic scales

Wave-induced fluid flow at microscopic and mesoscopic scales arguably constitutes the major cause of intrinsic seismic attenuation throughout the exploration seismic and sonic frequency ranges. The quantitative analysis of these phenomena is, however, complicated by the fact that the governing physical processes may be dependent. The reason for this is that the presence of microscopic heterogeneities, such as micro-cracks or broken grain contacts, causes the stiffness of the so-called modified dry frame to be complex-valued and frequency-dependent, which in turn may affect the viscoelastic behaviour in response to fluid flow at mesoscopic scales. In this work, we propose a simple but effective procedure to estimate the seismic attenuation and velocity dispersion behaviour associated with wave-induced fluid flow due to both microscopic and mesoscopic heterogeneities and discuss the results obtained for a range of pertinent scenarios.

The El Masnou Infralittoral sedimentary environment (Barcelona province, NW Mediterranean Sea): morphology and Holocene seismic stratigraphy

A detailed analysis of the morphology and the Holocene seismic and sequence stratigraphy and architecture of the infralittoral sedimentary environment of the El Masnou coast (Catalonia, NW Mediterranean Sea) was carried out using multibeam bathymetry and GeoPulse seismic data. This environment extends down to 26-30 m water depth, and is defined morphologically by two depositional wedges whose seafloor is affected by erosive furrows, slides, fields of large- and small-scale wavy bedforms, and dredging trenches and pits. Erosive terraces are also identified in the transition domain toward the inner continental shelf. The Holocene stratigraphy of the infralittoral environment is defined by two major seismic sequences (lower and upper), each one formed by internal seismic units. The sequences and units are characterised by downlapping surfaces made up of deposits formed by progradation of coastal lithosomes. The stratigraphy and stratal architecture, displaying a retrogradational arrangement with progradational patterns of minor order, were controlled by different sea-level positions. The stratigraphic division represents the coastal response to the last fourth-order transgressive and highstand conditions, modulated by small-scale sea-level oscillations (≈1-2 m) of fith to sixth order. This study also highlights the advantage of an integrated analysis using acoustic/seismic methods for practical assessment of the anthropogenic effects on infralittoral domains based on the association of marine geological observations.

Seismic characterization of heterogeneous sedimentary and fractured rocks based on Biot's theory of poroelasticity

Understanding and quantifying seismic energy dissipation, which manifests itself in terms of velocity dispersion and attenuation, in fluid-saturated porous rocks is of considerable interest, since it offers the perspective of extracting information with regard to the elastic and hydraulic rock properties. There is increasing evidence to suggest that wave-induced fluid flow, or simply WIFF, is the dominant underlying physical mechanism governing these phenomena throughout the seismic, sonic, and ultrasonic frequency ranges. This mechanism, which can prevail at the microscopic, mesoscopic, and macroscopic scale ranges, operates through viscous energy dissipation in response to fluid pressure gradients and inertial effects induced by the passing wavefield. In the first part of this thesis, we present an analysis of broad-band multi-frequency sonic log data from a borehole penetrating water-saturated unconsolidated glacio-fluvial sediments. An inherent complication arising in the interpretation of the observed P-wave attenuation and velocity dispersion is, however, that the relative importance of WIFF at the various scales is unknown and difficult to unravel. An important generic result of our work is that the levels of attenuation and velocity dispersion due to the presence of mesoscopic heterogeneities in water-saturated unconsolidated clastic sediments are expected to be largely negligible. Conversely, WIFF at the macroscopic scale allows for explaining most of the considered data while refinements provided by including WIFF at the microscopic scale in the analysis are locally meaningful. Using a Monte-Carlo-type inversion approach, we compare the capability of the different models describing WIFF at the macroscopic and microscopic scales with regard to their ability to constrain the dry frame elastic moduli and the permeability as well as their local probability distribution. In the second part of this thesis, we explore the issue of determining the size of a representative elementary volume (REV) arising in the numerical upscaling procedures of effective seismic velocity dispersion and attenuation of heterogeneous media. To this end, we focus on a set of idealized synthetic rock samples characterized by the presence of layers, fractures or patchy saturation in the mesocopic scale range. These scenarios are highly pertinent because they tend to be associated with very high levels of velocity dispersion and attenuation caused by WIFF in the mesoscopic scale range. The problem of determining the REV size for generic heterogeneous rocks is extremely complex and entirely unexplored in the given context. In this pilot study, we have therefore focused on periodic media, which assures the inherent self- similarity of the considered samples regardless of their size and thus simplifies the problem to a systematic analysis of the dependence of the REV size on the applied boundary conditions in the numerical simulations. Our results demonstrate that boundary condition effects are absent for layered media and negligible in the presence of patchy saturation, thus resulting in minimum REV sizes. Conversely, strong boundary condition effects arise in the presence of a periodic distribution of finite-length fractures, thus leading to large REV sizes. In the third part of the thesis, we propose a novel effective poroelastic model for periodic media characterized by mesoscopic layering, which accounts for WIFF at both the macroscopic and mesoscopic scales as well as for the anisotropy associated with the layering. Correspondingly, this model correctly predicts the existence of the fast and slow P-waves as well as quasi and pure S-waves for any direction of wave propagation as long as the corresponding wavelengths are much larger than the layer thicknesses. The primary motivation for this work is that, for formations of intermediate to high permeability, such as, for example, unconsolidated sediments, clean sandstones, or fractured rocks, these two WIFF mechanisms may prevail at similar frequencies. This scenario, which can be expected rather common, cannot be accounted for by existing models for layered porous media. Comparisons of analytical solutions of the P- and S-wave phase velocities and inverse quality factors for wave propagation perpendicular to the layering with those obtained from numerical simulations based on a ID finite-element solution of the poroelastic equations of motion show very good agreement as long as the assumption of long wavelengths remains valid. A limitation of the proposed model is its inability to account for inertial effects in mesoscopic WIFF when both WIFF mechanisms prevail at similar frequencies. Our results do, however, also indicate that the associated error is likely to be relatively small, as, even at frequencies at which both inertial and scattering effects are expected to be at play, the proposed model provides a solution that is remarkably close to its numerical benchmark. -- Comprendre et pouvoir quantifier la dissipation d'énergie sismique qui se traduit par la dispersion et l'atténuation des vitesses dans les roches poreuses et saturées en fluide est un intérêt primordial pour obtenir des informations à propos des propriétés élastique et hydraulique des roches en question. De plus en plus d'études montrent que le déplacement relatif du fluide par rapport au solide induit par le passage de l'onde (wave induced fluid flow en anglais, dont on gardera ici l'abréviation largement utilisée, WIFF), représente le principal mécanisme physique qui régit ces phénomènes, pour la gamme des fréquences sismiques, sonique et jusqu'à l'ultrasonique. Ce mécanisme, qui prédomine aux échelles microscopique, mésoscopique et macroscopique, est lié à la dissipation d'énergie visqueuse résultant des gradients de pression de fluide et des effets inertiels induits par le passage du champ d'onde. Dans la première partie de cette thèse, nous présentons une analyse de données de diagraphie acoustique à large bande et multifréquences, issues d'un forage réalisé dans des sédiments glaciaux-fluviaux, non-consolidés et saturés en eau. La difficulté inhérente à l'interprétation de l'atténuation et de la dispersion des vitesses des ondes P observées, est que l'importance des WIFF aux différentes échelles est inconnue et difficile à quantifier. Notre étude montre que l'on peut négliger le taux d'atténuation et de dispersion des vitesses dû à la présence d'hétérogénéités à l'échelle mésoscopique dans des sédiments clastiques, non- consolidés et saturés en eau. A l'inverse, les WIFF à l'échelle macroscopique expliquent la plupart des données, tandis que les précisions apportées par les WIFF à l'échelle microscopique sont localement significatives. En utilisant une méthode d'inversion du type Monte-Carlo, nous avons comparé, pour les deux modèles WIFF aux échelles macroscopique et microscopique, leur capacité à contraindre les modules élastiques de la matrice sèche et la perméabilité ainsi que leur distribution de probabilité locale. Dans une seconde partie de cette thèse, nous cherchons une solution pour déterminer la dimension d'un volume élémentaire représentatif (noté VER). Cette problématique se pose dans les procédures numériques de changement d'échelle pour déterminer l'atténuation effective et la dispersion effective de la vitesse sismique dans un milieu hétérogène. Pour ce faire, nous nous concentrons sur un ensemble d'échantillons de roches synthétiques idéalisés incluant des strates, des fissures, ou une saturation partielle à l'échelle mésoscopique. Ces scénarios sont hautement pertinents, car ils sont associés à un taux très élevé d'atténuation et de dispersion des vitesses causé par les WIFF à l'échelle mésoscopique. L'enjeu de déterminer la dimension d'un VER pour une roche hétérogène est très complexe et encore inexploré dans le contexte actuel. Dans cette étude-pilote, nous nous focalisons sur des milieux périodiques, qui assurent l'autosimilarité des échantillons considérés indépendamment de leur taille. Ainsi, nous simplifions le problème à une analyse systématique de la dépendance de la dimension des VER aux conditions aux limites appliquées. Nos résultats indiquent que les effets des conditions aux limites sont absents pour un milieu stratifié, et négligeables pour un milieu à saturation partielle : cela résultant à des dimensions petites des VER. Au contraire, de forts effets des conditions aux limites apparaissent dans les milieux présentant une distribution périodique de fissures de taille finie : cela conduisant à de grandes dimensions des VER. Dans la troisième partie de cette thèse, nous proposons un nouveau modèle poro- élastique effectif, pour les milieux périodiques caractérisés par une stratification mésoscopique, qui prendra en compte les WIFF à la fois aux échelles mésoscopique et macroscopique, ainsi que l'anisotropie associée à ces strates. Ce modèle prédit alors avec exactitude l'existence des ondes P rapides et lentes ainsi que les quasis et pures ondes S, pour toutes les directions de propagation de l'onde, tant que la longueur d'onde correspondante est bien plus grande que l'épaisseur de la strate. L'intérêt principal de ce travail est que, pour les formations à perméabilité moyenne à élevée, comme, par exemple, les sédiments non- consolidés, les grès ou encore les roches fissurées, ces deux mécanismes d'WIFF peuvent avoir lieu à des fréquences similaires. Or, ce scénario, qui est assez commun, n'est pas décrit par les modèles existants pour les milieux poreux stratifiés. Les comparaisons des solutions analytiques des vitesses des ondes P et S et de l'atténuation de la propagation des ondes perpendiculaires à la stratification, avec les solutions obtenues à partir de simulations numériques en éléments finis, fondées sur une solution obtenue en 1D des équations poro- élastiques, montrent un très bon accord, tant que l'hypothèse des grandes longueurs d'onde reste valable. Il y a cependant une limitation de ce modèle qui est liée à son incapacité à prendre en compte les effets inertiels dans les WIFF mésoscopiques quand les deux mécanismes d'WIFF prédominent à des fréquences similaires. Néanmoins, nos résultats montrent aussi que l'erreur associée est relativement faible, même à des fréquences à laquelle sont attendus les deux effets d'inertie et de diffusion, indiquant que le modèle proposé fournit une solution qui est remarquablement proche de sa référence numérique.

Developing a system for down-hole seismic testing together with the CPTU

This paper presents a system for performing down-hole seismic test together with the piezocone test in order to determine the shear wave velocity (Vs) and for calculating the maximum shear modulus (Go); a basic parameter for analyzing the dynamic soil behavior and a reference value of the soil stiffness. The system components are described and tests results for checking the geophone response are also presented, both before and after installation into the probe. The system was used in down-hole tests carried out at three experimental research sites located in the interior of Sao Paulo State, Brazil, where in situ seismic test results are available. The Vs values measured in down-hole tests carried out with this system were consistent with those determined in cross-hole tests and with a commercial seismic piezocone, which enabled to validate the developed system.

Long-term recovery patterns of arctic tundra after winter seismic exploration

In response to the increasing global demand for energy, oil exploration and development are expanding into frontier areas of the Arctic, where slow-growing tundra vegetation and the underlying permafrost soils are very sensitive to disturbance. The creation of vehicle trails on the tundra from seismic exploration for oil has accelerated in the past decade, and the cumulative impact represents a geographic footprint that covers a greater extent of Alaska’s North Slope tundra than all other direct human impacts combined. Seismic exploration for oil and gas was conducted on the coastal plain of the Arctic National Wildlife Refuge, Alaska, USA, in the winters of 1984 and 1985. This study documents recovery of vegetation and permafrost soils over a two-decade period after vehicle traffic on snow-covered tundra. Paired permanent vegetation plots (disturbed vs. reference) were monitored six times from 1984 to 2002. Data were collected on percent vegetative cover by plant species and on soil and ground ice characteristics. We developed Bayesian hierarchical models, with temporally and spatially autocorrelated errors, to analyze the effects of vegetation type and initial disturbance levels on recovery patterns of the different plant growth forms as well as soil thaw depth. Plant community composition was altered on the trails by species-specific responses to initial disturbance and subsequent changes in substrate. Long-term changes included increased cover of graminoids and decreased cover of evergreen shrubs and mosses. Trails with low levels of initial disturbance usually improved well over time, whereas those with medium to high levels of initial disturbance recovered slowly. Trails on ice-poor, gravel substrates of riparian areas recovered better than those on ice-rich loamy soils of the uplands, even after severe initial damage. Recovery to pre-disturbance communities was not possible where trail subsidence occurred due to thawing of ground ice. Previous studies of disturbance from winter seismic vehicles in the Arctic predicted short-term and mostly aesthetic impacts, but we found that severe impacts to tundra vegetation persisted for two decades after disturbance under some conditions. We recommend management approaches that should be used to prevent persistent tundra damage.

Interplay of Late Cenozoic Siliciclastic Supply and Carbonate Response on the Southeast Florida Platform

High-resolution seismic-reflection data collected along the length of the Caloosahatchee River in southwestern Florida have been correlated to nannofossil biostratigraphy and strontium-isotope chemostratigraphy at six continuously cored boreholes. These data are interpreted to show a major Late Miocene(?) to Early Pliocene fluvial– deltaic depositional system that prograded southward across the carbonate Florida Platform, interrupting nearly continuous carbonate deposition since early in the Cretaceous. Connection of the platform top to a continental source of siliciclastics and significant paleotopography combined to focus accumulation of an immense supply of siliciclastics on the southeastern part of the Florida Platform. The remarkably thick (> 100 m), sand-rich depositional system, which is characterized by clinoformal progradation, filled in deep accommodation, while antecedent paleotopography directed deltaic progradation southward within the middle of the present-day Florida Peninsula. The deltaic depositional system may have prograded about 200 km southward to the middle and upper Florida Keys, where Late Miocene to Pliocene siliciclastics form the foundation of the Quaternary carbonate shelf and shelf margin of the Florida Keys. These far-traveled siliciclastic deposits filled accommodation on the southeastern part of the Florida Platform so that paleobathymetry was sufficiently shallow to allow Quaternary recovery of carbonate sedimentation in the area of southern peninsular Florida and the Florida Keys.

Three dimensional seismic imaging and earthquake locations in a complex, segmented fault region in Southern Apennines (Italy)

The southern Apennines of Italy have been experienced several destructive earthquakes both in historic and recent times. The present day seismicity, characterized by small-to-moderate magnitude earthquakes, was used like a probe to obatin a deeper knowledge of the fault structures where the largest earthquakes occurred in the past. With the aim to infer a three dimensional seismic image both the problem of data quality and the selection of a reliable and robust tomographic inversion strategy have been faced. The data quality has been obtained to develop optimized procedures for the measurements of P- and S-wave arrival times, through the use of polarization filtering and to the application of a refined re-picking technique based on cross-correlation of waveforms. A technique of iterative tomographic inversion, linearized, damped combined with a strategy of multiscale inversion type has been adopted. The retrieved P-wave velocity model indicates the presence of a strong velocity variation along a direction orthogonal to the Apenninic chain. This variation defines two domains which are characterized by a relatively low and high velocity values. From the comparison between the inferred P-wave velocity model with a portion of a structural section available in literature, the high velocity body was correlated with the Apulia carbonatic platforms whereas the low velocity bodies was associated to the basinal deposits. The deduced Vp/Vs ratio shows that the ratio is lower than 1.8 in the shallower part of the model, while for depths ranging between 5 km and 12 km the ratio increases up to 2.1 in correspondence to the area of higher seismicity. This confirms that areas characterized by higher values are more prone to generate earthquakes as a response to the presence of fluids and higher pore-pressures.

Non linear response of planar asymmetric systems

The seismic behaviour of one-storey asymmetric structures has been studied since 1970s by a number of researches studies which identified the coupled nature of the translational-to-torsional response of those class of systems leading to severe displacement magnifications at the perimeter frames and therefore to significant increase of local peak seismic demand to the structural elements with respect to those of equivalent not-eccentric systems (Kan and Chopra 1987). These studies identified the fundamental parameters (such as the fundamental period TL normalized eccentricity e and the torsional-to-lateral frequency ratio Ωϑ) governing the torsional behavior of in-plan asymmetric structures and trends of behavior. It has been clearly recognized that asymmetric structures characterized by Ωϑ >1, referred to as torsionally-stiff systems, behave quite different form structures with Ωϑ <1, referred to as torsionally-flexible systems. Previous research works by some of the authors proposed a simple closed-form estimation of the maximum torsional response of one-storey elastic systems (Trombetti et al. 2005 and Palermo et al. 2010) leading to the so called “Alpha-method” for the evaluation of the displacement magnification factors at the corner sides. The present paper provides an upgrade of the “Alpha Method” removing the assumption of linear elastic response of the system. The main objective is to evaluate how the excursion of the structural elements in the inelastic field (due to the reaching of yield strength) affects the displacement demand of one-storey in-plan asymmetric structures. The system proposed by Chopra and Goel in 2007, which is claimed to be able to capture the main features of the non-linear response of in-plan asymmetric system, is used to perform a large parametric analysis varying all the fundamental parameters of the system, including the inelastic demand by varying the force reduction factor from 2 to 5. Magnification factors for different force reduction factor are proposed and comparisons with the results obtained from linear analysis are provided.