Fragility and reliability analyses of soil -pile -bridge pier interaction

U radu je prikazana procedura evaluacije seizmičkih performansi interakcije šip-tlo inkrementalnom nelinearnom dinamičkom analizom (INDA -Incremental Nonlinear Dynamic Analysis). Ulazni signal u sistemu je tretiran preko generisanih veštačkih akcelerograma, a koji su dodatno procesirani po slojevima tla do osnovne stene. Postprocesiranje INDA analiza izvršeno je posebno za stub, a posebno za šip, tako da su konstruisane krive PGA=f(DR) u kapacitativnom domenu. Za ovako konstruisane krive određeni su performansni nivoi, a na osnovu određenih DR i PGA parametara sprovedene su regresione analize. Krive povredljivosti su konstruisane na osnovu rešenja regresione analize i teorije verovatnoće log-normalne raspodele. Takođe, konstruisane su i krive pouzdanosti na osnovu rešenja analize povredljivosti. Metodološki postupak za analizu seizmičkih performansi, prezentovan u ovom istraživanju, omogućava integrisano kvantitativno-kvalitativno razmatranje i evaluaciju kompleksne interakcije konstrukcija-tlo (SFSI -Soil-Foundation-Structure Interaction).The purpose of this paper is to present the methodology for performance-based seismic evaluation of soil-pile-bridge pier interaction using the incremental nonlinear dynamic analysis (INDA). The system's input signal was treated through the generated artificial accelerograms which were subsequently processed by soil layers and for the bedrock. The INDA analysis was post processed separately for the pier and for the pile, so that the constructed PGA=f(DR) curves are in the capacitive domain. For these curves the authors identified the performance levels, while the regression analyses were conducted based on the specific DR and PGA parameters. Fragility curves were constructed based on the solutions of regression analysis and the probability theory of log-normal distribution. Based on the results of fragility analysis, reliability curves were also constructed. The methodological procedure for seismic performance analysis presented in this study provides an integrated quantitative-qualitative consideration and evaluation of the complex soil-foundation-structure interaction (SFSI)

Structural assessment and seismic vulnerability of earthen historic structures. Application of sophisticated numerical and simple analytical models

Adobe constructions account for a significant portion of the built heritage, associated with early building techniques, material accessibility and low-cost. Nonetheless, adobe buildings, due to their low mechanical properties and overturning resistance, are subject to early structural damage, such as cracking, separation of structural elements and, possibly, collapse in areas of high seismic hazard. The lack of maintenance and absence of adequate retrofitting techniques usually intensifies the loss of historic fabric. The current paper, aims at the structural assessment and seismic safety, in current conditions, of the Church of Kuno Tambo, a religious adobe structure of the 17th century, in Cusco region, in Peru. The inspection and diagnosis involved sonic testing and damage mapping, while ambient vibration tests revealed the modal response of the structure. The assessment of seismic vulnerability, together with the necessity of retrofitting measures were verified through nonlinear static and pushover parametric analyses, complemented with a macro-block limit analysis and a performance based assessment, under local seismic criteria. A more realistic response from dynamically induced ground motions was performed, by a nonlinear time history analysis, according to the Eurocode 8 framework. Through an integrated approach, in situ inspection, testing, numerical and analytical modelling are associated under the scope of reproducing the existing structural damage, the sequence of inelastic behavior and verification of the necessity of retrofitting measures.The current work is part of the Seismic Retrofitting Project, of the Getty Conservation Institute and was partly supported by FCT (Portuguese Foundation for Science and Technology), within the INFRARISK PhD program and ISISE, project UID/ECl/04029/2013.info:eu-repo/semantics/publishedVersio

Seismic assessment of the medieval Armenian church in Famagusta, Cyprus

This paper presents the analysis of the structural performance of the Armenian Church in Famagusta, Cyprus. Visual and photographic survey, non-destructive testing and numerical evaluation, using a tridimensional finite element model, are included. The non-linear behavior is analyzed under gravity and seismic loading, and pushover and time-history analysis are performed, including in both cases bidirectional loading. Results are studied in terms of capacity curves and damage patterns. The influence of the Church mass-distribution is also evaluated. As a result, the analysis estimates an adequate safety level of the building and various real damage features of the Church are justified and attributed to seismic actions.The authors acknowledge the support from Dr. Michael Walsh and the Nanyang Technological University of Singapore. Further credits are given to Solvotek Engineering, from Turkey, which performed the laser scans of the Church.During the realization of this study, the first author was an European Commission Erasmus Mundus grant holder of the Advanced Masters in Structural Analysis of Monuments and Historical Constructions, as well as grant holder of the Mexican Secretariat of Public Education (SEP) and the Mexican Government

Spectral Matching of Three-Component Seismic Ground Accelerations for Critical Structures

Abstract This study introduces a new procedure for three-component spectral matching of seismic ground acceleration records. The procedure involves preserving the phase angle and varying the Fourier amplitudes to match the spectra of the recorded records with a provided target spectrum that represents the spectral values of ground motions in three orthogonal directions. Several three-component recorded records were matched against two target response spectra to test the method. An accurate convergence was achieved in the velocity and displacement records. In addition, the scales among the components’ magnitude were maintained, creating realistic three-component ground motions. Such ground motion inputs are important to qualify critical structures sensitive to three-component motions. Our proposed procedure is novel because the present practices are mostly created based on one-component ground motions. Keywords: Spectral matching, three components, component scales, phase angle, critical structures. Abstrak Studi ini mengusulkan prosedur baru terkait penyesuaian spektra catatan percepatan gempa. Prosedur ini dilakukan dengan mempertahankan sudut phasa tetapi mengubah amplitudo Fourier catatan percepatan gempa demikian hingga spektrumnya sesuai dengan target. Target spektrum yang diinputkan mewakili tiga komponen gempa. Untuk mengujinya, beberapa catatan gempa tiga-komponen disesuaikan spektrumnya terhadap dua target spektra. Ketelitian yang cukup baik terlihat dari catatan kecepatan dan simpangan. Skala ketiga komponen juga dijaga sehingga dihasilkan percepatan gempa tiga-komponen yang realistis. Percepatan gempa sejenis ini diperlukan dalam proses kualifikasi struktur penting yang sensitif terhadap pengaruh gerakan tiga arah. Prosedur tersebut adalah baru karena teknik yang ada sekarang didasarkan pada penyesuain spektrum per-komponen. Kata-kata Kunci: Penyesuaian spektra, tiga komponen, skala komponen, sudut phasa, struktur penting.

Developing a methodology for the integrated numerical evaluation and performance assessment of soil-pile-pier

In the paper is developed a discrete numerical solid pile model with a discontinuity and defects. Model included performance-based seismic evaluation of the soil-pile-bridge pier interaction. The pile discontinuity and defects are modelled by reducing the specific finite elements and elastic modulus of concrete. The wave-propagation response of the pile was analyzed based on a step-by-step numerical integration using the Hilber-Hughes-Taylor (HHT) method in time domain (THA). The response analysis is performed with an integration of individual reflectograms into a reflectogram surface, which is generated in a 3D cylindrical coordinate system. Non-linear response of the system is considered using the incremental-iterative Newton-Raphson`s method, while the stability analysis is performed according to the modified geometrical nonlinearity analysis of stability. Determination of critical load and effective length of the pile are performed based on numerical solution and using regression analysis of the power function. The procedure of the soil-pile-bridge pier performance evaluation is based on the incremental nonlinear dynamic analysis (INDA). The system's input signal is treated through the generated artificial accelerograms, which were subsequently processed by soil layers and for the bedrock. Fragility curves were constructed based on solutions of the regression analysis and the probability theory of log-normal distribution, while the generation of reliability curves is based on a solution of vulnerability

Design Earthquakes Based on Probabilistic Seismic Hazard Analysis

Earthquake is one of the most destructive natural disasters leading to financial, environmental, and even human losses. The most effective approach to prevent losses induced by structural damage is seismic design for structures, in which the determination of design earthquakes, including seismic design spectra and seismic design ground motions, is of great importance. Probabilistic Seismic Hazard Analysis (PSHA) has been widely used for the determination and selection of design earthquakes. However, there are a number of issues on the engineering application of PSHA in obtaining the design earthquakes, which need to be addressed before it can be readily implemented into reliability- and performance-based seismic design. In this research, based on the PSHA, the generation of seismic design spectra and spectrum-compatible earthquake groundmotions is studied. The PSHA-based seismic design spectra mainly include Uniform Hazard Spectrum (UHS), predicted spectrum based on Ground-Motion Prediction Equations (GMPEs), and Conditional Mean Spectrum considering ε (CMS-ε). These existing design spectra, however, do not or only partially provide probabilistic knowledge about the simultaneous occurrence of spectral accelerations at multiple vibration periods. The lack of such probabilistic knowledge of the design spectra may prevent them from being incorporated into reliability- and performance-based seismic design. The purpose of this study is to bridge the gaps between seismological analyses and engineering applications, i.e., to find suitable representations of design earthquakes from the PSHA. A generalized approach is developed to generate seismic design spectra using both scalar and vector-valued PSHA,which overcomes the deficiencies and preserves certain advantages of the existing PSHA-based seismic design spectra. An approximate approach is also developed to simplify the approach to the generation of seismic design spectra so that they can be easily incorporated into structural design and further performance-based seismic design. On the other hand, spectrum-compatible earthquake ground motions, which are generated by manipulating recorded ground motions, have been widely used for seismic design verification and seismic qualification of structures. The existing spectral matching algorithms in frequency-domain, however, may distort the valuable information contained in recorded earthquake ground motions due to the deficiency of the transformation methodologies on which they are based. To properly preserve the frequency contents and nonstationary characteristics of recorded ground motions, a signal processing method called Hilbert-Huang Transform (HHT) is used to generate spectrum-compatible earthquake ground motions. In the proposed generation procedures, the strategy of the selection of recorded ground motions is based on the PSHA so that the generated ground motions reflect realistic seismic hazard for the site of interest

Multi-angle and non-uniform ground motions on cable-stayed bridges

The definition of the spatial variability of the ground motion (SVGM) is a complex and multi-parametric problem. Its effect on the seismic response of cable-stayed bridges is important, yet not entirely understood to date. This work examines the effect of the SVGM on the seismic response of cable-stayed bridges by means of the time delay of the ground motion at different supports and of the loss of coherency of the seismic waves. The focus herein is the effect of the SVGM on cable-stayed bridges with various configurations in terms of their length and of design parameters such a s t he pylon shape and the pylon–cable system configuration, combined with the influence of the incidence angle of the earthquakes. The aim of this paper is to provide general conclusions that are applicable to a wide range of cable-stayed bridges, instead of attempting to interpret the effect SVGM on a case-by-case basis, and to contribute to the ongoing effort to interpret and predict the effect of the SVGM. It has been found that the effect of the SVGM on the seismic response of cable-stayed bridges varies depending on the pylon shape, height and section dimensions, on the cable-system configuration and on the response quantity of interest. Furthermore, the earthquake incidence angle defines whether t he S VGM is important t o t he seismic response of the cable-stayed bridges. It is also observed that the SVGM excites vibration modes of the bridges that do not contribute to their seismic response when identical support motion is considered

Inelastic response of cable-stayed bridges subjected to non-uniform motions

This paper studies for the first time the effect of the Spatial Variability of Ground Motions (SVGM) with large intensities on the inelastic seismic response of the pylons which are responsible for the overall structural integrity of cable-stayed bridges. The svgm is defined by the time delay of the earthquake at different supports, the loss of coherency of the seismic waves and the incidence angle of the ground motion. An extensive study is conducted on cable-stayed bridges with ‘H’- and inverted ‘Y’-shaped pylons and with main spans of 200, 400 and 600 m. The svgm is most detrimental to the pylon of the 200-m span bridge owing to the large stiffness of this bridge compared to its longer counterparts. The stiff configuration of the inverted ‘Y’-shaped pylon makes it more susceptible against the multi-support excitation than the flexible ‘H’-shaped pylon, especially in the transverse direction of the response. Finally, the earthquake incidence angle is strongly linked with the svgm and should be included in the seismic design of cable-stayed bridges

Macro-element nonlinear dynamic analysis for the assessment of the seismic vulnerability of masonry structures

Doctoral Thesis Civil EngineeringEarthquakes constitutes one of the most devastating natural hazards since they lead to the collapse of buildings; and consequently, a significant number of human losses. Some typologies of buildings, namely historical and masonry constructions, are one of the most vulnerable elements at risk due to their weak performance when subjected to seismic actions. For instance, historic structures were built based on simple rules since seismic codes were not properly established at the time of their construction. On the other hand, many masonry structures, especially in developing countries, are usually constructed without taking into consideration the specifications provided by current seismic codes. These constructions are mainly characterised by a poor connection between orthogonal walls and between walls and horizontal diaphragms which lead to the occurrence of out-of-plane mechanisms. This behaviour is considered one of the most vulnerable, yet one of the most neglected failure mechanisms when assessing the seismic performance of these constructions. This thesis aims at the assessment of the out-of-plane behaviour as well as the seismic vulnerability of masonry structures with a predominant out-of-plane collapse. For this purpose, a simplified computational tool, based on a macro-element modelling approach, was extended into the dynamic field by the definition of cyclic constitutive laws and the introduction of a consistent mass matrix. This modelling approach is capable of accurately simulating the main in-plane and most importantly out-of-plane mechanisms of this type of constructions with a reduced computational burden. These features are validated by the comparison of the linear and nonlinear dynamic response of three case studies investigated by means of differential equations and sophisticated computational tools. This validation demonstrated the capability of this simplified modelling approach of accurately estimating dynamic properties and simulating the rocking motion of a rigid block and the nonlinear hysteretic behaviour of masonry structures. After validation, this modelling approach was employed for the assessment of the out-of-plane behaviour of two unreinforced masonry structures previously tested by means of shaking table tests. This investigation was carried out in the static and dynamic nonlinear fields by the application of a mass distributed lateral force and a recorded input from the experimental campaign. The unreinforced masonry structures were also investigated considering a more sophisticated numerical approach, namely Finite Element models. A comparison between these two numerical models was conducted in terms of maximum load capacity, post-elastic behaviour and hysteretic response demonstrating significant resemblance. An additional comparison was conducted taken into consideration numerical and experimental failure mechanisms. A good agreement was obtained when comparing the in-plane response of these structures. Nevertheless, the out-of-plane mechanisms were not successfully simulating evidencing the complexity of this behaviour, especially in a dynamic context. Based on these results, it was demonstrated that this simplified numerical tool can be considered as an alternative computational tool for the assessment of this type of structures since the computational burden was significantly reduced. Finally, the seismic vulnerability of one of these unreinforced masonry structures was investigated by the derivation of analytical fragility curves. For this purpose, the simplified model of such structure was subjected to a set of nonlinear dynamic analyses based on accelerogram artificially generated. In addition, three limit states, whose definition was based on an alternative procedure consisting of the application of nonlinear static analyses, were considered for the assessment of the seismic vulnerability of such structure. This was conducted concentrating the uncertainty initially on the seismic input (artificial accelerograms), and subsequently, on additional parameters such as mechanical properties, thickness of walls, and damping ratio by the definition of probabilistic models. From these assessments, it was possible to determine the probability of exceeding the three limit states due to the application of dynamic loading to masonry structures that are characterised by out-of-plane collapse mechanisms.Os sismos constituem um dos perigos naturais mais devastadores, uma vez que envolvem o colapso de edifícios e, consequentemente, um número significativo de perdas humanas. Algumas tipologias de edifícios, nomeadamente construções históricas e de alvenaria, são um dos elementos mais vulneráveis devido ao seu fraco desempenho quando submetidas a ações sísmicas. Por exemplo, as estruturas históricas foram construídas com base em regras simples, uma vez que os regulamentos sísmicos não foram adequadamente estabelecidos aquando a sua construção. Por outro lado, muitas estruturas de alvenaria, especialmente em países em desenvolvimento, são geralmente construídas sem ter em consideração as especificações fornecidas pelos códigos sísmicos atuais. Essas construções são caracterizadas, principalmente, por uma deficiente ligação entre paredes ortogonais e entre paredes e diafragmas horizontais que levam à ocorrência de mecanismos para fora do plano. Esse comportamento é considerado um dos mais vulneráveis e, também, um dos mecanismos de colapso mais negligenciados na avaliação do desempenho sísmico dessas construções. Esta tese tem como objetivos principais a avaliação quer do comportamento para fora do plano, quer a vulnerabilidade sísmica das estruturas de alvenaria com um colapso predominante para fora do plano. Para isso, uma ferramenta computacional simplificada, baseada numa abordagem de modelação de macro-elemento, foi alargada ao comportamento dinâmico pela definição de leis constitutivas cíclicas e a introdução de uma matriz consistente de massa. Esta abordagem de modelação é capaz de simular com precisão os mecanismos principais no plano e, mais importante, os mecanismos para fora do plano deste tipo de construções com um esforço computacional reduzido. Essas características foram validadas pela comparação da resposta dinâmica linear e não-linear de três casos de estudo investigados por meio de equações diferenciais e outras ferramentas computacionais sofisticadas. Esta validação demonstrou a capacidade da presente abordagem para estimar com precisão as propriedades dinâmicas e simular o movimento de corpo rígido e o comportamento histerético não-linear das construções em alvenaria. Após a validação, esta abordagem de modelação foi utilizada para a avaliação do comportamento para fora do plano de duas estruturas de alvenaria simples, previamente testadas numa mesa sísmica. Esta investigação foi realizada nos regimes estáticos e dinâmicos não-lineares pela aplicação de uma força de massa lateralmente distribuída e através de registos da campanha experimental. As estruturas de alvenaria simples também foram investigadas considerando uma abordagem numérica mais sofisticada, nomeadamente modelos de elementos finitos. Uma comparação entre esses dois modelos numéricos foi realizada em termos de capacidade de carga máxima, comportamento pós-elástico e resposta histerética demonstrando uma semelhança significativa. Uma comparação adicional foi realizada levando em consideração os mecanismos de rotura numéricos e experimentais. Foi obtido um bom acordo na comparação da resposta no plano dessas estruturas. No entanto, os mecanismos para fora do plano não foram simulados com sucesso, evidenciando a complexidade desse comportamento, especialmente num contexto dinâmico. Com base nesses resultados, foi demonstrado que esta ferramenta numérica simplificada pode ser considerada como uma ferramenta computacional alternativa para a avaliação deste tipo de estruturas, uma vez que o esforço computacional foi significativamente reduzido. Finalmente, a vulnerabilidade sísmica de uma dessas estruturas de alvenaria simples foi investigada através da obtenção analítica de curvas de fragilidade. Para esse fim, o modelo simplificado dessa estrutura foi submetido a um conjunto de análises dinâmicas não-lineares através de acelerogramas gerados artificialmente. Para a avaliação da vulnerabilidade sísmica foram considerados três estados limite, cuja definição foi baseada num procedimento alternativo baseado na aplicação de análises estáticas não-lineares. Esta avaliação foi conduzida concentrando, inicialmente, a incerteza na ação sísmica (acelerogramas artificiais) e, subsequentemente, em parâmetros adicionais como propriedades mecânicas, espessura das paredes e rácio de amortecimento com definição de modelos probabilísticos. A partir dessas análises, foi possível determinar a probabilidade de atingir os três estados limite devido à aplicação do carregamento dinâmico a estruturas de alvenaria que são caracterizadas por mecanismos de rotura para fora do plano.Los sismos constituyen uno de los peligros naturales más devastadores, ya que implican el colapso de edificios y, por consiguiente, un número significativo de pérdidas humanas. Algunas tipologías de edificios, en particular construcciones históricas y de albañilería, son uno de los elementos más vulnerables debido a su débil desempeño cuando se someten a acciones sísmicas. Por ejemplo, las estructuras históricas se construyeron sobre la base de reglas simples, ya que los códigos sísmicos no fueron establecidos adecuadamente en su construcción. Por otro lado, muchas estructuras de albañilería, especialmente en países en desarrollo, generalmente se construyen sin tener en cuenta las especificaciones proporcionadas por los códigos sísmicos actuales. Estas construcciones se caracterizan principalmente por una conexión deficiente entre muros ortogonales, y entre muros y diafragmas horizontales, lo que conlleva a la ocurrencia de mecanismos por fuera del plano. Este comportamiento es considerado uno de los más vulnerables y, también, uno de los mecanismos de colapso menos considerados en la evaluación del desempeño sísmico de esas construcciones. Esta tesis tiene como objetivos principales la evaluación tanto del comportamiento por fuera del plano como la vulnerabilidad sísmica de las estructuras de albañilería con un colapso predominante por fuera del plano. Para ello, una herramienta computacional simplificada, basada en un enfoque de modelado de macro-elemento, se ha ampliado al comportamiento dinámico mediante la definición de leyes constitutivas cíclicas y la introducción de una matriz consistente de masa. Este enfoque de modelado es capaz de simular con precisión los mecanismos principales en el plano y, más importante, los mecanismos por fuera del plano de este tipo de construcciones con una demanda computacional reducida. Estas características fueron validadas mediante la comparación de la respuesta dinámica lineal y no lineal de tres casos de estudio investigados por medio de ecuaciones diferenciales y otras herramientas computacionales sofisticadas. Esta validación demostró la capacidad del presente enfoque para estimar con precisión las propiedades dinámicas, y simular el movimiento de cuerpo rígido y el comportamiento histerético no lineal de las construcciones en albañilería. Después de la validación, este enfoque de modelado fue utilizado para evaluar el comportamiento por fuera del plano de dos estructuras de albañilería simples, previamente ensayadas en una mesa sísmica. Esta investigación se realizó en los regímenes estáticos y dinámicos no lineales mediante la aplicación de una fuerza de masa lateralmente distribuida, y por medio de registros de la campaña experimental. Las estructuras de albañilería simple también fueron investigadas considerando un enfoque numérico más sofisticado, en particular modelos de elementos finitos. Una comparación entre estos dos modelos numéricos se llevó a cabo en términos de capacidad de carga máxima, comportamiento post-elástico y respuesta histerética demostrando una semejanza significativa. Una comparación adicional fue realizada tomando en cuenta los mecanismos de falla numéricos y experimentales. Se obtuvo una buena correspondencia en la comparación de la respuesta en el plano de estas estructuras. Sin embargo, los mecanismos por fuera del plano no fueron simulados con éxito, evidenciando la complejidad de este comportamiento, especialmente en un contexto dinámico. En base a estos resultados, se ha demostrado que esta herramienta numérica simplificada puede considerarse como una herramienta computacional alternativa para la evaluación de este tipo de estructuras, ya que el esfuerzo computacional se redujo significativamente. Finalmente, la vulnerabilidad sísmica de una de esas estructuras de albañilería simple fue investigada a través de la obtención analítica de curvas de fragilidad. Para este fin, el modelo simplificado de esa estructura fue sometido a un conjunto de análisis dinámicos no lineales en base a acelerogramas generados artificialmente. Para la evaluación de la vulnerabilidad sísmica se consideraron tres estados límite, cuya definición se basó en un procedimiento alternativo basado en la aplicación de análisis estáticos no lineales. Esta evaluación se llevó a cabo concentrando inicialmente la incertidumbre en la acción sísmica (acelerogramas artificiales) y, posteriormente, en parámetros adicionales como propiedades mecánicas, espesor de las paredes y coeficiente de amortiguamiento, mediante la definición de modelos probabilísticos. A partir de estos análisis, fue posible determinar la probabilidad de alcanzar los tres estados límite debido a la aplicación de carga dinámica a estructuras de albañilería que se caracterizan por mecanismos de falla por fuera del plano

The effect of multi-angle spatially variable ground motions on the seismic behaviour of cable-stayed bridges

The definition of the Spatial Variability of the Ground Motion (SVGM) is a complex and multi-parametric problem that has caught the attention of the research community since the first accelerometer arrays were installed. The effect of this phenomenon on the seismic response of long and multiply supported structures in general, and on cablestayed bridges, in particular, has been studied by many researchers who have emphasised the significance of considering this phenomenon in the seismic analysis, but have also agreed that the SVGM is a multi-component phenomenon that needs more research. This work examines the effect of the multi-support excitation on the seismic response of cable-stayed bridges by means of the time delay of the earthquake at different supports and of the loss of coherency of the seismic waves. The focus herein is the effect of the SVGM on cable-stayed bridges with various configurations in terms of their length and in terms of design parameters, such as the pylon shape and the pylon-cable system configuration, combined with the influence of the incidence angle of the seismic waves. Furthermore, the SVGM is examined at higher levels of earthquake intensity in order to assess the vulnerability of the bridge when subjected to lower probability ground motions. The aim of this research is to provide general conclusions that are applicable to a wide range of cable-stayed bridges and to contribute to the ongoing effort to interpret and predict the effect of the SVGM. It has been found that the influence of the multi-support excitation on the seismic response of the bridges is strongly affected by the shape of the pylons, by the pylon-cable system configuration and by and the seismic incidence angle. The SVGM also excites vibration modes that do not contribute to the seismic response when identical support motions are considered and it increases the probability of failure in the constituent components of the bridge