Seismic behaviour of cable-stayed bridges : design, analysis and seismic devices
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(2) Seismic Behaviour of Cable-Stayed Bridges: Design, Analysis and Seismic Devices Doctoral Thesis Universidad Politécnica de Madrid Madrid, October 2011. The composition of the text has been made using LATEX and GNU applications. Author; Alfredo Cámara Casado Ingeniero de Caminos, Canales y Puertos (6-year MEng in Civil Engineering) Advisor; Miguel Ángel Astiz Suárez Prof. Dr. Ingeniero de Caminos, Canales y Puertos Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos Department of continuum mechanics and theory of structures Technical University of Madrid Profesor Aranguren s/n Madrid 28040 Phone: (+34) 91 336 5358 Fax: (+34) 91 336 6702 e-mail: [email protected] Home page: http://w3.mecanica.upm.es/∼alfredo.
(3) Tribunal nombrado por el Mgfco.. y Excmo.. Sr.. Rector de la Universidad. Politécnica de Madrid, el día. Examination panel appointed by the Rector of the Technical University of Madrid on. ..................................... Presidente / President. D.. .................................................... Vocal / Vowel. D.. .................................................... Vocal / Vowel. D.. .................................................... Vocal / Vowel. D.. .................................................... Secretario / Secretary. D.. .................................................... Suplente / Substitute. D.. .................................................... Suplente / Substitute. D.. .................................................... Realizado el acto de defensa y lectura de la Tesis el día (Once defended the Doctoral Thesis on) . . . . . . de . . . . . . . . . . . . . . . de 2011 en la (at ). E.T.S. de Ingenieros de. Caminos, Canales y Puertos de la U.P.M. Cali cación / Mark : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. EL PRESIDENTE. LOS VOCALES. President. Vowels. EL SECRETARIO Secretary.
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(5) Dedicado a mis padres y a mi hermana. The important thing is not to stop questioning. Albert Einstein.
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(7) Agradecimientos / Acknowledgements Muchas veces me he preguntado si merecía la pena tanto esfuerzo dedicado a la tesis doctoral, y en más de una ocasión me ha resultado difícil justi car el sacri cio necesario para alcanzar el objetivo.. Sin embargo, la satisfacción que produce ver. terminado el trabajo compensa con creces todo lo demás. Sirvan estas líneas para animar a cualquiera que comience el desafío del doctorado, y como reconocimiento a las personas que ya lo terminaron. Siempre he pensado que la parte más difícil de escribir serían los agradecimientos, y no me equivocaba, resulta imposible agradecer con unas simples palabras todo el apoyo que he recibido hasta aquí. Miguel Á. Astiz, gracias por guiar mi investigación en estos primeros años de dedicación, ha sido un auténtico honor tenerte como director de tesis. José M. Goicolea, aprovecho esta ocasión para agradecerte todo el apoyo que me has ofrecido, incluso desde antes de terminar la carrera. Ana M. Ruiz Terán y Peter J. Staord, sois el mejor ejemplo de lo que quiero llegar a ser, gracias por toda vuestra ayuda, tanto en Londres como aquí. M. a. Dolores G. Pulido, va a ser difícil que pueda devolverte todo lo que me has. dado, espero tener la ocasión de hacerlo. Gracias. Es injusto que me limite a nombrar simplemente a todas las personas que me han acompañado estos años en la 9. a. planta de nuestra escuela, espero que sepáis com-. prenderlo y perdonadme los que se me olviden, gracias por vuestra ayuda, comprensión y cariño; Inés Cano, Khanh Nguyen Gia, Cesar A. Polindara, Javier Oliva, Pablo Antolín, Mario Bermejo, Sergio Blanco, Mustapha El Hamdaoui, Óscar González, Yolanda Cabrero, Eloína Fernández y tantos y tantas otras.. Pasquale Dinoi, he. aprendido mucho trabajando contigo, gracias por tu excelente esfuerzo. Thank you very much Bradleys, you made me feel part of your great family during my time in London, sometimes di cult, but with your company always marvelous. Gracias a todos mis amigos y amigas, los que me quedan después de tanto tiempo dedicado en exclusiva al doctorado, por vuestra compresión y apoyo. Agradezco profundamente la ayuda económica e institucional de la Universidad Politécnica de Madrid, de la Universitat Politècnica de Catalunya y de la University of East London, sin la cual hubiera sido muy complicado realizar la tesis doctoral. Por último, y especialmente, gracias a mi familia. Donde estoy, y a donde quiera que llegue, os lo debo a vosotros. Parece que ha llegado el momento de apagar el ordenador y afrontar nuevos retos, lo hago con la ilusión del primer día. Espero estar en contacto con todos los que he citado aquí, los que se me han olvidado y los que me quedan por conocer. En Madrid, a 19 de Octubre de 2011.
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(9) Summary The social and economical importance of long-span bridges is extremely large; cablestayed bridges currently span distances ranging from 200 to even more than 1000 m, representing key points along infrastructure networks and requiring an outstanding knowledge of their seismic response. The objective of the study is three-fold; (i) to discern how project decisions aect the seismic behaviour of cable-stayed bridges; (ii) to shed light on appropriate analysis strategies in order to address their linear and nonlinear dynamic response; and (iii) to compare dierent control schemes with passive seismic devices disposed along the towers. The organization of the content follows a natural progression, starting with the motivation of the work and presenting the state of art on this topic, followed by the description of the framework where the research is developed; the studied structures and the simplifying assumptions employed throughout the document are clearly de ned.. Modal analysis of these bridges precedes the presentation of the seismic. action and the validation of the accelerograms which have been used. At this point, the discussion about the seismic results starts with a chapter completely devoted to the linear and nonlinear analysis procedures available to date, being followed by the comparison of the elastic and inelastic response of cable-stayed bridges, focused on the eect of dierent project decisions. Finally, several control strategies with seismic devices are addressed in order to maintain the towers in the elastic range. The main conclusions are drawn to close the thesis, and new lines of research are suggested. Satisfying the proposed objectives by means of rigorous nite element models, validated through the comparison with experimental tests conducted by other authors, the main contributions of the present work are highlighted:. •. The localization of plastic strain deformation in nite element models representing hollow-section reinforced concrete structures has been studied, observing that the plastic hinge length obtained by classical expressions, is twice the optimum dimension for the linear `beam'-type elements forming the towers.. •. A speci c procedure to generate synthetic accelerograms for nonlinear analysis with Rayleigh damping has been introduced and validated, imposing actions coherent with the design spectrum when damping varies with the frequency.. •. The seismic consequences of key features like the tower shape, main span. •. Dierent calculation methodologies are validated in the linear range, and new. length, cable-system arrangement and type of foundation soil are analyzed.. `pushover' procedures are proposed in order to study the nonlinear response of these exible and strongly coupled structures when three-dimensional seismic excitations are imposed..
(10) vi. SUMMARY •. Due to the large in uence of the transverse seismic reaction of the deck against the towers, analytical models are proposed and validated in order to predict this eect prior to the de nition of nite element models representing the full bridge, providing valuable information for the designer in the early stages of the project.. •. In light of the unacceptable damage recorded in several models of diamondtype pylons, specially in the lower part, the project of this element in bridges located in seismic areas has been also addressed; from these analyses, new design recommendations are obtained to minimize the inelastic demand in the towers and other features related to the economic cost of the foundations. Based on the energy balance, a scalar parameter that quanti es in a simple yet practical manner the damage caused by the earthquake has been proposed, facilitating the comparison between models.. •. Large inelastic excursions of the reinforcement rebars and the concrete, besides extensive cracking at key locations of the tower, have been observed in several models, which may compromise their structural integrity. In order to prevent such inadvisable behaviour, and trying to maintain the towers in the linear range, the incorporation of dierent devices to control the seismic behaviour is explored through parameters based on the extreme seismic response and the energy dissipation. Both yielding metallic dampers and viscous uid dampers have been considered with several designs and con gurations, obtaining relevant conclusions for the designer..
(11) Resumen La importancia social y económica de los grandes puentes atirantados es extremadamente elevada; sus vanos principales varían típicamente entre 200 y 600 metros, llegando incluso a sobrepasar los 1000 metros. Estas estructuras representan puntos clave en las redes de transporte y requieren un estricto conocimiento de su respuesta sísmica. El objetivo del presente estudio consta de tres partes, que dan nombre a la tesis doctoral: (i) obtener conclusiones sobre el efecto que tienen diferentes decisiones de proyecto en el comportamiento sísmico de los puentes atirantados; (ii) explorar los diversos procedimientos de análisis para abordar con garantías el comportamiento sísmico de estas estructuras, tanto en rango lineal como no lineal; y (iii) comparar diferentes estrategias de control sísmico con dispositivos pasivos colocados en las torres. La organización de los contenidos sigue una progresión natural, comenzando por la motivación del trabajo y presentando el estado del conocimiento sobre este tema, seguido de la descripción de las estructuras consideradas en el estudio y de las hipótesis que simpli can el problema. A continuación, se incluye el análisis modal y la acción sísmica al detalle, así como la validación de los acelerogramas sintéticos empleados. Llegados a este punto, comienza la presentación de los resultados del estudio sísmico con un capítulo dedicado en exclusiva a los procedimientos de análisis disponibles, tanto en rango lineal como no lineal, seguido por la comparación tipológica de la respuesta sísmica elástica e inelástica de todos los puentes atirantados analizados, con especial atención al efecto causado por diferentes decisiones de proyecto. Por último, se han abordado diversas estrategias de control con el objetivo de mantener la torre en rango elástico. El trabajo concluye recogiendo las principales conclusiones obtenidas y abriendo nuevas líneas de investigación que podrían continuar el estudio. Cumpliendo con los objetivos establecidos y empleando rigurosos modelos de elementos nitos, validados exhaustivamente con ensayos experimentales llevados a cabo por otros autores, deben destacarse las siguientes contribuciones de la presente tesis doctoral:. •. Se ha estudiado la localización de la deformación plástica en modelos de elementos nitos que representan estructuras de hormigón con secciones huecas, como las empleadas en las torres de atirantamiento, observando que la dimensión óptima en los elementos nitos lineales tipo `viga' es la mitad de la longitud de la rótula plástica en piezas de hormigón armado.. •. Un algoritmo de generación de acelerogramas sintéticos ha sido presentado y validado con el objetivo de obtener señales adecuadas y coherentes con el espectro de diseño en cálculos no lineales con amortiguamiento en función de la frecuencia, como el de Rayleigh..
(12) viii •. SUMMARY El efecto en la respuesta sísmica que tienen la forma de la torre, la luz principal del puente, el tipo de atirantamiento y la clase de terreno de cimentación, entre otros aspectos, ha sido analizado en detalle.. •. Se han estudiado diferentes procedimientos de análisis sísmico, tanto en régimen lineal como no lineal, y se han propuesto modi caciones de los métodos `pushover' para abordar el cálculo no lineal de unas estructuras tan exibles y fuertemente acopladas como los puentes atirantados cuando se someten a excitaciones sísmicas tridimensionales.. •. Dada la importancia del empuje transversal del tablero en la respuesta sísmica de las torres, se ha propuesto y validado un modelo analítico para poder predecir dicha acción sin necesidad de establecer un modelo de elementos nitos que represente el puente completo, lo cual puede ser de gran utilidad para el proyectista en las primeras fases de diseño.. •. Debido al inaceptable daño sísmico que ha sido registrado en varios modelos de torre con diamante inferior, se ha optimizado el diseño de este elemento y se han obtenido recomendaciones de diseño que minimizan tanto la disipación de energía por parte de la propia torre, como factores directamente relacionados con el coste de la cimentación.. Un parámetro escalar que cuanti ca sim-. pli cadamente el daño estructural en las torres debido al terremoto ha sido propuesto en función del balance energético, facilitando la comparación entre distintos modelos.. •. Se han observado relevantes incursiones inelásticas, tanto de las armaduras como del hormigón en varias estructuras, así como una importante suración en zonas clave para la seguridad de la torre y, por tanto, de todo el puente. Con el objetivo de evitar este inapropiado comportamiento, y de acercar la respuesta de las torres al rango puramente elástico, se ha estudiado la incorporación de dispositivos sísmicos. Para ello, la respuesta sísmica extrema y la energía disipada han sido contrastadas antes y después de incluir disipadores basados en la plasti cación de metales y amortiguadores de uidos viscosos. Han sido considerados en cada caso varios diseños y diversas posiciones de estos dispositivos en la torre, obteniendo conclusiones relevantes para el proyectista..
(13) Contents. 1 Motivation and scope. 3. 2 State of the art. 9. 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2. Seismic behaviour of cable-stayed bridges. . . . . . . . . . . . . . . .. 10. . . . . . . . . . . . . . . . . . . . . . . . . .. 12. 2.2.1. Vibration modes. 2.2.2. Damping. 2.2.3. Dynamic analysis procedures. 2.2.4. Seismic response of the towers. 2.2.5. Spatial variability: multi-component seismic excitation. 2.2.6. In uence of tower-deck connection on the seismic response. 2.2.7. Soil-structure interaction. 2.2.8. Seismic behaviour of multiple-span cable-stayed bridges. . . .. 32. 2.2.9. Near- eld earthquakes and vertical excitation . . . . . . . . .. 33. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.3. Capacity versus mitigation design. 2.4. Mitigation design. 2.5. 9. 17. . . . . . . . . . . . . . . . . . .. 20. . . . . . . . . . . . . . . . . .. 27. . . .. 27. .. 30. . . . . . . . . . . . . . . . . . . . .. 31. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.4.1. Energy-based design. 2.4.2. Passive dampers. 2.4.3. Active and semi-active devices in cable-stayed bridges. 2.4.4. Compendium of the seismic device typologies. 33 36. . . . . . . . . . . . . . . . . . . . . . . .. 38. . . . . . . . . . . . . . . . . . . . . . . . . .. 39. . . . .. 53. . . . . . . . . .. 54. Cable-stayed bridges constructed in seismic areas . . . . . . . . . . .. 54. 2.5.1. 56. Seismic failures reported in cable-stayed bridges . . . . . . . .. 3 Modelling and basic assumptions 3.1. Introduction. 3.2. Cable-stayed bridges description. 61. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61. . . . . . . . . . . . . . . . . . . . .. 62. . . . . . . . . . . . . . . . . . . . . . . . .. 62. 3.2.1. Geometric aspects. 3.2.2. Boundary conditions. 3.2.3. Deck-tower connection. 3.2.4. Prestress of the lower strut. . . . . . . . . . . . . . . . . . . . . . .. 65. . . . . . . . . . . . . . . . . . . . . .. 66. . . . . . . . . . . . . . . . . . . .. 70. 3.3. Materials and damping . . . . . . . . . . . . . . . . . . . . . . . . . .. 73. 3.4. Loading scheme and analysis. 76. 3.5. Finite element model description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 78. 3.5.1. Discretization of the towers: localization phenomena. . . . . .. 79. 3.5.2. Discretization of the cable-system: cable-structure interaction. 82. 3.5.3. Discretization of the deck. . . . . . . . . . . . . . . . . . . . .. 84. 3.5.4. Special-purpose elements . . . . . . . . . . . . . . . . . . . . .. 85. Spatial variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86. 3.7. Symmetry of the seismic response . . . . . . . . . . . . . . . . . . . .. 90. 3.8. Basic assumptions. 91. 3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
(14) x. Contents. 4 Modal analysis. 95. 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2. Fundamental vibration modes . . . . . . . . . . . . . . . . . . . . . .. 96. 4.3. Higher mode contribution. . . . . . . . . . . . . . . . . . . . . . . . .. 99. 4.4. Modal coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 104. 5 Seismic action. 95. 107. 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 107. 5.2. Eurocode 8 spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 108. 5.3. Natural records versus synthetic accelerograms. . . . . . . . . . . . .. 110. 5.4. Synthetic accelerograms description . . . . . . . . . . . . . . . . . . .. 111. 5.4.1. Signi cant duration. . . . . . . . . . . . . . . . . . . . . . . .. 111. 5.4.2. Calculation scheme . . . . . . . . . . . . . . . . . . . . . . . .. 112. 5.4.3. MRHA accelerograms: constant damping. 5.4.4. NL-RHA accelerograms: Rayleigh damping. 5.4.5. . . . . . . . . . . .. 115. . . . . . . . . . .. 115. Number of required records. . . . . . . . . . . . . . . . . . . .. 121. 5.5. Synthetic accelerograms validation. . . . . . . . . . . . . . . . . . . .. 122. 5.6. Basic assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 125. 6 Seismic analysis. 129. 6.1. Introduction. 6.2. Mathematical approach. . . . . . . . . . . . . . . . . . . . . . . . . .. 130. 6.3. Elastic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 130. 6.4. 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.3.1. Direct Response History Analysis: DRHA . . . . . . . . . . .. 131. 6.3.2. Modal Response History Analysis: MRHA. . . . . . . . . . .. 133. 6.3.3. Modal Response Spectrum Analysis: MRSA. . . . . . . . . .. 136. . . . . . . . . . . . . . . . . . . . . . . . . .. 140. 6.3.4. Previous studies. 6.3.5. Comparison of the results. Inelastic analysis. . . . . . . . . . . . . . . . . . . . .. 141. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 150. 6.4.1. Modal Pushover Analysis: MPA. 6.4.2. Extended Modal Pushover Analysis: EMPA. . . . . . . . . . . . . . . . .. 6.4.3. Coupled Nonlinear Static Pushover: CNSP. A new method. .. 161. 6.4.4. Discussion of the results . . . . . . . . . . . . . . . . . . . . .. 166. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 177. . . . . . . . . .. 7 Typological study of the elastic seismic behaviour 7.1. 129. 151 158. 183. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 183. 7.2. Transverse static response of the towers. . . . . . . . . . . . . . . . .. 184. 7.3. Extreme seismic response of the towers. . . . . . . . . . . . . . . . .. 185. . . . . . . . . . . . . . . . . . . . . .. 185. . . . . . . . . . . . . . . . . . . . . . .. 189. 7.3.1. Extreme seismic forces. 7.3.2. Extreme total stresses. 7.4. Eect of the accelerogram component. . . . . . . . . . . . . . . . . .. 195. 7.5. Eect of the transition between sections. . . . . . . . . . . . . . . . .. 197. 7.6. Extreme transverse deck reaction . . . . . . . . . . . . . . . . . . . .. 199.
(15) Contents. xi. 7.6.1. Full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 200. 7.6.2. Simpli ed model. . . . . . . . . . . . . . . . . . . . . . . . . .. 200. 7.6.3. Analytical model . . . . . . . . . . . . . . . . . . . . . . . . .. 208. 7.7. Performance of the lower strut. . . . . . . . . . . . . . . . . . . . . .. 218. 7.8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 223. 8 Inelastic seismic behaviour. 227. 8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8.2. Extreme seismic forces . . . . . . . . . . . . . . . . . . . . . . . . . .. 228. 8.3. Seismic demand of inelastic deformation. . . . . . . . . . . . . . . . .. 231. . . . . . . . . . . . . . . . . .. 233. . . . . . . . . . . . . . . . . . . . . . . . .. 235. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 238. 8.3.1. Damage in the lower diamond. 8.3.2. Typological study. 8.4. Speci c studies. 8.5. Dissipation factor. Ω:. an energetic approach. 227. . . . . . . . . . . . . . .. 239. 8.5.1. Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . .. 239. 8.5.2. Dissipation factor de nition. . . . . . . . . . . . . . . . . . .. 241. 8.5.3. Application . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 242. 8.6. Lower diamond optimization. . . . . . . . . . . . . . . . . . . . . . .. 243. 8.7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 247. 9 Seismic devices. 251. 9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 251. 9.2. Initial design considerations . . . . . . . . . . . . . . . . . . . . . . .. 253. 9.3. Mathematical background . . . . . . . . . . . . . . . . . . . . . . . .. 255. 9.4. Yielding Metallic Dampers: MD. 257. 9.5. . . . . . . . . . . . . . . . . . . . .. 9.4.1. Design basis. 9.4.2. Triangular plates: TADAS. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.4.3. Shear Links: SL. 9.4.4. Buckling-Restrained Braces: BRB. 257. . . . . . . . . . . . . . . . . . . .. 259. . . . . . . . . . . . . . . . . . . . . . . . . .. 270. Viscous uid Dampers: VD. . . . . . . . . . . . . . . .. 280. . . . . . . . . . . . . . . . . . . . . . . .. 282. 9.5.1. Design and optimization basis. 9.5.2. Optimization of the velocity exponent;. . . . . . . . . . . . . . . . . .. αd. 282. . . . . . . . . . . .. 287. 9.6. Comparison between dierent strategies. . . . . . . . . . . . . . . . .. 291. 9.7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 292. 10 Conclusions and further studies. 297. 10.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 298. 10.2 Further studies. 306. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. A Constructed cable-stayed bridges. 309. A.1. Rion-Antirion bridge . . . . . . . . . . . . . . . . . . . . . . . . . . .. 309. A.2. Memorial Bill Emerson bridge . . . . . . . . . . . . . . . . . . . . . .. 312. A.3. Tsurumi Fairway bridge. A.4. Yokohama Bay bridge. . . . . . . . . . . . . . . . . . . . . . . . . .. 314. . . . . . . . . . . . . . . . . . . . . . . . . . .. 315.
(16) xii. Contents A.5. Ting Kau bridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 319. A.6. Stonecutters bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 321. B Dimensions and characteristics of the models. 323. B.1. Tower height and span distribution. B.2. Cross-section of the deck. . . . . . . . . . . . . . . . . . . . . . . . .. 325. B.3. Dimensions of the towers. . . . . . . . . . . . . . . . . . . . . . . . .. 327. B.3.1. . . . . . . . . . . . . . . . . . .. Thickness of the tower sections. . . . . . . . . . . . . . . . . .. 336. . . . . . . . . . . . . . . . . . . .. 337. . . . . . . . . . . . . . . . . . . . . . . .. 341. B.4. Characteristics of the foundations. B.5. Cable-system arrangement. B.6. Characteristics of each stay. Cable cross-section. . . . . . . . . . . .. C Nonlinear nite element model C.1. C.2. C.3. C.4. Materials: constitutive properties and assumptions. . . . . . . . . . .. 347. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 347. C.1.1. Concrete. C.1.2. Reinforcement steel. . . . . . . . . . . . . . . . . . . . . . . .. 353. C.2.1. Motivation. 353. C.2.2. Plastic hinge length. . . . . . . . . . . . . . . . . . . . . . . .. 354. C.2.3. Solid cantilever under monotonic loads . . . . . . . . . . . . .. 355. C.2.4. Hollow cantilever under monotonic loads . . . . . . . . . . . .. 357. C.2.5. Full cable-stayed bridge FEM . . . . . . . . . . . . . . . . . .. 359. C.2.6. Conclusions and application rules . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Experimental veri cation employing cyclic tests on columns C.3.1. Takahashi and Iemura's tests. C.3.2. Sakai and Unjoh's tests. C.3.3. Orozco and Ashford's tests. Model optimization. 361. . . . . .. 363. . . . . . . . . . . . . . . . . . .. 363. . . . . . . . . . . . . . . . . . . . . .. 365. . . . . . . . . . . . . . . . . . . .. 366. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 368. 371. Background theory and analysis framework. D.2. Static procedures based on imposed displacements. . . . . . . . . . . . . . .. 372. . . . . . . . . . .. 373. D.2.1. Eurocode 8 (part 2) proposal. . . . . . . . . . . . . . . . . . .. 373. D.2.2. Spanish code proposal: NCSP . . . . . . . . . . . . . . . . . .. 375. D.2.3. Priestley's proposal . . . . . . . . . . . . . . . . . . . . . . . .. 376. D.2.4. Results obtained with the static procedures. 378. . . . . . . . . . .. Dynamic procedure based on delayed accelerograms . . . . . . . . . .. 378. D.3.1. Results obtained with the dynamic procedure . . . . . . . . .. 381. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 383. E Validation of synthetic accelerograms E.1. 351. Mesh sensitivity: localization in hollow sections . . . . . . . . . . . .. D.1. D.4. 342. 347. D Spatial variability. D.3. 323. De nition of earthquake scenarios. 385. . . . . . . . . . . . . . . . . . . .. 385. E.1.1. Empirical spectrum . . . . . . . . . . . . . . . . . . . . . . . .. 386. E.1.2. Speci c horizontal earthquake scenarios. 388. . . . . . . . . . . . ..
(17) Contents. xiii. E.2. Seismological features. . . . . . . . . . . . . . . . . . . . . . . . . . .. E.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 391. E.4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 395. F Pushover analysis in seismic codes and guidelines F.1. Eurocode 8 - Part 2: Bridges. F.2. ATC-40. F.3. FEMA-356. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. G Step-by-step description of advanced pushover . . . . . . . . . . . . . . . . . . . .. 389. 397 397 399 399. 401. G.1. Modal Pushover Analysis: MPA. G.2. Extended Modal Pushover Analysis: EMPA . . . . . . . . . . . . . .. 403. G.3. Coupled Nonlinear Pushover Analysis: CNSP. 406. . . . . . . . . . . . . .. 401. H Nonlinear SDOF integration. 409. I Transverse reaction of the deck. 413. J Tributary mass of the deck: tower model. 417. List of Abbreviations. 422. List of Symbols. 425. Bibliography. 435. Abstract. 456.
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(21) Chapter 1 Motivation and scope. Why is it important to study the seismic behaviour of cable-stayed bridges?. Concern about the safety of infrastructure networks under the earthquake strike has been historically a matter of great concern for the society, and specially for the civil engineering community. Along such network, bridges represent key points since their failure may be associated with the interruption of infrastructures, preventing the access of health personnel and the evacuation of injured, besides immense economical costs. Unfortunately examples of partially or completely damaged bridges are not an exception, as occurred in past Kobe (1995), Chi-Chi (1999) or recently in Tohoku (2011) earthquakes. The present thesis is focused on the seismic response of several classical cablestayed typologies, which may span natural barriers insurmountable in the past; the main span of these outstanding structures range competitively from 200 to 1000 meters.. Cable-stayed bridges represent iconic symbols for the region where they. are located and conform neuralgic points, their failure due to extreme events, like earthquake ground motions, is therefore inadmissible. Dierent analysis strategies, besides the eect of several project decisions and the incorporation of seismic devices to improve their behaviour are explored in the present research.. Contents organization In order to address the study, chapter 2 starts describing the current state of knowledge on the seismic behaviour of cable-stayed bridges and the incorporation of passive devices to control their response. Once the framework established by current experience is set, the description of the considered structures and their nonlinear nite element modelling follows in chapter 3, along with the basic assumptions employed thereafter.. A detailed. research on the section dimensions and proportions of several constructed cablestayed bridges has been performed to establish the extensive parametrization of proposed bridges; the main span ranges from 200 to 600 meters and is the key parameter which de nes completely the rest of the structure.. Furthermore, ve. types of tower shapes have been considered, besides two cable-system arrangements and two extremely dierent types of foundation soils; rocky or soft. Modal analysis of proposed structures is accomplished in chapter 4, shedding some light on the dynamic properties of cable-stayed bridges. The rest of the work is rooted in several conclusions obtained in this chapter about the modal characteristic of these bridges..
(22) 4. Chapter 1. Motivation and scope Eurocode 8 compliant seismic action is de ned in chapter 5, representing large. earthquakes that may occur in seismic prone areas worldwide.. An ad hoc code. is presented in order to obtain synthetic accelerograms coherent with the design spectra when the damping is constant or variable with the frequency. The discussion about the suitability of such arti cial signals compared with empirical predictions of natural ground motions is also included. The applicability of dierent analysis procedures in order to study the linear and nonlinear seismic behaviour of cable-stayed bridges is discussed in chapter 6. New nonlinear static procedures which extend the classical pushover to three-axially excited cable-stayed bridges and takes into account the characteristic modal coupling of these structures are proposed, and their results compared with the ones obtained by means of rigorous nonlinear response history analysis and other pushover strategies. Conclusions drawn from above comparison of analysis techniques serve as a starting point for linear elastic seismic analysis in chapter 7. First, the study of dierent structural typologies under the proposed seismic excitation is accomplished in elastic range, obtaining consequences of dierent project decisions on the seismic demand recorded along the towers, which assume the key role in the global integrity of the structure. Speci c research about the transverse reaction in the towers due to the deck and the eectiveness of the lower strut distributing such action between both sides are included, besides other studies.. An analytical procedure is proposed to. obtain approximately the dynamic strike of the deck against the towers, before the development of a full-bridge nite element model. A remarkable stress concentration was observed in several sections resulting from the elastic analysis, far exceeding cracking and yielding limit deformations. Chapter 8 takes the seismic analysis to its more advanced and rigorous expression, including nonlinear yielding and cracking eects, in order to obtain an accurate response which is rst compared with the elastic solution presented before to gain some information on the ductility demand along the studied types of towers.. A damage factor is. proposed in order to compare the dissipated hysteretic energy recorded in dierent typologies, which helps to optimize the parameters de ning the lower diamond in bridges located in seismic areas. Even optimizing the tower design, the seismic excitation studied is strong enough to cause relevant structural damage in several parts of this vital member, specially if central cable arrangement is employed and if the foundation soil stiness is reduced. Traditional capacity design relies on the dissipation of energy by means of such inelastic demand produced in the structure itself. Nowadays, this approach is deemed unsafe in cable-supported bridges, since the towers are critical members for the structural integrity and should remain practically elastic under strong ground shaking. Large cable-stayed bridges in seismic areas currently include seismic devices to centralize the earthquake demand, easier to repair (if required) than the great damaged towers sections resulting from betting on capacity design. Chapter 9 addresses the study of these innovative techniques to reduce the tower inelastic response, obtaining useful design recommendations and applicability ranges..
(23) 5 Chapter 10 sets the end of the present doctoral thesis about cable-stayed bridges, collecting the relevant conclusions and proposing further studies to continue the research. Several appendices are included at the end of the document with additional material, which have been separated from the text to facilitate the reading; appendix A collects technical information about some major cable-stayed bridges constructed in seismic zones; appendix B presents the research on real cable-stayed bridge dimensions, the proposed parametrization and complementary data; appendix C includes an elaborated review of the constitutive behaviour describing the employed materials, both in linear an nonlinear range, besides speci c studies validating the nite element discretization employed, furthermore, results about sensitivity studies dealing with the optimization of the parameters which de ne the model and the analysis are gathered in this section; appendix D addresses the eects caused by the spatial variability of the seismic action in the proposed cable-stayed bridges, considering different wave-propagation velocities, incidence angles and analysis strategies proposed by seismic codes or research works; appendix E presents a thorough veri cation of the synthetic records considered in this thesis; a revision of pushover in seismic codes and guidelines is collected in appendix F; appendix G summarizes the proposed advanced pushover methodologies in a step-by-step form; appendix H includes an integration algorithm for the dynamic response of a Single Degree Of Freedom system with combined isotropic/kinematic hardening; the analytical model and the complete expressions proposed to predict the extreme seismic reaction of the deck against the towers is included in appendix I; nally, appendix J studies the portion of the deck aecting the tower in transverse direction during the earthquake, in order to obtain simple and accurate models representing exclusively the towers..
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(27) Chapter 2 State of the art. Contents. 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Seismic behaviour of cable-stayed bridges . . . . . . . . . . 10 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8 2.2.9. Vibration modes . . . . . . . . . . . . . . . . . . . . . . . . Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic analysis procedures . . . . . . . . . . . . . . . . . Seismic response of the towers . . . . . . . . . . . . . . . . Spatial variability: multi-component seismic excitation . . In uence of tower-deck connection on the seismic response Soil-structure interaction . . . . . . . . . . . . . . . . . . . Seismic behaviour of multiple-span cable-stayed bridges . . Near- eld earthquakes and vertical excitation . . . . . . .. . . . . . . . . .. 12 17 20 27 27 30 31 32 33. 2.4.1 2.4.2 2.4.3 2.4.4. Energy-based design . . . . . . . . . . . . . . . . . . . Passive dampers . . . . . . . . . . . . . . . . . . . . . Active and semi-active devices in cable-stayed bridges Compendium of the seismic device typologies . . . . .. . . . .. 38 39 53 54. 2.5.1 Seismic failures reported in cable-stayed bridges . . . . . . .. 56. 2.3 Capacity versus mitigation design . . . . . . . . . . . . . . . 33 2.4 Mitigation design . . . . . . . . . . . . . . . . . . . . . . . . . 36 . . . .. . . . .. . . . .. 2.5 Cable-stayed bridges constructed in seismic areas . . . . . 54. 2.1 Introduction This chapter gives an overview about the state of knowledge on the seismic response of cable-stayed bridges, and the solutions that have been carried out in real cases constructed worldwide. Cable-stayed structural solution has been known for centuries, but only became practicable with the advent of high-strength steel wire and the proposal to permanently prestress the stays by Eduardo Torroja (1927) and Franz Dischinger (1934), thereby preventing the excessive de ections and aerodynamic instabilities suered by the deck until that time. Since then, this kind of bridges have been employed.
(28) 10. Chapter 2. State of the art. successfully with spans ranging from 100 to more than 1000 meters, and it seems that in the near future the limits will be pushed. Before looking at the seismic behaviour of these structures, their static response should be known since this base plays a key role in the design of their principal constitutive elements; towers, cable-system and deck, which governs in turn their dynamic behaviour.. The static response of cable-stayed bridges is a widely doc-. umented topic which is not included in the present thesis, the interested reader is directly referred to classical texts; [Gimsing 1998], [Walther 1988], [Manterola 2006], [Virlogeux 2001], [Fernandez-Troyano 2004], among others. Both dynamic and static responses of cable-stayed bridges may present relevant. 1. nonlinearities , whose origin has been observed by many authors; [Walther 1988], [Abdel-Ghaar 1991a], [Morgenthal 1999], [Ren 1999], [Ali 1995], [Ren 2005], among others. Two sources of nonlinearity have been clearly distinguished; material nonlinearity, which depends on the materials employed and the relationship between each other; and geometric nonlinearity, which is characteristic of cable-stayed bridges and is in turn composed of:. •. Geometric nonlinearity due to the response of cable-stays: It is produced by. •. Geometric nonlinearity due to second order moments (P. •. Geometric nonlinearity due to large movements: Cable-stayed bridges are very. the modi cation in the catenary shape because of changes in the cable stress.. pressed elements (the deck and the towers).. −∆. eects) in com-. exible structures and may undergo important movements.. The geometric nonlinearities introduced by stay-cables lead to the increase of the structural stiness when demanding forces are enlarged, presenting cable-stayed bridges a slight geometric hardening in the elastic range which distinguishes this typology from the rest [Fleming 1980] [Karoumi 1998]. During early loading stages, geometric nonlinearities domain the response and dierentiate cable-stayed bridges from other structures, however, material nonlinearity governs advanced demand stages and the stiness is degraded as in conventional structures. Figure 2.1 illustrates the dierence in the elastic response of cable-stayed bridges compared with other types of structures, and the importance of nonlinear calculations under strong loads.. 2.2 Seismic behaviour of cable-stayed bridges Due to the great exibility of cable-stayed bridges and consequent long fundamental vibration periods with low spectral accelerations associated, these structures present. 1 In the present work, it has been observed that material nonlinearities could become very important in the towers and seismic devices if included..
(29) 2.2. Seismic behaviour of cable-stayed bridges. 11. Figure 2.1. Schematic comparison between the elastic response of classical cable-stayed. bridges and structures without cable-system. An idea of the analysis accuracy with dierent linearization approaches depending on the stage of loading is included. O-B: area governed by geometric nonlinearity. Beyond B: area governed by material nonlinearity. 2. in principle a good seismic behaviour .. Furthermore, the number of supports is. reduced (abutments, towers and intermediate piers), allowing the large expected. 3. relative movements without introducing important forces . However, this important exibility, added to their light weight and low associated damping, causes large amplitude oscillations when they are excited by an earthquake or another dynamic action, thus reducing such vibrations with added dampers either passive, active, semi-active, or hybrid (combinations of the previous types) could be specially recommendable. As long as the main span of the bridge is increased, it becomes more susceptible to environmental dynamic actions like earthquakes or wind [He 2001]. Section 2.5.1 collects the most important failures reported in cablestayed bridges due to ground motions. The seismic response of cable-stayed bridges have focused the attention of the scienti c community since early 80's, being remarkable the work of Abdel-Ghaar and his co-authors [Abdel-Ghaar 1991a] [Abdel-Ghaar 1991b], with special attention on the nonlinear behaviour, the sensitivity to support conditions, cable-vibration phenomena and spatial variability. More recently, the concern about curved cablestayed bridges and the incorporation of seismic devices to control the seismic response have raised new research trends.. 2 The. problems caused by the seismic action in the deck are normally related to the horizontal component of the ground motion; the vertical action aects more to the cable-system and the towers because the deck is isolated from vibrations in this direction, behaving like a beam over elastic foundation [Walther 1988]. 3 The analysis of the spatial variability eect in the seismic action included in chapter 3, and discussed further in appendix D, suggests that dierential imposed movements at the foundations do not introduce severe loads generally, due to the large exibility of these structures..
(30) 12. Chapter 2. State of the art. 2.2.1 Vibration modes The study of the dynamic response of a cable-stayed bridge, particularly if it is subjected to earthquake excitations, requires a thorough previous analysis of the vibration properties (frequencies, modal deformations, participation factors and activated modal mass) [Walther 1988], even if the employed methodology does not accomplish the modal decomposition approach.. Modal analysis is important to. know the way the structure vibrates and to nd out if there are any element with inappropriate dynamic behaviour prior to time-consuming seismic calculations. Modal coupling is a characteristic feature of cable-stayed bridges, specially between the transverse exure of the deck and its torsional response, which dierentiates their dynamic response from suspension bridges. This coupling is governed to a large extent by the mass distribution through the deck cross-section and the geometry of the stay system.. Due to this modal interaction, the distinction be-. tween pure vertical, transverse and torsional movements is complicated and forces the designer to employ three-dimensional models in order to represent the structure [Abdel-Ghaar 1991a]. Modal coupling is stronger as long as the main span is longer (more generally speaking; as long as the exibility is higher). First vibration modes have long periods and are generally associated with the deck, followed by modes which excite the cable-system and may be coupled with the deck.. Higher frequencies appear deforming mainly the towers and their cou-. pling with the deck depends on the connection conditions between both elements [Morgenthal 1999]. Due to the complexity and modal couplings inherent in cablestayed bridges, a large number of modes are usually required in their dynamic analysis. Bruno and Leonardi [Bruno 1997] performed analytical and numerical studies about vibration modes presented in cable-stayed bridges, observing the small contribution of deck stiness and the negligible eect of the tower shape in the case of. 4. lateral cable arrangement , with the exception of torsional modes, which are clearly aected. A review of the main body of knowledge in vibration modes of cable-stayed bridges is summarized below, more information may be found in other research works [Walker 2009] [Morgenthal 1999] [Valdebenito 2005].. Pure vertical deck modes The vertical stiness of modern cable-stayed bridges with closely spaced stays and slender decks is dominated by the cable-system, being well characterized by geometric parameters like the main span length, the ratio between side and main span lengths, or the tower height. Wyatt [Wyatt 1991] proposed analytical expressions to estimate the fundamental vertical vibration of cable-stayed bridges, neglecting the stiness of the deck. Kawashima et al [Kawashima 1993] published the following expression for the. 4 This. result has been veri ed in the present work and is presented in chapter 4..
(31) 2.2. Seismic behaviour of cable-stayed bridges. 13. rst frequency associated with the vertical bending exure of the deck,. fv , correlated. from eld forced excitation tests conducted for thirteen constructed cable-stayed. 5. bridges in Japan :. fv = 33.8L−0.763 P LP. (2.1). being the main span in meters.. Higher vertical deck modes present more zero-displacements nodes along this element, reducing the in uence of axial deformation in the cable-system and increasing the importance of deck stiness.. Pure torsional deck modes Unlike vertical stiness, which is governed by the cable-system, torsional stiness may arise in equal measure from the cable-system geometry or the deck cross-section, and it is thus more readily in uenced by the form of the tower and deck sections. Kawashima et al [Kawashima 1993] also proposed an expression for the rst torsional frequency of the deck, rooted in the correlation of experimental results:. fθ = 17.5L−0.453 P. (2.2). The fundamental torsional frequency is dominated by the torsional rigidity of the deck if this is a box-girder and, consequently, the torsional rigidity `GJ '. 6. has a sig-. ni cant magnitude; this is the case of bridges with central cable plane arrangement, where the cables only provide up to 10-20 % additional rigidity [Virlogeux 1999]. Wyatt [Wyatt 1991] proposed the following expression to obtain the rst natural torsional frequency of the deck, neglecting the contribution of the stays (beam model), which is appropriate in bridges with moderate spans and central cable arrangement:. 1 fθ = 2. Torsional (beam model). Ltor m. s. GJ mr2 L2tor. (2.3). being the length of the deck between the points where torsion is prevented,. the mass of the deck per unit length and. r. the radius of gyration.. However, since the torsional stiness of the deck cross-section is. GJ/Ltor , in case. of open cross-sections (associated with lateral cable arrangements) and/or cablestayed bridges with long spans, the torsional stiness is governed by the cablesystem.. Gimsing [Gimsing 1998] proposed an idealized model with two springs. representing the cables in order to study (for lateral cable arrangements) the vertical and torsional fundamental frequencies of the deck neglecting its stiness;. Torsional (stay model). 5 New. 1 fv = 2π. r. 2Ccp 1 ; fθ = Md 2π. s. Ccp B 2 2Im,x. (2.4). proposals for the vertical and transverse fundamental frequencies of cable-stayed bridges have been proposed in this thesis (chapter 4). 6 `GJ ' is the torsional rigidity; G shear modulus, J torsion constant..
(32) 14. Chapter 2. State of the art Where. Ccp. is the vertical stiness of each cable plane and. Md , B , Im,x. are re-. spectively the mass of the deck, the width between both lateral cable planes and the deck torsional mass moment of inertia. Gimsing proposed two extreme idealized models in order to obtain. Im,x ,. presented in gure 2.2; (a) if two concentrated. masses are considered in the cable planes, hence. Im,x = Md B 2 /4;. (b) if uniformly. distributed mass of the deck is considered across the section width. Im,x =. B,. therefore. Md B 2 /12. In real structures, the mass distribution will be somewhere be-. tween both extremes and hence, considering expression 2.4 and both values of the ratio. fθ /fv. is seen to lie between 1 and. √. 3 ≈ 1.73. Im,x ,. (which is further increased. if the stiness of deck cross-section is included). Observed ratios. fθ /fv. are between. 1.5 and 1.6 in practice [Wyatt 1991], which agree with the proposal of Gimsing.. (a) Concentrated masses (lateral cable arrangement). (b) Distributed mass (central cable arrangement). Figure 2.2. Ideal dynamic models for the estimation of deck vertical and torsional frequencies. Proposed by [Gimsing 1998].. Therefore, cable-stayed bridges may present very closely spaced vertical and torsional frequencies, specially if: (1) the deck has negligible torsional and vertical stiness related to the cable-system, and (2) the mass of the deck is concentrated out to the edges. This suggests that lateral cable plane arrangement, which includes open deck sections with two lateral girders, could maximize couplings between the torsional and vertical exure, aecting the accuracy of modal combination rules in seismic analysis [Walker 2009] and also the critical speed for utter. The stiness of the cable-system,. Ccp , may be selected by the designer to some ex-. tent. In lateral cable arrangements, pure torsional mode requires anti-phase motion of the two cable planes and compatible motion of the towers; `H'-shaped towers allow dierential movements between both lateral cable planes, however `A'-, inverted `Y'- or single mast-shaped towers with two cable planes prevent such movements because these two planes meet at the same point and therefore pure torsional modes require axial extension of the stays, which is signi cantly stier than the dierential longitudinal movement of the legs in `H'-shaped towers..
(33) 2.2. Seismic behaviour of cable-stayed bridges. 15. Transverse deck modes The cable-system oers small transverse restraint to the deck, unless the cable planes are signi cantly inclined (which is only appreciable in bridges with moderate spans).. 7. Neglecting the contribution of the stays and the exibility of the towers , the transverse frequencies of the deck may be approximated from those of a continuous beam with the same span arrangement, being dominated by the transverse exural stiness of the deck,. EIH. (respectively. E. is the Young's elasticity modulus and. the transverse moment of inertia of the deck).. IH. Wyatt [Wyatt 1991] proposed the. following expression for the rst transverse frequency related to the deformation of the deck:. Transverse (beam model). Cy. 1 fy = 2. s. π 2 Cy EIH mL4P. (2.5). being a factor depending on the main to side span length. The transverse. frequencies of the deck are not easily controlled by the designer since. m. and. EIH. are governed by the width of the deck (imposed by infrastructure requirements). Kawashima et al [Kawashima 1993] again employed experimental results to correlate an expression for the rst transverse frequency of cable-stayed bridges in terms. 8. of the main span :. fy = 482L−1.262 P. (2.6). The coupling between the transverse exure and torsional response of the deck is characteristic in cable-stayed bridges, as it was already stated, and this interaction is important if. fy. and. fθ. are close to each other, observing no signi cant reduction. in the natural frequencies themselves, otherwise this coupling is weak [Wyatt 1991].. Tower modes The cable-system strongly coerces the tower in longitudinal direction whereas it causes a negligible eect along the transverse axis (unless the cable planes are signi cantly inclined).. Therefore, pure transverse tower modes, which may be ap-. proximated by cantilever models of the tower, appear before the longitudinal ones, estimated through encastred-pinned beam models; encastred at the foundations and pinned at the top due to the constraint exerted by the cables, specially the ones anchored to the points of the deck with prevented vertical movements, i.e. the abutments and intermediate piers.. 7 Section. 7.6 proposes a methodology to estimate the transverse reaction of the deck against the tower, which includes an analytical expression to obtain the rst frequency associated with the transverse exure of the deck, neglecting its torsion and the cable-system contribution, but taking into account the exibility of the towers. 8 An alternative expression for the rst transverse frequency related to the deck, f , (also in y terms of the main span length exclusively) has been proposed in chapter 4..
(34) 16. Chapter 2. State of the art. Cable-structure interaction Another distinctive property of cable-stayed bridges is the transferred energy between local cable and global modes, which is usually referred as cable-structure interaction. This eect was rst studied by Leonhardt and Zellner [Leonhardt 1980] and developed before by many others.. Normally, cable-structure interaction plays a. bene cial role in the seismic response of cable-stayed bridges, however, when subjected to earthquakes with speci c dominant frequencies, the eect could lead to a signi cant increase of the seismic forces. The discretization employed to represent the cable-system could be therefore im-. 9. portant in terms of the accuracy of the dynamic analysis in cable-stayed bridges . If only one nite element per stay-cable is employed (sometimes referred in the literature as OECS; One Element Cable-System), their local vibration is neglected and the dynamic interaction between the cable-system and the deck is also ignored, which may be important in the seismic analysis of cable-stayed bridges [Abdel-Ghaar 1991a].. The cable vibration could introduce a signi cant amount. of energy through higher modes, which are relevant in terms of seismic force contributions but in lesser extent in terms of displacements [Abdel-Ghaar 1991a] [Abdel-Ghaar 1991b]. A large number of new vibration modes with low frequencies appears if multiple elements are employed in each cable (referred as MECS; Multiple Element CableSystem).. Many of these modes are associated with pure local cable in-plane or. out-of-plane lateral exure, and do not change the structural response of the bridge subjected to a broadband seismic excitation. However, a smaller number of coupled deck-cable modes arises, modifying pre-existing global modes and hence aecting the dynamic response.. Thuladar et al [Tuladhar 1995] con rmed this conclusion,. observing important eects if the rst natural frequencies of the cables overlap with the rst frequencies of the bridge. Caetano et al [Caetano 2000] compared analytical OECS and MECS models with experimental shaking-table results of Jindo bridge physical model (South Korea). No signi cant dierences between models with single or multiple elements per cable were observed and, hence, the physical model was modi ed to bring local and global modes together. Such `arti cial' model exhibited low seismic eects, precisely due to cable-structure interaction, when it was subjected to broadband earthquakes, but modifying the signal and employing narrow-band records tuned to the rst global and local cable modes, signi cant eects arise. Simpli ed models consisting of mass-cable systems were later studied by Caetano [Caetano 2007] applying `external-. 10 ,. excitation' perpendicular to the stay axis. and verifying that, if the structure has. global modes with frequencies close to the fundamental cable ones (or twice this value), cable-structure interaction may be important.. 9A. speci c study about the eect of cable modelling has been conducted in the present thesis and is presented in chapter 3. 10 On the other hand, the excitation parallel to the cable chord is termed `parametric-excitation'..
(35) 2.2. Seismic behaviour of cable-stayed bridges. 17. Cable-structure interaction is likely to be bene cial if broadband excitation strikes the bridge, reducing the seismic forces due to moderate cable vibrations. However, it could be unfavourable if the structure is subjected to narrow-band earthquakes with important energy associated with coupled frequencies, presenting large cable oscillations which may act to increase the overall response. The eect of local cable vibrations in the global behaviour of the bridge has been widely studied and is now a source of major research.. Sometimes special devices. are installed in the cables in order to control the vibration caused by rain and wind eects, subsequently improving their seismic eect. Such devices increase the energy dissipation in the stays, which otherwise present reduced damping by themselves, about 0.05 % to 4 % [Abdel-Ghaar 1991b]. Several authors have veri ed the improvement in the accuracy of the recorded seismic demand if multiple elements per cable are employed [Abdel-Ghaar 1991b], [Tuladhar 1995], [Au 2003] [Ko 2001]. These authors recommend the discretization of cables with multiple elements in the seismic analysis of cable-stayed bridges, however Ni et al [Ni 2000] studied the modal properties of Ting Kau bridge (China) and veri ed that the eect of multi-element discretization is only appreciable in the longest cables, 465 m long, whilst the rest of the stays could be modeled with one element per cable without losing accuracy, which is also defended by Wilson and Gravelle [Wilson 1991]. How many elements should be included per element to take into account realistically the cable-structure interaction?.. Caetano [Caetano 2007] conducted a. sensitivity analysis on this topic studying Vasco da Gama bridge (Portugal), concluding that 9 elements per cable yields errors less than 5 %, even in the longest cables with 226 m long, being able to obtain the rst three local vibration modes of these stays accurately.. 2.2.2 Damping Cable-stayed bridges present low damping and assuming standard values for the fraction of critical damping (ξ ) of 5 % falls on the unsafe side [Kawashima 1991]. The estimation of damping is important, but also complicated since it depends on the relative damping of each constitutive element (towers, cable-system and deck) and even on their con guration and interaction between each other. Broadly speaking, the eect of damping has been implemented by means of three procedures in previous research works and also in the present doctoral thesis:. •. Through the realistic representation of the nonlinearity sources that may be developed during the earthquake (e.g. hysteresis loops in structural members and seismic devices, radiation of energy by means of supports, etc.). This is the most precise methodology in order to take into account the dissipation of energy under large seismic events, strong enough to develop such nonlinearities. It could be implemented in nonlinear dynamic calculations, or by means of reduced spectra (which is less accurate)..
(36) 18. Chapter 2. State of the art •. Using Rayleigh or Caughey damping theory to decompose the structure damping matrix. c and perform modal dynamic analysis.. Dierent damping is asso-. ciated with each vibration mode, and previous modal analysis is required in order to nd the range of frequencies with the most signi cant contribution in the response. Values of. ξ ≈ 2−4. % are imposed on the borders of such inter-. val, obtaining higher values of damping associated with higher modes, whose participation is assumed negligible and may cause numerical instabilities in dynamic calculations. However, Yamaguchi and Furukawa [Yamaguchi 2004], in their work about Yokohama Bay cable-stayed bridge (Japan), concluded that Rayleigh (or Caughey) damping is not appropriate in the seismic analysis of this structure due to the special connection which is established between the deck and the towers (see appendix A). This approach is further developed in chapter 5.. •. Considering a fraction of the critical damping constant for all vibration modes,. ξ =. constant. ≈ 2−4. %.. This is the simplest procedure and has been. adopted by many researchers (e.g.. Ali and Ghaar [Ali 1995], Morgenthal. [Morgenthal 1999]), seismic codes and guidelines. This approach can only be directly used in calculations based on modal decomposition (spectrum analysis or modal dynamics) and, therefore, is valid for bridges behaving in linear range without seismic devices. The research of Kawashima and Unjoh [Kawashima 1991] concluded that damping is strongly in uenced by the considered vibration mode, since dierent constitutive elements are excited depending on the mode shape, which in turn present dierent damping values. Furthermore, it depends on modal coupling, velocity of wave propagation, dimensions of the foundations and the direction of the studied. 11 ,. response. among other parameters. These authors proposed a methodology to es-. timate the damping by dividing the structure in elements with the same dissipation mechanism, obtaining the contribution of each sub-structure and aggregating the results to approximate the modal damping. The study demonstrated that damping is largely aected by the amplitude of ground excitation, and thus harp cable-system arrangement presents higher damping values associated with longitudinal oscillations than analogous bridges with fan cable-system. are illustrated in gure 2.3.. These cable arrangements. Such dependency of the rst modes on the ampli-. tude of the seismic excitation has been experimentally contrasted in constructed cable-stayed bridges, as it was published in the research of Siringoringo and Fujino [Siringoringo 2005] about Yokohama Bay cable-stayed bridge (Japan). An improved damping de nition is achieved in structures with passive seismic devices (chapter 9) since dissipation in these elements is realistically de ned through their constitutive properties, and are responsible for most of the dissipated energy [Soong 1997].. 11 For example, the towers have lower damping values in transverse direction due to the negligible bending stiness of the cable-system, however such cables in longitudinal direction involve the response of the tower and the deck, resulting higher damping values..
(37) 2.2. Seismic behaviour of cable-stayed bridges. 19. Figure 2.3. Cable layout solutions in elevation view for cable-stayed bridges.. 2.2.2.1 Damping mechanisms Walker [Walker 2009] presented a practical revision of damping sources in classical cable-stayed bridges, here, the concluding remarks are included:. Structural damping The cyclic demand of materials in the structure dissipates energy due to the hysteresis loops associated with plastic deformation, which may arise even below the material elastic limit because of stress concentrations that occur at microscopic localized plastic ow.. However, this type of damping increases dramatically when. the plastic limit is exceeded, governing the dissipation of the bridge. This source of damping depends on the vibration amplitude but not on its frequency [Chopra 2007].. Friction at bearings When relative movements between the deck and the abutments (incorporating support devices) are recorded, damping is produced due to coulomb friction, or by means of hysteresis loops if support devices include a lead core (LRB). Such damping depends on the amplitude and the vibration mode to the extent this relative movement is achieved.. Cable-slip in the cable-system Energy may be dissipated through internal slip between wires forming the stays if a threshold amplitude is exceeded, overcoming its internal friction. Consequently, this kind of damping depends on the amplitude of vibration and type of cables employed.. Foundation radiation damping Vibrations in the foundations originated by the seismic excitation cause energy dissipation due to the radiation to surrounding subsoil, which could lead to higher damping levels than those associated with the superstructure, hence highly damped substructures are often treated as a secondary dynamic system. clearly depends on the vibration mode.. This damping.
(38) 20. Chapter 2. State of the art. Aerodynamic damping Superstructure vibration is coerced by the surrounding air, providing a resistance proportional to the square of the relative velocity; air damping is viscous, i.e. ratedependent. This kind of damping is typically modest in conventional cable-stayed bridges due to the low air density, the reduced superstructure surface area and the large associated inertia forces involved in its seismic movement.. System damping It is due to the interaction between the deck, cable-system and towers. A signi cant amount of energy may be dissipated by means of this source of damping in classical cable-stayed bridges if vibration modes of the stay-cables are similar to the ones associated with the deck and coupling eects arise [Caetano 2000], as it has been already discussed in section 2.2.1.. 2.2.2.2 Practical simulation of damping sources Energy dissipation, despite represents a key issue in any dynamic analysis, is far from being accurately simulated in the engineering practice due to the complex and miscellaneous nature of damping processes. Conventional bridges and buildings typically avoid speci c considerations about any of the dissipation sources presented above, and simply consider viscous damping through constant factors (ξ ) provided by relevant codes, hence preventing the introduction of additional uncertainties in the model and obtaining solutions on the safe side, since speci c dissipation mechanisms are ignored. 12 .. 2.2.3 Dynamic analysis procedures Several seismic analysis strategies may be employed to obtain the response of a structure subjected to earthquake excitations, with inherent advantages and disadvantages which may support or discourage their use depending on the type of structure, expected nonlinearities and type of response measure extracted. Wethyavivorn and Fleming [Wethyavivorn 1987] and Abdel-Ghaar [Abdel-Ghaar 1991a] published reference works on the seismic analysis procedures and their viability in the study of cable-stayed bridges. In this section, the available seismic analysis techniques are brie y discussed, introducing their key features in order to better understand the limits of applicability and sources of errors associated with each procedure. description of each method is included in chapter 6.. A detailed mathematical For a thorough treatment,. the works of Chopra [Chopra 2007], Clough and Penziem [Clough 1993], Villaverde [Villaverde 2009] or García Reyes [García-Reyes 1998] are strongly recommended.. 12 In the present thesis, the structural damping is rigorously included in nonlinear analysis through realistic material properties (chapters 8 and 9), being the most important dissipation mechanism. The other sources of damping are deliberately ignored, which is a safe assumption..
(39) 2.2. Seismic behaviour of cable-stayed bridges. 21. N -degree of freedom structure subjected to the earthquake action is expressed by means of the following system of N coupled dierential The dynamic response of a. equations (hereinafter referred as system of dynamics):. mü + cu̇ + fS (u, u̇) = −mιüg Where. u(t). is the relative displacement vector,. mass and damping matrices of the structure,. fS. m. (2.7). and. c. are respectively the. is the stiness component of the. force vector in the structure and de nes the relationship between force and displacement vectors; while the structure behaves in the linear elastic range. k. the elastic stiness matrix of the structure.. Finally,. ι. fS = ku,. being. is the in uence matrix. connecting the degrees of freedom of the structure and the imposed accelerogram directions. üg (t),. which generally, and neglecting imposed rotations at the founda-. tions, is a vector with three components. Y Z üTg (t) = (üX g , üg , üg ),. each one related to. the accelerogram component along each principal direction.. Inelastic seismic analysis procedures There are several ways to face the general nonlinear dynamic problem, next ordered from more to less time-demanding procedures.. •. Non-Linear Response History Analysis (NL-RHA): One rigorous way to address the dynamic response of the structure is to solve directly the complete coupled system of dynamics (2.7); Non-Linear Response History Analysis (NL-RHA) integrates this system step-by-step, considering the tangent stiness at each iteration in order to linearize the problem. Several algorithms may be employed to solve directly the coupled system, being the most commonly used the HHT scheme [Hilber 1977]. NL-RHA is the most accurate methodology to predict the inelastic seismic demand in a structure; the procedure may fully take into account the geometric and material nonlinearities (e.g. cyclic stiness degradation, hysteretic dissipation), and is able to analyze realistically the eect of seismic devices if they are equipped.. However, a set of three-axial representative accelerograms is. required, besides mathematical models capable of representing adequately the cyclic load-deformation characteristics of all the important elements, and e cient computing tools in order to deal with time-consuming calculations, which are not justi ed in every structure and engineering o ce [Krawinkler 1998], [Bommer 2003], [Chopra 2007]. Uncertainties arise when the nonlinear cyclic behaviour of concrete, steel and bonding interfaces have to be described.. Priestley [Priestley 1996] recom-. mended direct nonlinear dynamics only when speci c aspects of the design of the bridge need to be veri ed. Furthermore, the integration scheme introduces phase errors, which are higher as long as the ratio of the step-time (∆t) over the considered vibration period (T ) is increased [Hilber 1977], as it may be.
(40) 22. Chapter 2. State of the art appreciated in gure 2.4.. There are seismic regulations which preclude this. procedure, like the National Annex of Eurocode 8 in Germany [EC8 2011]. NL-RHA is largely the most demanding method available but is gaining traction nowadays, since earthquake engineering is moving towards performancebased design and away from force-based design, thus reliable methods are required in order to obtain the realistic nonlinear seismic demand. Moreover, nonlinear dynamic analysis bene ts from the improvement in computers capabilities. Incremental Dynamic Analysis (IDA, sometimes referred as `dynamic pushover') [Vamvatsikos 2002] performs sets of nonlinear dynamic calculations by scaling several accelerograms with dierent intensities, in order to obtain an accurate sight of the nonlinear response of the bridge. It is commonly considered the most precise way to explore the behaviour of structures under large ground motion excitations. However, its computational cost is di cult to justify in all cases, being used as reference result in many research works in order to stress. Period elongation / T. the accuracy of simpli ed methods, like pushover analysis.. a a. Figure 2.4. Relative phase error in terms of the ratio of the step-time and the vibration. period (∆t/T ) in several direct integration methods; Hilber - Hughes - Taylor (HHT) with dierent numerical damping values αa ; Wilson; Newmark; and Houbolt. Taken from Hilber et al [Hilber 1977]. •. Non-linear Static Procedures (NSP): In recent years, Nonlinear Static Procedures (NSP), commonly named pushover methods, have received a great deal of research, specially since seismic design guidelines ATC-40 [ATC 1996] and FEMA 273 [fem 1997] were published. 13 .. Their main goal is to estimate the nonlinear seismic response by means of static calculations, pushing the structure up to certain target displacement using load patterns which try to represent the distribution of inertia forces.. 13 Normative. pushover strategies are presented in appendix F..
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