Tema 5 respuesta sismica sistemas inelasticos

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Universidad de Costa Rica 1

Dinámica de Estructuras

Tema 5

Respuesta sísmica de sistemasinelásticos

II Semestre 2013, 20 setiembre 2013

Espectros elásticos

• Se ha mostrado que el cortante basal máximoinducido por un movimiento sísmico en un sistema elástico lineal es Vb = (A/g)w, donde wes el peso del sistema y A es la ordenada del espectro de pseudo‐aceleracióncorrespondiente al periodo natural de oscilación y el porcentaje de amortiguamientodel sistema.

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Sin embargo, …

• La mayoría de las edificaciones son diseñadaspara cortantes basales menores que el cortante basal elástico asociado con el movimiento más fuerte que puede ocurrir en un sitio.

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• Esto puede verse en la figura 7.1, en donde el coeficiente basal A/gproveniente del espectro de diseño de la figura 6.9.5, escalado para 0.4 para obtener correspondenciacon la aceleración pico de 0.4g, es comparado con el coeficiente de cortantebasal especificado en el International Building Codede 2000.

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Figura 7.1 Comparación de coeficientes de cortante basal de un espectro elástico de diseño y el del International Building Code.

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Figura 7.2 The six-story Imperial County Services Building was overstrained by the Imperial Valley, California, earthquake of October 15, 1979. The building is located in El Centro, California, 9 km from the causative fault of the Magnitude 6.5 earthquake; the peak ground acceleration near the building was 0.23g. The first-story reinforced-concrete columns were overstrained top and bottom with partial hinging. The four columns at the right end were shattered at ground level, which dropped the end of the building about 6 in.; see detail. The building was demolished. (Courtesy of K. V. Steinbrugge Collection, Earthquake Engineering Research Center, University of California at Berkeley.)

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Figura 7.3 The O’Higgin’s Tower, built in 2009, is a 21-story reinforced-concrete building with an unsymmetric (in plan) shear wall and column-resisting system that is discontinuous and highly irregular over height. Located in Concepcion, 65 miles from the point of the initial rupture of the fault causing the Magnitude 8.8 Offshore Maule Region, Chile, earthquake of February 27, 2010, the building experienced very strong shaking. The dam-age was so extensive—including collapse of its 12th floor—that the building is slated to be demolished: (a) east face; (b) southeast face; (c) south face; and (d) southeast face: three upper floors and machine room. (Courtesy of Francisco Medina.)

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Figura 7.4 Psychiatric Day Care Center: (a) before and (b) after the San Fernando, California, earthquake, Magnitude 6.4, February 9, 1971. The structural system for this two-story reinforced-concrete building was a moment-resisting frame. However, the masonry walls added in the second story increased significantly the stiffness and strength of this story. The first story of the building collapsed completely. (Photograph by V. V. Bertero in W. G. Godden collection, National Information Service for Earthquake Engineering, University of California, Berkeley.)

Daño severo que causa colapso.

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Por tanto, …

• La respuesta de estructuras deformándosedentro del rango inelástico durantemovimiento sísmico fuerte es de máximaimportancia en la ingeniería sísmica.

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Pruebas de laboratorio

• California– Berkeley– San Diego

• Japón– U de Tokio/Tsukuba

• Nueva Zelanda– Park, Paulay, Priestley/Christchurch

• México– Unam/Cenapred

• Costa Rica– Lanamme

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Figura 7.1.1 Acero estructural(Krawinkler, Bertero & Popov, 1971)

Relaciones de Fuerza – Deformación para componentes estructurales de diferentes materiales

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Figura 7.1.1 Concreto reforzado (Popov & Bertero, 1977)

Relaciones de Fuerza – Deformación para componentes estructurales de diferentes materiales

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Figura 7.1.1 Force–deformation relations for structural components in different materials: (a) structural steel (from H. Krawinkler, V. V. Bertero, and E. P. Popov, “Inelastic Behavior of Steel Beam to Column Subassemblages,” Report No. EERC 71-7, University of California, Berkeley, 1971); (b) reinforced concrete [from E. P. Popov and V. V. Bertero, “On Seismic Behavior of Two R/C Structural Systems for Tall Buildings,” in Structural and Geotechnical Mechanics (ed. W. J. Hall), Prentice Hall, Englewood Cliffs, N.J., 1977]; (c) masonry [from M. J. N. Priestley, “Masonry,” in Design of Earthquake Resistant Structures (ed. E. Rosenblueth), Pentech Press, Plymouth, U.K., 1980].

Relaciones de Fuerza – Deformaciónpara componentes estructurales de diferentes materiales

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Idealización Elastoplástica

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Figura 7.1.2 Curva fuerza–deformación durante carga inicial: curva real e idealización elastoplástica.

Areas bajo la curva son iguales para desplazamiento máximo.

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Figura 7.1.3 Relación fuerza–deformación elastoplástica.

La relación cíclica fuerza‐deformación depende del camino seguido.

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Figura 7.1.4 Sistema elastoplástico y su correspondiente sistema lineal.

Para amplitudes grandes del movimiento el periodo natural no existe para el sistema inelástico.

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Resistencia normalizada a la fluencia, factor de reducción de resistencia de fluencia y factor de ductilidad

y yy

o o

f uf

f u

1o oy

y y y

f uR

f u f Ry = factor de reducción

de resistencia de fluencia

factor de ductilidadm

y

u

u

resistencia de fluencianormalizada

yf

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Relación entre deformación máxima y deformación elástica equivalente

my

o y

uf

u R

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Ecuación del movimiento y parámetrosgobernantes

( ) ( )S gmu cu f u mu t

22 ( ) ( )n n y S gu u u f u u t

( ), , ( )

2S

n Sn y

f uk cf u

m m f

2 2 ( )2 ( ) g

n n S ny

yy

u tf

a

fa

m

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Figura 7.3.1 Relaciones fuerza-deformación normalizadas.

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Figura 7.4.1 Respuesta de un sistema lineal con Tn = 0.5 s y ζ = 0 para terremoto de El Centro.

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Figura 7.4.2 Respuesta de sistema elastoplástico con Tn = 0.5 s, ζ = 0, y ƒy = 0.125 para terremotode El Centro: (a) deformación; (b) fuerza resistente y aceleración; (c) intervalos de tiempo de fluencia; (d) relación fuerza-deformación.

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Figura 7.4.3 Respuesta de deformación y fluencia para cuatro sistemas debidasal terremoto de El Centro; Tn = 0.5 s, ζ = 5%; y ƒy = 1, 0.5, 0.25, y 0.125._

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Figura 7.4.4 (a) Peak deformations umand uo of elastoplastic systems and corresponding linear system due to El Centro ground motion; (b) ratio um/uo. Tn is varied;ζ = 5% and ƒy = 1, 0.5, 0.25, and 0.125.

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Figura 7.4.5 Ductility demand for elastoplastic system due to El Centro ground motion; ζ = 5% and ƒy = 1, 0.5, 0.25, and 0.125, or Ry = 1, 2, 4, and 8._

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Figura 7.5.1 Relationship between normalized strength (or reduction factor) and ductility factor due to El Centro ground motion; ζ = 5%.

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Figura 7.5.2 Espectro de respuesta de ductilidad constante para sistemaselastoplásticos sometidos al terremoto de El Centro; μ = 1, 1.5, 2, 4, and 8; ζ = 5%.

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Figura 7.5.3 Espectros de respuesta de ductilidad constante para sistemaselastoplásticos sometidos al terremoto de El Centro; μ = 1, 1.5, 2, 4, and 8; ζ = 5%.

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Figura 7.7.1 Resistencia normalizada ƒy de sistemas elastoplásticos comofunción del periodo natural de oscilación Tn para μ = 1, 1.5, 2, 4, y 8; ζ = 5%; terremoto de El Centro.

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Figure 7.8.1 Espectros de respuesta para sistemas elastoplásticos para el terremoto de El Centro; ζ = 2, 5, y 10% y μ = 1, 4, y 8.

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Figura 7.9.1 Time variation of energy dissipated by viscous damping and yielding, and of kinetic plus strain energy; (a) linear system, Tn = 0.5 sec, ζ = 5%; (b) elastoplastic system, Tn = 0.5 sec, ζ = 5%, ƒy = 0.25._

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Figura 7.10.1 (a) Fluid viscous damper: schematic drawing; (b) force–displacement relation; and (c) diagonal bracing with fluid viscous damper. [Credits: (a) Cameron Black; (b) Cameron Black; and (c) Taylor Devices, Inc.]

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Figura 7.10.2 (a) Buckling restrained brace (BRB): schematic drawings; (b) force–displacement relation; and (c) diagonal bracing with BRB. [Credits: (a) Ian Aiken; (b) Cameron Black; and (c) Ian Aiken.]

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Figura 7.10.3a, b (a) Schematic diagram of slotted bolted connection (SBC); (b) force–displacement diagram of an SBC. (Adapted from C. E. Grigorian and E. P. Popov, 1994.)

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Figura 7.10.3c, d (c) SBC at top of chevron brace in test structure; (d) test structure with 12 SBCs on the shaking table at the University of California at Berkeley. (Courtesy of K. V. Steinbrugge Collection, Earthquake Engineering Research Center, University of California at Berkeley.)

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Figura 7.11.1 Yield-strength reduction factor Ry for elastoplastic systems as a function of Tn for μ = 1, 1.5, 2, 4, and 8; ζ = 5%: (a) El Centro ground motion; (b) LMSR ensemble of ground motions (median values are presented).

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Figura 7.11.2 Valores de diseño de factores de reducción de resistencia de fluencia.

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Figura 7.11.3 Construcción del espectro de diseño inelástico.

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Figura 7.11.4 Espectro de diseño inelástico (84.1 percentil) para movimientosísmico con ügo = 1g, ugo =48 in./s, and ugo = 36 in.; μ = 1.5, 2, 4, 6, y 8; ζ = 5%.

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Figura 7.11.5 Espectro de diseño inelástico (pseudo-acceleración, 84.1 percentil) para movimientossísmicos con ügo = 1g, ugo = 48 in./s, y ugo = 36 in; μ = 1.5, 2, 4, 6, y 8; ζ = 5%.

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Figura 7.11.6 Inelastic (pseudo-acceleration) design spectrum (84.1th percentile) for ground motions with ügo = 1g, ugo = 48 in./sec, and ugo = 36 in; μ = 1.5, 2, 4, 6, and 8; ζ = 5%.

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Figura 7.11.7 Inelastic (deformation) design spectrum (84.1th percentile) for ground motions with ügo = 1g, ugo = 48 in./sec, and ugo = 36 in; μ = 1.5 ,2 ,4, 6, and 8; ζ = 5%..

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Figura 7.11.8 Ratio um/uo of peak deformations um and uo of elastoplastic system and corresponding linear system plotted against Tn ; μ = 1, 1.5, 2, 4, 6, and 8.

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Example 7.1

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Example 7.1 (continued)

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Example 7.2

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Figura 7.12.1 Sistema UGL idealizado.

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Example 7.3

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Figura E7.3

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TABLA E7.3 PROCEDIMIENTO ITERATIVO PARA DISEÑO DIRECTO BASADO EN DESPLAZAMIENTO

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Figura 7.13.1 Comparison of standard design spectrum (ügo = 0.319g) with response spectrum for El Centro ground motion; μ = 1 and 8; ζ = 5%.

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Problema 7.1 The lateral force–deformation relation of the system of Example 6.3 is idealized as elastic– perfectly plastic. In the linear elastic range of vibration this SDF system has the following properties: lateral stiffness, k = 2.112 kips/in., and ζ = 2%. The yield strength ƒy = 5.55 kips and the lumped weight w = 5200 lb.(a) Determine the natural period and damping ratio of this system vibrating at amplitudes smaller than uy.(b) Can these properties be defined for motions at larger amplitudes? Explain your answer.(c) Determine the natural period and damping ratio of the corresponding linear system.(d) Determine ƒy and Ry for this system subjected to El Centro ground motion scaled up by a factor of 3.

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Problema *7.2 Determine by the central difference method the deformation response u(t) for 0 < t < 10 sec of an elastoplastic undamped SDF system with Tn = 0.5 sec and ƒy= 0.125 to El Centro ground motion. Reproduce Fig. 7.4.2, showing the force–deformation relation in part (d) for the entire duration.

*Denotes that a computer is necessary to solve this problem.

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Problema *7.3 For a system with Tn = 0.5 sec and ζ = 5% and El Centro ground motion, verify the following assertion: “doubling the ground acceleration üg(t) will produce the same response μ(t) as if the yield strength had been halved.” Use the deformation–time responses available in Fig. 7.4.3a–c and generate similar results for an additional system and excitation as necessary.


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Problema *7.4 For a system with Tn = 0.5 sec and ζ = 5% and El Centro ground motion, show that for ƒy = 0.25 the ductility factor μ is unaffected by scaling the ground motion.


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Problema 7.5 From the response results presented in Fig. 7.4.3, compute the ductility demands for ƒy = 0.5, 0.25, and 0.125.

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Problema 7.6 For the design earthquake at a site, the peak values of ground acceleration, velocity, and displacement have been estimated: ügo = 0.5g, ugo = 24 in./sec, and ugo = 18 in. For systems with a 2% damping ratio and allowable ductility of 3, construct the 84.1th percentile design spectrum. Plot the elastic and inelastic spectra together on (a) four-way log paper, (b) log-log paper showing pseudo-acceleration versus natural vibration period, Tn, and (c) linear-linear paper showing pseudo-acceleration versus Tn from 0 to 5 sec. Determine equations A(Tn) for each branch of the inelastic spectrum and the period values at intersections of branches.

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Problema 7.7 Consider a vertical cantilever tower that supports a lumped weight w at the top; assume that the tower mass is negligible, ζ = 5%, and that the force–deformation relation is elastoplastic. The design earthquake has a peak acceleration of 0.5g, and its elastic design spectrum is given by Fig. 6.9.5 multiplied by 0.5. For three different values of the natural vibration period in the linearly elastic range, Tn = 0.02, 0.2, and 2 sec, determine the lateral deformation and lateral force (in terms of w) for which the tower should be designed if (i) the system is required to remain elastic, and (ii) the allowable ductility factor is 2, 4, or 8. Comment on how the design deformation and design force are affected by structural yielding.

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Problema 7.8 Consider a vertical cantilever tower with lumped weight w, Tn = 2 sec, and ƒy = 0.112w. Assume that ζ = 5% and elastoplastic force–deformation behavior. Determine the lateral deformation for the elastic design spectrum of Fig. 6.9.5 scaled to a peak ground acceleration of 0.5g.

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Problema 7.9 Solve Example 7.3 for an identical structure except for one change: The bents are 13 fthigh.

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Tema 5 respuesta sismica sistemas inelasticos