Tema 5 respuesta sismica sistemas inelasticos by Guillermo Santana

IC‐0905

II‐2013

Dinámica de Estructuras Tema 5 Respuesta sísmica de sistemas inelásticos II Semestre 2013, 20 setiembre 2013

Espectros elásticos • Se ha mostrado que el cortante basal máximo inducido por un movimiento sísmico en un sistema elástico lineal es Vb = (A/g)w, donde w es el peso del sistema y A es la ordenada del espectro de pseudo‐aceleración correspondiente al periodo natural de oscilación y el porcentaje de amortiguamiento del sistema. IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

Sin embargo, … • La mayoría de las edificaciones son diseñadas para cortantes basales menores que el cortante basal elástico asociado con el movimiento más fuerte que puede ocurrir en un sitio.

IC‐0905/II‐2013

• Esto puede verse en la figura 7.1, en donde el coeficiente basal A/g proveniente del espectro de diseño de la figura 6.9.5, escalado para 0.4 para obtener correspondencia con la aceleración pico de 0.4g, es comparado con el coeficiente de cortante basal especificado en el International Building Code de 2000. IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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

IC‐0905/II‐2013

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 reinforcedconcrete 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.)

IC‐0905/II‐2013 Daño severo

Universidad de Costa Rica

que impide reparación económicamente factible.

IC‐0905

II‐2013

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 damage 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.)

IC‐0905/II‐2013 Daño severo

que impide reparación económicamente factible.

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. IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

Por tanto, … • La respuesta de estructuras deformándose dentro del rango inelástico durante movimiento sísmico fuerte es de máxima importancia en la ingeniería sísmica.

IC‐0905/II‐2013

Pruebas de laboratorio • California – Berkeley – San Diego

• Japón – U de Tokio/Tsukuba

• Nueva Zelanda – Park, Paulay, Priestley/Christchurch

• México – Unam/Cenapred

• Costa Rica – Lanamme IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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

Relaciones de Fuerza – Deformación para componentes estructurales de diferentes materiales IC‐0905/II‐2013

Figura 7.1.1 Concreto reforzado (Popov & Bertero, 1977)

Relaciones de Fuerza – Deformación para componentes estructurales de diferentes materiales IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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 – Deformación para componentes estructurales de diferentes materiales

IC‐0905/II‐2013

Idealización Elastoplástica

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

Figura 7.1.2

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

Areas bajo la curva son iguales para desplazamiento máximo. IC‐0905/II‐2013

Figura 7.1.3

Relación fuerza–deformación elastoplástica.

La relación cíclica fuerza‐deformación depende del camino seguido. IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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. IC‐0905/II‐2013

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

fy 

Ry 

fy fo



uy uo

f o uo 1   f y uy f y



f y  resistencia de fluencia normalizada

Ry = factor de reducción de resistencia de fluencia

um  factor de ductilidad uy

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

Relación entre deformación máxima y deformación elástica equivalente

um    fy  uo Ry

IC‐0905/II‐2013

Ecuación del movimiento y parámetros gobernantes mu  cu  f S (u )  mug (t )

u  2nu  n2u y fS (u )  ug (t ) f (u ) k c ,  , fS (u )  S m 2mn fy

n 

  2n   n2 fS (  )  n2 ay 

ug (t ) ay

fy m

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

Figura 7.3.1

Relaciones fuerza-deformación normalizadas.

IC‐0905/II‐2013

Figura 7.4.1

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

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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

IC‐0905/II‐2013

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

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

Figura 7.4.4 (a) Peak deformations um and 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.

IC‐0905/II‐2013

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.

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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

IC‐0905/II‐2013

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

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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

IC‐0905/II‐2013

_ Figura 7.7.1 Resistencia normalizada ƒy de sistemas elastoplásticos como función del periodo natural de oscilación Tn para μ = 1, 1.5, 2, 4, y 8; ζ = 5%; terremoto de El Centro. IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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.

IC‐0905/II‐2013

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. IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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.]

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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.]

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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.)

IC‐0905/II‐2013

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.)

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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).

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

Figura 7.11.2

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

IC‐0905/II‐2013

Figura 7.11.3

Construcción del espectro de diseño inelástico.

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

Figura 7.11.4 Espectro de diseño inelástico (84.1 percentil) para movimiento sísmico con ügo = 1g, ugo =48 in./s, and ugo = 36 in.; μ = 1.5, 2, 4, 6, y 8; ζ = 5%. . IC‐0905/II‐2013

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

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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%. . IC‐0905/II‐2013

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%. . IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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.

IC‐0905/II‐2013

Example 7.1

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

Example 7.1 (continued)

IC‐0905/II‐2013

Example 7.2

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

Example 7.2 (continued)

IC‐0905/II‐2013

Figura 7.12.1 Sistema UGL idealizado.

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

Example 7.3

IC‐0905/II‐2013

Example 7.3 (continued)

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

Example 7.3 (continued)

IC‐0905/II‐2013

Figura E7.3

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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

IC‐0905/II‐2013

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%.

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

IC‐0905/II‐2013

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.

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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.

IC‐0905/II‐2013

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

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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

IC‐0905/II‐2013

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.

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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 pseudoacceleration 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.

IC‐0905/II‐2013

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.

IC‐0905/II‐2013

Universidad de Costa Rica

IC‐0905

II‐2013

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.

IC‐0905/II‐2013

Problema 7.9 high.

Solve Example 7.3 for an identical structure except for one change: The bents are 13 ft

IC‐0905/II‐2013

Universidad de Costa Rica