Initial cost and seismic vulnerability functions for buildings with energy-dissipating devices

Transcription

1 Safety and Security Engineering 161 Initial cost and seismic vulnerability functions for buildings with energy-dissipating devices J. García-Pérez, M. Zenteno & O. Díaz Instituto de Ingeniería, Mecánica Aplicada, Universidad Nacional Autónoma de México, Mexico Abstract The current optimization process stated within the framework of decision theory computes design values by minimizing the total cost of a structure, including the initial cost as well as the costs due to earthquakes. Initial cost functions are described in terms of design parameters, usually the seismic design coefficient. These functions are obtained in this study by analyzing four different types of structures. Each structure is represented by a reinforced concrete frame comprised of beams and columns, with hysteretic energy-dissipating devices installed as braces. The structures studied are hypothetical buildings built at a soft site in the Valley of Mexico with five, ten, fourteen and twenty-stories high. Cost analyses obtained for these systems are compared with those obtained for a conventional frame made up of only beams and columns. From those structures studied here, vulnerability functions (drift-seismic intensity) are obtained for both systems, that is, the conventional system and the system with dissipating devices. These vulnerability functions together with the cost analyses performed are used to find cost of damage-seismic intensity relationships. The results show that the use of systems with energy-dissipating devices gives a better cost-benefit behavior when the system is under high seismic intensities. Moreover, these results are appropriate for performing long-term cost-benefit analyses. Keywords: initial cost, energy-dissipating devices, reinforced concrete buildings, vulnerability functions. 1 Introduction The current seismic design philosophy is based on the energy-dissipation capability of structures by means of elastic or inelastic deformations (which

2 162 Safety and Security Engineering cause damage to structural or nonstructural elements) that allow for developing more efficient levels of structural control. Thus, one approach of earthquake engineering to the control of structural damage due to earthquakes is the use of energy dissipating-devices (EDDs). Earthquake-resistant design criteria for reinforced concrete buildings provided with hysteretic EDDs rely on information about site seismicity, structural response under specific earthquakes, estimation of structural dynamic properties and economic considerations. Moreover, decisions regarding design values must be made in terms of a cost-benefitanalysis that takes into account initial costs, as well as costs caused by earthquakes. This paper starts by introducing the variables used in the seismic design approach proposed by Campos [1], which is applied in this study to the analysis and design of buildings provided with EDDs. The proposed methodology gives a criterion for distributing seismic shear forces for the conventional frame as well as for the frame reinforced by EDDs. The foregoing provides values of the comparative parameters between the proposed methodology and the established in current seismic codes. The EDDs are usually installed as external elements to the conventional reinforced concrete frame. Theses devices are manufactured in a factory, maintaining appropriate quality control, therefore, they will have a stable hysteretic behavior under a high number of deformation cycles that gives a high energy dissipating capability. Their corresponding failure conditions are determined in laboratory tests by studying their load-deformation relations. The design methodology presented is applied to four different types of buildings provided with EDDs. For all those cases studied here, cost analyses are carried out for the conventional frame and the frame with EDDs, thereby obtaining functions relating initial costs and seismic design coefficients or structural periods. These curves are used later for computing vulnerability functions of the systems studied. 2 Earthquake resistant design variables The methodology proposed by Campos [1] is based on a criterion of optimum performance-based design by means of allowable ductilities, involving a group of parameters that significantly influence the behavior of structural systems, and can be related between them during the different steps of analysis and design. The amount of energy induced to a structure depends on different factors, which are directly related to soil motion, structural dynamic properties as the fundamental period of vibration, damping, stiffness and strength, among other things, and geotechnical characteristics. Structural response under seismic excitations improves if input seismic energy is reduced (by reduction factors) or if dissipating energy is increased by either viscous or hysteretic damping. Because of this, energy-dissipating elements are selected to be studied. These elements are steel eccentric diagonals which reduce the input energy induced by earthquakes, thus improving structural system behavior.

3 Safety and Security Engineering Control variables The control variables proposed by Campos [1] are those design parameters defining the mechanical properties of the structural system, as well as maintenance and repairing policies. These variables are: rk = k d / K, the ratio between the stiffness of the dissipating device and the total stiffness K in each building story. ϕ = δ yd / δ yc is the ratio between the yield deformation for the energy-dissipating device and the conventional frame, respectively. µ is the ductility factor pertaining to the inelastic pseudo-acceleration spectrum of the design earthquake. rr = Rd / R is the ratio between the lateral strength of the dissipating device and total lateral strength. ε = µ d / µ c = 1/ ϕ if we consider the same maximum lateral deformation for both the conventional frame and the frame provided with the EDDs. 2.2 Design spectrum The design spectrum used in this study is an average spectrum normalized with respect to the spectral intensity for a family of five accelerograms, with statistical properties similar to those of the EW component of the SCT (Mexico City) record of September 19, 1985 (SCT850919EW), as shown in fig. 1. S a /g Period (sec) Figure 1: Design spectrum for soft soil in Mexico City. The spectral intensity of each normalized acceleration spectrum is given by T1 I = (1/ 2π ) S ( T, ε ) dt, where ( T, ε ) is the acceleration spectral ordinates 0 a S a applicable to period T, with damping ε = and T1 = 3. 5 for earthquakes on soft soil.

4 164 Safety and Security Engineering 2.3 Basic equations The basic equations obtained from the definitions of control variables are given by: α = r k /( 1 rk ) = kd / kc (1a) β = ϕα = R d / R c (1b) where k c and R c are the lateral stiffness and strength of the conventional frame. From here we can obtain total stiffness and strength given by: K = kc + kd = k( 1+ α) (2a) R = Rc + Rd = Rc ( 1 + β ) (2b) To apply a superposition of the two structural systems one must satisfy the following equation: µ = δ δ = δ K R (3) max / y max / where δ y and δ max stand for the relative yield and maximum deformations for the frame provided with energy dissipating devices. In order to determine the reduction factor for spectral ordinates, Q ( µ, T ), dependent from a given ductility and from the structural natural period T, we use the following equation: Q ( µ, T ) = Se ( T ) / S( µ, T ) = ce / ci (4) where Q relates the elastic ( c e ) and inelastic ( c i ) design coefficients obtained from the design spectrum. 3 Design process The design process for buildings provided with energy-dissipating devices consists of the preliminary design and the final design, as proposed by Campos [1]. 3.1 Preliminary design In the first step, values for α and β are computed considering a more adequate level of ductility. Maximum story displacements δ max are obtained which, according to the RCDF [2], must be of the height of the story. The fundamental period of the building is estimated by applying the thumb rule N/10 where N represents the total number of stories. With this period, elastic and inelastic coefficients c e and ci are obtained from the design spectrum. Now, by using a static seismic method, seismic forces and the corresponding design shear force ( R e ) are determined without reducing by ductility for each level of the building. Once all the variables are known, the reduced shear design R = Re / Q = Re / γµ is obtained. Based on the data obtained above, values for the control variables are proposed as percentages of stiffness and strength that the energy-dissipating

5 Safety and Security Engineering 165 system and conventional frame must provide. Thus a relation for ductilities is found which will be the limit parameter for comparison between the structural design of the conventional frame and the frame provided by EDDs. Moreover, it is considered that under lateral load, each story presents two significant displacements, a relative horizontal, δ H, and a rotation as a rigid body, θ, as result of the elongation and contraction of the columns, thereby defining the story deformation as δ = δ H θ h where h is the height story. Preliminary dimensions of structural elements of the frame with EDDs are obtained in terms of the stiffness and strength required according to the percentage that the concrete must provide, and then successive iterations are performed until adequate dimensions (those producing similar displacements to those allowed) are found. In order to complete the first stage, a comparison between the story stiffness is made by means of coefficient C k, defined as C k = ka / kc with ka as the new lateral stiffness of the story and C 1 ε. 3.2 Final design k After determining the dimensions of the structural elements and the EDDs comprising the frames, the design is carried out. The structural elements are designed by bending and shear with the established criteria for elastic design. The EDDs, which in this case are eccentric diagonals, are designed with the allowable stresses design criteria. Finally relative displacements at each story are checked to comply with the allowable limits indicated by the RCDF [2]. 3.3 Analysis and design of buildings The methodology described above was applied to a group of buildings with a height of five, ten, fourteen and twenty-stories, respectively. The configuration for a ten-story building is shown in fig. 2a-b, but this configuration was used for all buildings studied here. The bay number in all frames studied is four. They have a square plan area of 400 m 2. The height of the first story is 3.5 m and all upper stories have 3 m. It is assumed that they will be built at a soft site in the Valley of Mexico where local soil conditions are similar to those of the recording site of the accelerogram SCT850919EW mentioned above. The structural analysis was done by using plane frames comprised of rigid nodes connecting flexible bars of finite stiffness, thus resembling behavior of real buildings, because they take into account deformations by axial load in columns, as well as deformations due to bending and shear in beams and columns. Eccentric braces were modeled as equivalent diagonal bars subjected to axial deformations caused by lateral forces. Loads and load combinations applied to each structural system were determined according to the RCDF [2]. Dead load intensities for floor and roof levels are 420 kg/m 2 and 560 kg/m 2, respectively. Live load intensities for floor and roof levels are 100 kg/m 2 and 250 kg/m 2, respectively. Detailed information of the buildings analyzed here is presented in [3].

6 166 Safety and Security Engineering a) b) Figure 2: a) Conventional frame and b) Frame with EDDs. 3.4 Design of the EDDs Energy-dissipating systems are comprised of steel diagonals A-36 which stiffen the frame and are diagonally located in each story in the array previously determined as shown in fig. 2b. Accordingly, the foregoing takes into account that EDDs must produce efficient structural behavior and must be beneficial. These systems also have tension anchors and base plates for beams and columns. The steel diagonals give stiffness K D, which together with the concrete frame stiffness K C, provide the stiffness of the structural system required by the design. The cross-sectional area of the diagonal element is obtained by A = K D L /( ED D cos 2 θ ), where L is the diagonal length, E D is the material elasticity modulus, D stands for the number of diagonals in each story, and θ is the angle between the diagonal and the horizontal. Once a preliminary crosssection area is obtained, the design forces of the dissipating system are computed. The diagonal elements work under axial loads either in tension or in compression, thus their design is made by the allowable stress criterion. It is assumed for the connection design that the design force induced by seismic motion acts in the direction of the diagonal. The anchors are designed by tension, and they are checked by shear and crushing.

7 4 Applications to buildings with EDDs The methodology described above is applied to the buildings studied here. After establishing service and seismic loads, a conventional seismic analysis is performed to obtain the weights of each story, the shear forces per story and at the building base, the eccentricities and torsion effects. Detailed computations for the different buildings presented here are found in [3]. A value of α = 0. 5 is proposed and rk is determined from eqn (1a) to be From eqn (2a) we find that rk = kd / K, then k d = and k c = Now we propose β = 0. 5, and we obtain R c = and R d = from eqn (2b). The ratio of yielding displacements between the dissipating device and of each story is given by β / α = 1, therefore the ratio between the lateral strength of the EDD and the total strength of the structural system of each story is rk = Rd / R = ψ rk /[1 + ( ψ 1) rk ] = The ductility ratios between the two systems is given by ε = µ d / µ c = 1/ ψ which becomes 1 if we assume that µ d = µ c = 3. Superposition of both systems requires that µ = δ δ = δ k R. max / y max / Safety and Security Engineering 167 Ic (Thousands of dollars) T (sec) Figure 3: Natural period vs costs (frames+ EDDs). 4.1 Initial cost function After designing the conventional buildings and the buildings with EDDs, a cost analysis was performed in order to know the total costs of each type of building. This cost analysis only takes into account the plain cost of the structure (no indirect costs or finishing). Initial cost functions obtained ( I c ) are described in terms of the period of vibration of each building. In addition, for buildings with EDDs it gives I c = T, and for conventional buildings this function is I c = T. Where I c is in thousands of dollars and T in seconds, figures 3 and 4 show the applicable initial cost functions.

8 168 Safety and Security Engineering Ic (Thousands of dollars) T (sec) Figure 4: Natural period vs costs (conventional frame). 5 Vulnerability functions Seismic vulnerability functions describe quantitatively probable damages of a system in terms of the motion intensity which originate them. These functions can be expressed by physical damage indicators or by their economic consequences. The vulnerability function in terms of economic consequences can be written as δ E ( y) = δ E ( y S)(1 pf ( y)) + δ EF pf ( y) (5) In this equation a certain amount of the expected damage costs is associated with the probable ultimate failure of a system (collapse, total failure), likewise another part of these costs is related to damage occurred in survival conditions of the system. δ E (y) is the expected cost value of damages caused by an earthquake of intensity y, δ E (y S) is the expected value of this cost, conditional to the survival of the system, δ EF and p F (y) are costs of collapse and probability of collapse, respectively. These functions are estimated here from theoretical models of dynamic responses of structural systems, and from assumptions of relationships between damage levels and characteristics of structural responses [4]. Eqn (5) is evaluated through an estimation of the physical damage indicator, determined by the maximum absolute value of angular distortion that a segment of the system undergoes during its response to ground motion. These physical damage indicators are transformed into economic damage indicators normalized with respect to initial cost of the complete system. Moreover, these indicators are estimated by considering direct costs of repairing or retrofitting as well as indirect costs such as interruption of activities, malfunctions, and logistic activities. Direct costs are computed taking into account a repairing threshold or a physical damage value under which all damages are considered so small that they do not deserve to be repaired. It is also assumed that replacement costs of a structural or nonstructural element turn out to be in general greater than those of their initial manufacture, due to extra activities that must be done [5]. The probability of collapse p F (y) is estimated from the probability density function of the ratio between global distortion of the system and the corresponding deformation capacity.

9 Safety and Security Engineering 169 The variables explained above were obtained doing a step-by-step analysis of nonlinear response of the system under study, subjected to a set of seismic ground motions with intensity y. Total cost of the structure comprises initial costs of infill walls, floor systems, finishing works, installations, as well as initial costs given in figs. 3 and 4. The former were taken here doing some reasonable approximations as suggested by Sierra [6] and Ismael [7]. Figure 5 shows vulnerability functions obtained for the ten-stories building for both cases, that is, the conventional system and the system provided by energy-dissipating devices. These curves display expected cost value of damages δ E (y), and seismic intensities (normalized with respect to gravity) given by the ordinates of the elastic response spectrum, for a five per cent damping and for the fundamental period of the system. Results show that for this type of structure, the use of braces as dissipating devices provides systems with less damage and cost than those obtained for the conventional system, as long as they are subjected to same seismic intensities Conventional system System with dissipating devices δe(y) y/g Figure 5: Vulnerability functions. 6 Conclusions Initial cost functions were obtained from the analysis of different types of buildings with and without EDDs. Buildings five, ten, fourteen and twentystories high were analyzed. From those structures studied, vulnerability functions were found. The analysis show that cross sections of the elements designed by current criteria in the codes for the five-story building, o not vary significantly from those obtained by using the method adopted here for buildings with EDDs. However, for higher structures there is indeed a significant difference, thus giving less material and reinforcement, as well as structures with less weight and less cost. The initial costs functions obtained were used to compute vulnerability functions which show that structures provided by energy-dissipating devices present less cost and damage than the conventional frames.

10 170 Safety and Security Engineering References [1] Campos, D (2005), Optimization criteria for the design of buildings with hysteretic energy-dissipating devices, doctoral thesis, National University of Mexico (in Spanish) [2] Federal District Building Code and its Complementary Technical Norms (RCDF), 1993, Diario Oficial de la Federación, México, DF, August, (in Spanish) [3] Zenteno, M (2004), Seismic analysis and design of buildings with energydissipating devices: variation of the initial cost, bachelor thesis, University of Puebla (in Spanish) [4] Ismael, E., Díaz, O and Esteva, L. (2004), Seismic vulnerability analysis for optimum design of multistory reinforced concrete buildings, Proc. 13WCEE, Paper No. 514, Vancouver, B. C., Canada, August, 1-6. [5] Ismael, E. y Esteva, L. (2003), Seismic vulnerability functions for optimum design of frame-wall systems, XIV National Congress on Earthquake Engineering, León, Guanajuato, (in Spanish) [6] Sierra, G. M. (2002), Life-cycle optimization in the establishment of seismic design parameters, Master thesis, National University of Mexico, (in Spanish) [7] Ismael, E. (2003), Seismic vulnerability functions for optimum design of frame-wall systems, Master thesis, National University of Mexico, (in Spanish)