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2 Hajirasouliha I, Pilakoutas K (202) General seisic load distribution for optiu perforance-based design of shear-buildings. Journal of Earthquake Engineering, 6(4), General seisic load distribution for optiu perforancebased design of shear-buildings Ian Hajirasouliha Departent of Civil Engineering, The University of Nottingha, Nottingha, UK Kypros Pilakoutas Departent of Civil & Structural Engineering, The University of Sheffield, Sheffield, UK Abstract An optiisation ethod based on unifor daage distribution is used to find optiu design load distribution for seisic design of regular and irregular shear-buildings to achieve iniu structural daage. By using 75 synthetic spectru-copatible earthquakes, optiu design load distributions are obtained for different perforance targets, dynaic characteristics and site soil classifications. For the sae structural weight, optiu designed buildings experience up to 40% less global daage copared to code-based designed buildings. A new general load distribution equation is presented for optiu perforance-based seisic design of structures which leads to a ore efficient use of structural aterials and better seisic perforance. Key words: Optiisation; Perforance-based seisic design; Irregular structures; Lateral forces; Site soil classification; Ductility. Address correspondence to Ian Hajirasouliha, Departent of Civil Engineering, University of Nottingha, University Park, Nottingha NG7 2RD, UK; E-ail: i.hajirasouliha@nottingha.ac.uk, Tel: +44 (0) Fax: +44 (0)

3 . Introduction The preliinary design of ost buildings is norally based on equivalent static forces obtained fro seisic design guidelines and codes of practice. The height-wise distribution of these static forces is iplicitly based on the first-ode dynaic response of elastic structures [Hart, 2000; Chopra, 200]. As structures exceed their elastic liits in severe earthquakes, the use of inertia forces corresponding to the elastic odes ay not lead to the unifor distribution of ductility deands [Chopra, 200]. The seisic behaviour of different code-based designed structural systes have been extensively investigated over the last two decades [Anderson et al., 99; Gilore and Bertero, 993; Martinelli et al., 2000; Lee and Foutch, 2002; Goulet et al., 2007]. The results of these studies showed that, in general, buildings designed based on new seisic design guidelines satisfy the collapse prevention and iediate occupancy perforance levels. However, the lateral load distribution used by current seisic design guidelines does not always lead to the optiu use of structural aterials [Chopra, 200; Moghadda and Hajirasouliha, 2006]. In an attept to find optiu distribution of structural properties, Takewaki (996, 997) developed an analytical ethod to find stiffness (and strength) distribution that leads to a constant storey-ductility deand for a shear-building structure subjected to a given design spectru. This ethod is based on an elastic equivalent linearization technique, and the results showed that for tall buildings it does not lead to a unifor ductility deand distribution when the structure is subjected to a tie-history excitation. Gantes et al. (2000) used the Euler Bernoulli bea theory to find optiu bending and shear distribution of ulti-storey steel fraes to obtain unifor height-wise drift distribution. Although their proposed ethod is siple and practical, it is not capable of considering the non-linear behaviour of structures. Lee and Goel (200) and Chao et al. (2007) analyzed a series of steel oent and braced fraes subjected to various earthquake excitations. They showed that in general there is a discrepancy between the earthquake induced shear forces and the forces deterined by assuing code-based design load distribution patterns. Based on the results of their studies, they suggested a new lateral force distribution for seisic loads to address the influence of increasing higher ode 2

4 effects in the inelastic state. However, the effects of ground otion characteristics and the degree of nonlinearity were not considered in their suggested load distribution. Moghadda and Hajirasouliha (2006) and Hajirasouliha and Moghadda (2009) developed an effective optiisation ethod to find optiu lateral load distribution for seisic design of regular shear-building structures to obtain unifor storey ductility. They showed that, for the sae target storey-ductility deand, structures designed with the average of optiu load patterns for a set of earthquakes with siilar characteristics, have relatively lower structural weight copared to those designed conventionally. This lead to the following equation for preliinary design of height-wise regular shear-buildings based on the results of twenty earthquake excitations recorded on soft rock: Ki φ i =, () n K j = j K i cit + d = ( a ) ( it + bi µ t 00 i ) where φ i is the ratio of optiu design force at i th storey to the base shear for a regular structure with fundaental period of T and axiu ductility deand of µ t. a i, b i, c i, and d i are constant coefficients at i th storey that should be calculated for each set of design earthquakes. The above load pattern is a function of structural perforance level (i.e. storey ductility), and therefore, is suitable for perforance-based seisic design of structures. However, the load pattern adopted cannot be used directly in practical design of structures, as the utilized seisic records were not copatible with odern building code design spectra (such as Eurocode 8 and IBC-2009), and the effects of height-wise irregularity and site soil profile were not considered in the above equation. This ay lead to structures with unacceptable seisic perforance under design spectrucopatible earthquakes, and hence, needs further developent. In this study, the above entioned optiisation ethod is further developed to optiize both regular and height-wise irregular structures to exhibit iniu structural daage under a design spectru. By using 200 shear-building odels subjected to 75 synthetic earthquakes copatible with 3

5 IBC-2009 design spectra, the effects of using different daage criteria, height-wise irregularity and site soil classification are investigated. Based on the results of this study, a ore realistic lateral design load distribution is proposed and its efficiency is assessed by using a design exaple. 2. Modelling and assuptions In spite of soe drawbacks, shear-building odels have been widely used to study the seisic response of ulti-storey buildings [Diaz et al., 994] because of siplicity and low coputational effort that enables a wide range of paraetric studies. In shear-building odels, each floor is assued to be a luped ass that is connected by perfect elastic-plastic springs which only have shear deforations when subjected to lateral forces as shown in Fig.. All paraeters required to define a shear-building odel corresponding to a full-frae odel can be deterined by perforing a single pushover analysis [Hajirasouliha and Doostan, 200]. The shear-building odel is capable of considering both non-linear behaviour and higher ode effects for the first few effective odes, and therefore, can represent well the actual behaviour of several types of ulti-storey buildings [Diaz et al., 994]. In shear-building odels, the strength of each floor is obtained fro the corresponding storey shear force, and therefore, the height-wise distribution of storey strength can be easily converted to the height-wise distribution of lateral forces. This akes shear-buildings a very suitable odel for calculating the optiu seisic design load pattern for ulti-storey structures with different dynaic characteristics and perforance targets. In the present study, 200 regular and irregular 5, 0 and 5-storey shear-building odels with fundaental period ranging fro 0. sec to 3 sec, and axiu ductility deand equal to,.5, 2, 3, 4, 5, 6 and 8 are utilized. Prior studies by Hajirasouliha and Moghadda (2009) showed that, for a specific fundaental period and target ductility deand, optiu load pattern can be considered independent of nuber of stories. Therefore, 5, 0 and 0 storey shear-building odels with different fundaental periods can be representative of a wider range of structural systes. The range of the fundaental period considered in this study is wider than that of real structures to cover special cases. The Rayleigh daping odel with a constant daping ratio of 0.05 is assigned to the first ode and to the first ode at which the cuulative ass participation exceeds 95%. To predict the seisic response of the shear-building 4

6 odels, nonlinear tie-history analyses were carried out using coputer progra DRAIN-2DX [Parakash et al., 992]. To investigate the effects of different soil profiles on the optiu design load distributions, five sets of spectru-copatible synthetic earthquakes were generated using the SIMQKE progra [Vanarke, 976] to represent the elastic design response spectra of IBC-2009 (and ASCE 7-05) soil types A, B, C, D and E (Table ). These design response spectra are assued to be an envelope of the any possible ground otions that could occur at the site. To include the ground otion variability, each set of synthetic earthquakes consists of 5 generated seisic excitations, with a PGA of 0.4g. It is shown in Fig. 2 that the average acceleration response spectru of each set of synthetic earthquakes copares well with its corresponding IBC-2009 design spectru. 3. Optiu seisic load distribution for a design earthquake In this study, the optiisation target is to obtain a seisic design load that leads to iniu structural daage (i.e. optiu distribution of structural aterial) using a fixed aount of structural aterial. In shear-building structures, any increase in structural aterial is norally accopanied by an increase in storey strength. Therefore, total structural weight could be considered proportional to the su of all storey shear strengths. Consequently, the storey shear strength can be considered as a design variable to optiise the seisic behaviour of shear-building structures. 3.. Optiisation ethodology During strong earthquakes the deforation deand (that is corresponding to structural and nonstructural daage) in code-based designed structures is not expected to be unifor [Chopra, 200]. As a result, in soe parts of the structures the axiu level of seisic capacity is not necessarily utilized. If the strength of underused eleents is decreased increentally, for a ductile structure, it is expected to eventually obtain a status of unifor daage distribution. In such a case, the dissipation of seisic energy in each structural eleent is axiized and the aterial capacity is fully exploited. Therefore, in general, it can be assued that a status of unifor distribution of structural daage is a 5

7 direct consequence of the optiu use of aterial. The optiisation of a non-linear structure subjected to a dynaic excitation is a coplex proble; however, the use of the concept of unifor daage distribution siplifies the atheatics of the optiisation algorith to a large extent. In the present study, in an attept to reach unifor daage distribution through the structure, the following optiisation procedure is adopted: - The initial structure is designed for seisic loads based on a design guidelines, such as IBC The distribution of storey shear strength along the structure is then deterined. 2- A odel of the structure is subjected to the design seisic excitation, and a suitable local daage index (such as storey ductility, inter-storey drift and cuulative daage) is calculated for all stories. 3- The Coefficient of Variation (COV) of daage indices of all stories is calculated. If this COV is sall enough (e.g. less than 0.), the structure is considered to be practically optiu. Otherwise, the optiisation algorith proceeds to iterations. 4- During the iterations, the distribution of storey shear strength is odified. The shear strength is reduced in the stories with lower-than-average daage index and increased in the stories which experienced higher-than-average daage. To obtain convergence in nuerical calculations, this alteration needs to be applied increentally using the following equation: DI a S i ( S i ) n+ = ( i ) n (2) DI ave where (S i ) n is the shear strength of the i th storey at n th iteration, DI i and DI ave are daage index for the i th storey and average of daage indices for all stories, respectively. a is convergence paraeter ranging fro 0 to. Analyses carried out in this study on different odels and seisic excitations indicated that an acceptable convergence is usually obtained by using a values of 0. to 0.2. The results presented in this paper are based on a value of The shear strength of all stories are scaled such that the su of storey shear strengths (and structural weight) reains unchanged. The optiisation procedure is then repeated fro step 2 6

8 until the COV of daage indices becoe sall enough. The final solution is considered to be practically optiu. Analyses carried out by the authors showed that the optiu distribution of storey shear strengths is independent of the seisic load distribution used for initial design. The concept of unifor daage distribution can also be used to find the optiu distribution of storey shear strengths for a specific perforance target (DI target ). In this case, the equation (2) in the optiisation process should be replaced by the following equation, and there is no need to scale the su of storey shear strengths in the step 5. ( DI + = S (3) i S i ) n ( i ) n DI t arg et a In perforance-based design ethods, design criteria are expressed in ters of achieving specific perforance targets during a design level earthquake. Perforance targets could be satisfied by controlling the level of stress, displaceent or structural and non-structural daage. The proposed ethod can optiise the design using different types of perforance paraeters as discussed in the following sections Miniu storey ductility Storey ductility has been widely used to assess the level of daage in non-linear shear-building structures (Chopra 200). In this section, storey ductility is considered as the daage index in the optiisation process (Equation 2). To show the efficiency of the proposed ethod, the above optiisation algorith is used for the optiu design of a 0-storey shear-building with fundaental period of. sec subjected to a ground otion recorded at the Canoga Park station in the Northridge earthquake 994 (CNP96). Based on the concept of unifor daage distribution, the proposed optiisation ethod is expected to lead to a structure with iniu storey ductility (selected response paraeter). Fig. 3 shows the variation of axiu storey ductility and COV of storey ductility deands fro IBC-2009 to optiu designed odel. It is shown that decreasing the COV was always accopanied by 7

9 reduction of axiu storey ductility, and the proposed ethod practically converged to the optiu solution in less than 7 steps without any fluctuation. Fig. 4-a copares the storey ductility distribution of IBC-2009 and optiu designed odel (iniu storey ductility). It is shown that the proposed optiisation ethod resulted in a design with an alost perfectly unifor storey ductility distribution. The results indicate that, for the sae structural weight, the optiu designed structure experienced 52% less axiu storey ductility (i.e. less structural daage) copared to the conventionally designed structure. As it is entioned before, the final height-wise distribution of storey strength in a shear-building odel can be easily converted to the height-wise distribution of design lateral forces. Such pattern ay be regarded as the optiu distribution pattern for seisic design forces (Opt eq ). The lateral force distribution of IBC-2009 and optiu designed odels are copared in Fig. 4-b. The results indicate that to iprove the seisic perforance under this specific earthquake, the above entioned odel should be designed based on an equivalent lateral load distribution different fro one used by conventional code patterns. However, this optiu load distribution ay not be suitable for other cases as it depends on the characteristics of the structure and seisic excitation Miniu cuulative daage To investigate the effect of selected daage criteria on the optiu design load pattern, the daage index proposed by Baik et al. (988) based on the classical low-cycle fatigue approach is used in the optiisation process (Equation 2). The inter-storey inelastic deforation is chosen as the basic daage quantity, and the cuulative daage index after N excursions of plastic deforation is calculated as: DIi = N pj j= δ y Dδ c (4) where DI i is the cuulative daage index at i th storey, ranging fro 0 for undaaged to for severely daaged stories, N is the nuber of plastic excursions, Dδ pj is the plastic deforation of i th 8

10 storey in j th excursion, δ y is the noinal yield deforation, and c is a paraeter that accounts for the effect of plastic deforation agnitude which is taken to be.5 [Krawinkler and Zohrei, 984]. To assess the daage experienced by the full structure, the global daage index is obtained as a weighted average of the daage indices at the storey levels, with the energy dissipated being the weighting function. DI g n i= = n DI W i= i W pi pi (5) where DI g is the global daage index, W pi is the energy dissipated at i th storey, DI i is the daage index at i th storey, and n is the nuber of stories. The previous 0-storey exaple was solved by considering the cuulative daage (Equation 4) as the local daage index. As shown in Fig. 5, the optiu designed structure in this case exhibits iniu cuulative daage during the design earthquake. Based on the results, for the sae structural weight, 0-storey buildings optiised for iniu storey-ductility and cuulative daage experience on average 40% less global daage index as copared to the IBC-2009 designed structure. The results shown in Figures 4 and 5 indicate that, in general, changing the daage assessent criteria does not have a ajor effect on the optiu design load distribution, as well as the axiu storey-ductility deand and the cuulative daage of optiu designed structures. This conclusion has been confired by analysis of different structures and ground otion records. 4. Optiu seisic design load distribution for building code design spectra Based on the work presented in the previous sections, it was found that for every building there is a specific optiu load distribution that leads to optiu seisic perforance during the design 9

11 earthquake (Opt eq ). This optiu pattern depends on the characteristics of the design earthquake, and therefore, varies fro one earthquake to another. However, there is no guarantee that the structure will experience seisic events with the exact characteristics of the design ground otion. Therefore, for practical applications, appropriate design load distributions should be developed for typical building code design spectra. Using the proposed optiization algorith, the optiu load distribution patterns for the 200 shear-building odels presented earlier were calculated for the five sets of selected synthetic earthquakes representing different soil types (5,000 optiu load patterns). The average of the optiu load patterns for each set of synthetic records was then used to design new shear-buildings (Opt ave ). For each seisic excitation, the required structural weight to obtain a target storey-ductility deand was deterined for 60,000 shear-buildings designed with: a) optiu load pattern corresponding to the design earthquake (Opt eq ); b) IBC 2009 design; c) Hajirasouliha and Moghadda (2009) load pattern; and d) the average of optiu load patterns (Opt ave ). As expected, the results of this study showed that, for the sae storey-ductility deand, structures designed with the optiu load patterns corresponding to the design earthquake (Opt eq ) always have less structural weight (optiu structural weight) copared to the other structures. However, these optiu load patterns are specific to the particular design earthquake, and therefore, are not appropriate for general design purposes which rely on a design spectru. To copare the adequacy of different design load patterns, Fig. 6 copares the ratio of required to optiu structural weight (based on Opt eq ) for structures with fundaental period of 0.5 and sec and axiu ductility deands of to 8. This figure is based on the average weights required for each of the 5 synthetic earthquakes representing soil type C. It is shown that in the elastic range of response (i.e. µ t =), the total structural weight for odels designed based on IBC-2009 load distribution is on average around 8% above the optiu value. Therefore, it is confired that using conventional loading patterns leads to acceptable designs for elastic structures. However, the efficiency of the code load distribution deteriorates increasingly in the non-linear range of behaviour. In the low ductility range (i.e. µ t <3), Hajirasouliha and Moghadda (2009) load pattern leads to structures with less structural weight copared to IBC-2009 designed odels. However, this load 0

12 pattern gets worse in the high ductility range as it can result in structural weights up to 80% ore than the optiu values. This is attributed to the fact that Hajirasouliha and Moghadda (2009) load pattern was ainly developed based on a liited nuber of seisic records rather than a group of design spectru-copatible earthquakes. Structures designed with the average of optiu load distributions for a set of spectrucopatible earthquakes (Opt ave ) always have less (up to 37%) structural weights copared to IBC designed structures. The results indicate that the average of the optiu load distributions can be used for seisic design of buildings in a wide range of target ductility deands (i.e. different perforance targets). However, calculation of the average load patterns requires a lot of coputational effort, and therefore, for practical design purposes it is necessary to develop a siple ethod to estiate the average of optiu load patterns for different structures and perforance targets. The results of this study show that the general for of Equation () can be adopted to represent the average of optiu load patterns corresponding to different building code design spectra (Opt ave ). For this purpose, the constant coefficients a i, b i, c i, and d i in Equation () should be calculated based on the average of the results for a set of synthetic spectru-copatible earthquakes representing a specific design spectru. For exaple, Table 2 shows the constant coefficients corresponding to the design response spectru of IBC-2009 soil type C. These coefficients can be obtained at each level of the structure by interpolating the values given in Table 2. Fig. 6 shows that structures designed with the odified coefficients (Table 2) always require less structural weight copared to siilar structures designed with IBC-2009 and Hajirasouliha and Moghadda (2009) load patterns. The results also indicate that structures designed with the proposed equation behave very siilar to those designed with the average of optiu load patterns (Opt ave ). Fig. 7 copares the new general load distributions (calculated by using Equation and odified coefficients given in Table 2) and the corresponding load distributions obtained fro nonlinear dynaic analysis. The results of the proposed equation copare very well with the analytical results, and the equation works well for different periods and ductility deands.

13 5. Efficiency of the proposed design load pattern The adequacy of different design load patterns can be assessed by evaluating their correlation with the average of optiu load patterns corresponding to the typical building code design spectra. For this purpose, the following efficiency factor is defined in this study: EF = n i= [( φ ) ( φ ) i design n i ave ] 2 (6) where n is the nuber of stories, and (φ i ) design and (φ i ) ave are the scaled lateral load pattern at i th storey calculated based on the selected design load pattern and the average of optiu load patterns, respectively. Fig. 8 copares the EF factor for structures designed with IBC-2009, Hajirasouliha and Moghadda (2009) load pattern, and the general load pattern proposed in this study. This figure shows the average of the results for ten 0-storey structures with fundaental periods between 0. to 3 sec. It is shown that the general load pattern has better agreeent with the average of the optiu load patterns copared to IBC-2009 and Hajirasouliha and Moghadda (2009) load patterns. The efficiency of the proposed equation is further assessed by using a design exaple in the upcoing sections. 6. Effect of site soil profile on optiu design load distribution To investigate the effect of site soil classification on the optiu seisic design load distribution, five sets of 5 synthetic earthquakes were considered as introduced in section 2. For each synthetic ground otion record, the optiu design load distribution was derived for the 200 shear-building odels with different fundaental periods and target ductility deands. Using the suggested forula for optiu design load distributions, the constant coefficients a i, b i, c i, and d i were deterined for each group of synthetic earthquakes representing a site soil classification, and copared in Fig. 9. 2

14 The results indicate that the optiu load distributions for structures with siilar fundaental period and axiu ductility deand sited on soil profiles type A, B, C and D are practically identical. However, the optiu load distributions for soft soil profiles (type E) are slightly different. Therefore, for practical applications, it is suggested to provide two sets of coefficients a i, b i, c i, and d i for hard rock to stiff soil profiles and for soft soil. 7. Design load distribution for height-wise irregular structures To investigate the effect of height-wise irregularity on the design load distribution, six 0-storey shear-buildings with different ass distribution patterns were considered as shown in Fig. 0. Using the proposed optiisation algorith, the buildings were designed to have a fundaental period of sec and unifor storey ductility of 4 when subjected to the Northridge earthquake of 994 (CNP96). For each height-wise ass distribution, there is a specific load distribution that leads to a unifor storey ductility deand. Using the proposed optiisation ethod, the optiu seisic design load distribution for the irregular shear-building odels (type A to F) were calculated as shown in Fig.. To evaluate the efficiency of the proposed optiu design load distributions, the shear-building odels shown in Fig. 0 were also designed with the IBC-2009 load distribution using the sae structural weight as the optiu designed odels. Storey ductility distribution of IBC-2009 and optiu designed odels are copared in Fig. 2. The results indicate that, for the sae structural weight, optiu designed odels experience less axiu storey ductility (up to 55% less), and therefore, less structural daage during the design earthquake (Northridge earthquake of 994). Lateral seisic design load in ost of design guidelines is considered to be proportional to the storey weight (UBC-97, Eurocode 8, IBC-2009 and ASCE 7-05). In this study, a siilar concept is used to noralize the optiu design load distribution of height-wise irregular structures. Fig. 3 copares the optiu design load distributions of different height-wise irregular structures (shown in Fig. 0) after being noralized to the relative storey weight. Despite the big difference between height-wise ass distributions of the exained shear-building structures, the results indicate that the noralized optiu design loads are alost identical for buildings with siilar fundaental period 3

15 and axiu ductility deand. The sall difference between the noralized optiu design load distributions ay be due to the effect of higher odes. By knowing the optiu load distribution for a height-wise regular structure, the optiu load distribution for an irregular structure with siilar fundaental period and axiu ductility deand can be calculated by using the following equation: ' i i φ i = n (7) wφ j= w φ j j where φ i and φ' i are the ratio of optiu design force at i th storey to the base shear (load distribution pattern) of regular and irregular structures, respectively; w i is the weight of the i th storey; and n is the nuber of stories. To calculate the optiu load distribution for an irregular shearbuilding, first the optiu load distribution for a regular building with siilar fundaental period and axiu ductility deand should be calculated by using Equation () with appropriate coefficients for site soil classification (shown in Fig. 9). Subsequently, Equation (7) should be used to convert the optiu design load distribution for the equivalent irregular structure. 8. Verification using an irregular shear-building design exaple The efficiency of the general load pattern proposed in this study is deonstrated through the seisic design of an irregular five-storey shear-building shown in Fig. 4-a. The building is assued to be located on a soil type B of IBC-2009 with fundaental period of 0.6 sec and axiu storey ductility deand of 6. The proposed design load pattern was calculated by using Equations () and (6) as explained in the previous section. Two shear buildings were designed based on the IBC-2009 and the proposed load pattern (shown in Fig. 4-b), and subjected to 5 synthetic earthquakes representing the IBC-2009 soil type B design spectru. For each seisic excitation, the required structural weight (i.e. su of storey shear strengths) was calculated to obtain target ductility deand of 6. The average and 95 th percentile (average plus.65 ties the standard deviation) of the required storey shear strengths were 4570 kn and 6290 kn for the IBC-2009 structure, and 3660 kn and 470 4

16 kn for the structure designed based on the proposed load pattern, respectively. Therefore, for the sae axiu ductility deand, the shear-building designed by the proposed load pattern requires considerably less (up to 34% less) structural aterial. The average and envelope of axiu storey ductility deands for the IBC-2009 and the structure designed by the proposed load pattern are copared in Fig. 5. It is shown that by using the proposed load pattern, the seisic capacity of the designed structure was fully utilized as the axiu ductility deand of all stories reached the target ductility of 6 at least at one earthquake. The results indicate that, in general, the proposed load pattern leads to shear-buildings with a ore unifor storey ductility deand. 9. Verification using concentrically braced fraes The efficiency of the proposed load pattern is further exained for the seisic design of three concentrically braced steel fraes of 5, 0 and 5 stories (shown in Fig. 6). The buildings were assued to be located on a soil type C of IBC-2009 (and ASCE 7-05) category, with the design spectral response acceleration at short and -sec periods equal to.g and 0.64g, respectively. Ordinary concentrically braced fraes (OCBF) were designed to support gravity and lateral loads in accordance with the iniu requireents of ANSI/AISC and ANSI/AISC Siple bea to colun connections were used such that no oent is transitted fro beas to supporting coluns. In all odels, the top storey was considered to be 25% lighter than the rest. IPB (wide flange I-section), IPE (ediu flange I-section) and UNP (U-Channel) sections, according to DIN-025, were chosen for coluns, beas and bracings, respectively. To eliinate the effect of discrete section sizes, auxiliary sections were artificially developed by assuing a continuous variation of section properties based on DIN-025. In the code based designed odels, once the ebers were seized, the entire design was checked for the code drift liitations and refined to eet the code requireents when necessary. A bea-colun eleent which allows for the foration of P-M hinges near its ends was eployed to odel the coluns. The post-buckling behaviour of brace ebers was taken into account by utilizing the hysteretic odel suggested by Jain et al. (980). In this section, shear inter- 5

17 storey drift is considered as the ain perforance paraeter to assess the level of structural and nonstructural daage as suggested by Bertero et al. (99) and Moghadda et al. (2005). The 5, 0 and 5 storey concentrically braced fraes were designed based on IBC-2009 and the proposed load patterns (Figures 7-a to 9-a) and subjected to 5 synthetic earthquakes representing the IBC-2009 soil type C design spectru. The proposed design load pattern for each odel was scaled to obtain the sae structural weight as the IBC-2009 fraes. The average and envelope of axiu shear inter-storey drifts for the IBC-2009 and the structures designed by the proposed load pattern are copared in Figures 7-b to 9-b. The results indicate that the efficiency of the proposed load pattern for seisic design of a concentrically braced frae is less than for shear-building odels as the distribution of shear storey drift is not fully unifor. However, for the sae structural weights, concentrically braced fraes designed with the proposed load pattern always undergo lower shear inter-storey drifts (up to 20%) under design earthquakes, and therefore, exhibit an overall better seisic perforance copared to IBC-2009 designed odels. 0. Application of the proposed design load pattern In perforance-based design ethods, different ultiple liit states (e.g. service event, rare event, very rare event) are usually considered. The optiu design for a specific liit state does not guarantee the optiu behaviour in other conditions. In this case, it is usually accepted to use the very rare event as the governing criterion for the initial design, and then check the design for other liit states. The results of this study indicate that the general loading pattern proposed in this paper is efficient for structural systes that exhibit shear-building like behaviour, such as buckling-restrained braced fraes and oent resisting fraes with high bea-to-colun stiffness ratio. The efficiency of the proposed load pattern reduces slightly for conventional concentrically braced fraes, since the seisic behaviour of the fraes is significantly influenced by the slenderness of the brace eleents (Karavasilis et al. 2007). However, the proposed load pattern can still iprove the seisic perforance of the designed fraes, and should prove useful in the conceptual design phase. 6

18 Initial studies show that the proposed loading pattern cannot be directly applied to soe structural systes such as structural walls, as they behave substantially different fro shear-building type of structures. Further research is required to extend the proposed load pattern to different structural systes and different perforance targets.. Conclusions A ethod based on the concept of unifor daage distribution is adopted for optiu seisic design of regular and irregular structures subjected to a design seisic excitation. It is shown that, for the sae structural weight, structures designed with the optiu load distribution experience up to 50% less axiu storey ductility and 40% less global daage copared to code-based designed structures. It is shown that optiu design load distribution, storey ductility deand and global daage index for buildings optiised for iniu storey ductility and iniu cuulative daage are relatively siilar. For a set of synthetic earthquakes representing a typical building code design spectru, optiu seisic design load distributions were deterined. It is shown that structures designed with the average of optiu load distributions have up to 37% less structural weight copared to siilar conventionally designed structures. The results indicate that the optiu load distributions for structures with siilar fundaental period and axiu ductility deand sited on IBC-2009 soil profiles type A, B, C and D (hard rock to stiff soil) are nearly identical. However, the optiu load distributions for soft soil profiles (type E) are slightly different. Based on the results of this study, a general load distribution is introduced for seisic design of height-wise regular and irregular structures that is a function of soil type, fundaental period of the structure and axiu ductility deand. It is shown that using the proposed loading pattern leads to a ore efficient use of structural aterials, and therefore, better seisic perforance for shear- 7

19 building like structures and concentrically braced fraes. Further work is required to extend the proposed design load pattern to other types of structural systes and perforance targets. References ANSI/AISC [2005] Seisic Provisions for Structural Steel Buildings, Aerican Institute of Steel Construction, Inc., Chicago, IL. ANSI/AISC [2005] Specification for Structural Steel Buildings, Aerican Institute of Steel Construction, Inc., Chicago, IL. Anderson, J. C., Miranda, E. and Bertero, V. V. [99] "Evaluation of the seisic perforance of a thirty-story RC building" UCB/EERC-9/6, Berkeley: Earthquake Engineering Research Centre, University of California. ASCE 7-05 [2006] Miniu Design Loads for Buildings and Other Structures, Aerican Society of Civil Engineers, Reston, Virginia. Baik, S. W., Lee, D. G. and Krawinkler, H. [988] "A siplified odel for seisic response prediction of steel frae structures," In: The 9 th WCEE, Vol. 5, Japan. Bertero, V. V., Anderson, J. C., Krawinkler, H. and Miranda E. [99] Design guidelines for ductility and drift liits, Report No. UCB/EERC-9/5. University of California, Earthquake Eng Center, Berkeley, CA. CEN Eurocode 8 [2004] design provisions of structures for earthquake resistance. European Coittee for Standardization. Brussels. Chao, S. H., Goel, S. C. and Lee, S. S. [2007] "A seisic design lateral force distribution based on inelastic state of structures," Earthquake Spectra 23(3),

20 Chopra, A. K. [200] Dynaics of structures: theory and applications to earthquake engineering. 2 nd Edition, Prentice-Hall: Englewood Cliffs, NJ. Diaz, O., Mendoza, E. and Esteva, L. [994] "Seisic ductility deands predicted by alternate odels of building fraes," Earthquake Spectra 0(3), DIN 025 [995] Hot rolled I and H sections: Diensions, ass and static paraeters, DIN Deutsches Institut Fur Norung EV, Berlin. Gantes, C. J., Vayas, I., Spiliopoulos, A. and Pouangare, C. C. [2000] "Optiu Bending and Shear Stiffness Distribution for Perforance Based Design of Rigid and Braced Multi-Story Steel Fraes," In: 3 rd International Conference STESSA Behaviour of Steel Structures in Seisic Areas, F.M. Mazzolani and R. Treblay (Editors), Montreal, Canada, Gilore, T. A. and Bertero, V. V. [993] "Seisic perforance of a 30-story building located on soft soil and designed according to UBC 99," UCB/EERC-93/04, Berkeley, EERC, University of California. Goulet, C. A., Haselton, C. B., Reiser, J. M., Beck, J. L., Deierlein, G. G., Porter, K. A. and Stewart, J. P. [2007] "Evaluation of the seisic perforance of a code-conforing reinforced-concrete frae building-fro seisic hazard to collapse safety and econoic losses," Earthquake Eng Struc 36(3), Hajirasouliha, I. and Doostan, A. [200] "Siplified odel for seisic response prediction of concentrically braced fraes," Advances in Engineering Software 4 (3), Hajirasouliha, I. and Moghadda, H. [2009] "A new lateral force distribution for seisic design of structures," ASCE J. Structural Eng 35 (8), Hart, G. C. [2000] "Earthquake forces for the lateral force code," The Structural Design of Tall Buildings 9(), IBC-2009 [2009] International Building Code. International Code Council, Country Club Hills, USA. 9

21 Jain, A. K., Goel, S. C. and Hanson, R. D. [980] "Hysteretic cycles of axially loaded steel ebers", J Structural Division ASCE, 06, Karavasilis, T.L., Bazeos, N. and Beskos, D.E. [2007] "Estiation of seisic drift and ductility deands in planar regular X-braced steel fraes," Earthquake Engineering and Structural Dynaics (5), Krawinkler, H. and Zohrei, M. [984] "Cuulative daage in steel structures subjected to earthquake ground otions," Coputers & Structures 6 (-4), Lee, K. and Foutch, D. A. [2002] "Perforance evaluation of new steel frae buildings for seisic loads," Earthquake Eng Struc 3(3), Lee, S. S. and Goel, S. C. [200] "Perforance based seisic design of structures using target drift and yield echanis," In: U.S Japan Seinar on Advanced Stability and Seisicity Concept for Perforance Based Design of Steel and Coposite Structures, Kyoto, Japan. Martinelli, L., Perotti, F. and Bozzi, A. [2000] "Seisic design and response of a 4-story concentrically braced steel building," in: Behavior of Steel Structures in Seisic Areas. Mazzolani F and Treblay R (eds), Balkea, Rotterda, Moghadda, H. and Hajirasouliha, I. [2006] "Toward ore rational criteria for deterination of design earthquake forces," Int J Solids Struct 43(9), Moghadda, H., Hajirasouliha, I. and Doostan, A. [2005] "Optiu seisic design of concentrically braced steel fraes: concepts and design procedures," J Constr Steel Res 6 (2), Prakash, V., Powell, G. H. and Filippou, F. C. [992] DRAIN-2DX: base progra user guide. UCB/SEMM- 92/29, Earthquake Engineering Research Centre, University of California, Berkeley. Takewaki, I. [996] "Design-oriented approxiate bound of inelastic responses of a structure under seisic loading," Coput Struct 6 (3),

22 Takewaki, I. [997] "Design-oriented ductility bound of a plane frae under seisic loading," J Vib Control 3 (4), Unifor Building Code, UBC [997] International Conference of Building Officials, vol. 2. Vanarke, E.H. [976] SIMQKE: A Progra for Artificial Motion Generation, User s Manual and Docuentation. Departent of Civil Engineering, Massachusetts Institute of Technology. 2

23 Table - Site soil classifications according to IBC-2009 Site class Soil profile nae Soil shear wave velocity A Hard rock > 500 /s B Rock 760 to 500 /s C Very dense soil and soft rock 370 to 760 /s D Stiff soil profile 80 to 370 /s E Soft soil profile < 80 /s Table 2- Modified coefficients for Equation () as a function of relative height (site class C) Relative Height a b c d

24 Fig. - Typical shear-building odel.2 Spectral Acceleration (g) Type A Type D Type B Type C IBC-2009 Design Spectru Average of 5 Synthetic Eq Type E Period (Sec) Fig. 2- Coparison between IBC-2009 design spectru and average response spectra of 5 synthetic earthquakes COV (x0) & Max Storey Ductility COV (x0) Maxiu storey ductility Steps Fig. 3- Variation of axiu storey ductility and COV of storey ductility deands fro IBC-2009 to optiu odel 23

25 Storey (a) IBC-2009 Miniu Storey Ductility Miniu Cuulative Daage Storey Ductility Storey (b) IBC-2009 Miniu Storey Ductility Miniu Cuulative Daage Lateral Force / Base Shear Fig. 4- (a) Storey ductility, and (b) lateral force distribution of IBC-2009 odel and optiu odels designed for iniu storey ductility and cuulative daage Global Daage Index (%) IBC-2009 Min Ductility Min Daage Fig. 5- Global daage index for IBC-2009 odel and optiu odels designed for iniu storey ductility and cuulative daage 24

26 Required to optiu structural weight IBC Hajirasouliha & Moghadda (2009) Average of load patterns This study T= 0.5 sec Target Ductility Required to optiu structural weight IBC Hajirasouliha & Moghadda (2009) Average of load patterns This study T= sec Target Ductility Fig. 6- The ratio of required to optiu structural weight for structures designed with IBC-2009, Hajirasouliha and Moghadda (2009), average of optiu load patterns (Opt ave ), and the general load pattern proposed in this study, average of 5 synthetic earthquakes (soil type C) 25

27 0.9 Height / Total Height T=0.2, µ= (Equation) 0.6 T=0.2, µ= (Analytical) 0.5 T=0.6, µ=3 (Equation) 0.4 T=0.6, µ=3 (Analytical) 0.3 T=, µ=5 (Equation) 0.2 T=, µ=5 (Analytical) T=2, µ=8 (Equation) 0. T=2, µ=8 (Analytical) Lateral Force / Base Shear Fig. 7- Correlation between the proposed equation and analytical results 20 5 EF (x 000) 0 5 IBC 2009 Hajirasouliha & Moghadda (2009) This study Target Ductility Fig. 8- Efficiency Factor (EF) for structures designed with IBC-2009, Hajirasouliha and Moghadda (2009), and the general load pattern proposed in this study 26

28 Relative Height Type-A 0.6 Type-B 0.5 Type-C 0.4 Type-D 0.3 Type-E Coefficient a i 0.9 Relative Height Type-A Type-B Type-C Type-D Type-E Coefficient b i Relative Height 0.9 Type-A 0.8 Type-B 0.7 Type-C 0.6 Type-D 0.5 Type-E Coefficient c i Relative Height 0.9 Type-A 0.8 Type-B 0.7 Type-C 0.6 Type-D 0.5 Type-E Coefficient d i Fig. 9- Constant coefficients a i, b i, c i, and d i (Equation ) for different site soil classifications 27

29 Type- A Type- B Type- C Type- D Type- E Type- F Fig. 0- Shear-building odels with different height-wise ass distribution Storey Type-A Type-B Type-C Type-D Type-E Type-F Lateral Force / Base Shear Fig. - Optiu seisic design load distribution for shear-buildings with different height-wise ass distribution Storey Type-A Type-B Type-C Type-D Type-E Type-F Optiu (all types) Storey Ductility Fig. 2- Storey ductility distribution of IBC-2009 and optiu designed buildings having different height-wise ass distribution 28

30 Storey Type-A Type-B Type-C Type-D Type-E Type-F Relative Lateral Force / Base Shear Fig. 3- Noralized optiu seisic design load distribution for shear-buildings with different heightwise ass distributions (a) 750 kn 5 (b) 000 kn kn Storey 3 IBC-2009 Proposed Load 000 kn 000 kn Lateral Force / Base Shear Fig. 4- (a) Storey weight, and (b) Coparison between IBC-2009 and the proposed load pattern 5 Storey Storey Ductility IBC-2009 (Average) Proposed load (Average) IBC-2009 (Maxiu) Proposed load (Maxiu) Fig. 5- Maxiu and average of storey ductility deands for 5 synthetic earthquakes representing IBC-2009 design spectru 29

31 Top storey load: 45 kn/ Storey load: 60 kn/ 3 = 5 3 = 30 3 = 45 6 = 30 6 = 30 6 = 30 Fig. 6- Typical geoetry of concentrically braced fraes Storey T= 0.6 µ t = 3 IBC Proposed Load Storey IBC (Average) Proposed load (Average) IBC (Maxiu) 2 2 Proposed Load (Maxiu) (a) (b) Lateral Force/ Base Shear Shear Storey Drift (c) Fig. 7- (a) IBC-2009 and optiu seisic design load distribution, and (b) Maxiu and average of shear storey drifts for the 5-storey concentrically braced frae Storey T=.2 µ t = 2 IBC Proposed Load (a) Lateral Force/ Base Shear Storey IBC (Average) Proposed load (Average) 4 IBC (Maxiu) 2 (b) Proposed Load (Maxiu) Shear Storey Drift (c) Fig. 8- (a) IBC-2009 and optiu seisic design load distribution, and (b) Maxiu and average of shear storey drifts for the 0-storey concentrically braced frae 30

32 Storey T=.8 µ t =.5 IBC Proposed Load (a) Lateral Force/ Base Shear Storey IBC (Average) Proposed load (Average) IBC (Maxiu) 3 Proposed Load (b) (Maxiu) Shear Storey Drift (c) Fig. 9- (a) IBC-2009 and optiu seisic design load distribution, and (b) Maxiu and average of shear storey drifts for the 5-storey concentrically braced frae 3