Probabilistic seismic performance evaluation of steel moment frame using high-strength and high-ductility steel

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Probabilistic seismic performance evaluation of steel moment frame

using high-strength and high-ductility steel

Kuo-Wei Liao

a,⇑

_{, Yu-I Wang}

b

_{, Cheng-Cheng Chen}

c a

Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei 106, Taiwan

b

CTCI Corporation, Taipei, Taiwan

c

Department of Civil & Construction Engineering, National Taiwan University of Science & Technology, Taiwan

h i g h l i g h t s

Using ductile and high-strength steels, a damaged-controlled frame is designed. Kinematic hardening and two surface theories are coded as an ABAQUS UMAT for BRB. A design recommendation is provided for engineers based on the optimal structure. The economic feasibility of using high-strength steel in a frame is revealed.

a r t i c l e

i n f o

Article history: Received 18 May 2016

Received in revised form 6 January 2018 Accepted 23 June 2018

Keywords: High strength steel High ductility steel UMAT

Kinematic hardening Two surfaces

a b s t r a c t

To shorten a steel building recovery time after an earthquake, a dual and damaged-controlled system is proposed, in which the seismic energy is absorbed by a highly ductile buckling restrained brace (BRB) and the gravity load is resisted by high-strength column. Because the seismic energy is mainly dissipated by the BRB, to accurately simulate the BRB hysteretic behavior is essential. Thus, kinematic hardening and two surface theories are adopted and coded as an ABAQUS user-defined material (UMAT). Particle swarm optimization (PSO) is used to find the optimal design equivalent to a corresponding traditional structure. The performance of the optimal frame is verified by nonlinear time history and fragility analysis. Based on the found optimum, a practical design guideline is recommended. The performance of high-strength steel and a high-ductility structural system such as the inter-story drift, maximum roof acceleration, property of BRB hysteresis, strength ratio between the main frame and BRB and cumulative fatigue damage are investigated. In addition, the economic feasibility of using high-strength steel in the structural system is compared to that using traditional steel materials.

1. Introduction

A dual system is often employed indesigning a building to reduce the seismic damage potential. The dual system consists of two major components: non-dissipative and dissipative members. Based on this concept, many different designs have been devel-oped. For example, Zhang and Zirakian[1] employed low yield point (LYP) steel plate shear walls (SPSWs) as the primary dissipa-tive members in moment resisting frames resulting in a better seis-mic performance. Tenchini et al.[2]proposed using high-strength steel (HSS) in non-dissipative members and mild carbon steel (MCS) in dissipative zones. Perri et al. [3]investigated the cost and low-cycle fatigue characteristics of Y-shaped steel bracing that

is often adopted for retrofitting buildings in medium-to-low seismic intensity area. Marshall and Charney[4]studied the seis-mic response of steel frame structures using a hybrid passive con-trol system. Experimental and numerical studies of buckling restrained braces (BRB) have revealed that BRB, all-steel or con-crete filled, is an effective dissipative structural member [5,6]. The performance of high- and low-strength steel in a moment frame has drawn much attention and is the focus of the current study. To be specific, this study addresses HT690, SN490 and A36 for use in building construction and includes a probabilistic seis-mic assessment. The performance of the aforementioned hybrid structure under seismic excitations is investigated and compared to that of a building constructed from traditional steel having nor-mal yield strength. The motivation of using such a hybrid structure is that yield occurs only in the dissipative components, and through the quick-recovery characteristic of the dissipating

⇑ Corresponding author.

E-mail address:[email protected](K.-W. Liao).

Construction and Building Materials 184 (2018) 151–164

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system, structures are restored to their original function in the shortest time possible.

Because the majority of seismic energy is absorbed by the BRB, accuracy in modeling the hysteretic behavior of BRB is essential. Budaházy and Dunai[5]emphasized that the behavior of the BRB is complex. Instead of being numerically simulated, it is often stud-ied by experimental tests. For example, Chen et al.[7]conducted cyclic loading tests with low yield steel (LYS 100) as the main load-bearing element of the BRB and combined the BRB with duc-tile concentrically braced frames (DCBF) for a reduced-size struc-ture shaking table test. This combination aims to achieve a lower strength ratio in the main frame system and a higher strength ratio in the braces to ensure that plastic deformation occurs on the BRB. The strength ratio is the quotient of the strength demand (e.g., flex-ural and shear) from external force to the strength capacity pro-vided by the considered structural member. The results indicate that compared to traditional braces, the BRB provides relatively good strength capacity, ductility and energy dissipation behavior, the DCBF exhibits good seismic behavior and the BRB reduces structure acceleration response and provides effective control in terms of the inter-story drift after yielding. The inter-story drift is a ratio of the relative translational displacement between two consecutive stories to height of that floor. Although much BRB research has been conducted through experiments, some numeri-cal models have been proposed. For example, OpenSees provides a material called Steel BRB that can be used to investigate the hys-teretic properties of BRB core material[8]. Cofie and Krawinkler[9]

used a bounding surface model based on monotonic and cyclic stress-strain curves to simulate the nonlinear behavior of struc-tural steel. In their model, the upper and lower stress bounds are controlled by hardening, softening, and mean stress relaxation, the strain amplitude of the last excursion and the previous stress-strain history. Wang [10] adopted the bounding surface model developed by Cofie and Krawinkler[9]to establish an all-steel BRB hysteretic model under a DRAIN-2D environment, in which three different steel materials, LYS, A572Gr.50 and TMCP, were used as the main load-bearing elements of BRB. The bounding surface model was also adopted by Ling et al.[11]to simulate the mechanical behavior of geosynthetic reinforcements that often exhibit large plastic strains, highly nonlinear and hysteretic behav-iors under cyclic loading.

To accurately describe the nonlinear behavior of BRB subjected to seismic excitations, this study develops a uniaxial equivalent constitutive model that incorporates strain softening and harden-ing. The hysteretic model is constructed based on experimental results of monotonic and cyclic stress-strain curves and imple-mented in ABAQUS through its user subroutine interface - user-defined material (UMAT). After calibration, the established UMAT is applied to BRB members of steel moment frames for nonlinear time history analysis.

The structural members of the hybrid frame are minimized using particle swarm optimization (PSO) with performance con-straints to ensure it is equivalent to that of the dual moment frame using normal strength steel. A probabilistic seismic performance evaluation and a fragility analysis are conducted for the optimal frame, and the performance measurements of interest in this study are the inter-story drift, maximum roof acceleration, BRB hys-teretic behavior, strength ratio between main frame and BRB and cumulative fatigue damage. In addition to the structural perfor-mance, the economic feasibility of using high-strength steel in the structural system is also compared to that of using traditional steel materials. Based on these research results, this study provides a design guideline that includes the strength ratio values for col-umn, beam and brace to leverage the results of the adopted opti-mization process and streamline the design process. Because the non-linear behavior of the structure system mainly occurs in the

BRB members, the common beam element with potential plastic hinges at two ends is adopted to simulate the main structure frame system. The details of this study are provided below.

2. BRB tests and test results

The Structural Engineering Laboratory, Department of Civil and Construction Engineering, National Taiwan University of Science and Technology successively completed over 40 large-size all-steel BRB load-bearing tests[7]. The main load-bearing elements of the test specimens ranged from 10 to 20 mm in thickness and approximately 2800 mm in length and are tested through displacement-controlled cyclic loading. The main load-bearing ele-ments are fabricated from LYP, A36, A572 Gr. 50 and SM570 steel. The details of a tested BRB and test arrangement of the BRB spec-imens are shown inFig. 1. The loading cycle is divided into two groups (LV and LC). The LV loading group used a gradual increment of displacement amplitude as shown inFig. 2, and its results can provide the necessary statistics for establishing the hysteretic model. The LC series specimens, in contrast, used a fixed displacement amplitude to provide the parameters needed to

(a) The overall view of a tested BRB

(b) The load-carrying element of a tested BRB

A-A B -B C-C (c) The section views of a BRB

Steel Bar 200mmLVDT Strain Gage 2860 Machine Head Strong Floor Adaptor Cyclic Loading End Plate LYS-BIB Specimen

(d) Experiment setup for BRB specimen

Fig. 1. (a) The overall view of a BRB (b) The load-carrying element of a BRB (c) The section views of a BRB (d) Experiment setup for BRB specimen.

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establish the fatigue life. The displacement amplitudes for LC1 to LC6 were ±2Dy, ±4Dy, ±8Dy, ±11Dy, ±16Dy and ±32Dy; among

these, y = 1.25 mm was the yield displacement for the LYS-BRB element.

Fig. 3shows the load-displacement hysteresis loops for the LV group specimens. A total of 941, 163, 89 and 63 cycles were com-pleted for specimens from LV1 to LV4, respectively.Fig. 3 shows the stable hysteresis loops generated by LYS-BRB. No strength degradation occurred before failure; the specimen exhibited strain hardening with increasing displacement load. In addition, the hys-teresis loops are saturated without displaying pinching, and there

is no significant degradation between the hysteresis loops. All specimens exhibited stable increase in strength after the yield point and the yield potential was considered favorable.

A total of 14,337 displacement cycles were completed for LC1 (±2Dy), but the displacement cycles decreased with increasing

dis-placement amplitude. For example, only 18 disdis-placement cycles were completed for LC6 (±32Dy). All of the testing results for LC

specimen are displayed inTable 1. With information revealed in

Table 1, the strain fatigue curve can be built based on the ‘‘BRB damage index” section.Fig. 4shows the hysteresis loops of partial LC specimens. It can be observed that energy dissipation in each cycle increases as the number of displacement loops increases as a result of strain hardening behavior. The most significant strain hardening phenomenon was detected in the first three cycles, but the phenomenon was attenuated in later cycles. For example, for the load at LC2 (±2Dy) (Fig. 4(a)) the maximum load in the first

cycle is +512 kN and is 555 kN in the last cycle. The differential is 43 kN. For the load at LC5 (±16Dy) (Fig. 4(c)) the maximum load in -40 -20 0 20 40 -40 -20 0 20 40 y (a) V1 (b) V2 (c) V3 -40 -20 0 20 40 (d) V4 Deformation( ) Part I Part II Part I Part II Part I Patrt II 0 15 30 45 60 75 No. of Cycles -40 -20 0 20 40 Deformation ( y ) Deformation ( y ) Deformation ( y )

Fig. 2. LV1 to LV4 loading cycles.

-40 -20 0 20 40 Deformation (mm) -800 -400 0 400 800 Load (KN) -800 -400 0 400 800 Load (KN) (a) LV1 (c) LV3 -40 -20 0 20 40 Deformation (mm) (b) LV2 (d) LV4

Fig. 3. Hysteresis loops of specimens subjected to LV1 to LV4.

Table 1

The detailed testing results of LC series. Test name Loading condition Cycles at failure Absorbed energy Max. tension Max. compression LC1 2Dy 14,337 31,296 441 436 LC2 4Dy 3899 25,962 594 599 LC3 8Dy 1084 18,754 698 764 LC4 11Dy 714 20,490 768 813 LC5 16Dy 115 4147 828 838 LC6 32Dy 18 1859 906 1022 -800 -400 0 400 800

Load

(KN)

-800 -400 0 400 800

Load

(KN

)

-800 -400 0 400 800

Load

(KN)

(a) LC2 (b) LC3 -40 -20 0 20 40

Deformation (mm)

(c) LC5

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the first cycle is +648 kN and is 828 kN in the last cycle. The differ-ential is 180 kN.Fig. 5shows the hysteresis loops of LV1, LV2 and LV3 specimens from part I to part II loading cycles. As shown in

Fig. 2, the load of LV1 in the second stage dropped from 30Dyto

8Dy(from the 64th to 941st cycle), LV2 from 32Dyto 14Dy(the

57th to 163rd cycle) and LV3 from 32Dy to 22Dy (the 64th to

89th cycle). The maximum load of all three specimens decreased with an increasing number of displacement cycles when entering the part II phase, and strain softening was detected. The strain soft-ening phenomenon is less obvious when the displacement decrease is smaller (LV3 inFig. 5) and more obvious when the dis-placement decrease is larger (LV1Fig. 5). The dotted lines inFig. 5

are skeleton curves.Fig. 5indicates that the strain softening phe-nomenon caused the maximum load of each loop to lean toward the skeleton curve as the number of loops increased, but the imum load of each loop did not meet the skeleton curve. The max-imum load of LV1, LV2 and LV3 exceeded the skeleton curves by 15, 11 and 5%, respectively, after strain softening.

3. BRB mathematical model, ABAQUS UMAT and damage index

All-steel BRB exhibits stable hysteresis loops and, as displace-ment increases, the strain hardening phenomenon becomes more obvious. When the displacement decreases, strain softening becomes clearly detectable. In addition, the yielding stress shifts because of the displacement change as shown inFigs. 3–5. To accu-rately simulate these BRB behaviors, the bounding surfaces method and kinematic hardening model are adopted based on the results of monotonic and cyclic experiments. For example, the cyclic test is used to construct the skeleton curve that is one of the major factors determining the locations of stress bounds, and the monotonic test can be used to determine the initial yield-ing stress. Existyield-ing finite element software often does not replicate this mechanical behavior. Therefore, this study develops a nonlin-ear user defined material provided in ABAQUS to simulate BRB hysteresis behaviors. The details of establishing UMAT are described below.

3.1. Bounding surfaces model

¼ 6455 ð

e

Þ þ 348 for

e

> 0:00242 in compression ð4Þ

2. Determination of the initial stress bounds: The initial values of the upper and lower bounds are set as

r

ywith a slope of 0.05E,

where

r

yis the yield stress and E is the elastic Young’s modulus.

3. Calculation of the strain-hardening parameter (En p):E

n

p is

calcu-lated using Eq.(5)

Enp¼ E 0 pþ ðE E 0 pÞ d din 2 ð5Þ in which E0

Pis the initial strain-hardening parameter equal to 0.05E

and EnPis the updated strain-hardening parameter. The definitions of

din, E0Pand E n

Pare provided inFig. 6.

4. Updating the upper and lower stress bounds: Based on Cofie and Krawinkler [9], the stress bounds are updated based on the strain (

-800 -400 0 400 800 Load (KN ) (a)L V1 -40 -20 0 20 40 Deformation (mm) -800 -400 0 400 800 (c) LV3 (b) LV2 -800 -400 0 400 800

Fig. 5. Softening phenomenon of specimens subjected to LV1, LV2 and LV3.

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3.2. ABAQUS UMAT and simulation results

The aforementioned BRB mod el is formulated as an ABAQUS UMAT. Prior to each ABAQUS load increment, the UMAT subroutine is executed to provide the necessary information such as the ele-ment contribution [Ke] to the global stiffness matrix [K] and the

element solution-dependent state variables (e.g., the plastic energy dissipation). The ABAQUS main program starts the next increment if the resulting structural load has converged to the applied load in the current load increment. The calculation flow chart of the UMAT applied in BRB is described inFig. 7. The responses of ABAQUS BRB

subjected to loadings of LV1 LV4 are displayed inFig. 8. Compar-ing to the experimental results as shown inFig. 3, a very promising outcome is identified.

3.3. BRB damage index

The failure of BRB has been identified as a low-cycle fatigue

[10]. Thus, a strain fatigue curve similar to the stress fatigue curve of the high-cycle fatigue is then established based on test results of LC cyclic loadings, as shown in Eq.(6).

D

e

p¼

e

fð2NfÞc ð6Þ

in whichD

e

pis the range of plastic strain, 2Nfrepresents the

num-ber of cycles that causes fatigue failure and

e

fand c are regression

parameters. Because the loading sequence in low-cycle fatigue is not as important as in high-cycle fatigue, this study adopts Miner

[12]to establish the linear cumulative damage index for BRB, as shown in Eq.(7): D¼X k i ni Nfi¼ ni Nf 1þ n2 Nf 2þ . . . þ nk Nfk ð7Þ

in which niis the number of cycles of strain i, Nfiis the number of

cycles for fatigue failure when a constant strain of i is applied, and if D is greater than 1, the BRB has reached fatigue damage. Eqs.(8) and (9)are strain fatigue curves of LYS all-steel BRB estab-lished through regression based on the test results.

N¼ 3:8 104_ð

_e

prÞ3:078 when

e

prP 0:00863 ð8Þ N¼ 0:12ð

e

prÞ1:867 when

e

pr< 0:00863 ð9Þ Balance of equation Input E, Y,and others Solution

P incremental steps starting

Strain

strain increment and the current stress at the integration points are given

BRB UMAT

Jacobian matrix, updated bounds and stresses are calculated and state variables are stored Global stiffness

Element stiffness is obtained and integrated with global stiffness

Start

End the current increment and start the next increment

NO

YES

Fig. 7. The built BRB UMAT in ABAQUS calculation.

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in which

e

pris the plastic strain range of one half cycle, as shown in

Fig. 9. The parameter N is the number of cycles to cause fatigue fail-ure corresponding to

e

pr.Table 2shows the damage index for

LYS-BRB by numerical simulation. Because the number of specimens is insufficient, the values of damage index shown inTable 2deviate from the reference value (i.e., 1.0) to a certain degree. Nevertheless, these values are considerably closer to the expected value, indicat-ing that the linear Miner cumulative index (Mindex) provides an acceptable prediction.

4. Numerical studies

4.1. Structural configuration, loading and design requirements As shown inFig. 10, the demonstrated building is a 20-story steel moment frame with a floor plan of 5 3 bays, in which an equal bay spacing of 9 m is used. Except for the first floor, which has a story height of 4.5 m, the story height of the remaining floors is 4 m [13]. The BRB in the demonstrated building is arranged based on engineering practice. For example, the number of BRB in the lower stories is often greater than that of the higher stories. In addition, an open space is usually provided in the middle span of the outer frame. Therefore, for stories that are less than twelve story, the BRBs are installed on the two side spans of the outer frame along the X direction, and the second and fourth spans on the inner frames along the Y direction. For stories that are greater than twelve story, the number of BRB decreases from two to one in both X and Y directions as indicated inFigs. 10.1–10.3, respec-tively. Two types of columns are used: C1 connected to braces and C2 unconnected to braces. The floors are 12 cm in thickness and are fabricated with 4000 psi concrete.

The static load is 1.1 tf/m2_{for floors 1 to 7, 1.0 tf/m}2_{for floors 8}

to 14 and 0.9 tf/m2for floors 14 to roof. The live load is 0.2 tf/m2for floors 1 to 19 and 0.3 tf/m2_{for the roof. The total weight of the}

structure is approximately 24421.5 ton. It is assumed that the building is located in Zone II of Taipei and the buildings are designed for a peak ground acceleration (PGA) of 0.291 g based on a Taiwan seismic hazard curve corresponding to a 10% excee-dance probability in 50 years. The design criteria are:

1. Under the design earthquake, the maximum inter-story drift of the buildings may not exceed the 0.5% limit.

2. The strength ratio of column, beam and brace must be less than 0.95.

Fig. 9. Illustration of plastic strain range.

Table 2

BRB damage index via simulation.

LV1 LV2 LV3 LV4 Ave.

Damage index 1.25 0.85 0.94 1.07 1.03

Fig. 10.1. Plan view of the studied frame.

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3. The moment structure should be able to sustain 25% seismic force alone.

4. The depth of the beams may not be less than 1/20 of the span. 5. Beams in both directions beams are separately designed. Ide-ally, the beam depth should be identical in both directions. In case they are different, the differential may not be smaller than 10 mm to facilitate installation of the stiffener inside the box-type columns.

4.2. Preliminary designs and materials

In the preliminary study, three buildings are designed under the same criteria (section 4.1) and are termed SNSN, SNHT and HTHT.

SNSN indicates SN490 is used for both beam and column, and SNHT indicates that SN490 and HT690 are used for beam and col-umn, respectively. HTHT indicates that HT690 is used for both beam and column. For a fair comparison, these three buildings are designed to have a similar strength ratio. Nonetheless, there are some exceptions. For example, the depth of beams is strictly not permitted to be less than 1/20 of the span by Taiwan practices. In such a situation, the strength ratio may not be the major control-ling factor, making it impossible for all three buildings to have an identical strength ratio. One of the focuses in these preliminary designs is to identify the performance of the SNSN building that is used as the design target (or constraints) in the subsequent opti-mization study. Please note that in practice, the built-up box is often used for columns and all beam-to-column joints are designed as moment-resisting connections in Taiwan.

The core material in BRB is A36. The minimum cross-section area for the braces is 30 cm2_{with an increment of 5 cm}2_{. SN 490}

and HT 690 are used for beam and column, respectively. If the plate thickness of the beams is larger than 40 mm, the material is chan-ged from SN490B to SN490C and a uniform girder size is used for all stories (BH400 300 12 25). Periods of the three buildings are displayed inTable 3. As expected, the HTHT frame has a lower stiffness as a result of the smaller section and therefore has the highest period.Table 4shows the properties of SN490B, SN490C, HT690 and A36 steel.

4.3. Evaluation process

Both static and dynamic analyses are conducted in the struc-tural evaluation process. The key evaluations in the static analysis are inter-story drift, strength ratio and whether the moment-resisting frame has the ability to sustain 25% of the seismic force without the presence of the BRB. The key evaluations in the dynamic analysis are inter-story drift, top-floor acceleration ampli-fication factor and maximum/cumulative damage index of the BRB. For dynamic analysis, the design regulations in Taiwan require a minimum of three earthquake records, and their response spec-trums must be scaled to fit the design response spectrum. To be specific, for a damping ratio of 5%, the response spectrum of the selected earthquakes falling in between 0.2 T and 1.5 T (T as the fundamental period) may not be lower than 90% of the correspond-ing design spectral acceleration. In addition, the average value of the response spectrum within the designated period range may not be lower than the average value of the corresponding design spectral accelerations. This study selects eight seismic records and scales them based on the above regulations, as shown in

Fig. 11.

4.4. Results of static analysis for the preliminary buildings

Table 5 displays the maximum inter-story drifts of these three buildings. As expected, to satisfy the demands of gravity loading, the SNSN frame needs to have a greater column size because its material strength is lower than others, resulting to a greater structural stiffness and a smaller inter-story drift.

Fig. 10.3. Elevation view of the 20-story building in Y direction.

Table 3

Periods of three preliminary buildings.

SNSN SNHT HTHT

1st period 2.56 s 2.68 s 2.84 s

2nd period 2.49 s 2.56 s 2.70 s

3rd period 1.81 s 1.88 s 1.95 s

Table 4

Mechanical properties for the steel materials used in case study.

Steel Yield stress (N/mm2

) Ultimate stress (N/mm2 ) SN490B 65 t 5 12 125 t 5 16 165 t 5 40 t > 40 490–610 325 325–445 325–445 295–415 SN490C 165 t 5 40 405 t 5 50 t > 50 490–610 325–445 295–415 295–415 A36 300 400–550 HT690 550 690

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Although the inter-story drifts of SNHT and HTHT frames are not up to the level of SNSN, they still meet the requirements (0.5%). Note that the BRBs in these three buildings are at the same loca-tions but have different section areas. Table 6 shows the total steel weight used in these three buildings. Compared to the SNSN frame, the HTHT and SNHT frames can reduce the total steel weight by 19% and 11%, respectively. Table 7 displays the strength ratios of three buildings only with a moment resisting

frame (i.e., no BRB) under 25% seismic loading, indicating that all three buildings meet the standards.

4.5. Results of dynamic analysis for the preliminary buildings

Eight sets of earthquake records are selected from the NGA-West2 Ground Motion Database, namely TCU065, TCU084, TCU129, CHY028, TAP103, Kobe, Loma Prieta and Northridge.

Fig. 11displays the time history and response spectrum of the scaled Kobe earthquake corresponding to the 475-year design response spectrum. The PGAs of all of the scaled earthquakes are shown inTable 8.

Fig. 12shows the X direction inter-story drift of each floor for three buildings under the scaled Kobe earthquake excitation. It is seen that for all buildings, the BRB arrangement has a significant influence on the drift values. For example, the inter-story drifts between the 11th and 12th stories are abruptly heightened because of the reduced number of braces (from 2 to 1) on the 12th floor.

Table 9exhibits the maximum inter-story drift of three build-ings for the eight earthquake excitations in both directions. Similar to the results of static analysis, the SNSN frame has the smallest inter-story drift. Analysis of variance (ANOVA) is adopted to ana-lyze the inter-story drift performance among three buildings. As indicated inTable 9, the difference among them is significant. Nev-ertheless, the inter-story drifts of three buildings all meet the requirement of collapse prevention in FEMA356 (2%).

To evaluate the fatigue performance for three buildings, the fatigue damage index (Mindex) is used. Two types of Mindex are used as the measurement for comparison: the maximum value of Mindex among all BRB members in a building (denoted as Dmax hereafter) and the cumulative value of M_{index for all} BRB members in a building (denoted as Cdamage hereafter).

Table 10shows the Dmax in X direction and the corresponding story number for three buildings. It is seen that Dmax is prone to occur on higher stories because of the vertical layout of BRB.

Table 11shows the Cdamage in the X direction. The results of ANOVA show that differentials in X-axis Dmax among the three buildings are not significant, but that the differentials in Y-axis Dmax, X-axis Cdamage and Y-axis Cdamage are significant. The all-steel BRB of all three buildings has not yet reached fatigue fail-ure point under the selected earthquake excitations. Among them, the BRB of the SNSN frame absorbs the highest level of energy and SNHT follows.

The ‘‘roof acceleration amplification factor” refers to the max ratio of the roof acceleration to the ground acceleration through the entire dynamic analysis duration.Table 12shows the X-axis (in positive direction) roof acceleration amplification factor of the three buildings under the eight sets of scaled seismic excitations. The results indicate that the roof acceleration amplification factors of all three building all meet the code requirement. In addition, the F value suggests that the differentials among the three buildings in terms of roof acceleration factor are not significant.

4.6. Optimal design using PSO and its structural responses

To increase the benefits and structural performance of using high-strength steel, this study used PSO to find an optimized design (O_SNHT) having the equivalent structural performance of an SNSN

Fig. 11. Example of the Kobe earthquake and the scaled spectrum compatible with the design spectrum of 475 years.

Table 5

Maximum inter-story drifts (%) of the three buildings under static analysis.

Frame X-Dir. Y-Dir.

SNSN 0.38% 0.33%

SNHT 0.45% 0.39%

HTHT 0.46% 0.48%

Table 6

Total steel weight used in the three buildings.

Frame Total steel

content (Ton) Reduction rate* SNSN 3720 – SNHT 3310 11.01% HTHT 2986 19.71% * Compared to SNSN Table 7

Strength ratios of the three moment resisting frames under 25% seismic loading.

Frame Strength ratio of beam Strength ratio of column

SNSN 0.3–0.65 0.3–0.45

SNHT 0.35–0.55 0.35–0.75

HTHT 0.3–0.45 0.25–0.7

Table 8

PGA of the eight scaled earthquakes.

Earthquake TCU065 TCU084 TCU129 TAP103 CHY028 Kobe Loma Prieta Northridge

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frame that is confirmed by dynamic and fragility analysis. In addi-tion, the strength ratios of the each element in the optimization design are suggested as a design reference for engineers. Factors con-sidered in the optimization process include the following: maximum inter-story drift should be no greater than the maximum inter-story drift of SNSN frame, BRB strength ratio must fall between 0.95 and

0.7 and the column size in lower floors may not be smaller than that in higher floors. The design variables are the width and thickness of box column and BRB core area. Because the column size is designed with two floors in a unit, the number of design variables is 79. It should be noted that the dynamic analysis is not included in optimization because of its higher computational cost.

Fig. 12. X-Dir. inter-story drifts under the scaled Kobe earthquake excitation.

Table 9

Maximum inter-story drifts of three buildings for the eight earthquakes.

X-direction Y-direction D_SNSN D_SNHT D_HTHT D_SNSN D_SNHT D_HTHT TCU065 1.27% 1.50% 1.93% 1.42% 1.35% 1.63% TCU084 1.29% 1.62% 1.62% 1.24% 1.41% 1.35% TCU129 1.17% 1.36% 1.32% 1.14% 1.31% 1.34% TAP103 1.35% 1.83% 1.84% 1.28% 1.54% 1.59% CHY028 1.29% 1.56% 1.57% 1.13% 1.33% 1.45% Kobe 1.53% 1.66% 1.64% 1.44% 1.59% 1.66% LomaPrieta 1.56% 1.65% 1.57% 1.40% 1.59% 1.62% Northridge 1.34% 1.77% 1.65% 1.17% 1.47% 1.44% Average 1.35% 1.62% 1.64% 1.28% 1.45% 1.51% STD 0.00132 0.001487 0.00184 0.00128 0.00115 0.00130 ANOVA F value 8.62 7.46 Critical value 3.47 3.47 Table 10

Dmax in X direction for three buildings.

SNSN Story SNHT Story HTHT Story

TCU065 0.0035 16 0.0023 13 0.0021 17 TCU084 0.0036 16 0.0035 17 0.0031 17 TCU129 0.0034 16 0.0013 17 0.0011 17 TAP103 0.0026 16 0.0028 17 0.0028 17 CHY028 0.0023 16 0.0023 17 0.0021 17 Kobe 0.0031 16 0.0027 17 0.0023 17 LomaPrieta 0.0031 16 0.0029 17 0.0026 17 Northridge 0.0030 17 0.0043 17 0.0041 17 Average 0.0031 0.0028 0.0025 STD 4.46104 8.86104 8.73104 ANOVA F value 1.045 Critical value 3.466

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In summary, the optimization is displayed in Eq.(10):

Min: the total weight of building

s:t:

story drift 6 Maximum story drift of SNSN

column sizeiP column sizeiþ1

0:7 6 Brace stress ratio 6 0:95

particle lower bound 6 particle ðsection sizeÞ

6 particle upper bound

stress ratio of beam and column 6 0:95

0 B B B B B B B B @ 1 C C C C C C C C A where i is the story number from 1 to 20

ð10Þ

In the optimization process, a parametric structural model is developed for evaluating the building responses in an automatic fashion. The SAP2000 Application Programming Interface (API) is able to carry out both pre- and post-processing such as assigning the section area and capturing the strength ratio, respectively. Thus, the API is adopted here to prepare the building model and deliver the analysis and design outcomes in each optimization iter-ation. PSO is used as the optimization engine by taking advantage of not being trapped in a local minimum because of the inherent random searching procedure. A new position of a particle, which represents a set of section sizes in the current study, is generated using Eq.(11), as shown below:

~xiðt þ 1Þ ¼ ~

v

iðt þ 1Þ þ~xiðtÞ ð11Þ

where~xiðt þ 1Þ denotes the position of the ithparticle in the next

iteration,~xiðtÞ denotes the position of the ithparticle in the current

iteration and~

v

iðt þ 1Þ denotes the velocity of the ithparticle in the

current iteration. The velocity of the ith_{particle is determined by Eq.}

(12), as shown below:

~

v

iðt þ 1Þ ¼ w ~

v

iðtÞ þ r1c1ð~xpBest~xiðtÞÞ þ r2c2ð~xgBest~xiðtÞÞ ð12Þ

where w is the inertia factor,~

v

iðtÞ is the velocity of at previous

iter-ation, ri(i = 1–2) are random numbers between 0 and 1, and c1and

c2are the cognition factor and the social factor, respectively. The

parameter~xpBestis the particle position with the minimum objective

value in the ith_{population, and}_~x

gBestis the particle position with the

minimum objective value among all populations. Detailed informa-tion for the PSO used here can be found inTable 13. The flowchart of PSO-based optimization is shown inFig. 13.

Fig. 14shows the convergence history of the objective function. The maximum X-axis inter-story drift of the O_SNHT frame is 0.38% (SNSN: 0.38%), and the maximum Y-axis inter-story drift is 0.32% (SNSN: 0.33%). The total steel weight for O_SNHT is 3211 tons, which achieves a saving of 13% steel compared to that of SNSN frame. It can be observed that the proposed optimization algorithm reduces the inter-story drift (from 0.45% to 0.38% and from 0.39% to 0.32% for the X-axis and Y-axis, respectively) as well as the amount of steel. Based on the analysis results of O_SNHT, the strength ratios of C1 and C2 columns, beams and braces are recom-mended to be the vicinities of 0.9, 0.8, 0.8 and 0.95, respectively, to ensure that the design frame fits the concept of damage control.

Although the dynamic analysis is not included in optimization, the performances of the optimal frame (O_SNHT) under seismic excitations are examined and displayed inTable 14. Compared to the results of SNSN, inter-story drifts and top-floor acceleration amplification factors of O_SNHT in both X and Y directions are sim-ilar to those of SNSN. In the BRB damage index, significant differ-ences are found in all indices except Y-axis Dmax. It is seen that although the dynamic analysis is not considered in optimization, most of the indices indicate that the structural performance of O_SNHT is similar to that of SNSN. Please note that all responses of O_SNHT meet the standards.

4.7. Results of fragility analysis

In addition to evaluate the deterministic performance of the investigated buildings, this study further assessed the probabilistic performance of two frames, O_SNHT and SNSN, through the fragi-lity analysis. In general, the inter-story drift is used as the struc-tural performance index, and therefore the relationship between spectral acceleration and maximum inter-story drift is established through Eq.(13).

D¼ aðSaÞb ð13Þ

in which D is the demand of the inter-story drift, Sais the spectral

acceleration and a and b are constants derived from regression analysis.

The fragility curve is a conditional probability computation, representing a system failure probability for a given intensity mea-surement. For example, when Sais given, assuming that the

capac-ity and demand of the building are lognormally distributed, the corresponding failure probability can be calculated through Eq.

(14):

Table 12

X-axis top-level acceleration amplification factor (positive).

Earthquake SNSN SNHT HTHT TCU065 2.2 2.3 2.2 TCU084 3.0 3.3 3.2 TCU129 2.1 2.1 2.3 TAP103 2.2 2.2 2.1 CHY028 3.0 2.9 2.7 Kobe 2.4 2.4 2.3 LomaPrieta 3.2 3.2 3.2 Northridge 3.1 3.1 3.3 Average 2.6 2.7 2.7 STD 0.46 0.49 0.50 ANOVA F value 0.012 Critical value 3.466 Table 13

Parameters of PSO used in this study.

Number of particle 10

Number of iteration 40

Column width upper bound 1500

Column width lower bound 100

Column thickness upper bound 70

Column thickness lower bound 16

BRB area upper bound 300

BRB area lower bound 10

Upper bound of w 0.9

Lower bound of w 0.2

c1 2

c2 2

Table 11

Cdamage in X direction for three buildings.

SNSN SNHT HTHT TCU065 0.12 0.10 0.10 TCU084 0.14 0.11 0.11 TCU129 0.14 0.04 0.04 TAP103 0.13 0.09 0.09 CHY028 0.07 0.05 0.05 Kobe 0.13 0.01 0.01 LomaPrieta 0.13 0.11 0.11 Northridge 0.10 0.11 0.10 Average 0.121 0.077 0.076 STD 0.0239 0.0388 0.0377 ANOVA F value 4.247 Critical value 3.466

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Fig. 13. Flow chart of PSO-based optimization. 3100 3300 3500 3700 3900 4100 4300 4500 4700 4900 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

T

o

tal weight (T

on)

No. of Iteration

(13)

PfðC > DjSa¼ xÞ ¼ 1 / lnðC DÞ b ¼ 1 / lnð C axbÞ b " # ð14Þ

in which C is the mean value of the capacity, D (aðSaÞb) is the mean

value of demand in terms of inter-story drift, is the standard devi-ation with respect to the limit state, and is the probability density function of standard normal. With reference to the FEMA356, two structural performance levels are adopted: life safety (LS) and col-lapse prevention (CP). That is, the values of C are 1.5% and 2% for LS and CP, respectively. The parameter is calculated by Eq.(15):

b ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2DjSaþ b 2 c q ð15Þ

in whichcis the standard deviation of capacity for a given Sa. The

value falls between 0.2 and 0.4[14]. The parameter bDjSais the stan-dard deviation of demand corresponding to a given Sa, representing

the uncertainty in inter-story drift under excitations with a given Sa.

To calculate the value of bDjSa, similar to D, this study establishes the relationship between the standard deviation of the inter-story drift and spectral acceleration from eight sets of earthquake excitations and from Eq.(16), as shown below.

bDjSa¼ cðSaÞ

f

ð16Þ

in which c and f are constants derived from regression analysis. Parameters of a, b, c, and f are displayed inTable 15.

Fig. 15shows the fragility curves of O_SNHT and SNSN (COV of capacity are 0.2, 0.3 and 0.4) targeting on the performance levels of LS and CP. It is seen that the trends of LS and CP fragility curves of the two buildings are similar. When Sais larger than 0.8 g - 1.2 g,

the fragility curve of O_SNHT rises gradually above that of the SNSN frame. In other conditions, the performance of O_SNHT frame is slightly better than the SNSN frame. Although dynamic analysis is not part of the optimization, the results show that the structural

Table 14

Dynamic performance comparison between O_SNHT and SNSN frames.

O_SNHT SNSN X_drift Average 1.31% 1.35% STD 0.0006 0.00132 Z-value 0.79 (1.96) Y_drift Average 1.17% 1.28% STD 0.0011 0.00128 Z-value 1.81 (1.96) X_Dmax Average 0.0020 0.0031 STD 0.0013 0.00043 Z-value 2.25 (1.96) Y_Dmax Average 0.0029 0.0028 STD 0.0009 0.00048 Z-value 1.64 (1.96) X_ Cdamage Average 0.05 0.12 STD 0.03 0.02 Z-value 5.38 (1.96) Y_ Cdamage Average 0.06 0.15 STD 0.03 0.04 Z-value 4.77 (1.96) X-axis top-floor acceleration

amplification factor (positive)

Average 2.6 2.6

STD 0.56 0.47

Z-value 0.21 (1.96) X-axis top-floor acceleration

amplification factor (negative)

Average 2.5 2.4

STD 0.24 0.27

Z-value 0.43 (1.96) Y-axis top-floor acceleration

amplification factor (positive)

Average 2.8 2.8

STD 0.63 0.40

Z-value 0.19 (1.96) Y-axis top-floor acceleration

amplification factor (negative)

Average 2.6 2.5

STD 0.40 0.43

Z-value 0.54 (1.96)

Table 15

Parameters used in fragility analysis.

Regression parameter O_SNHT SNSN

a 3.65 3.84

b 0.932 0.924

c 0.37 0.43

f 0.960 0.986

Fig. 15.1. Fragility curves for O_SNHT and SNSN with COV of 0.2.

Fig. 15.2. Fragility curves for O_SNHT and SNSN with COV of 0.3.

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performance of the optimal frame (O_SNHT) is indeed fairly close to the performance of SNSN.

5. Conclusion

This study explored the safety and economic benefits of a dual-steel moment frame constructed using a combination of high-strength and high-ductility steel. An ABAQUS UMAT was estab-lished to simulate the BRB nonlinear behavior based on test results. The all-steel BRB damage index was investigated through Miner’s suggestion [12]. Three numerical examples, based on 20-story dual-steel moment frames (SNSN, SNHT and HTHT) located in Zone II Taipei, were used to obtain detailed information with respect to the issues of safety and economic cost. To ensure that damage occur in the brace system first, different strength ratio are assigned for the main and secondary systems in the design stage. Pseudo-static, dynamic and fragility analyses were conducted to compare the structural performances among three frames using ANOVA. The structural performances investigated included the inter-story drift, maximum roof acceleration, property of BRB hysteresis, strength ratio between main frame and BRB and cumulative fati-gue damage. To ensure that the SNHT structure can provide similar structural performance as structures built with regular steel but with minimized use of steel materials, a PSO-based optimization was carried out. Fragility analyses confirmed that the structural performance of the optimization design was similar to the SNSN frame. From the results of the abovementioned analyses, it was found that all structures constructed from regular or high-strength steel met the standards in both static and dynamic anal-ysis. Compared to SNSN, SNHT saved 11% steel; however, the dynamic analysis indicated that control of the maximum inter-story drift of SNHT was not as favorable as SNSN. After performing optimization, O_SNHT saved approximately 13% of steel compared to SNSN and simultaneously improved the structural performance of SNHT up to the level comparable with SNSN or even higher. The fragility curve of O_SNHT was similar to that of SNSN in both LS and CP. In particular, if the spectrum acceleration is smaller than 0.8 g, O_SNHT is more capable in terms of satisfying the demands in LS and CP. Based on the results of optimization, for a 20-story dual-steel moment frame, the strength ratios of C1 and C2 col-umns, beams and braces are recommended to be the vicinities of 0.9, 0.8, 0.8 and 0.95, respectively, to ensure that the designed frame can provide similar performance to the optimal design iden-tified in this study.

Conflict of interest No conflict of interest.

Acknowledgements

The authors would like to thank the China Steel Corporation (Taiwan) for financially supporting this study.

Appendix A: Establishment of the hysteretic model

The two bounding surfaces method is combined with kinematic hardening to form the complete hysteretic modeling as described below.

1. For i = 1 to the required increment number,

D

S¼ ED

e

;

r

iþ1_¼

_r

i_þ

_D

_S;

a

i_{¼ E}i

p

e

p

where i is the ith_{increment, is the strain increment, E is the elastic}

modulus, S is the stress increment,

r

i

is the stress at ithincrement, is the shifted stress value, Ei

pis the strain-hardening parameter at the

ithincrement, and

e

p

is the accumulated plastic strain. 2. If status(i) = elastic, perform the following calculations:

Ifj

r

iþ1

_a

_{j 6}

_r

yr

r

iþ1_¼

_r

iþ1

else

statusði þ 1Þ ¼ yielded b ¼ ryjriaij jriþ1_jj_ri_ai_j End if else go to step 3 end if

3. If status (i) = yielded, perform the following calculations:

If

r

i_D_s_{< 0}

statusði þ 1Þ ¼ elastic

r

iþ1_¼

_r

iþ1 else b ¼ 0 End if else go to step 4 End if 4. Compute

r

iþ1_¼

_r

i_{þ bDS}_þ EE i p Eþ Ei p ð1 bÞD

e

p_¼

e

p_þ 1 b 1þ Ei p=E

D

the hardening coefficient and Fsis the softening coefficient.

7. End for References

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