Optimal Seismic Design of Reinforced Concrete Shear Wall-Frame

KSCE Journal of Civil Engineering (0000) 00(0):1-10 Copyright ⓒ2014 Korean Society of Civil Engineers DOI 10.1007/s12205-014-0640-x

Structural Engineering

pISSN 1226-7988, eISSN 1976-3808 www.springer.com/12205

TECHNICAL NOTE

Optimal Seismic Design of Reinforced Concrete Shear Wall-Frame Structures Ali Kaveh* and Pooya Zakian** Received December 26, 2012/Revised September 7, 2013/Accepted December 16, 2013/Published Online

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Abstract Seismic design of Reinforced Concrete (RC) dual systems is performed as an optimization problem for which the charged system search algorithm is utilized as an optimizer. An efficient structural modeling is also presented for this purpose. Here, first databases are created based on ACI seismic criteria for beams, columns and shear walls. Formulations for optimal seismic design of dual systems (shear wall-frame) are proposed. With some modifications on these formulations, optimal seismic design of RC moment frames can also be performed. This procedure is along with ordinary design constraints and effective seismic design constraints. These constraints consist of beams, columns, shear wall design criteria, and some other seismic provisions. Cost of the structure is considered as the objective function. According to the results of the numerical example, the proposed methodology can be considered as a suitable practical approach for optimal seismic design of reinforced concrete shear wall-frame structures. Keywords: seismic design, modeling and optimization of dual systems, RC shear wall, RC moment frame, charged system search, discrete structural cost optimization ·······································································································································································

1. Introduction Earthquake induced damages to structures are unavoidable. But it can be reduced by observance of seismic design provisions. Seismic design of structures often leads to construction of expensive structures thus it is reasonable to optimize the cost of the building constructions whereas the design criteria are satisfied. Economical design of structures can be performed via structural optimization. Clearly, in this process objective function must be selected in a way to lead to minimum structural cost. Ordinary objective functions are weight or cost of the structures. Obviously, the cost of the structures usually is a function of structural weight. Walls that mainly withstand lateral loads due to the wind or earthquakes acting on the building are called structural walls or shear walls. Design guidelines emphasis that in the regions with moderate or high seismic risk level, special reinforcement is required so as to have desirable performance of concrete structures against earthquake hazards. Shear wall regions comprising of concentrated and tied reinforcement are known as boundary elements, irrespective of whether or not they are thicker than the rest of the wall. Thus these shear walls are called special shear walls, Wight and Macgregor (2012), and Kaveh and Zakian (2012). Dual system is a combinatorial structural system that resists lateral loads with simultaneous performance of different struc-

tural elements, and transfers them into the supports. In a dual system the following requirements must be fulfilled: I) Gravity loads should mainly be carried out by the building frame components. II) Lateral loads are resisted by simultaneous reaction of shear walls or braced frames together with moment frames. There are some publications on seismic design of RC shear walls and buildings. Wallace (1995a and 1995b) presented new code format for seismic design of RC structural walls, a performance design methodology relative to concrete structural walls was proposed by Sasani (1998). Boulaouad and Amour (2011) presented a displacement-based seismic design method for reinforced concrete structures. Kowalsky (2001) presented RC structural walls designed according to the UBC and displacement-based methods. There are many researchers who have concerned with the optimum design of seismic structures, but a few studies exist on optimal seismic design of shear walls or shear wall-frame structures incorporating new seismic design provisions. Saka (1991) proposed optimum design of multistory structures with shear walls, In this work an algorithm was presented which considers displacement, ultimate axial load and bending moment and minimum size limitations and utilizes optimality criteria approach to handle sway limitation. Fadaee and Grierson (1998) accomplished design optimization of 3D reinforced concrete structures having shear walls, Campetal. (2003) developed a

*Professor, Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Scienceand Technology, Tehran-16, Iran (Corresponding Author, E-mail: [email protected]) **Post Graduate Student, School of Civil Engineering, Iran University of Science and Technology, Tehran, Iran (E-mail: [email protected]) 1

Ali Kaveh and Pooya Zakian

design procedure implementing a genetic algorithm for discrete optimization of reinforced concrete frames and their objective was to minimize the material and construction costs of the structure subjected to main requirements described by the ACI Code. Lee and Ahn (2003) employed beam and column databases and then performed flexural design of reinforced concrete frames by genetic algorithm, Kwak and Kim (2008) fulfilled an optimum design of reinforced concrete plane frames based on predetermined section database. Zou and Chan (2005) performed an optimal resizing technique for dynamic drift design of concrete buildings under response spectrum and time history loadings. Kaveh and Sabzi (2011) accomplished a comparative study of two meta-heuristic algorithms so-called heuristic big bang-big crunch (HBB-BC) and a heuristic particle swarm ant colony optimization (HPSACO) to discrete design optimization of reinforced concrete frames based on ACI 318-08 code. Kaveh and Zakian (2012) presented an optimization problem for performance based optimal seismic design of special reinforced concrete shear walls via charged system search algorithm by generating shear wall section database in order to have adiscrete optimization. In this article, optimal seismic design of reinforced concrete dual systems and moment frames is proposed as an optimization problem and charged system search algorithm is utilized as the meta-heuristic optimizer. First, beams, columns and shear wall databases are created utilizing the ACI criteria. Formulations for optimal seismic design of dual systems (shear wall-frame) are then presented. With some modifications on these formulations, optimal seismic design of RC moment frames can also be performed. This procedure is along with ordinary design constraints and effective seismic design constraints. These constraints consist of beams, columns and shear wall design criteria and some other seismic provisions. In this procedure some main provisions of FEMA (2000, 2006) and ACI (2011) are incorporated. Cost of the structure is considered as an objective function. According to the results of the numerical example, the proposed methodologies can be considered as a suitable practical approach for optimal seismic design of reinforced concrete shear wall-frame structures.

2. Optimization Algorithm: Charged System Search 2.1 A Brief Description of the CSS Algorithm Charged System Search (CSS) is an optimization algorithm that was proposed by Kaveh and Talatahari (2010). This metaheuristic algorithm was inspired from the laws of Gauss and Coulomb from electrostatics, and the Newtonian laws of classic mechanics. The rules utilized for establishing the CSS are as follows (Kaveh and Talatahari (2012)): Rule 1: The CSS is a population-based algorithm, i.e. in each iteration a certain number of agents are utilized to seek the domain and the magnitude of the charge for each agent or CP, and the separation distance between two charged parti-

cles is defined in Kaveh and Talatahari (2010). Rule 2: The initial positions of the CPs are determined randomly in the search space and the initial velocities of the charged particles are chosen as zero.  Rule 3: Electric forces between any two charged particles are supposed to be attractive. Rule 4: All good CPs can attract bad CPs and only some of the bad agents attract good agents, employing a probability function. Rule 5: The value of the resultant electrical force affecting a CP is determined using equation defined in Kaveh and Talatahari (2010). Rule 6: The new position and the velocity of each CP are determined. Rule 7: The CSS utilizes a memory so-called Charged Memory (CM) which saves the best CP vectors and their related objective function values. Rule 8: The agents violating the limits of the variables are regenerated using the harmony search-based handling approach. Rule 9: The iterative procedure of the CSS algorithm continues untilthe target terminating criterion is satisfied. Here, a constant number of 200 iterations is considered for the terminating criterion. 2.2 Some Points for Discrete Optimization Discrete optimization problems sometimes are used in structural optimization in order to consider practical aspects of an optimal design. Using meta-heuristic algorithms that are inherently for continuous problems needs some modifications. This is due to the reduction of the exploration ability of the algorithm after transformation of continuous variable space to discrete variable space. Firstly, one of this modifications is a way of considering discrete database arrays as variables. In this article, the created databases are sorted by unit costs and then the cost is a variable as a representative. In each iteration, first we permit the algorithm to update and check the variable domain, after that a function is defined to round the variable to the nearest permissible value in its corresponding database. Observance of this sequence results in maintaining the exploration ability of the algorithm. Secondly, it is the exploration parameter that must be tuned appropriately

3. Construction of Structural Element Databases In order to accomplish a discrete optimization, cross-section databases of structural members need to be generated. Some of the researches who used the structural element databases concept, are Lee and Ahn (2003), Kwak and Kim (2008), Kaveh Sabzi (2011), and Kaveh & Zakian (2012). Cost per unit length of column and beam sections, cost per unit height of the shear wall sections are calculated and stored in an ascending order. These costs are then employed as variables corresponding to specific sections. Here, constant sizes of reinforcement bars are

2

KSCE Journal of Civil Engineering

Optimal Seismic Design of Reinforced Concrete Shear Wall-Frame Structures

terval value; 300 < Dc < 1400 is the depth domain of columns having aspect ratios between 1-2, depending on their width. Hence, 432 column cross-sections are generated in the database. Table 1 shows the details of each column section.

used for column, beam and shear wall cross-sections, obviously, other sizes may be applied in RC structures. 3.1 Columns Creation of column database is similar to that of Lee & Ahn (2003). However, a modification must be imposed for seismic criterion. This criterion is reinforcement ratio of the column that must be limited between 0.01 and 0.06. Here, the interaction diagram of column contains the characteristics as shown in Fig. 1. Concrete cover thickness is taken as 40 mm. Reinforcement bar diameter is 25 mm. Width, depth, and their interval values are as: 300 < Wc < 1200 is the width domain of columns with 50 mm in

3.2 Beams Here, some modifications are made on the creation of the database of the beams. However, other aspects of the generation are similar to those of Lee & Ahn (2003). Firstly, the number of compressive reinforcements at both ends of the beams are variable. Secondly, effects of the compressive reinforcements at both ends of a beam are added on its bending capacity. In the other hand, the number of compressive reinforcements at the middle of beams is constant and is equal to 2. Concrete cover thickness is taken as 40 mm. Reinforcement bar diameter is 22 mm. Width, depth, and their interval values are as: 300 < Wb < 450 is the width domain of beams with 50 mm interval value; 450 < Db < 900 is the depth domain of the beams having aspect ratios between 1.5-2.5 depending on their width. Therefore, 1092 beam cross-sections are generated in the database. Table 2 shows the details of these beam sections. 3.3. Shear wall Database for shear wall sections are generated utilizing the formulations presented in Kaveh and Zakian (2012). Here, the following restrictions are imposed: The lw=hw+2×tf is the length of the shear wall which is equal to 6.0 m; 200 < tw < 400 mm is assumed as the range of the wall thickness; 600 < tf < 1000 mm

Fig. 1. Interaction Diagram of a Column.

Table 1. Column Cross-section database Column number 1 2 3 4 ---429 430 431 432

Width (mm) 300 300 300 300 ---1200 1200 1200 1200

Depth (mm) 300 300 350 350 ---1300 1300 1350 1400

Number of bars ( 25) 4 6 4 6 ---32 34 34 34

P0 (kN) 1643.3 1880.7 1838.1 2075.5 ---24053.1 24290.5 25069.5 25848.5

P0(max) (kN) 1314.7 1504.6 1470.5 1660.4 ---19242.5 19432.4 20055.6 20678.8

Pnb (kN) 415.5 387.9 546.3 534.1 ---9790.5 9784.8 10184.7 10585.9

P0 

(kN) 692.7 1039.1 692.7 1039.1 ---5541.8 5888.1 5888.1 5888.1

MP

n max

(kN.mm) 21399.2 24903.9 31452.5 36827.1 ---1887846.9 1922739.8 2056287.7 2193922.6

Mnb (kN.mm) 79944.1 98835.5 110519.9 136747.5 ---5597098.9 5741316.8 6104528.0 6477469.3

M0 (kN.mm) 69585.2 98852.6 86903.3 124829.7 ---3282040.5 3484055.6 3631258.8 3778462.1

Table 2. Beam cross-section database Beam number

Width (mm)

Depth (mm)

1 2 3 4 ---1089 1090 1091 1092

300 300 300 300 ---450 450 450 450

450 450 450 450 ---900 900 900 900

Vol. 00, No. 00 / 0000 0000

Center reinforcement number of bars ( 22) tension compressive 2 2 2 2 2 2 2 2 ------10 2 10 2 10 2 10 2 3

End reinforcement number of bars ( 22) tension compressive 2 2 3 2 4 2 5 2 ------7 5 8 5 9 5 10 5

Factored moment resistance (kN.mm) Center End 96595.6 96595.6 96595.6 139891.1 96595.6 179851.8 96595.6 216477.6 ------1008663.5 739211.7 1008663.5 841316.9 1008663.5 942683.2 1008663.5 1043139.0


Table 3. Shear Wall Cross-section Database Shear wall number 1 2 3 4 ---1845 1846 1847 1848

tw (mm) 200 200 200 200 ---400 400 400 400

tf (mm) 600 600 600 600 ---1000 1000 1000 1000

bf (mm) 400 400 400 400 ---1000 1000 1000 1000

Ssh (mm) 300 350 400 450 ---400 400 400 400

Nrl (30) 4 4 4 4 ---14 16 18 20

Where As,ij is the total reinforcement bars’ area , Lij is the length, Wc,ij is the width and Dc,ij is the depth of the j th column from i th column group. Here ncg is the number of column group, and nc(i) is the number of member of the i th group.

Nrl (30) 6 6 6 6 ---14 16 18 20

nbg nb  i 

Cbeams =

∑ ∑ {[Ase ij    Lij + Ase  ij    Lij + As ij   1 – 2    Lij

i=1 j=1

 A s ij   1 – 2    Lij ]  s  UCs +  Wb ij  Db ij  Lij   UCc } (2)

Fig. 2. A Special Shear Wall Cross Section.

is assumed as the range of the length of the wall flange; 200 < bf < 1000 mm is assumed as the range of the width of the wall flange; 300 < Ssh < 450 mm based on ACI 318-11 (2000) code is the distance of the vertical and horizontal shear bars which are considered identical in order to reduce the discrete section numbers. Diameter of these bars is taken as 16 mm. be is the diameter of reinforcement bars of each flange (boundary element) selected as 30 mm. Intervals of the dimensions are considered as 100 mm, and intervals of the Ssh are taken as 50 mm. Concrete cover thickness is taken equal to 60 mm. Asfmin=0.01 × tf × bf is the minimum reinforcement area of one flange. Asfmax = 0.04 × tf × bf is the maximum reinforcement area of one flange. Thus, 1848 shear wall cross-sections are generated in the database. Table 3 shows the details of different shear wall sections. A typical shear wall cross section is illustrated in Fig. 2 and dimension notations are introduced. Nrl and Nud are defined in Kaveh and Zakian (2012). This figure shows the arrangements of the reinforcement bars in the wall and its flanges.

4. Procedure of Seismic Design Optimization

In the above relationship, cutoff points of the beam reinforcement are included. As an empirical design formula (Wight and Macgregor (2012)), it includes a constant coefficient  that is equal to 0.3 times of the considered beam span length. Considering j th beam from i th group, Ase,ij and Ase  ij are the tension and compressive total reinforcement areas at each end of the beam, respectively. As,ij and As ij are tension and compressive total reinforcement areas at the center of the beam. Lij is the length, Wb,ij is the width and Db,ij is the depth of the beam. nbg is the number of beam group and nb(i) is the number of member of the i th beam group. Cshear wall =   2  m1  Asf + m2  Asw   Hw + 2  lw  m3  Asw   s  UCs +  2  b f  tf + h w  tw – 2  m1  Asf –m2  Asw   Hw  UCc (3) Asf is the cross-section area of each reinforcement bar in the flange of the wall; Asw is the cross-section area of each longitudinal (vertical) reinforcement bar in the web of the wall, for longitudinal and transversal shear reinforcement considered as one cross-section area;hw is the length of the shear wall's cross-section web; Hw is the total height of the wall; m1 is the number of reinforcement bars in each flange; m2 is the number of reinforcement bars in the web; m3 is equal to integer part of [Int(Hw/Ssh)]. Constant values: UCc = 60$/m3 is the unit cost of the concrete; UCs = 0.9$/kg is the unit cost of the steel; UCf = 18$/m2 is the unit cost of the formwork which is assumed identical for all structural elements;s = 7850$/m3 is the density of the steel; Total cost of structural members’ formwork is calculated by: ncg nc  i 

Cf =

4.1 Formulation of Optimal Seismic Design In this section, a seismic design optimization problem for reinforced concrete dual systems and moment frames is proposed. More information about design of reinforced concrete shear walls and frame structures can be found in Wight and Macgregor (2012). Cost of a RC structure includes the cost of concrete, steel and formwork. Thus cost of columns, beams and shear wall are defined as follows: Ccolumns =

ncg nc  i 

∑∑

∑ ∑ {2   Wc ij + Dc ij   Lij]  UCf } c

i=1 j=1

ncg nc  i 

+∑

∑ {  Wb ij + 2  Db ij   Lij ]  UCf } b

+  4   bf + tf – 0.5  tw   Hw + 2  hw  Hw   UCf

Cost = Ccolumns + Cbeams + Cshear well + Cf

(1)

w

Finally, total cost of the structure is expressed as:

  As ij  Lij   s  UCs +  Wc ij  D c ij  Lij   UCc 

i=1 j=1

(4)

i=1 j=1

(5)

Objective function that should be optimized is formulated as follows:

4


Optimal Seismic Design of Reinforced Concrete Shear Wall-Frame Structures 

Minimize Ob(X) Ob(X) = Cost (X) × fpenalty(X) X = [x1, x2,...., xnv] i = 1, 2,...., nv d xi  R

(6)

The following penalty function is employed to handle the constraints: 2

fpenalty  X  =  1 + 1    , v =

n

∑ max  0 vi 

(7)

i=1

Here, the parameters 1 and 2 are selected as 1 and 2, respectively. v shows the sum of the violated constraints. Design constraints consist of seven ordinary design constraints and nine seismic design constraints, respectively, as follows: First, the strength constraint of column is defined that is determined based on the parameters provided in Table 1. Assuming that B is a point corresponding to loading of the column in the interaction diagram and A is an intersection point of OB line segment with interaction diagram curve, see also Fig. 1 exhibiting these items. At point B, the length and slope of the line segment OB are determined as: B=

Mu

Mul is the negative moment demand at the left side of beam; and  Mur is the negative moment demand at the right side of the +  beam; Mu is the nominal resistant positive moment; Mn is the nominal resistant negative moment, and  is a the strength reduction factor. ACI code expresses that the positive-moment strength of the beam section at the face of the beam-column joint should not be less than half the negative-moment strength. This leads to the following constraint which permits the beam to develop large curvatures at hinging zones and greatly improves the ductility of the ends of the beams.

 g8 = ----------- – 1  0 0.5 

  and  denote the compressive and tension reinforcement ratios, respectively. In order to obtain the axial force, shear forces and moment acting on the shear wall in a dual system, it is necessary to attain the shear wall support reactions for imposing shear wall design constraints. Special shear wall design optimization constraints that were presented in (Kaveh and Zakian, 2012) are as follows: Vu – 1  0 g9 = --------Vn

(8)

Pu 2

2

2

 t req

Pu ⎞ mOB = arctan ⎛ -------⎝ Mu ⎠

t -–10 g10 = ---------

Therefore, the column strength constraint is: L g1 = -----u – 1  0 La

(9)

Obviously, in the design of well-known building frames, column dimensions are reduced by increasing of the story level, i.e. it is necessary to reduce the column dimensions of higher story level (Width WcT and Depth DcT) to make it smaller or identical to that of lower story level (WcB and DcB), because of the over loading value reduction. Thus, these constraints are defined: WcT DcT - – 1  0 , g3 = -------–10 g2 = -------WcB DcB

(10)

Also, the number of reinforcement bars of higher level column (ncT)issmalleroridenticaltothatoflowerlevelcolumn(ncB): ncT -–10 g4 = -----ncB

(11)

Strength constraints of beams are as below: +





Mu – 1  0 , g = Mul – 1  0 , g = Mur – 1  0 (12) g5 = -------------------- ----------- 6 7  Mu+  Mu  Mu +

Mu is the positive moment demand at the center of beam; Vol. 00, No. 00 / 0000 0000

(14)

Vu is the shear force acting at the base of the wall and Vn is the nominal shear capacity.

OB = Lu =  Pu  +  Mu  , OA = La =   Pu  +   Mu  , 2

(13)

(15)

t is the existing transversal reinforcement ratio and t req is the required transversal reinforcement ratio of the wall. Due to tallness of the wall and neglecting the longitudinal (vertical) shear reinforcement, with a reasonable approximation one can convert the demand moment Mu to a couple of compressive force Cu and tension force Tu, Wight and Macgregor (2012). Hence, one can design flanges similar to a column by the following force demands: M M Tu = ------u , Cu = Pu + ------u z z

(16)

Tua, Cua and Pu are the allowable tension, compressive and axial forces, respectively; z is the distance between the center of two flanges. Tua = t As fy , Cua = c  0.85fc   Ag – As  + As fy 

(17)

t and c are the strength reduction factors being equal to 0.90 and 0.65 according to the ACI, respectively; As and Ag are reinforcement area and gross area of each flange, respectively; fy is the yield stress of steel and fc  is the compressive strength of concrete. Tension and compressive strength constraints of the flanges are as follows:

5


Fig. 3. Nominal resisting moments at a beam-column intersection joint (Wight and Macgregor (2012)).

Tu – 1  0 , g = Cu – 1  0 g11 = -----------12 Tua Cua

(18)

Next constraint is for the flange length limitation: tf -–10 g13 = --------tt mim

(19)

tf is the current flange length and tt mim is the lower bound of it. Additional computational information about all the above shear wall constraints can be found in Kaveh and Zakian (2012). If plastic hinges form in columns, the stability of the structure may be under hazard. The design of ductile moment-resisting frames endeavors to force the structure to respond in a way that strong column-weak beam action in which the seismic forces induced plastic hinges form at the ends of the beams rather than in the columns. The hinging zones are detailed to hold their shear capacity while the plastic hinges undergo flexural yielding in both positive and negative bending . Since the dead load must always be transferred down via the columns, thus the damage of columns should be minimized. ACI Code requires the use of a strong column-weak beam design. Strong column-weak beam behavior is made more likely by requiring (Fig. 3) that:

∑ Mnc g14 = ---------------------- – 1  0 1.2 ∑ Mnb

(20)

where Mnc is the nominal flexural capacity of the columns and Mnb is the nominal flexural capacity of the girders connected to that joint. Notice that for all load combination this constraint must be checked, because loading capacity of a column is a function of its loading. From another point of view, when a frame sway to right one side of beam utilizes its compressive moment capacity and the other side uses its tension moment capacity, when the frame sway to left, this is vice versa. Therefore, this consideration should also be implemented. Drift constraint is imposed at each story: DR g15 = ----------i –1  0 DRa

Fig. 4. (a) Structural modeling of a dual system (shear wallframe), (b) Structural model of a frame component (after removing the shear wall).

(21)

DRi is the drift ratio of the ithstoryand DRa is its admissible value.

In dual systems, the frame component (by removing the shear wall) must resist nearly 25% of the lateral seismic loads. Thus columns and beams of the structure must have enough strength to withstand such a load. According to FEMA-451 (2006), in the frame component model of a structure, after removing the shear wall from a dual system, the boundary elements (flanges) remain in the model as a column. Hence, the last constraint is defined as: g16 = g25%(g1, g5, g6, g7)  1  0

(22)

This constraint implies that columns and beams of the frame component must sustain 25% of the lateral seismic loading, so it is called 25% of lateral load constraint. This means that we must check the strength constraints of beams and columns for the second model which is called frame component model or frame only model. 4.2 Topology of Structural Model Dual system modeling is accomplished by rigid links (rod members). In this model, the shear wall is divided into three component containing the flanges and web of the wall as shown in Fig. 4(a). In these links only transitional degrees of freedom (DOFs) are considered which restrict those of web to those of flanges of the wall. Master nodes for rigid link application in both sides are of the wall web. For modeling the frame component (while the shear wall is removed), a model as shown in Fig. 4(b) is introduced for the analysis of system under 25% of lateral loads. As mentioned before, the flanges remain in the model, as depicted in this figure. 4.3 Fundamental Steps of Optimization 4.3.1. Reinforced Concrete Dual System The procedures for seismic design optimization of RC dual system are presented in the flowchart, Fig. 6. For this optimization problem sixteen constraints are defined and employed.

6



Fig. 5. Flowchart of the Proposed Optimization Procedure.

It should be mentioned that in each iteration of the optimization process two structural models are analyzed, Fig. 4(a) and Fig 4(b). In order to design the shear wall its support reactions are determined. 4.3.2. Reinforced Concrete Moment frame The flowchart of RC dual system optimization can be used for RC moment frame optimization by performing some modifications in the objective function and neglecting the shear wall constraints and the 16th constraint (the last one).

5. Design Example

Fig. 6. Geometry of the 20-story Dual RC Frame. Vol. 00, No. 00 / 0000 0000

This example is studied to demonstrate the proposed methodologies. Equivalent static analysis is performed for determining structural demands via OpenSees (2012) software. P-Delta effect of columns and shear wall, reinforcement arrangements and cutoff points of reinforcements in beams are considered at this stage. For considering reinforcement arrangements, fiber sections are defined by employing nonlinear beam-column elements with elastic materials. These are programmed to change adaptively for each section. Cutoff points are included by defining four nodes for every beam leading to three finite elements. All other computer programs are coded in MATLAB (2007) software. Hence, the interference between OpenSees and MATLAB is accomplished for the optimization process. Dead load and live load are considered as 16.5 kN/m and 7


Case (2-3): Case (4-5):

Fig. 7. Progress History of the CSS Algorithm for the 20-story RC Dual Frame Optimization.

7.2 kN/m, respectively. Axial loads of the shear wall in the dual system are considered as concentrated loads equivalent to the distributed loads. Material properties are as: Concrete Steel

fc  = 23.5 MPa fc = 392 MPa

U = 1.2D + 1.0L ± 1.4E U = 0.9D ± 1.4E

Allowable drift ratio is considered as 0.0045 according to (ASCE 7-10). The CSS algorithm is utilized for optimization of a 20-story RC dual system with 4 column groups, 5 beam groups and one shear wall. Therefore, 10 design variables as unit cost of members are included. Fig. 6 indicates geometry of the structure and its lateral seismic loading details. Progress curve of the optimization algorithm are shown in Fig. 7. The obtained optimized structural members are provided in Table 4. Here, constraints are violated by a small amount. This violation is for beam stress constraint. Values of the main constraints for the obtained optimum solution are provided in Figs. 8-11. Fig. 8 illustrates the values of strong column-weak beam constraints for all beam-column joints and all the load combinations. Stress ratios of critical column of each group and critical beam of each group are plotted in Fig. 9. Moreover, Fig. 10 indicates these values for the frame component which should withstand 25% of the lateral seismic loads. Gravity loads are imposed on the frame component model for efficient consideration of the PDelta effect. Drift ratios of each story of frame are plotted in Fig. 11 for the critical load combination.

6. Discussions and Conclusions and load combinations are based on the ACI 318-11 code (2000) as: Case 1:

U = 1.2D + 1.6L

It can be seen that the drift values are far from the allowable value, this is because of the effects of other constraints that selected members in this way. Furthermore, stress ratio of beams

Table 4. Details of Optimal Structural Members for the 20-Story Dual RC Frame Member category

Sectional Dimensions (mm) Width Depth

Group tag

Story level

Section number

Bg1

1-4

601

400

700

Bg2

5-8

485

400

600

Bg3

9-12

207

300

600

Bg4

13-16

460

350

650

Bg5

17-20

1036

450

900

Column

Cg1 Cg2 Cg3 Cg4

1-5 6-10 11-15 16-20

427 384 321 263

Shear Wall

Wg1

1-20

1580

Beam

1150 950 800 700 tw tf 400 900

1350 1200 1150 1100 bf 800

Computational time 10 hours

Reinforcements Details Compressive(C) / Tension (T) End 22 T6 C3 End 22 T5 C4 End 22 T5 C3 End 22 T6 C3 End 22 T5 C4

Center 22 T6 C2 Center 22 T8 C2 Center 22 T6 C2 Center 22 T7 C2 Center 22 T8 C2 32  25 26  25 20  25 1625 Boundary Element 30 Nrl 14 Nud 16

Wall Web 16 Ssh 400

Sum of violated constraint value 0.0018 Optimal Cost 116012.35 $ 8



Fig. 8. Values of the Strong Column-weak Beam Constraint for the Attained Optimum 20-story RC Dual Frame.

Fig. 9. Stress Ratios of the Critical Member of Each Group for Optimum Solution (a) for Columns (b) for Beams

Fig. 10. Stress Ratios of Each Group of Frame Only Model for Optimum Solution (a) for Critical Column (b) for Critical Beam

are at high level near their allowable boundaries, however, these values for columns are at lower level relative to their allowable boundaries. This is because of strong column-weak beam constraint. In order to satisfy this constraint, a beam and a column must be employed in a way that we have strong column and weak beam. Thus, in the procedure of optimization, beam Vol. 00, No. 00 / 0000 0000

and column constraints work together in order to satisfy strong column-weak beam constraint, apart from their stress constraints. An efficient optimal design is a solution that makes the constraints values to move to near their boundaries, minimize (or maximize for some problems) the objective function, and simultaneously satisfying the design constraints. In this paper,

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Fig. 11. Drift Ratios of the Attained Optimum 20-story Dual RC Frame

we have almost achieved this scope, but some constraints such as strong column-weak beam ones make the procedure harder as descried before. In this research, methods are developed for optimal seismic design of RC dual systems. With some modification the proposed methodologies can be incorporated in the RC frames seismic design optimization. An efficient structural model is also presented for this purpose. In order to perform a practical optimization, section databases of columns, beams and shear walls are generated. The main ordinary and seismic design provisions such as FEMA and ACI guidelines are imposed for the proposed methodologies. This procedure helps the engineers to not only reduce the cost of the structure and avoid the tedious try and error procedures (taking a great deal of computational effort), but also carry out automatically an optimum seismic design. The Charged System Search algorithm is utilized as an optimizer. However, other efficient algorithms can also be incorporated for optimization in this process. The results demonstrate that presented methods can be good practical approaches for optimal seismic design of shear wall-frame structures.

Acknowledgement The first author is grateful to the Iran National Science Foundation for the support.

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