research article optimum performance-based seismic design...

Research ArticleOptimum Performance-Based Seismic DesignUsing a Hybrid Optimization Algorithm

S Talatahari1 A Hosseini2 S R Mirghaderi2 and F Rezazadeh2

1 Department of Civil Engineering University of Tabriz Tabriz Iran2Department of Civil Engineering Faculty of Engineering University of Tehran Tehran Iran

Correspondence should be addressed to S Talatahari siamaktalatgmailcom

Received 16 May 2013 Accepted 21 November 2013 Published 16 February 2014

Academic Editor Yudong Zhang

Copyright copy 2014 S Talatahari et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A hybrid optimization method is presented to optimum seismic design of steel frames considering four performance levels Theseperformance levels are considered to determine the optimum design of structures to reduce the structural cost A pushover analysisof steel building frameworks subject to equivalent-static earthquake loading is utilizedThe algorithm is based on the concepts of thecharged system search in which each agent is affected by local and global best positions stored in the charged memory consideringthe governing laws of electrical physics Comparison of the results of the hybrid algorithm with those of other metaheuristicalgorithms shows the efficiency of the hybrid algorithm

1 Introduction

Many of engineering problems can be modeled into opti-mization problems Therefore developing new optimizationtechniques to solve these types of problems becomes highlysignificant For this case a large number of optimizationalgorithms as optimization techniques have already beenproposed and applied in solving engineering problems suchas ant colony optimization (ACO) [1] ABC (artificial beecolony) [2 3] cuckoo search (CS) [4ndash6] bat algorithm(BA) [7 8] genetic programming (GP) [9] ES (evolutionarystrategy) [10] GA (genetic algorithm) [11] HS (harmonysearch) [12 13] biogeography-based optimization (BBO) [14ndash16] differential evolution (DE) [17ndash19] particle swarm opti-mization (PSO) [20 21] electromagnetism-like mechanism(EM) [22] and the charged system search algorithm (CSS)[23ndash25]

In 2010 Kaveh and Talatahari have firstly proposed arobust metaheuristic search technique namely CSS algo-rithm [23] for possibly nonlinear functions The governinglaws from the physics initiate the base of the CSS algo-rithm CSS is a multiagent algorithm in which each agent is

considered as a charged sphere Since these agents are treatedas charged particles that can affect each other according to theCoulomb and Gauss laws from electrostatics they are calledcharged particles (CPs) After determining the resultant forceaffected on each CP the Newtonian motion law is utilizedto determine the movement of the agents The successivemoving of CPs considering the resultant forces directs theagents toward optimum solutions

The contribution of this paper is to present a hybrid CSS-based algorithm to find a seismic optimum design of steelframes considering four performance levels The nonlinearanalysis is required to reach the structural response at variousperformance levels Therefore the refined plastic hinge anal-ysis method is developed to estimate the nonlinear behaviorof the entire structural system and members effectively

The organization of this paper is as follows Section 2and Section 3 describe the statement of the problem and theutilized analyses method respectively Our proposed CSS-based hybrid method is described in detail in Sections 4 and5 Subsequently the merits of our method are verified bynumerical examples in Section 6 At last Section 7 summa-rizes our work

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 693128 8 pageshttpdxdoiorg1011552014693128

2 Mathematical Problems in Engineering

2 Statement of Seismic Design of Frames

Themathematical formulation of the structural optimizationproblems can be expressed asminimizing the weight of struc-tures as the cost function without taking into considerationother influencing tributary parameters

Minimize 119882 (119883) =

119899119890

sum

119895=1

120588 sdot 119871119895sdot 119860119895 (1)

where119882(119883) is the weight of the structure 119883 is the vector ofdesign variables taken from W-shaped sections found in theAISC design manual [26] 119899119890 is the number of members 120588 isthe material mass density and 119871

119895and 119860

119895are the length and

the cross-sectional area of the member119895 respectivelyLateral deflections of a building may cause human dis-

comfort and minor damage of nonstructural componentsExtreme inelastic lateral deflections due to a severe earth-quake can cause the failure of mechanical electrical andplumbing systems or suspended ceilings and equipment tofall thereby posing threats to the human life This matter isconsidered as the constraint functions in this paper as [27]

OP Level ΔOP(119883) le Δ

OP

IO Level ΔIO(119883) le Δ

IO

LS Level ΔLS(119883) le Δ

LS

CP Level ΔCP(119883) le Δ

CP

(2)

whereΔlevel is the lateral drift and Δlevel is the allowable lateraldrift (04 07 25 and 5 of the height of the buildingare taken as the allowable roof drifts for theOP IO LS andCPperformance levels resp) Here OP IO LS and CP are thedifferent performance levels Operational (OP) ImmediateOccupancy (IO) life safety (LS) and collapse Prevention(CP) (FEMA-273 1997) [28] are building performance levelsThe operational level is that at which a building has sustainedminimal or no damage to its structural and nonstructuralcomponents and the building is suitable for normal occu-pancy or use a building at the immediate occupancy levelhas sustainedminimal or no damage to its structural elementsand onlyminor damage to its nonstructural components andis safe to be reoccupied immediately a building at the lifesafety level has experienced extensive damage to its structuraland nonstructural components and while the risk to life islow repairs may be required before reoccupancy can occurthe collapse prevention level is when a building has reached astate of impending partial or total collapse where the buildingmay have suffered a significant loss of lateral strength andstiffness with some permanent lateral deformation but themajor components of the gravity load carrying system shouldstill continue to carry gravity load demands

3 Pushover Analysis forPerformance-Based Design

There are various methods of static pushover analyses topredict the seismic demands on building frameworks underequivalent static earthquake loading [29ndash35] however herea developed computer-based pushover analysis procedure isutilized [27] which was originally conceived for the elasticanalysis of steel frameworks with semirigid connections[36 37] The analysis process is inspired from second-orderinelastic analysis of semi-rigid framed structures that rigidityfactor is replaced with plasticity factor in stiffness matrixFictitious plastic-hinge connections are necessary at the twoends of beam-column elements and semi-rigid analysis tech-niques were modified for the nonlinear load-deformationanalysis of building frameworks under increasing seismicloads The value of plasticity factor 119901 is conceived fromrigidity factor used in semi-rigid analysis This factor 119903

119894

defines the rotational stiffness of the connection and can beinterpreted as the ratio of the end-rotation 120572

119894of the member

to the combined rotation 120579119894of the member as

119903119894=

120572119894

120579119894

=

1

1 + (3EI119877119871)(119894 = 1 2) (3)

where119877 is the rotational stiffness of connection 119894 and EI and 119871are the bending stiffness and length of the connectedmemberrespectively In fact upon replacing connection rotationalstiffness 119877 with section postelastic flexural stiffness in (3)the degradation of the flexural stiffness of a member sectionexperiencing postelastic behavior can be characterized by theplasticity factor

119901 =

1

1 + (3EI119877119901119871) (4)

where119877119901 = 119889119872119889120601 is the section postelastic flexural stiffnessand 119901 is the plasticity factor

plasticity = 100 (1 minus 119901) (5)

Here the elastic stiffness matrix is comprised of both thefirst-order and the second-order geometric properties

119870 = 119878119890119862119890+ 119878119892119862119892 (6)

The matrix119870 consists of two parts the first part is conceivedfrom Monfortoon and Wursquos method [38] that employs therigidity factor concept to develop a first-order elastic analysistechnique for semi-rigid frames (ie 119878

119890times119862119890) and the second

part is conceived from Xursquos method [36] that considers therigidity-factor concept to develop a second-order elastic anal-ysis technique for semi-rigid frames (ie 119878

119892times119862119892) Here 119878

119890and

119878119892are the standard first-order elastic and the second-order

geometric stiffness matrices respectively when the memberhas rigid moment-connections 119862

119890and 119862

119892are the corre-

sponding correction matrices which account for the reducedrotational stiffness of the semi-rigid moment-connectionsThe flowchart of pushover analysis for performance-baseddesign is shown in Figure 1

Mathematical Problems in Engineering 3

Selection of primary parameters of structure loading and location

Gravity analysis

Determination of nodal displacement and elements force and update of axial force and elastic period of structure

Determination of nodal displacement axial force (N) moment (M) and correction stiffness matrix

End

No

No

No

No

Yes

Yes

Yes

Determination of roof driftfor per hazard level

Yes

Start

Increase lateral loading and next iteration (k = k + 1)

Assign slight lateral load V = 00008 g times W

VIO minus [k times (step of lateral load) + initial share] le step of lateral load

VLS minus [k times (step of lateral load) + initial share] le step of lateral load

VCP minus [k times (step of lateral load) + initial share] le step of lateral load

VCP minus [k times (step of lateral load) + initial share] le 0

Figure 1 Flowchart of pushover analysis for performance-based design

4 Utilized Algorithms

A review of utilized metaheuristic algorithms is presented inthe following subsections

41 Charged System Search Algorithm The charged systemsearch (CSS) algorithm is based on the Coulomb and Gausslaws from electrical physics and the governing laws ofmotionfrom the Newtonian mechanics This algorithm can beconsidered as a multiagent approach where each agent is acharged particle (CP) Each CP is considered as a charged

spherewith radius119886 having a uniformvolume charge densityand is equal to [23]

119902119895=

119882119895minus119882worst

119882best minus119882worst 119895 = 1 2 119873 (7)

where119882best and119882worst are the minimum and the maximumweight among all the particles 119882

119895represents the weight of

the agent 119894 and119873 is the total number of CPsCPs can impose electrical forces on the others The kind

of the forces is attractive and its magnitude for the CP


located in the inside of the sphere is proportional to theseparation distance between the CPs and for a CP locatedoutside the sphere is inversely proportional to the square ofthe separation distance between the particles

F119895= 119902119895sum

119894119894 = 119895

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)119901119894119895(X119894minus X119895)

119895 = 1 2 119873

1198941= 1 119894

2= 0 lArrrArr 119903

119894119895lt 119886

1198941= 0 119894


119894119895ge 119886

(8)

where F119895is the resultant force acting on the 119895th CP and 119903

119894119895is

the separation distance between two charged particles whichis defined as follows

119903119894119895=

10038171003817100381710038171003817X119894minus X119895

10038171003817100381710038171003817

10038171003817100381710038171003817(X119894+ X119895) 2 minus Xbest

10038171003817100381710038171003817+ 120576

(9)

where X119894and X

119895are the positions of the 119894th and jth CPs

respectively Xbest is the position of the best current CPwith the minimal weight and 120576 is a small positive numberThe initial positions of CPs are determined randomly in thesearch space and the initial velocities of charged particles areassumed to be zero 119875

119894119895determines the probability of moving

each CP toward the others as

119901119894119895=

1

119882119894minus119882best

119882119895minus119882119894

gt rand or 119882119895gt 119882119894

0 otherwise(10)

The resultant forces and the motion laws determine thenew location of the CPs At this stage each CP moves towardto its new position considering the resultant forces and itsprevious velocity as

X119895new = rand

1198951sdot 119896119886sdot

F119895

119898119895

sdot Δ1199052

+ rand1198952sdot 119896V sdot V119895old sdot Δ119905 + X

119895old

V119895new =

X119895new minus X

119895old

Δ119905

(11)

where 119896119886is the acceleration coefficient 119896V is the velocity

coefficient to control the influence of the previous velocityand rand

1198951and rand

1198952are two random numbers uniformly

distributed in the range of (0 1) If each CP exits from theallowable search space its position is corrected using theharmony search-based handling approach as described byKaveh andTalatahari [39] In addition to save the best designa memory (charged memory) is considered containing theCMS number of positions for the so far best agents

Both CSS and EM [22] are based on the governing lawsfrom the electrical physics however themovement strategiesthe resultant force for each agent and deification of electricalcharges for agents are different The CSS algorithm utilizes

a velocity term while in the EM we have no term of avelocity The EM just uses the Coulomb law to determine theforces while the CSS approach uses the Coulomb law as wellas Gaussrsquos law to explore the search space more efficientlyAfter evaluating the total force vector in the EM each agentis moved in the direction of the force by a random steplength (being uniformly distributed between 0 and 1) whilethe movements in the CSS are based on the governing lawsof motion from the Newtonian mechanics The potency ofthe EM is summarized to find the direction of an agentrsquomovement while in the CSS not only the directions but alsothe amount of movements are determined

From the above discussion it can be concluded that theCSS algorithm is a general form of the EM which contains itssuperiorities and avoids its disadvantages

42 Particle Swarm Optimization The particle swarm opti-mization (PSO) is motivated from the social behavior ofbird flocking and fish schooling which has a populationof individuals called particles that adjust their movementsdepending on both their own experience and the populationrsquosexperience [20] In other words each particle in the PSOalgorithm continuously focuses and refocuses on the effortof its search according to both local best and global best InPSO the position of each agentX119896

119894 and its velocityV119896+1

119894 are

calculated as

X119896+1119894= X119896119894+ V119896+1119894

V119896+1119894= 120596V119896119894+ 11988811199031∘ (P119896119894minus X119896119894) + 11988821199032∘ (P119896119892minus X119896119894)

(12)

where 120596 is an inertia weight to control the influence of theprevious velocity 119903

1and 1199032are two random vectors uniformly

distributed in the range of (0 1) and 1198881and 119888

2are two

acceleration constants and the sign ldquo∘rdquo denotes element-by-element multiplicationThe abovementioned formulations ofthe PSO algorithm can be combined and rewritten as

X119896+1119894= X119896119894+ 120596V119896119894+ 11988811199031∘ (P119896119894minus X119896119894) + 11988821199032∘ (P119896119892minus X119896119894)

(13)

In some previous studies to improve the performance ofthe algorithm another term is added to the above formulaeas

X119896+1119894= X119896119894+ 120596V119896119894+ 11988811199031∘ (P119896119894minus X119896119894)

+ 11988821199032∘ (P119896119892minus X119896119894) +

119899119890

sum

119895=1

119888119895119903119895∘ (R119896119895minus X119896119894)

(14)

where 119888119895 similar to 119888

1and 119888

2 is a constant value and 119903

119895

is a random vector 119899119890 denotes the number of extra termsconsidered in the algorithm and R119896

119895is defined based on the

type of the algorithm being used

5 A Hybrid Optimization Algorithm

In the present hybrid algorithm the advantage of the PSOcontaining utilizing the local best and the global best is added


to the CSS algorithm The charged memory (CM) for thehybrid algorithm is treated as the local best in the PSO andthe CM updating process is defined as

CM119894new =

CM119894old 119882 (X

119894new) ge 119882(CM119894old)

X119894new 119882 (X

119894new) lt 119882(CM119894old)

(15)

in which the first term identifies that when the new positionis not better than the previous one the updating does notperformwhile when the new position is better than the storedso far good position the new solution vector is replacedIn the first iteration the vector stored in CM and the firstpositions of the agents will be identical Considering theabovementioned new charged memory the electric forcesgenerated by agents are modified as

F119895= sum

119894isin1198781

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)(CM

119894old minus X119895)

+ sum

119894isin1198782

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)119886119903119894119895119901119894119895(X119894minus X119895)

(16)

where 1198781and 1198782are defined as follows

1198781= 1199051 1199052 119905

119899| 119902 (119905) gt 119902 (119895) 119895 = 1 2 119873 119895 = 119894 119892

(17)

1198782= 119878 minus 119878

1 (18)

in which 1198781determines the set of agents utilized from CM 119899

denotes the number of CM agents 119878 is utilized as a set of allagentsrsquo number and thus 119878

2will be the set of current agents

used for directing the agent 119895 Here in the primary iterations119899 is set to two continuing the number of the best stored so faragent among all CPs (global best) and 119895th agent stored in theCM which is treated as local best Then the number of usedagents from CM is increased linearly and finally it reached119873 in the last iterations In this hybrid algorithm CM

119894old willbe treated similar to P119896

119894in the PSOThe other modification is

that the forces can be attractive or repulsive and 119886119903119894119895is added

to fulfill this aim which determines the kind of the force as

119886119903119894119895=

+1 wp 119896119905

minus1 wp 1 minus 119896119905

(19)

where ldquowprdquo represents the abbreviation for ldquowith the proba-bilityrdquo and 119896

119905is a parameter to control the effect of the kind of

forces Comparing to (10) this new formula (18) considers thebest so far location of agents and the best local position of thecurrent agent in addition to the location of other agents Alsohere119898

119895is assumed to be 119902

119895and therefore (12) is simplified as

X119895new = 119896119886 sdot 1199031 sdot F119895 + 119896V sdot 1199032 sdot V119895old + X

119895old (20)

The pseudocode of the hybrid algorithm can be summarizedas follows

Step 1 (initialization) The magnitude of the charge for eachCP is defined by (7) The initial positions of the CPs aredetermined randomly and the initial velocities of chargedparticles are assumed to be zero

Step 2 (CM creation) The position of the initial agents andthe values of their corresponding objective functions aresaved in the charged memory (CM)

Step 3 (the forces determination) The probability of movingeach CP towards the others (119901

119894119895) and the kind of forces

(119886119894119895) are determined using (10) and (19) respectively and the

resultant force vector for each CP is calculated using (18)

Step 4 (solution construction) Each CP moves to the newposition according to (20)

Step 5 (CM updating) CM updating is performed accordingto (15)

Step 6 (terminating criterion control) Steps 3ndash5 are repeatedfor a predefined number of iterations

6 Design Examples

Two building frameworks are selected for seismic optimumdesign using the metaheuristic algorithm [27] These frameshave previously been used to illustrate the pushover analysistechnique by Hasan et al [42] and Talatahari [40]

The expected yield strength of steel material used forcolumn members is 120590ye = 397MPa while 120590ye = 339MPais considered for beammembersThe constant gravity load119908is accounted for a tributary-area width of 457m and dead-load and live-load factors of 12 and 16 respectively For eachexample 30 independent runs are carried out using the newhybrid algorithms and compared with other algorithms Thenumber of 20 individuals for CPs is used and the values ofconstants 119896V and 119896119886 are set to 04

61 Four-Bay Three-Story Steel Frame The configurationgrouping of the members and applied loads of the four-baythree-story framed structure are shown in Figure 2 [27] The27 members of the structure are categorized into five groupsas indicated in the figureThemodulus of elasticity is taken as119864 = 200GPa The constant gravity load of 119908

1= 32 kNm is

applied to the first and second story beams while the gravityload of 119908

2= 287 kNm is applied to the roof beams The

seismic weight is 4688 kN for each of the first and secondstories and 5071 kN for the roof story

The performance-based optimum results for the meta-heuristic algorithm are summarized in Table 1 The hybridCSS HPACO ACO and GA need 4500 4500 3900 and6800 analyses to reach a convergence while 8500 analysesrequired by the PSO The best hybrid CSS design results ina frame that weighs 2737 kN which is lighter than the designof Gall optimization algorithm The result of conventionaldesign [41] is approximately 50 more than the result ofnew algorithm In a series of 30 different design runs theaverage weight of the hybrid CSS designs is 2867 kN with astandard deviation of 5651 kN while the average weight ofthe PSACO PSO and ACO designs is 2904 kN 3024 kNand 2943 kN respectively The standard deviation values are645 kN 1045 kN and 756 for the PSACO PSO and ACOrespectively


w2

w1

w1

P3

P2

P1

1

1

1

2 2 2

2

2 2 2

22

3 3 3 3

4 4 4 4

5 5 5 5

430998400(914m)

1

1

1

313998400(396m)

Figure 2 Three-story steel moment-frame

Table 1 The statistical information of performance-based optimum designs for the 4-bay 3-story frame

Algorithm Hybrid CSS PSACO [40] PSO [40] ACO [27] GA [27] A conventional design [41]Best weight (kN) 2737 2792 2863 2834 3039 4129 kNAverage weight (kN) 2867 2904 3024 2943 3215 mdashWorst weight (kN) 2978 2985 3107 3032 3397 mdashStd dev (kN) 5651 6453 10453 7566 14332Average number of analyses 4500 4500 8500 3900 6800 mdash


Algorithm Hybrid CSS PSACO [40] PSO [40] ACO [27] GA [27]Best weight (kN) 156866 160132 168263 163183 17231Average weight (kN) 162632 165055 172536 16962 17914Worst weight (kN) 172536 175965 181325 178694 19432Std dev (kN) 3035 3852 6635 4933 7833Average number of analyses 5000 6000 12500 5600 9700

62 Five-Bay Nine-Story Steel Frame A five-bay nine-storysteel frame is considered as shown in Figure 3 The materialhas a modulus of elasticity equal to 119864 = 200GPa The 108members of the structure are categorized into fifteen groupsas indicated in the figure The constant gravity load of 119908

1=

32 kNm is applied to the beams in the first to the eighthstory while119908

2= 287 kNm is applied to the roof beamsThe

seismic weights are 4942 kN for the first story 4857 kN foreach of the second to eighth stories and 5231 kN for the roofstory In this example each of the five beam element groupsis chosen from all 267 W-shapes while the eight columnelement groups are limited to W14 sections (37 W-shapes)

Table 2 presents the statistical results obtained by themetaheuristic algorithmsThe best hybrid CSS design resultsin a frame weighing 156866 kN which is 19 70 38and 95 lighter than the PSACO PSO ACO and GA Inorder to converge to a solution for the hybrid CSS algorithmapproximately 5000 frame analyses are required which areless than the 6000 12500 and 9700 analyses necessary forthe PSACO PSO and GA respectively The ACO needs only5600 analyses to find an optimum result

7 Conclusion Remarks

The problem of optimum design of frame structures is for-mulated to minimize the weight of the structure considering

the required constraints specified by design codes Forseismic design of structures two main points should beconsidered structural costs and structural damages As aresult it is essential to control the lateral drift of buildingframeworks under seismic loading at various performancelevels To fulfill this aim in this paper a hybrid optimizationmethod is presented The algorithm is based on the CSSalgorithm CSS is a multiagent algorithm in which eachagent is considered as a charged sphere Since these agentsare treated as charged particles that can affect each otheraccording to theCoulomb andGauss laws from electrostaticsin the present hybrid algorithm the advantage of the PSOcontaining utilizing the local best and the global best is addedto the CSS algorithm The charged memory for the hybridalgorithm is treated as the local best in the PSO and the CMupdating process is redefined to adapt the new requirements

A simple computer-based method for push-over analysisof steel building frameworks subject to equivalent-staticearthquake loading is utilizedThemethod accounts for first-order elastic and second-order geometric stiffness propertiesand the influence that combined stresses have on plasticbehavior and employs a conventional elastic analysis proce-dure modified by a plasticity-factor to trace elastic-plasticbehavior over the range of performance levels for a structure[27] Two examples are optimized using the new algorithm as


w2

w1

w1

w1

w1

w1

w1

w1

P3

P2

P1

11

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

9

9

9

9

9

9

9

9

9

9

13

1

1

1

12

11

13

12

11

13

12

11

13

12

11

13

12

2 2 2 2

3 3 3 3

3 3 3 3

4

4 5 5 5 5

5 5 5 5

7 7 7 7

w17 7 7 7

6 6 6 6

6 6 6 6

530998400(914m)

11

4

4

6

6

8

8

6

6

8

8

1

813998400(396m)

P6

P9

P8

P7

P5

P4

18998400

(548m)

Figure 3 Nine-story steel moment-frame

well as some advanced metaheuristic algorithms to investi-gate the capability of the newmethodThe genetic algorithmant colony optimization particle swarm optimization andparticle swarm ant colony optimization method as well asthe new hybrid method are utilized to find optimum seismicdesign of examplesThe obtained results indicate that the newalgorithm compared to GA ACO PSO and PSACO can findbetter optimum seismic design of structures

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004

[2] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[3] X Li and M Yin ldquoParameter estimation for chaotic systems byimproved artificial bee colony algorithmrdquo Journal of Computa-tional and Theoretical Nanoscience vol 10 no 3 pp 756ndash7622013

[4] A H Gandomi X-S Yang and A H Alavi ldquoCuckoo searchalgorithm a metaheuristic approach to solve structural opti-mization problemsrdquo Engineering with Computers vol 29 no 1pp 17ndash35 2013

[5] G Wang L Guo H Duan H Wang L Liu and M ShaoldquoA hybrid meta-heuristic DECS algorithm for UCAV three-dimension path planningrdquo The Scientific World Journal vol2012 Article ID 583973 11 pages 2012

[6] X T Li and M H Yin ldquoParameter estimation for chaoticsystems using the cuckoo search algorithm with an orthogonallearning methodrdquo Chinese Physics B vol 21 no 5 Article ID050507 2012

[7] X S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[8] GWang L Guo H Duan L Liu andHWang ldquoPath planningfor UCAV using bat algorithm with mutationrdquo The ScientificWorld Journal vol 2012 Article ID 418946 15 pages 2012

[9] A H Gandomi and A H Alavi ldquoMulti-stage genetic pro-gramming a new strategy to nonlinear system modelingrdquoInformation Sciences vol 181 no 23 pp 5227ndash5239 2011

[10] H-G BeyerThe theory of evolution strategies Natural Comput-ing Series Springer Berlin Germany 2001

[11] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1998

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013

[14] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[15] H DuanW Zhao GWang and X Feng ldquoTest-sheet composi-tion using analytic hierarchy process and hybrid metaheuristicalgorithmTSBBOrdquoMathematical Problems in Engineering vol2012 Article ID 712752 22 pages 2012

[16] X Li J Wang J Zhou and M Yin ldquoA perturb biogeographybased optimization with mutation for global numerical opti-mizationrdquo Applied Mathematics and Computation vol 218 no2 pp 598ndash609 2011


[17] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuous spacesrdquoJournal ofGlobalOptimization An International JournalDealingwith Theoretical and Computational Aspects of Seeking GlobalOptima and Their Applications in Science Management andEngineering vol 11 no 4 pp 341ndash359 1997

[18] A H Gandomi X-S Yang S Talatahari and S Deb ldquoCoupledeagle strategy and differential evolution for unconstrained andconstrained global optimizationrdquo Computers amp Mathematicswith Applications vol 63 no 1 pp 191ndash200 2012

[19] X Li andMYin ldquoSelf-adaptive constrained artificial bee colonyfor constrained numerical optimizationrdquo Neural Computingand Applications 2012

[20] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995

[21] S Talatahari M Kheirollahi C Farahmandpour and A HGandomi ldquoA multi-stage particle swarm for optimum designof truss structuresrdquo Neural Computing amp Applications vol 23no 5 pp 1297ndash1309 2013

[22] S Birbil and S-C Fang ldquoAn electromagnetism-like mechanismfor global optimizationrdquo Journal of Global Optimization vol 25no 3 pp 263ndash282 2003

[23] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3 pp 267ndash289 2010

[24] S Talatahari A Kaveh and N Mohajer Rahbari ldquoParame-ter identification of Bouc-Wen model for MR fluid dampersusing adaptive charged system search optimizationrdquo Journal ofMechanical Science and Technology vol 26 no 8 pp 2523ndash2534 2012

[25] AKaveh and S Talatahari ldquoAn enhanced charged system searchfor configuration optimization using the concept of fields offorcesrdquo Structural and Multidisciplinary Optimization vol 43no 3 pp 339ndash351 2011

[26] Manual of steel construction Load and Resistance FactorDesign American Institute of Steel Construction Chicago IllUSA 2001

[27] A Kaveh B Farahmand Azar A Hadidi F Rezazadeh Sorochiand S Talatahari ldquoPerformance-based seismic design of steelframes using ant colony optimizationrdquo Journal of ConstructionalSteel Research vol 66 no 4 pp 566ndash574 2010

[28] Federal Emergency Management Agency FEMA-273 NEHRPGuideline for the Seismic Rehabilitation of Buildings BuildingSeismic Safety Council Washington DC USA 1997

[29] R S Lawson V Vance and H Krawinkler ldquoNonlinear staticpush-over analysis why when and howrdquo in Proceedings of 5thUS National Conference on Earthquake Engineering vol 1 pp283ndash292 EERI Chicago Ill USA 1994

[30] AC Biddah andNNaumoski ldquoUse of pushover test to evaluatedamage of reinforced concrete frame structures subjected tostrong seismic groundmotionsrdquo in Proceedings of 7th CanadianConference on Earthquake Engineering Montreal Canada 1995

[31] A S Moghadam and W K Tso ldquo3-D pushover analysis foreccentric buildingsrdquo in Proceedings of 7th Canadian Conferenceon Earthquake Engineering Montreal Canada 1995

[32] A Ferhi and K Z Truman ldquoBehaviour of asymmetric buildingsystems under a monotonic loadmdashIrdquo Engineering Structuresvol 18 no 2 pp 133ndash141 1996

[33] A Ferhi and K Z Truman ldquoBehaviour of asymmetric buildingsystems under a monotonic loadmdashIIrdquo Engineering Structuresvol 18 no 2 pp 142ndash153 1996

[34] J M Bracci S K Kunnath and A M Reinhorn ldquoSeismicperformance and retrofit evaluation of reinforced concretestructuresrdquo Journal of Structural Engineering vol 123 no 1 pp3ndash10 1997

[35] V Kilar and P Fajfar ldquoSimple push-over analysis of asymmetricbuildingsrdquo Earthquake Engineering and Structural Dynamicsvol 26 no 2 pp 233ndash249 1997

[36] L Xu ldquoGeometrical stiffness and sensitivity matrices for opti-mization of semi-rigid steel frameworksrdquo Structural Optimiza-tion vol 5 no 1-2 pp 95ndash99 1992

[37] L Xu and D E Grierson ldquoComputer-automated design ofsemirigid steel frameworksrdquo Journal of Structural EngineeringNew York NY vol 119 no 6 pp 1740ndash1760 1993

[38] G RMonfortoon and T SWu ldquoMatrix analysis of semi-rigidlyconnected steel framesrdquo Journal of Structural Engineering vol89 no 6 pp 13ndash42 1963

[39] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[40] S Talatahari ldquoOptimum performance-based seismic design offrames using metaheuristic optimization algorithmsrdquo in Meta-heuristics in Water Geotechnical and Transport Engineering X-S Yang A H Gandomi S Talatahari and A H Alavi EdsElsevier 2012

[41] R Hassan B Cohanim O DeWeck and G Venter ldquoA compar-ison of particle swarm optimization and the genetic algorithmrdquoin Proceedings of the 46th AIAAASMEASCEAHSASC Struc-tures Structural Dynamics and Materials Conference pp 18ndash21April 2005

[42] R Hasan L Xu and D E Grierson ldquoPush-over analysis forperformance-based seismic designrdquo Computers and Structuresvol 80 no 31 pp 2483ndash2493 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


2 Statement of Seismic Design of Frames

Themathematical formulation of the structural optimizationproblems can be expressed asminimizing the weight of struc-tures as the cost function without taking into considerationother influencing tributary parameters

Minimize 119882 (119883) =

119899119890

sum

119895=1

120588 sdot 119871119895sdot 119860119895 (1)

where119882(119883) is the weight of the structure 119883 is the vector ofdesign variables taken from W-shaped sections found in theAISC design manual [26] 119899119890 is the number of members 120588 isthe material mass density and 119871

119895and 119860

119895are the length and

the cross-sectional area of the member119895 respectivelyLateral deflections of a building may cause human dis-

comfort and minor damage of nonstructural componentsExtreme inelastic lateral deflections due to a severe earth-quake can cause the failure of mechanical electrical andplumbing systems or suspended ceilings and equipment tofall thereby posing threats to the human life This matter isconsidered as the constraint functions in this paper as [27]

OP Level ΔOP(119883) le Δ

OP

IO Level ΔIO(119883) le Δ

IO

LS Level ΔLS(119883) le Δ

LS

CP Level ΔCP(119883) le Δ

CP

(2)

whereΔlevel is the lateral drift and Δlevel is the allowable lateraldrift (04 07 25 and 5 of the height of the buildingare taken as the allowable roof drifts for theOP IO LS andCPperformance levels resp) Here OP IO LS and CP are thedifferent performance levels Operational (OP) ImmediateOccupancy (IO) life safety (LS) and collapse Prevention(CP) (FEMA-273 1997) [28] are building performance levelsThe operational level is that at which a building has sustainedminimal or no damage to its structural and nonstructuralcomponents and the building is suitable for normal occu-pancy or use a building at the immediate occupancy levelhas sustainedminimal or no damage to its structural elementsand onlyminor damage to its nonstructural components andis safe to be reoccupied immediately a building at the lifesafety level has experienced extensive damage to its structuraland nonstructural components and while the risk to life islow repairs may be required before reoccupancy can occurthe collapse prevention level is when a building has reached astate of impending partial or total collapse where the buildingmay have suffered a significant loss of lateral strength andstiffness with some permanent lateral deformation but themajor components of the gravity load carrying system shouldstill continue to carry gravity load demands

3 Pushover Analysis forPerformance-Based Design

There are various methods of static pushover analyses topredict the seismic demands on building frameworks underequivalent static earthquake loading [29ndash35] however herea developed computer-based pushover analysis procedure isutilized [27] which was originally conceived for the elasticanalysis of steel frameworks with semirigid connections[36 37] The analysis process is inspired from second-orderinelastic analysis of semi-rigid framed structures that rigidityfactor is replaced with plasticity factor in stiffness matrixFictitious plastic-hinge connections are necessary at the twoends of beam-column elements and semi-rigid analysis tech-niques were modified for the nonlinear load-deformationanalysis of building frameworks under increasing seismicloads The value of plasticity factor 119901 is conceived fromrigidity factor used in semi-rigid analysis This factor 119903

119894

defines the rotational stiffness of the connection and can beinterpreted as the ratio of the end-rotation 120572

119894of the member

to the combined rotation 120579119894of the member as

119903119894=

120572119894

120579119894

=

1

1 + (3EI119877119871)(119894 = 1 2) (3)

where119877 is the rotational stiffness of connection 119894 and EI and 119871are the bending stiffness and length of the connectedmemberrespectively In fact upon replacing connection rotationalstiffness 119877 with section postelastic flexural stiffness in (3)the degradation of the flexural stiffness of a member sectionexperiencing postelastic behavior can be characterized by theplasticity factor

119901 =

1

1 + (3EI119877119901119871) (4)

where119877119901 = 119889119872119889120601 is the section postelastic flexural stiffnessand 119901 is the plasticity factor

plasticity = 100 (1 minus 119901) (5)

Here the elastic stiffness matrix is comprised of both thefirst-order and the second-order geometric properties

119870 = 119878119890119862119890+ 119878119892119862119892 (6)

The matrix119870 consists of two parts the first part is conceivedfrom Monfortoon and Wursquos method [38] that employs therigidity factor concept to develop a first-order elastic analysistechnique for semi-rigid frames (ie 119878

119890times119862119890) and the second

part is conceived from Xursquos method [36] that considers therigidity-factor concept to develop a second-order elastic anal-ysis technique for semi-rigid frames (ie 119878

119892times119862119892) Here 119878

119890and

119878119892are the standard first-order elastic and the second-order

geometric stiffness matrices respectively when the memberhas rigid moment-connections 119862

119890and 119862

119892are the corre-

sponding correction matrices which account for the reducedrotational stiffness of the semi-rigid moment-connectionsThe flowchart of pushover analysis for performance-baseddesign is shown in Figure 1



Gravity analysis



End

No

No

No

No

Yes

Yes

Yes


Yes

Start












119902119895=

119882119895minus119882worst

119882best minus119882worst 119895 = 1 2 119873 (7)







F119895= 119902119895sum

119894119894 = 119895

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)119901119894119895(X119894minus X119895)

119895 = 1 2 119873

1198941= 1 119894


119894119895lt 119886

1198941= 0 119894


119894119895ge 119886

(8)


119894119895is


119903119894119895=

10038171003817100381710038171003817X119894minus X119895

10038171003817100381710038171003817

10038171003817100381710038171003817(X119894+ X119895) 2 minus Xbest

10038171003817100381710038171003817+ 120576

(9)

where X119894and X





119901119894119895=

1

119882119894minus119882best

119882119895minus119882119894

gt rand or 119882119895gt 119882119894

0 otherwise(10)


X119895new = rand

1198951sdot 119896119886sdot

F119895

119898119895

sdot Δ1199052


119895old

V119895new =

X119895new minus X

119895old

Δ119905

(11)



1198951and rand








119894 are

calculated as

X119896+1119894= X119896119894+ V119896+1119894


(12)




2are two



(13)


X119896+1119894= X119896119894+ 120596V119896119894+ 11988811199031∘ (P119896119894minus X119896119894)

+ 11988821199032∘ (P119896119892minus X119896119894) +

119899119890

sum

119895=1

119888119895119903119895∘ (R119896119895minus X119896119894)

(14)

where 119888119895 similar to 119888

1and 119888


119895








CM119894new =

CM119894old 119882 (X

119894new) ge 119882(CM119894old)

X119894new 119882 (X

119894new) lt 119882(CM119894old)

(15)


F119895= sum

119894isin1198781

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)(CM

119894old minus X119895)

+ sum

119894isin1198782

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)119886119903119894119895119901119894119895(X119894minus X119895)

(16)


1198781= 1199051 1199052 119905

119899| 119902 (119905) gt 119902 (119895) 119895 = 1 2 119873 119895 = 119894 119892

(17)

1198782= 119878 minus 119878

1 (18)









119886119903119894119895=

+1 wp 119896119905


(19)







119895old (20)











6 Design Examples




1= 32 kNm is






w2

w1

w1

P3

P2

P1

1

1

1

2 2 2

2

2 2 2

22

3 3 3 3

4 4 4 4

5 5 5 5

430998400(914m)

1

1

1

313998400(396m)







1=










w2

w1

w1

w1

w1

w1

w1

w1

P3

P2

P1

11

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

9

9

9

9

9

9

9

9

9

9

13

1

1

1

12

11

13

12

11

13

12

11

13

12

11

13

12

2 2 2 2

3 3 3 3

3 3 3 3

4

4 5 5 5 5

5 5 5 5

7 7 7 7

w17 7 7 7

6 6 6 6

6 6 6 6

530998400(914m)

11

4

4

6

6

8

8

6

6

8

8

1

813998400(396m)

P6

P9

P8

P7

P5

P4

18998400

(548m)





References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Gravity analysis



End

No

No

No

No

Yes

Yes

Yes


Yes

Start












119902119895=

119882119895minus119882worst

119882best minus119882worst 119895 = 1 2 119873 (7)







F119895= 119902119895sum

119894119894 = 119895

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)119901119894119895(X119894minus X119895)

119895 = 1 2 119873

1198941= 1 119894


119894119895lt 119886

1198941= 0 119894


119894119895ge 119886

(8)


119894119895is


119903119894119895=

10038171003817100381710038171003817X119894minus X119895

10038171003817100381710038171003817

10038171003817100381710038171003817(X119894+ X119895) 2 minus Xbest

10038171003817100381710038171003817+ 120576

(9)

where X119894and X





119901119894119895=

1

119882119894minus119882best

119882119895minus119882119894

gt rand or 119882119895gt 119882119894

0 otherwise(10)


X119895new = rand

1198951sdot 119896119886sdot

F119895

119898119895

sdot Δ1199052


119895old

V119895new =

X119895new minus X

119895old

Δ119905

(11)



1198951and rand








119894 are

calculated as

X119896+1119894= X119896119894+ V119896+1119894


(12)




2are two



(13)


X119896+1119894= X119896119894+ 120596V119896119894+ 11988811199031∘ (P119896119894minus X119896119894)

+ 11988821199032∘ (P119896119892minus X119896119894) +

119899119890

sum

119895=1

119888119895119903119895∘ (R119896119895minus X119896119894)

(14)

where 119888119895 similar to 119888

1and 119888


119895








CM119894new =

CM119894old 119882 (X

119894new) ge 119882(CM119894old)

X119894new 119882 (X

119894new) lt 119882(CM119894old)

(15)


F119895= sum

119894isin1198781

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)(CM

119894old minus X119895)

+ sum

119894isin1198782

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)119886119903119894119895119901119894119895(X119894minus X119895)

(16)


1198781= 1199051 1199052 119905

119899| 119902 (119905) gt 119902 (119895) 119895 = 1 2 119873 119895 = 119894 119892

(17)

1198782= 119878 minus 119878

1 (18)









119886119903119894119895=

+1 wp 119896119905


(19)







119895old (20)











6 Design Examples




1= 32 kNm is






w2

w1

w1

P3

P2

P1

1

1

1

2 2 2

2

2 2 2

22

3 3 3 3

4 4 4 4

5 5 5 5

430998400(914m)

1

1

1

313998400(396m)







1=










w2

w1

w1

w1

w1

w1

w1

w1

P3

P2

P1

11

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

9

9

9

9

9

9

9

9

9

9

13

1

1

1

12

11

13

12

11

13

12

11

13

12

11

13

12

2 2 2 2

3 3 3 3

3 3 3 3

4

4 5 5 5 5

5 5 5 5

7 7 7 7

w17 7 7 7

6 6 6 6

6 6 6 6

530998400(914m)

11

4

4

6

6

8

8

6

6

8

8

1

813998400(396m)

P6

P9

P8

P7

P5

P4

18998400

(548m)





References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











F119895= 119902119895sum

119894119894 = 119895

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)119901119894119895(X119894minus X119895)

119895 = 1 2 119873

1198941= 1 119894


119894119895lt 119886

1198941= 0 119894


119894119895ge 119886

(8)


119894119895is


119903119894119895=

10038171003817100381710038171003817X119894minus X119895

10038171003817100381710038171003817

10038171003817100381710038171003817(X119894+ X119895) 2 minus Xbest

10038171003817100381710038171003817+ 120576

(9)

where X119894and X





119901119894119895=

1

119882119894minus119882best

119882119895minus119882119894

gt rand or 119882119895gt 119882119894

0 otherwise(10)


X119895new = rand

1198951sdot 119896119886sdot

F119895

119898119895

sdot Δ1199052


119895old

V119895new =

X119895new minus X

119895old

Δ119905

(11)



1198951and rand








119894 are

calculated as

X119896+1119894= X119896119894+ V119896+1119894


(12)




2are two



(13)


X119896+1119894= X119896119894+ 120596V119896119894+ 11988811199031∘ (P119896119894minus X119896119894)

+ 11988821199032∘ (P119896119892minus X119896119894) +

119899119890

sum

119895=1

119888119895119903119895∘ (R119896119895minus X119896119894)

(14)

where 119888119895 similar to 119888

1and 119888


119895








CM119894new =

CM119894old 119882 (X

119894new) ge 119882(CM119894old)

X119894new 119882 (X

119894new) lt 119882(CM119894old)

(15)


F119895= sum

119894isin1198781

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)(CM

119894old minus X119895)

+ sum

119894isin1198782

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)119886119903119894119895119901119894119895(X119894minus X119895)

(16)


1198781= 1199051 1199052 119905

119899| 119902 (119905) gt 119902 (119895) 119895 = 1 2 119873 119895 = 119894 119892

(17)

1198782= 119878 minus 119878

1 (18)









119886119903119894119895=

+1 wp 119896119905


(19)







119895old (20)











6 Design Examples




1= 32 kNm is






w2

w1

w1

P3

P2

P1

1

1

1

2 2 2

2

2 2 2

22

3 3 3 3

4 4 4 4

5 5 5 5

430998400(914m)

1

1

1

313998400(396m)







1=










w2

w1

w1

w1

w1

w1

w1

w1

P3

P2

P1

11

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

9

9

9

9

9

9

9

9

9

9

13

1

1

1

12

11

13

12

11

13

12

11

13

12

11

13

12

2 2 2 2

3 3 3 3

3 3 3 3

4

4 5 5 5 5

5 5 5 5

7 7 7 7

w17 7 7 7

6 6 6 6

6 6 6 6

530998400(914m)

11

4

4

6

6

8

8

6

6

8

8

1

813998400(396m)

P6

P9

P8

P7

P5

P4

18998400

(548m)





References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











CM119894new =

CM119894old 119882 (X

119894new) ge 119882(CM119894old)

X119894new 119882 (X

119894new) lt 119882(CM119894old)

(15)


F119895= sum

119894isin1198781

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)(CM

119894old minus X119895)

+ sum

119894isin1198782

(

119902119894

1198863119903119894119895sdot 1198941+

119902119894

1199032

119894119895

sdot 1198942)119886119903119894119895119901119894119895(X119894minus X119895)

(16)


1198781= 1199051 1199052 119905

119899| 119902 (119905) gt 119902 (119895) 119895 = 1 2 119873 119895 = 119894 119892

(17)

1198782= 119878 minus 119878

1 (18)









119886119903119894119895=

+1 wp 119896119905


(19)







119895old (20)











6 Design Examples




1= 32 kNm is






w2

w1

w1

P3

P2

P1

1

1

1

2 2 2

2

2 2 2

22

3 3 3 3

4 4 4 4

5 5 5 5

430998400(914m)

1

1

1

313998400(396m)







1=










w2

w1

w1

w1

w1

w1

w1

w1

P3

P2

P1

11

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

9

9

9

9

9

9

9

9

9

9

13

1

1

1

12

11

13

12

11

13

12

11

13

12

11

13

12

2 2 2 2

3 3 3 3

3 3 3 3

4

4 5 5 5 5

5 5 5 5

7 7 7 7

w17 7 7 7

6 6 6 6

6 6 6 6

530998400(914m)

11

4

4

6

6

8

8

6

6

8

8

1

813998400(396m)

P6

P9

P8

P7

P5

P4

18998400

(548m)





References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










w2

w1

w1

P3

P2

P1

1

1

1

2 2 2

2

2 2 2

22

3 3 3 3

4 4 4 4

5 5 5 5

430998400(914m)

1

1

1

313998400(396m)







1=










w2

w1

w1

w1

w1

w1

w1

w1

P3

P2

P1

11

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

9

9

9

9

9

9

9

9

9

9

13

1

1

1

12

11

13

12

11

13

12

11

13

12

11

13

12

2 2 2 2

3 3 3 3

3 3 3 3

4

4 5 5 5 5

5 5 5 5

7 7 7 7

w17 7 7 7

6 6 6 6

6 6 6 6

530998400(914m)

11

4

4

6

6

8

8

6

6

8

8

1

813998400(396m)

P6

P9

P8

P7

P5

P4

18998400

(548m)





References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










w2

w1

w1

w1

w1

w1

w1

w1

P3

P2

P1

11

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

9

9

9

9

9

9

9

9

9

9

13

1

1

1

12

11

13

12

11

13

12

11

13

12

11

13

12

2 2 2 2

3 3 3 3

3 3 3 3

4

4 5 5 5 5

5 5 5 5

7 7 7 7

w17 7 7 7

6 6 6 6

6 6 6 6

530998400(914m)

11

4

4

6

6

8

8

6

6

8

8

1

813998400(396m)

P6

P9

P8

P7

P5

P4

18998400

(548m)





References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article optimum performance-based seismic design...

Documents