Embodied Energy and Cost Optimization of RC Beam under

Research ArticleEmbodied Energy and Cost Optimization of RCBeam under Blast Load

Runqing Yu Diandian Zhang and Haichun Yan

State Key Laboratory of Disaster Prevention amp Mitigation of Explosion amp Impact PLA University of Science and TechnologyNanjing Jiangsu 210007 China

Correspondence should be addressed to Runqing Yu jkggkljqqcom

Received 13 October 2016 Accepted 12 February 2017 Published 22 March 2017

Academic Editor Sergii V Kavun

Copyright copy 2017 Runqing Yu 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

Reinforced concrete (RC) structures not only consume a lot of resources but also cause continuing pollution However sustainabledesign could make RC structures more environmental-friendly One important index for environmental impact assessment isembodied energy The aim of the present study is to optimize the embodied energy and the cost of RC beam subjected to the blastloads First a general optimization procedure was describedThen the optimization procedure was used to optimize the embodiedenergy and the cost of RC beams Optimization results of the cost and the embodied energy were compared It was found thatthe optimization results were influenced by the cost ratio 119899119862 (ratio of price of steel to price of concrete per unit volume) and theembodied energy ratio 119899119864 (ratio of embodied energy of steel to embodied energy of concrete per unit volume) An optimal designthat minimized both embodied energy and cost simultaneously was obtained if values of 119899119862 and 119899119864 were very close

1 Introduction

Worldwide RC structures consume a lot of energy [1] andgive off large amounts of greenhouse gases [2] In additionthe maintenance repair and refurbishment during the lifeof buildings also consume a lot of energy Even when thebuilding is abandoned quantities of solid construction wasteare difficult to recycle All of those burden the environment

In order to make RC structures more friendly to environ-ment a lot of efforts have been made to reduce the influenceof RC buildings on environment during the service life [3]Recently many researchers focused on the optimization ofRC structures considering environment factors [4ndash7] Paya-Zaforteza et al [4] minimized the carbon dioxide (CO2)emissions and the cost using simulated annealing Theirresearch showed that the CO2 emissions optimization andthe cost optimization were highly related Yeo and Gabbai[5] explored the implications of using the embodied energyas the objective function to be minimized from the point ofview of cost Their results showed that a reduction of 10in the embodied energy leads to an increase of 10 in thecost In their study certain values of the embodied energy

were used which was not true in practice because the exactvalue of embodied energy was not known [8 9] Medeirosand Kripka [6] proposed an optimal method to minimize themonetary and environmental costs of RC column In theirstudy four environmental impact assessment indicesmdashCO2global warming potential (GWP) energy consumption andEco-indicatormdashwere taken into consideration but the costwas not considered Yepes et al [7] suggested a designmethod to optimize the cost and theCO2 emission of precast-prestressed concrete U-beam road bridges Their analysisshowed that a reduction of costs by 1 Euro can save up to175 kg in CO2 emissions Park et al [10] provided structuraldesign guidelines that reduced the CO2 emission and thecost of RC columns Parametric study was conducted toinvestigate the influences of design factors on the CO2emission and the cost

In conclusion those researches optimized RC structuresin normal operation condition considering environmentfactors Extreme loading conditions such as explosionearthquake and hurricane were not considered In the lastcentury the design that used to resist the extreme loadswas not common With the rapid development of economy

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 1907972 8 pageshttpsdoiorg10115520171907972

2 Mathematical Problems in Engineering

the design to resist the extreme loads was widely acceptedby engineering field An increasing number of buildingswere designed considering the extreme loading conditionsTherefore studies on the sustainable design of RC structuresin extreme loading conditions are necessary

In this study a sustainable design of RC beam underblast loads was discussed through minimizing the embodiedenergy Meanwhile the cost was used as objective functionfor comparison First a general optimal design procedurewas presented Then parametric studies were conducted toanalyze the influences of the material strength and sectionalgeometry on the optimal design The cost and embodiedenergy of particular building materials (reinforcements andconcrete) were taken as variables and their influences onthe optimal design were presented The optimization resultsof cost and the embodied energy were compared and thecorrelation between them was discussed which may behelpful for sustainable deign of RC structural membersagainst blast

2 Methodology

21 Problem Description In this study an interior beamsubjected to air blast load was considered This problem wasan illustrative example of UFC 3-340-02 [11] Both ends ofthe beam were fixed The beam was assumed to deform ina flexural shape The cross section is shown in Figure 1 Auniform blast load is applied on the upper surface of thebeam The beam was designed according to the restrictionsand guidelines found in the UFC 3-340-02 code

There are two response indices that are usually used todefine the damage levels of blast-loaded structural compo-nents in a flexural response regime the support rotation angle(120579) and the ductility ratio (120583) [12] In the present study bothof the two indices were used as design constraints

Four design parameterswere considered in this studyThefour parameters were the height of the beam h the width bthe ratio of longitudinal reinforcement 1205881 and the ratio ofstirrup 120588222 Objective Functions The total cost and the total embod-ied energy were taken as objective function to be minimizedThe total cost is defined as [13]

1198621015840 = 119862119862119881119862 + 119862119878 (1198811198781 + 1198811198782) (1)

where 1198621015840 is the total cost of beam 119862119862 is the cost of concreteper unit volume 119881119862 is the volume of concrete 119862119878 is thecost of reinforcing steel per unit volume and 1198811198781 and 1198811198782are the volume of longitudinal reinforcement and shearreinforcement respectively

It should be noted that the optimal values were affectedby the relative values of the objective function only Equation(1) divided by 119862119878 is

11986210158401015840 = 119881119862119899119862 + (1198811198781 + 1198811198782) (2)

where 119899119862 = 119862119904119862119888 11986210158401015840 = 1198621015840119862119878

8

84

Unit in

120

b

h

Figure 1 Cross section

The value of 119899119862 varies from country to country thus (2)was more convenient to study the minimum cost of RC beamthan (1) [13]

Similarly to the cost objective function the embodiedenergy objective function is written as follows

11986410158401015840 = 119881119862119899119864 + (1198811198781 + 1198811198782) (3)

where 11986410158401015840 = 1198641015840119864119878 1198641015840 is the total energy 119864119878 is the embodiedenergy of steel per unit volume 119899119864 = 119864119878119864119862 is the energyratio and 119864119862 is the embodied energy of concrete per unitvolume

Comparing (2) with (3) a similarity can be found Theform of the energy objective function and the cost objectivefunction is very similarThus the cost objective function andthe energy objective function are represented by (4) for theconvenience of optimization calculation

119882 = 119881119862119899 + (1198811198781 + 1198811198782) (4)

where 119899 denotes 119899119862 and 119899119864 and119882 denotes 11986210158401015840 and 11986410158401015840In the following analysis the optimal designs for the cost11986210158401015840 and the embodied energy 11986410158401015840 were unified to the optimal

design for119882

23 Design Constraints The RC beam under blast loadis simplified to SDOF system (see Figure 2) The designconstraints were given in mathematical expressions and theoptimal design was transform into a constrained optimiza-tion problem

The design constraints were defined in accordance withthe UFC 3-340-02 code [11] The constraints for RC rectan-gular section beam under blast load were expressed

(1) Performance limits are as follows

120579 le 120579119886120583 le 120583119886 (5)

Mathematical Problems in Engineering 3

K

Me

(a) Equivalent sys-tem

y

Rm

ye

(b) Resistance curve

P0

td

(c) Blast load

Figure 2 The simplified SDOF model of RC beam under blast load

(2) Constraints of the ratio of longitudinal reinforcement1205881 and the ratio of stirrup 1205882 are as follows075(08511987011198911015840dc119891dy )( 8700087000 + 119891dy) ge 1205881

ge max(3radic1198911015840119888119891119910 200119891119910 )1205882 ge 00015

(6)

(3) Limit of the direct shear stress is as follows

0181198911015840dc119887119889 le 1198771198982 119897 (7)

(4) Limit of the diagonal tension stress is as follows(1198972 minus 119889) 119877119898119887119889 le 10 (1198911015840dc)05 (8)

(5) Limit of the unreinforced web shear capacity is asfollows

19 (1198911015840dc)05 + 25001205881 le 35 (1198911015840dc)05 (9)

where 120579119886 is the allowable support rotation angle 120583119886 isthe allowable ductility ratio 1198911015840119888 is the concrete compressivestrength 1198911015840dc is the dynamic concrete compressive strength119891119910 is the yield stress of steel 119891dy is the dynamic yield stress ofsteel 1198701 is a coefficient associated with 1198911015840119888 119887 is the width ofbeam 119889 is the distance from the extreme compression fiberto the centroid of the longitudinal tension reinforcement 119877119898is the bending resistance of RC beam and 119897 is the span length

Some explanations of design constraints are presentedhere Equations (5) are the deformation requirements Thevalue of 120579119886 and 120583119886 can be found inmany literatures [11 14 15]Equations (7)sim(9) are the requirements that ensure shearfailure does not occur

It should be noticed the parameters shown in (5)sim(9)wereinterrelatedThis interrelated relationshipwas reflected in the

calculation of 1205882 120579 119877119898 and 120583 Calculations of 1205882 120579 119877119898 and120583 were present

(1) 1205882 1205882 was computed by

1205882 = max (V119906 minus V119888 V119888)085119891dy (10)

where V119888 and V119906 are calculated by (11) and (12) respectively

V119888 = 19 (1198911015840119888 )05 + 25001205881 (11)

V119906 = (1198972 minus 119889) 119877119898119887119889 (12)

(2) 120579 120579 is defined in

120579 = arctan(2119910119898119897 ) (13)

where 119910119898 = 120583119910119890 119910119898 is the maximum midspan displacementand 119910119890 is the elastic midspan displacement119910119890 is calculated by

119910119890 = 119877119898119870 (14)

where 119870 is the equivalent elastic stiffness119870 is given by (15) considering clamped boundary

119870 = 3071198641198621198681198861198974 (15)

where 119864119862 is the concrete modulus of elasticity and 119868119886 is theaverage moment of inertia of the beams119868119886 is given by

119868119886 = 05 (119887ℎ312 + 1198661198871198893) (16)

The coefficient 119866 in (16) is evaluated by [16]

119866 = (3320312058831 minus 1819812058821 + 586241205881) ( 1198641198787119864119862)07 (17)

where 119864119878 is the steel modulus of elasticity


(3) 119877119898 The resistance 119877119898 is given by [11]

119877119898 = 8 (119872119873 +119872119875)1198972 (18)

where 119872119873 is ultimate negative moment capacity at thesupport and 119872119875 is ultimate positive moment capacity at themidspan

Values of119872119873 and119872119875 are calculated by (19) consideringsymmetrical reinforced concrete beam

119872119873 = 119872119875 = 1205881119891dy1198871198892 (1 minus 1205881119891dy1712058811198911015840119888 ) (19)

(4) 120583 120583 is calculated by [17]

2120596119905119889radic2120583 minus 1 + 2120583 minus 12120583 (1 + (14120587120596119905119889)) = 1198750119877119898 (20)

where 120596 is the natural frequency 119905119889 is the duration time and1198750 is the peak pressure120596 is calculated by

120596 = radic 119870(119870LM119898) (21)

where 119898 is the mass of the beam plus 20 of the slabs spanperpendicular to the beam and119870LM is the load-mass factor

24 Optimization Technique and Verification Manymethodswere used successfully in optimal design of RC structuressuch as Generalized Reduced Gradient (GRG) method [13]heuristic optimization methods [18 19] discretized contin-uum-type optimality criteria (DCOC) [20] and Lagrangemultiplier method [21] In this study the sequential quadraticprograming (SQP) method was used since SQP was effi-cient proved by an extensive comparative study done bySchittkowski [22] Besides SQP was easy to realize bythe constrained nonlinear optimization solver ldquofminconrdquo inMATLAB

In order to get the global optimummany randomstartingpoints were usedThe random starting points were generatedby Latin hypercube sampling (LHS) [23] LHS was a matrixof 119894 times 119895 order 119894 was the number of sampling points to beexamined 119895 was the number of design parameters Each of119895 columns of matrix that contained sampling points 1 2 119894was coupled to form the Latin hypercube This generatedrandom sample points which ensured that all portions ofdesign space were represented

The flowchart of optimum design method is shown inFigure 3

3 Numerical Examples and Discussions

31 Example 1 Verify the Effectiveness of Optimum DesignMethod Values of design parameters in this example arelisted in Table 1 where 120588 is density of concrete

Start

Find optimal point by SQP

Optimal design

End

Set variables and objective function

Set design space

Obtain design points by using LHS

Objective functions and design constraints

Is optimal point within design space

Yes

No

Figure 3 Flowchart of optimization technique

Table 1 Values of design parameters

Design parameters ValuesL (in) 2401198911015840119888 (psi) 4000119864119862(psi) 38 times 106120588 (lbsft3) 150119891119889119910 (psi) 66000119864119878 (psi) 29 times 1061198750 (psi) 72119905119889 (ms) 607120579119886 1∘120583119886 15119899 50

The ranges of 119887 and ℎ are given as follows in this example

20 le ℎ119887 le 35ℎ le 40 in (22)


Table 2 Optimal design solution

Standard solution Optimal solution119887 (in) 18 125ℎ (in) 30 4001205881 00045 0003181205882 000235 000228120583 90 150120596 0248 0322119872119906 (insdotlbs) 445 times 106 429 times 106119882 41273 34297

Table 3 Energy ratio

Reference [24] [25] [26]Concrete 111MJkg 3180MJm3 13MJkgReinforcing bar 353MJkg 89MJkg 111MJkg119899119864 101 218 272

Table 4 Cost ratio

Reference [27] [28] [6]Concrete 55 mm3 4393 EURm3 6565USDm3Reinforcing bar 05 mkg 53453 EURton 582630USDm3119899119862 709 949 889

The design variables obtained from the standard designapproach solution and the optimal design solution are shownin Table 2

From Table 2 it is found that ℎ and 120583 are larger afteroptimization This is explained that an increase of ℎ and 120583increases the bending resistance capacity of RC beam Byoptimal design W is smaller by 169 If 119882 is taken as thecost it will save 169 of the cost after optimal design whichis very economical The optimal design result is shown inTable 2 which proves the effectiveness of the optimal designmethod mentioned before

32 Example 2 Effect of Section Size (119887 ℎ) on the OptimalDesign Energy ratio and cost ratio vary from country tocountryThen ranges of the energy ratio 119899119864 and the cost ratio119899119862 should be gotten before parameter analysis Consideringthe uncertainties of energy ratio and cost ratio it is reasonableto extend the ranges that are shown in Tables 3 and 4Therefore 60 le 119899119862 le 200 and 10 le 119899119864 le 150 were usedin this study

Set 120579 = 2∘ and 120583119898 le 10 and other design parametersare the same as presented in Example 1 When analyzing theeffect of 119887 on optimal design set ℎ = 20 in and 10 inle ble 20 in(Example 2(a)) When analyzing the effect of ℎ on optimaldesign set b = 15 in and 18 in le h le 25 in (Example 2(b))

In order to compare the optimal results expedientlyoptimal design results for 119887 = 10 and ℎ = 10 were selected asthe reference results Then optimal results were normalizedby (23) where119882119877 represents the reference results Thus the

10 12 14 16 18 20minus10

0

10

20

30

40

b (in)

Embodied energy rationE

Cost rationC

ΔW

()

n = 10

n = 30

n = 60

n = 100

n = 150

n = 180

n = 200

Figure 4 Effect of 119887 on Δ119882 (Example 2(a) ℎ = 20 in 10 in le b le20 in)

18 19 20 21 22 23 24 25minus20

minus10

0

10

20

30

h (in)

ΔW

()

Embodied energy ratio

nE

Cost rationC

n = 10

n = 30

n = 60

n = 100

n = 150

n = 180

n = 200

Figure 5 Effect of ℎ on Δ119882 (Example 2(b) 119887 = 15 in 18 in le h le25 in)

smaller the value of Δ119882 the more effective the optimizationNormalized optimal results are shown in Figures 4 and 5

Δ119882 = 119882 minus119882119877119882119877 times 100 (23)

From Figures 4 and 5 it is observed that the effect ofcross section on optimization effectiveness Δ119882 varies withthe value of 119899 When the value of 119899 is small (119899 = 10) Δ119882decreases with the increase of values of 119887 and ℎ As the valueof 119899 increases effects of 119887 and ℎ change Ranges of 119899119864 and119899119862 are also indicated in Figures 4 and 5 When the differencebetween the value of 119899119864 and the value of 119899119862 is big an increasein costs results in a reduction in embodied energy Howeverwhen the value of 119899119864 is close to the value of 119899119862 the costoptimal results reduce the embodied energy simultaneously

It shall be noticed that two parameters were used as defor-mation constraints 120579 and 120583 Only one parameter reached


10 11 12 13 14 15 16 17 18 19 20b (in)

07

08

09

10

11

120583120583a

120579120579a

Figure 6 Values of 120583120583119886 and 120579120579119886

E

D

A

B

C

y

120583a120579a

Rm

Figure 7 Trends in different range

the allowed value in most cases Take Example 2(a) as anexample the optimal values of 120583120583119886 and 120579120579119886 are shown inFigure 6 When the value of 119887 is small the allowed value120579119886 is reached As the value of 119887 increases the value of 120579decreases while the value of 120583 increases until it reaches theallowed value120583119886 Two rangeswere defined 120579119886 range (inwhich120579 = 120579119886) and 120583119886 range (in which 120583 = 120583119886) Trends are differentin different ranges shown in Figure 7 As the value of 119887increases the resistance changes from curve A to curve ECurve A to curve C is 120579119886 range and curve C to curve E is 120583119886range In 120579119886 range the value of 119877119898 decreases as the value of 119887increases In 120583119886 range the value of 119877119898 increases as the valueof 119887 increases Those show that the optimal results are quitedifferent if different deformation constraints are adopted Inorder to obtain a safe design constraints of 120579 and 120583 should beconsidered simultaneously

4000 5000 6000

Cos

t

Embo

died

ener

gy

n = 10

n = 60

n = 100

n = 150

n = 200minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

ΔW

()

f998400c (psi)

Figure 8 Effect of 1198911015840119888 on cost and embodied energy

60000 65000 70000minus12

minus10

minus8

minus6

minus4

minus2

0

Cos

t

Embo

died

ener

gy

n = 10

n = 60

n = 100

n = 150

n = 200

ΔW

()

fy (psi)


33 Example 3 Effect of Material Strength on the OptimalDesign Values of design parameters were the same as thoseof Example 2(a) Optimal design results when 119899 = 10 werechosen as the reference results Then optimal design resultswere normalized by (23) First an investigation was per-formed to study the effect of concrete compression strength1198911015840119888 on optimal effectiveness Δ119882 (see Figure 8) It is clear tofind that the optimal effectiveness closely relates to 119899 How-ever the relationship between 1198911015840119888 and optimal effectivenessis not clear which is different from the conclusion underconventional load [29]

Then an investigation was performed to investigate theeffects of the yield stress of steel 119891119910 on optimal effectivenessΔ119882 (see Figure 9) As the yield stress of steel increasesΔ119882 decreases regardless of 119899 which means that the optimaldesign is very effective if the yield stress of steel is small


4 Conclusions

The sustainable design and the blast-resistance design werecombined in this paper A general optimization procedureto minimize the cost and the embodied energy of RC rect-angular cross-section beam was present The optimal designwas turned into a nonlinearly constrained optimizationThe optimal design was conducted using Latin hypercubesampling and the sequential quadratic programing (SQP)method Several examples were present to investigate theoptimization effectiveness Several major conclusions aredrawn as follows

(1) The optimal results are different if different deforma-tion constraints are adopted In order to obtain a safedesign constraints of 120579 and 120583 should be consideredsimultaneously

(2) The optimization results are closely related to the costratio 119899119862 and the embodied energy ratio 119899119864 When thevalue of 119899119864 is close to the value of 119899119862 the cost optimalresults reduce the embodied energy simultaneously

(3) The optimal design is more effective if the yieldstress of steel is small In the present study whenthe yield stress of steel is decreased (from 70000 psito 60000 psi) the efficiency of optimal design willsignificantly increase for both cost (from 56 to115) and embodied energy (from 42 to 98)

Competing Interests

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

Acknowledgments

Theauthors acknowledge the financial support fromNationalBasic Research Program of China (973 Program Grant no2015CB058003) National Natural Science Foundation ofChina (Grant nos 51478467 51378016 and 51210012)

References

[1] International Energy AgencyKeyWorld Energy Statistics Inter-national Energy Agency Paris France 2005

[2] Organization for Economic Cooperation and DevelopmentEnvironmentally Sustainable Buildings Challenges and PoliciesOrganization for Economic Cooperation and DevelopmentParis France 2003

[3] World Business Council for Sustainable Development EnergyEfficiency in BuildingsmdashBusiness Realities and OpportunitiesWorld Business Council for Sustainable Development GenevaSwitzerland 2008

[4] I Paya-Zaforteza V Yepes A Hospitaler and F Gonzalez-Vidosa ldquoCO2-optimization of reinforced concrete frames bysimulated annealingrdquo Engineering Structures vol 31 no 7 pp1501ndash1508 2009

[5] D Yeo and R D Gabbai ldquoSustainable design of reinforcedconcrete structures through embodied energy optimizationrdquoEnergy and Buildings vol 43 no 8 pp 2028ndash2033 2011

[6] G F De Medeiros and M Kripka ldquoOptimization of reinforcedconcrete columns according to different environmental impactassessment parametersrdquo Engineering Structures vol 59 pp 185ndash194 2014

[7] V Yepes J VMartı and T Garcıa-Segura ldquoCost and CO2 emis-sion optimization of precast-prestressed concrete U-beam roadbridges by a hybrid glowworm swarm algorithmrdquo Automationin Construction vol 49 pp 123ndash134 2015

[8] C K Chau T M Leung and W Y Ng ldquoA review on life cycleassessment life cycle energy assessment and life cycle carbonemissions assessment on buildingsrdquoApplied Energy vol 143 pp395ndash413 2015

[9] F H Abanda J H M Tah and F K T Cheung ldquoMathematicalmodelling of embodied energy greenhouse gases waste time-cost parameters of building projects a reviewrdquo Building andEnvironment vol 59 pp 23ndash37 2013

[10] H S Park H Lee Y Kim T Hong and SW Choi ldquoEvaluationof the influence of design factors on theCO2 emissions and costsof reinforced concrete columnsrdquo Energy and Buildings vol 82pp 378ndash384 2014

[11] UFC 3-340-02 Structures to Resist the Effects of AccidentalExplosions Unified Facilities Criteria (UFC) Washington DCUSA 2008

[12] D O Dusenberry Handbook for Blast-Resistant Design ofBuildings Wiley Press New York NYUSA 2010

[13] F Fedghouche and B Tiliouine ldquoMinimum cost design of rein-forced concrete T-beams at ultimate loads using Eurocode2rdquoEngineering Structures vol 42 pp 43ndash50 2012

[14] US Army Corps of Engineers Methodology Manual for theSingle-Degree-of Freedom Blast Effects Design Spreadsheets(SBEDS) US Army Corps of EngineersWashington DC USA2008

[15] Protective Design Center Single Degree of Freedom StructuralResponse Limits for Antiterrorism Design (PDC TR-06-08) USArmy Corps of Engineers Protective Design Center OmahaNeb USA 2008

[16] P Olmati F Petrini and K Gkoumas ldquoFragility analysis for thePerformance-Based Design of cladding wall panels subjected toblast loadrdquo Engineering Structures vol 78 pp 112ndash120 2014

[17] W L Bounds and P T King Design of Blast-Resistant Buildingsin Petrochemical Facilities ASCEPublications Reston Va USA1997

[18] C Perea J Alcala V Yepes F Gonzalez-Vidosa and AHospitaler ldquoDesign of reinforced concrete bridge frames byheuristic optimizationrdquo Advances in Engineering Software vol39 no 8 pp 676ndash688 2008

[19] I Paya V Yepes F Gonzalez-Vidosa and A Hospitaler ldquoMulti-objective optimization of concrete frames by simulated anneal-ingrdquo Computer-Aided Civil and Infrastructure Engineering vol23 no 8 pp 596ndash610 2008

[20] A Adamu and B L Karihaloo ldquoMinimum cost design of RCframes using theDCOCmethod Part I columns under uniaxialbending actionsrdquo Structural Optimization vol 10 no 1 pp 16ndash32 1995

[21] M H F M Barros R A F Martins and A F M Barros ldquoCostoptimization of singly and doubly reinforced concrete beamswith EC2-2001rdquo Structural and Multidisciplinary Optimizationvol 30 no 3 pp 236ndash242 2005

[22] K Schittkowski onlinear Programming Codes InformationTests Performance vol 183 of Lecture Notes in Economics andMathematical Systems Springer-Verlag Berlin Germany 1980


[23] M D McKay R J Beckman andW J Conover ldquoA comparisonof three methods for selecting values of input variables in theanalysis of output from a computer coderdquo Technometrics vol21 no 2 pp 239ndash245 1979

[24] G P Hammond and C I Jones ldquoEmbodied energy and carbonin construction materialsrdquo Proceedings of Institution of CivilEngineers Energy vol 161 no 2 pp 87ndash98 2008

[25] CBPR Table of Embodied Energy Coefficients Centre for Build-ing Performance Research (CBPR) Wellington New Zealand2003

[26] O F Kofoworola and S H Gheewala ldquoLife cycle energyassessment of a typical office building in Thailandrdquo Energy andBuildings vol 41 no 10 pp 1076ndash1083 2009

[27] M G Sahab A F Ashour and V V Toropov ldquoCost optimi-sation of reinforced concrete flat slab buildingsrdquo EngineeringStructures vol 27 no 3 pp 313ndash322 2005

[28] C Ji THong andH S Park ldquoComparative analysis of decision-making methods for integrating cost and CO2 emissionmdashfocuson building structural designrdquo Energy and Buildings vol 72 pp186ndash194 2014

[29] D H Yeo and F A Potra ldquoSustainable design of reinforced con-crete structures throughCO2 emission optimizationrdquo Journal ofStructural Engineering vol 141 no 3 2013

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the design to resist the extreme loads was widely acceptedby engineering field An increasing number of buildingswere designed considering the extreme loading conditionsTherefore studies on the sustainable design of RC structuresin extreme loading conditions are necessary

In this study a sustainable design of RC beam underblast loads was discussed through minimizing the embodiedenergy Meanwhile the cost was used as objective functionfor comparison First a general optimal design procedurewas presented Then parametric studies were conducted toanalyze the influences of the material strength and sectionalgeometry on the optimal design The cost and embodiedenergy of particular building materials (reinforcements andconcrete) were taken as variables and their influences onthe optimal design were presented The optimization resultsof cost and the embodied energy were compared and thecorrelation between them was discussed which may behelpful for sustainable deign of RC structural membersagainst blast

2 Methodology

21 Problem Description In this study an interior beamsubjected to air blast load was considered This problem wasan illustrative example of UFC 3-340-02 [11] Both ends ofthe beam were fixed The beam was assumed to deform ina flexural shape The cross section is shown in Figure 1 Auniform blast load is applied on the upper surface of thebeam The beam was designed according to the restrictionsand guidelines found in the UFC 3-340-02 code

There are two response indices that are usually used todefine the damage levels of blast-loaded structural compo-nents in a flexural response regime the support rotation angle(120579) and the ductility ratio (120583) [12] In the present study bothof the two indices were used as design constraints

Four design parameterswere considered in this studyThefour parameters were the height of the beam h the width bthe ratio of longitudinal reinforcement 1205881 and the ratio ofstirrup 120588222 Objective Functions The total cost and the total embod-ied energy were taken as objective function to be minimizedThe total cost is defined as [13]

1198621015840 = 119862119862119881119862 + 119862119878 (1198811198781 + 1198811198782) (1)

where 1198621015840 is the total cost of beam 119862119862 is the cost of concreteper unit volume 119881119862 is the volume of concrete 119862119878 is thecost of reinforcing steel per unit volume and 1198811198781 and 1198811198782are the volume of longitudinal reinforcement and shearreinforcement respectively

It should be noted that the optimal values were affectedby the relative values of the objective function only Equation(1) divided by 119862119878 is

11986210158401015840 = 119881119862119899119862 + (1198811198781 + 1198811198782) (2)

where 119899119862 = 119862119904119862119888 11986210158401015840 = 1198621015840119862119878

8

84

Unit in

120

b

h

Figure 1 Cross section

The value of 119899119862 varies from country to country thus (2)was more convenient to study the minimum cost of RC beamthan (1) [13]

Similarly to the cost objective function the embodiedenergy objective function is written as follows

11986410158401015840 = 119881119862119899119864 + (1198811198781 + 1198811198782) (3)

where 11986410158401015840 = 1198641015840119864119878 1198641015840 is the total energy 119864119878 is the embodiedenergy of steel per unit volume 119899119864 = 119864119878119864119862 is the energyratio and 119864119862 is the embodied energy of concrete per unitvolume

Comparing (2) with (3) a similarity can be found Theform of the energy objective function and the cost objectivefunction is very similarThus the cost objective function andthe energy objective function are represented by (4) for theconvenience of optimization calculation

119882 = 119881119862119899 + (1198811198781 + 1198811198782) (4)

where 119899 denotes 119899119862 and 119899119864 and119882 denotes 11986210158401015840 and 11986410158401015840In the following analysis the optimal designs for the cost11986210158401015840 and the embodied energy 11986410158401015840 were unified to the optimal

design for119882

23 Design Constraints The RC beam under blast loadis simplified to SDOF system (see Figure 2) The designconstraints were given in mathematical expressions and theoptimal design was transform into a constrained optimiza-tion problem

The design constraints were defined in accordance withthe UFC 3-340-02 code [11] The constraints for RC rectan-gular section beam under blast load were expressed

(1) Performance limits are as follows

120579 le 120579119886120583 le 120583119886 (5)


K

Me


y

Rm

ye


P0

td

(c) Blast load



ge max(3radic1198911015840119888119891119910 200119891119910 )1205882 ge 00015

(6)


0181198911015840dc119887119889 le 1198771198982 119897 (7)



19 (1198911015840dc)05 + 25001205881 le 35 (1198911015840dc)05 (9)





(1) 1205882 1205882 was computed by

1205882 = max (V119906 minus V119888 V119888)085119891dy (10)


V119888 = 19 (1198911015840119888 )05 + 25001205881 (11)

V119906 = (1198972 minus 119889) 119877119898119887119889 (12)

(2) 120579 120579 is defined in

120579 = arctan(2119910119898119897 ) (13)


119910119890 = 119877119898119870 (14)


119870 = 3071198641198621198681198861198974 (15)


119868119886 = 05 (119887ℎ312 + 1198661198871198893) (16)


119866 = (3320312058831 minus 1819812058821 + 586241205881) ( 1198641198787119864119862)07 (17)




119877119898 = 8 (119872119873 +119872119875)1198972 (18)



119872119873 = 119872119875 = 1205881119891dy1198871198892 (1 minus 1205881119891dy1712058811198911015840119888 ) (19)

(4) 120583 120583 is calculated by [17]



120596 = radic 119870(119870LM119898) (21)







Start


Optimal design

End


Set design space




Yes

No





20 le ℎ119887 le 35ℎ le 40 in (22)






Table 4 Cost ratio







10 12 14 16 18 20minus10

0

10

20

30

40

b (in)


Cost rationC

ΔW

()

n = 10

n = 30

n = 60

n = 100

n = 150

n = 180

n = 200


18 19 20 21 22 23 24 25minus20

minus10

0

10

20

30

h (in)

ΔW

()


nE

Cost rationC

n = 10

n = 30

n = 60

n = 100

n = 150

n = 180

n = 200



Δ119882 = 119882 minus119882119877119882119877 times 100 (23)




10 11 12 13 14 15 16 17 18 19 20b (in)

07

08

09

10

11

120583120583a

120579120579a


E

D

A

B

C

y

120583a120579a

Rm



4000 5000 6000

Cos

t

Embo

died

ener

gy

n = 10

n = 60

n = 100

n = 150

n = 200minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

ΔW

()

f998400c (psi)


60000 65000 70000minus12

minus10

minus8

minus6

minus4

minus2

0

Cos

t

Embo

died

ener

gy

n = 10

n = 60

n = 100

n = 150

n = 200

ΔW

()

fy (psi)





4 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










K

Me


y

Rm

ye


P0

td

(c) Blast load



ge max(3radic1198911015840119888119891119910 200119891119910 )1205882 ge 00015

(6)


0181198911015840dc119887119889 le 1198771198982 119897 (7)



19 (1198911015840dc)05 + 25001205881 le 35 (1198911015840dc)05 (9)





(1) 1205882 1205882 was computed by

1205882 = max (V119906 minus V119888 V119888)085119891dy (10)


V119888 = 19 (1198911015840119888 )05 + 25001205881 (11)

V119906 = (1198972 minus 119889) 119877119898119887119889 (12)

(2) 120579 120579 is defined in

120579 = arctan(2119910119898119897 ) (13)


119910119890 = 119877119898119870 (14)


119870 = 3071198641198621198681198861198974 (15)


119868119886 = 05 (119887ℎ312 + 1198661198871198893) (16)


119866 = (3320312058831 minus 1819812058821 + 586241205881) ( 1198641198787119864119862)07 (17)




119877119898 = 8 (119872119873 +119872119875)1198972 (18)



119872119873 = 119872119875 = 1205881119891dy1198871198892 (1 minus 1205881119891dy1712058811198911015840119888 ) (19)

(4) 120583 120583 is calculated by [17]



120596 = radic 119870(119870LM119898) (21)







Start


Optimal design

End


Set design space




Yes

No





20 le ℎ119887 le 35ℎ le 40 in (22)






Table 4 Cost ratio







10 12 14 16 18 20minus10

0

10

20

30

40

b (in)


Cost rationC

ΔW

()

n = 10

n = 30

n = 60

n = 100

n = 150

n = 180

n = 200


18 19 20 21 22 23 24 25minus20

minus10

0

10

20

30

h (in)

ΔW

()


nE

Cost rationC

n = 10

n = 30

n = 60

n = 100

n = 150

n = 180

n = 200



Δ119882 = 119882 minus119882119877119882119877 times 100 (23)




10 11 12 13 14 15 16 17 18 19 20b (in)

07

08

09

10

11

120583120583a

120579120579a


E

D

A

B

C

y

120583a120579a

Rm



4000 5000 6000

Cos

t

Embo

died

ener

gy

n = 10

n = 60

n = 100

n = 150

n = 200minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

ΔW

()

f998400c (psi)


60000 65000 70000minus12

minus10

minus8

minus6

minus4

minus2

0

Cos

t

Embo

died

ener

gy

n = 10

n = 60

n = 100

n = 150

n = 200

ΔW

()

fy (psi)





4 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877119898 = 8 (119872119873 +119872119875)1198972 (18)



119872119873 = 119872119875 = 1205881119891dy1198871198892 (1 minus 1205881119891dy1712058811198911015840119888 ) (19)

(4) 120583 120583 is calculated by [17]



120596 = radic 119870(119870LM119898) (21)







Start


Optimal design

End


Set design space




Yes

No





20 le ℎ119887 le 35ℎ le 40 in (22)






Table 4 Cost ratio







10 12 14 16 18 20minus10

0

10

20

30

40

b (in)


Cost rationC

ΔW

()

n = 10

n = 30

n = 60

n = 100

n = 150

n = 180

n = 200


18 19 20 21 22 23 24 25minus20

minus10

0

10

20

30

h (in)

ΔW

()


nE

Cost rationC

n = 10

n = 30

n = 60

n = 100

n = 150

n = 180

n = 200



Δ119882 = 119882 minus119882119877119882119877 times 100 (23)




10 11 12 13 14 15 16 17 18 19 20b (in)

07

08

09

10

11

120583120583a

120579120579a


E

D

A

B

C

y

120583a120579a

Rm



4000 5000 6000

Cos

t

Embo

died

ener

gy

n = 10

n = 60

n = 100

n = 150

n = 200minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

ΔW

()

f998400c (psi)


60000 65000 70000minus12

minus10

minus8

minus6

minus4

minus2

0

Cos

t

Embo

died

ener

gy

n = 10

n = 60

n = 100

n = 150

n = 200

ΔW

()

fy (psi)





4 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Table 4 Cost ratio







10 12 14 16 18 20minus10

0

10

20

30

40

b (in)


Cost rationC

ΔW

()

n = 10

n = 30

n = 60

n = 100

n = 150

n = 180

n = 200


18 19 20 21 22 23 24 25minus20

minus10

0

10

20

30

h (in)

ΔW

()


nE

Cost rationC

n = 10

n = 30

n = 60

n = 100

n = 150

n = 180

n = 200



Δ119882 = 119882 minus119882119877119882119877 times 100 (23)




10 11 12 13 14 15 16 17 18 19 20b (in)

07

08

09

10

11

120583120583a

120579120579a


E

D

A

B

C

y

120583a120579a

Rm



4000 5000 6000

Cos

t

Embo

died

ener

gy

n = 10

n = 60

n = 100

n = 150

n = 200minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

ΔW

()

f998400c (psi)


60000 65000 70000minus12

minus10

minus8

minus6

minus4

minus2

0

Cos

t

Embo

died

ener

gy

n = 10

n = 60

n = 100

n = 150

n = 200

ΔW

()

fy (psi)





4 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10 11 12 13 14 15 16 17 18 19 20b (in)

07

08

09

10

11

120583120583a

120579120579a


E

D

A

B

C

y

120583a120579a

Rm



4000 5000 6000

Cos

t

Embo

died

ener

gy

n = 10

n = 60

n = 100

n = 150

n = 200minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

ΔW

()

f998400c (psi)


60000 65000 70000minus12

minus10

minus8

minus6

minus4

minus2

0

Cos

t

Embo

died

ener

gy

n = 10

n = 60

n = 100

n = 150

n = 200

ΔW

()

fy (psi)





4 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Embodied Energy and Cost Optimization of RC Beam under