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INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATIONOptimum Shape in Brick Masonry ArchesUnder Static And Dynamic LoadsKAVEH KUMARCI, ARASH ZIAIE, MEHRAN KOOHIKAMALI, ARASH KYIOUMARSIAbstract - The objective of this study is to determine brickmasonry arches under dynamic and static loads. In this paper,considerable attention is given to arches, their importance, modelingstages, dynamic analysis, static analysis and arch optimization usingANSYS11 software. A multiple stage analysis framework wasconducted for semicircular arch:1- The study of optimum shape for semicircular arch on thebase of minimize of arch weight.2- Determination of linear and nonlinear analysis limits byincrease of density.3- The study of optimum shape in semicircular arch by linearand nonlinear analysis.All of these stages have been conducted for obtuse angel arches,four- centered pointed arch, tudor arch, ogee arch, equilateral arch,catenary arch, lancet arch, four-centered arch (normal, diminishedand steep). The main purpose has been study of arch optimum shapefor minimize of weight: Finally, according to the results, the optimumshape in arches under dynamic load has been determined.II. MODELING, ANALYSIS AND OPTIMIZATION OFARCH SHAPEArch modeling has been conducted by ANSYS11 software. Alsodynamic analysis has been conducted by north-south horizontalaccelerations of Elcentro earthquake in 1940.In this earthquake thetime, maximum acceleration, maximum velocity and maximumdisplacement were 31.98 sec, 0.31g, 33 cm/sec and 21.4cm,respectively. The element which used in this analysis was SOLID 65.Arch shape optimization emphasized on the minimizing of archweight. So, the base and top thickness, maximum tensile stress andweight of structure have been defined as design variable, statevariable and objective function, respectively Optimization has beenconducted in Design Optimum Processing. [5,6,8,10]A. Geometrical Modeling:According to optimization of design variables, such as basethickness (t0) and top thickness (t1) as parameters, all of key pointsare defined as follow. [9]In order to study of this material, semicircular arch is defined by keypoints as parameters (fig.I).Point 1: (0, 0)Point (2): (R, 0) Point3: (-R, 0) Pint4: (0, R)Point 5(R t0, 0) Point6: (-R-t0, 0)Point 7: (0, R t1)Keywords- optimum shape- arch- dynamic load- linear and nonlinear analysis- tensile stressI. INTRODUCTIONBEFORE, arch was defined as a part of circle or bow. If we wantto define it, we can say it is a curve surface for covering, that it’sspan is higher than it’s depth .Overall, arches are classified to threegroups:1- circular arches and similar to that2- obtuse angle arches3- decorative archesTime dynamic analysis is an analytical method to determineresponses in each time section, especially for earthquake that astructure is under accelerations of earth motion (accelerograph) in thebase level. In this model, structure dynamic response is function oftime and calculated by number integral in equation of structuremotion. [1,10,14,15, 16]fig. 1: semicircular archfigI: semicircular archManuscript received Febr.3, 2008: Revised Received June 12, 2008. This work issupport in part by Sama organization, affiliated with Islamic Azad University,Shahrekord branch.Kaveh Kumarci is with Sama Organization (affiliated with Islamic AzadUniversity) , Shahr-e-kord branch e-mail: Kumarci kaveh@yahoo.comfax:00983812226183 ; Tel:00989133811911Arash Ziaie is with civil engineer department university of bahonar kerman e-mail:Ziaie111@yahoo.comMehran Koohikamali is with Sama organization (afiiliated with Islamic AzadUniversity ), Sahr-e-kord branch e-mail: Mehran koohikamali@yahoo.comArash Kyiomarsi is with Electic Engeenering Department, university of isfahane-mail: Kiumarsi Ar@yahoo.comIssue 2, Volume 2, 2008In arch modeling, the tolerance increases because the thicknessdecreases from base to top. We should remember that in modeledarch, the thickness decrease from base (t0) to top (t1) linearly. Also,arch thickness in direction of length axis is 20 cm. The motion ofsupport nodes is zero, and dynamic force has no effect on them. Also,brick masonry is made by brick and mortar as homogenous material(table I). The efficient factors in inelastic nonlinear analysis show in(table II). [4,7,12]171
INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATIONTable I: Brick masonry specificationdensity( ρ )Elastic moduluskg1460 [2]m3NmSpan Length85 10 [3]2Allowable tension stress(ft) NTableIV: specification of optimum shape for semicircular archwith various spans under static load.50.5 10 [2,3,4]m2Poisson ratio ( υ 82.0909W /H445.684.80645032650982528155110048430(σ t )max N / mTable II: Effective coefficient in non elastic and nonlinear2analysismotion coefficient for open crack0.1 [5]motion coefficient for close crack0.9 [5]allowable tension stressNallowable compressive stress Nm2m2(ft)5 10 [2,3,4]45 10 [2,3,4]5(fc)Fig II: semicircular arch modeling by ansysIII. EVALUATION OF OPTIMUM SHAPE INSEMICIRCULAR ARCHIV. EVALUATION OF DIFFERENT ARCH ANDTHEIR OPTIMUM SHAPEThe analysis conducted for semicircular arch in fivespans: 4,5,6,7 and meters (TableIII,Table IV,Fig II).Here, in addition to semicircular arch, the obtuse angel, four centeredpointed, tudor ogee arch, equilateral catenary, four centered, lancedarches have been studied. Analyzed arches were studied in threespans: 4, 5 and 6 meters. In each span, dynamic force, maximumtension stress, arch optimum dimensions and stability factor arecalculated. Also, Obtus angel, four centered pointed tudor and ogeearch, arches have been analyzed in 3 levels: normal, diminished andsteep (Table V-XI, Fig III-XI ). [1,2,3,8,9]TableIII: specification of optimum shape for semicircular archwith various spans under dynamic 4052t1/R.1381.1127.099.091.0909W /H.4347.9175.68.435.80645098248072528155160048430N / m2(σ t )maxFig III: Catenary arch modeling by ansysIssue 2, Volume 2, 2008172
INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATIONTable V: Comparison of optimum arches under dynamic loadW /HL(m)t (m)t (m)K0(σt 39.28849.25002.5447095Lancet .3931.3143.22561.784890551.272561.2126Fig IV: Lancet arch modeling by ansysTable VII: Comparison of optimum arches underdynamic loadt0(m)t1(m)KW /H(σt 984Equilateral.8969normalCatenary archOgee 878Table VI: Comparison of optimum arches underdynamic load45363(σt )mat0(m)t1(m)KW .2056.74653990L(m)diminished6normalTudor arch456steep6arch.32669.2546archsteep.32409Fig V: Obtuse angel arch modeling by ansys8Issue 2, Volume 2, 2008173
INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATIONFig VI: Tudor arch modeling by ansysFig VII: Catenary arch modeling by ansysFig IX: Fourcentered arch modeling by ansysFig VIII: equilateral arch modeling by ansysFig X: Four centered pointed arch modeling by ansysIssue 2, Volume 2, 2008Fig XI: Ogee arch modeling by ansys174
INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATIONTable VIII: Comparison of optimum arches (dynamic load)(σt )maxL(m)t0(m)t1(m)KW /HTable X: Comparison of optimum arches(static 98992.37287.3765.0149506Table IX: Comparison of optimum arches(static load)diminishednormal(σ t 122.4501016.69.33.47.23.112.451211Table XI: Comparison of optimum arches(static load)L((σt )maW /H(σt 0/Rt1/RW 02565.8038.276.34.32.14.549400.81272878In this part, linear and nonlinear analysis ofsemicircular arches with span of 5m and obtuse angle arch .225span of 4 m has been studied. Also, the density is applied toevaluation of linear and nonlinear analysis. This was also noticed thatin which limits the maximum tension stress (the arch optimizationfactor) can change (table XII). steepW /Ht0(m)t0(m)L(m)Obtuse angel archnormal51732steepsteep.428four centred pointed arch(static analysis)normalFour centered pointed arch.3equilateral archdiminishedObtuse angel arch4.3diminishedL(m)156Issue 2, Volume 2, 2008V. DETERMINATION OF LIMITS IN LINEAR AND NONLINEAR ANALYSIS BY INCREASE OF DENSITYB.A. Evaluation And Comparison Of Linear And NonlinearLimits In Semi Circular And Obtuse Angel Arches ByDensity Factor175
INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATIONTable XII: Comparison between linear and nonlinear limits bydensity factor(dynamic load)ρ 1460 kg / m 3ρ1.5 ρ2ρ3ρTable XIII: Comparison of optimum shape in semicircular andObtus angle arches with of 4m spans by linear and nonlinearanalysis (dynamic 1.3324ρNon Linear176.1798ρNon LinearObtuse angel archIssue 2, Volume 2, 2008.1854.96941.609B.B. Evaluation And Comparison Of Optimum Shape InSemicircular And Obtus Angle Arch By Linear And NonLinear AnalysisThe optimum shape of semicircular arch and obtus arch withspans of 4m have been calculated by linear and nonlinear analysisand density of 4 ρ .Then the results compared to the optimum shapeof semicircular and obtus by linear analysis and density of ρ(TableXIII). [8,12,16].1798ρ183337According to results of test and error (table 2), if density ishigher than 4 ρ , the response of linear and nonlinear stress isdifferent. So for linear analysis, increase of density to 4 ρ isineffective. [6,9,10].21681.541Linear211944.33444ρNon nalysis24830760169.2763ρLinear Analysis26731794944.3317.8328(σ t )maxNon Linear AnalysisObtus angel archLinear Analysis856833148307Semicircular arch225149.2763ρNon 8307analysis(σ t )maxNon Linear AnalysisSemicircular archLinear Analysis212921Kind of
INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATIONContinue of Table XIV: Comparison of optimum shape insemicircular and Obtus angle arches with of 4m spans bylinear and nonlinear analysis. (dynamic load)HW /H(σ t )max1057.8.4347LinearNon Linear20760748891153890911800006550468max(N / m s the results show(table 10-3), for densitis which are higher than 1/6, the linear and nonlinear stresses are diffrent to each other. Also, inanalysis of semicircular arches , the place of maximum tensile stressis around of inner shield, near base of arch and in the middle of archlenghth.Also,maximum compressive stress is around of outter shieldnear base of arch(figXII).[11,12,14]FigXII:semicircular arch with 4m spam and the place ofstresses( N/ m2 )5781Non LinearVII. ESTIMATION OF BASE THRUST FORCE INX DIRECTIONLinearρNon Linear411884889112Analysisρ1188207607(σ )tAnalysisObtuse angel arch4LinearρNon .2Analysis4AnalysisSemicircular archρρ50982Linear917.2The results are as below:Table XV: the results of study of linear and nonlinear analysis bydensity. (dynamic load)1.6 1460 kg / m 3ρ2ρ1.5 ρ1.7 ρanalysisWρLinearAnalysisρKind ofanalysisanalysisdensityVI. THE SYUDY AND COMPARISION LINEAR ANDNONLINEAR ANALYSIS OF SEMICIRCULAR VAULTSWITH SPAN OF 5M BY DENSITY64835012.57652853According to this point that W / H (the weight of half of arch tothrust force in one side) is a main criteria in arch resistence, the wayof thrust force estimation is very important. Because of in modelling,we suppose that all of supports are restrained, so all of joints inY 0 has a horizontal force that its source is earthquake force thatIssue 2, Volume 2, 20085221.6253541is stimated byReaction Solutionprocessor and estimated inANSYS software. For example, for estimation of thrust force forhalf of arch span (radius 2m), is shown in( fig.XII ). [5,8,15]177
INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATION[2] The results of Kharagan tower and Emamzadeh Mansoor brickstests in Qazvin, Soil technical and mechanical laboratory inQazvin, No.002/21/002 2[3] Bsthe,K,J., Common Rules for Reinforced and UnreinforcedMasonry Structures, Part 1, Design of Masonry Structures,Eurocode 6,1996.[4]Drysdale,R.G., Hamid,A.A., Baker,L.R., , Masonry StructuresBehavior and Design, Prentice Hall, New Jersey, USA,1994[5]General Rules, Seismic Actions and Rules for Buildings, Design ofStructures for Earthquake Resistance, Part 1, Draft No.6,Eurocode 8, 2003.[6]Bsthe,K,J., The element procedures in engineering analysis,Prentice -Hall, Englewood Cliffs, 1982.[7]Hughes,T.J.R., The finite element method linear static anddynamic finite element analysis, Prentice-hall, Inc, EnglewoodCliffs,NJ,1987.[8] Rao.SS., Optimization and applications,Wiley Eastern, NewDelhi, 1984.[9] Gratte,N.Vanderplaates, Numerical optimization techniques forengineering design, McGraw-Hill publishing CO.[10]Hill,R.The Mathematical Theory of Plasticity, OxfordUniversity,1983.[11] K.V.Mital, Optimization Methods in Operations Research andSystems Analysis,2th edition, Wiley Eastern Limited, NewDelhi,1983.[12] R.K. Livesley, Linear programming in struturl analysis anddesign, Gallagher and O.C.Zienkiewicz, 1973.[13] U.Kirsch, Optimal Structural Design, New York, McGraw-HillBook Company, 1981.[14] Garbriele Milani, A Simple Equilibrated HomogenizationModel For The Limit Analysis of Masonry Structures, university offerrara (Italy), WSEAS transactions on applied and theoreticalmechanics, 2007.[15] Javad Marzbanrad, Thickness and Mterial Yield Strength Effectsof Thin Sheets on Dent Resistance ,university of science andtechnology ,Iran , WSEAS transactions on applied and theoreticalmechanics , 2007.[16] L.Pomante, P.di flice, Ad-hoc HW/SW Architecture for DBMSs:a Co-Design Approach, the 6th WSEAS int.conf.on artificialintelligence , knowledge engineering and data basis , corfu island,Greece , 2007FigXIII-a:estimation of thrust force at x directionFigXIII-b:estimation of thrust force at x directionVIII.CONCLUSIONConsidering to optimum shape in arches under dynamic load,several conclusions can be surmised from the results as follow:1-With increase of masonry density, the difference betweenmaximum tensile stress in linear and nonlinear analysis reveals. Itmeans that the increase of density to 4 ρ for linear and non linearanalysis is ineffective.2- The limit for increase of base thickness in linear and nonlinearanalysis for 4 ρ : ρ is 36 to 93%.3- The limit for increase of top thickness in linear and nonlinearanalysis for 4 ρ : ρ is 66 to 116%.First Author; Kaveh kumarci, is born in shahrekord-iran.Hestudied in the fild of civil engineer (structure) and got his certificationfrom Najaf Abad Azad university –iran He is full time member , Theassistence, The Manager of Searchin of Sama Technical andProfessional Organization in Shahrekord . Now. Mr.Kumarci ismember of Iran Engineering Diciplinary.Second Author; Arash Ziaie, is born in Isfahan, Iran in 1969 andgot his M.S. degree in civil engineering in 1994. He studied in thefild of civil engineering (structure) and got his certification fromShahid Bahonar Kerman university of Iran He is full time memberand the assistence of Kerman university. Mr. .Ziaie is a member ofIran Engineering Diciplinary now.Third Author; Mehran Koohikamali, is born in shahrekord-iran.He studied in the fild of civil engineering (structure). He is a memberof Sama Technical and Professional Organization in Shahrekord now.Mr. Koohikamali is a member of Iran Engineering Diciplinary.Fourth Author; Arash Kiyoumarsi, is born in shahrekord, iran in1972.He studied in the fild of electric engineering and got his Ph.D.degree from Isfahan university of Thecnology in 2001 in Iran. Thenhe continued his post doctoral studies in the same field in Germany.He is full time member, and the Professor of Isfahan university now.4-Increase of ϖ / H in linear and nonlinear analysis for 4 ρ : ρis 12%.5- Increase of arch base thickness in nonlinear analysis of 4 ρ tolinear analysis of 4 ρ is 21%.6- Increase of arch top thickness in linear analysis of 4 ρ to linearanalysis of 4 ρ is 30%.REFERENCES[1] Choopra, Anil, translated by Shapoor Tahooni, The structuraldynamics and determination of earthquake loadIssue 2, Volume 2, 2008178
1- The study of optimum shape for semicircular arch on the base of minimize of arch weight. 2- Determination of linear and nonlinear analysis limits by increase of density. 3- The study of optimum shape in semicircular arch by linear and nonlinear analysis. All of these stages have been conducted for obtuse angel arches,
