Curved steel buildings are frequently designed to supply the users of the structure with
ordinary light with a sense of capaciousness as well as grandness in public facilities such as
stations, buying malls, leisure centres and airports.
This paper presents a method for analysis and optimum design of 2D and 3D curved roof trusses
subjected to static loading and specified set of constraints. Here the optimization refers to
minimization of total weight of curved roof structures such that they can resist applied forces
(stress constraint) and don’t exceed certain deformations (displacement constraints). The finite
element formulations is developed and implemented for the static analysis of curved roof trusses
to determine the stresses and displacements.
The use of a reliable and competitive procedure for finding the optimum solutions for problems
involving continuous design variables based on genetic algorithms is demonstrated and used in
this study .The performance of genetic algorithms is affected by various factors such as
coefficients and constants, genetic operators, parameters and some strategies. Member grouping
and initial population strategies are also important factors.
Optimization is an automated design procedure in which the computers are utilized to obtain the
best results. The numerical methods of structural optimization with applications of computers
automatically generate a near optimal design (converge to solve) in interactive manner. A
program was modified and used to automate analysis and optimization of the structure written in
FORTRAN language based Finite Element analysis and Genetic Algorithm optimization
technique. The developed method is tested on several examples and compared with previous researches or SAP2000 results. It is concluded that this method can serve as a useful tool in
engineering design and optimization of curved roofs

Hussein, Ghaedan

Saber, Galawezh

Tayşi, Nildem

Epoka University

12nd International Balkans Conference on Challenges of Civil Engineering, BCCCE, 23-25 May 2013, Epoka University, Tirana, Albania.Analysis and Optimum Design of Curved Roof StructuresGalawezh Saber1, Nildem Tayşi2, Ghaedan Hussein31 Department of Civil Engineering, Gaziantep University, Turkey2 Department of Civil Engineering, Gaziantep University, Turkey3Department of Civil Engineering, Gaziantep University, TurkeyABSTRACTCurved steel buildings are frequently designed to supply the users of the structure withordinary light with a sense of capaciousness as well as grandness in public facilities such asstations, buying malls, leisure centres and airports.This paper presents a method for analysis and optimum design of 2D and 3D curved roof trussessubjected to static loading and specified set of constraints. Here the optimization refers tominimization of total weight of curved roof structures such that they can resist applied forces(stress constraint) and don’t exceed certain deformations (displacement constraints). The finiteelement formulations is developed and implemented for the static analysis of curved roof trussesto determine the stresses and displacements.The use of a reliable and competitive procedure for finding the optimum solutions for problemsinvolving continuous design variables based on genetic algorithms is demonstrated and used inthis study .The performance of genetic algorithms is affected by various factors such ascoefficients and constants, genetic operators, parameters and some strategies. Member groupingand initial population strategies are also important factors.Optimization is an automated design procedure in which the computers are utilized to obtain thebest results. The numerical methods of structural optimization with applications of computersautomatically generate a near optimal design (converge to solve) in interactive manner. Aprogram was modified and used to automate analysis and optimization of the structure written inFORTRAN language based Finite Element analysis and Genetic Algorithm optimizationtechnique. The developed method is tested on several examples and compared with previous12nd International Balkans Conference on Challenges of Civil Engineering, BCCCE, 23-25 May 2013, Epoka University, Tirana, Albania.researches or SAP2000 results. It is concluded that this method can serve as a useful tool inengineering design and optimization of curved roofs.Keywords: curved roof, static analysis, optimum weight, genetic algorithm finite elementmethod.INTRODUCTIONWide-span space structures have been more and more popular in covering large open areaswith few intermediate supports. Successful arched structure applications exist all complete theworld exhibition canters, bridges, public halls, covering stadiums, and other buildings.Curved roofs of course have a number of important benefits, as well as they can be asuperlative and long-lasting choice. That is where the attractiveness and price of the curved roofcome into play. Balanced to the standard flat roof instatement, an arched roof can be far moredurable, providing for superior charge for you.In structural design, it is needed to obtain a suitable form in a structure so that it can carry therequired loads safely and profitably. Traditional approaches to the job of discovery such shapesfor structures have been using experimental models or by intuition with experience. Arch'sstructures supply inexpensive results for crossing great spans with bear higher loads for apresumption volume of material when correctly shaped, balanced with beams shorter cross partscan be used in arches, like the membrane forces are dominant, [1].It is widely impossible to get analytical mathematical results, for problems requiring complexgeometries, loadings, plus material properties; analytical solutions are those presumptions by anarithmetical representation that yields the rates of the wanted unknown amounts at any positionin an individual with are thus well-founded for an infinite number of positions in the body.These analytical solutions widely need the solution of regular or partial differential equationsthat, sense of the difficult material properties, geometries, loadings, and are not generallyavailable. For this reason, we require to depend on numeric approaches, such like the FEapproach, for correct solutions. The FE formulation of the problem results in a technique ofsimultaneous algebraic equations for a result, rather than needing the solution of differentialequations. Briefly, the solution to constructional problems normally relates to determining thedisplacements at each intersection also the stresses inside each member construction up to thebuilding that is subjected to apply loads Logan, in 2007 [2].Although the subject of truss and arch roof analysis with optimization had been conversedfrequently complete current years, this topic was contained to show the validity of an analysisprogram, which is used in the GA optimization program. Although some structural optimizationmethods can deal with discontinuous search spaces, they support a native lack of generality andhence, can not be readily extended to different types of structures. The GA, for its part, is aproblem independent.12nd International Balkans Conference on Challenges of Civil Engineering, BCCCE, 23-25 May 2013, Epoka University, Tirana, Albania.The principal characteristics of a GA are established on the principles of endurance of the fittestwith adaptation. Since its establishment liked an intuitive idea, [3].Many inventors have explored the applicability of GA as well as advanced many applicablesupplements such as elitist. GA. Gero, et al[ 4], improved augmented Lagrangian GA, Adeli, H.and Cheng, N.T, [5], hybrid algorithms of GA with fuzzy system .Tan, L.P.et al [6] and withneural network, Grierson, D.E. and Hajela, P, [7].Various scientists have tried to solve the archproblem by different methods. It seems that FE has been the major tool in this research.2 Matrix analysis of trussesStatic analysis of trusses can be carried out accurately; also the equations of evencomplex trusses can be collected in a matrix shape amenable to numeric solution. Thisapproximation, now and then named “matrix analysis,” provided the basis of early FEadvancement.By considering the stiﬀness of each truss element one at a time matrix analysis of trusses acts,and after that applying these stiﬀnesses by the displacements of the joints, generally named“nodes” in FE to determine the forces that are set up in the truss. Afterwards noting that the forcethat is externally contributed by each element to a node must equal the sum of force that isapplied to that node, we can assemble a sequence. Of linear algebraical equations in which theapplied nodal forces are known amounts, also the nodal displacements are the unknowns. Theseequations are comfortably written in matrix shape, which gives the system its name.321321333231232221131211FFFdddkkkkkkkkk(1)Here Fi and dj indicate the force at the ith node and the deﬂection at the jth node (these wouldactually be vector quantities, with subcomponents along each coordinate axis). The Kij is globalstiﬀness matrix, with the ij component. The matrix equations can be abbreviated asKij dj = Fi (2)3 Analysis and optimum design examples12nd International Balkans Conference on Challenges of Civil Engineering, BCCCE, 23-25 May 2013, Epoka University, Tirana, Albania.3.1 Static analysis of 2D and 3D truss.Analysis is done by the FE method coded program for analysis of 2 and 3D curved truss roofing,results are compared with source program (SAP2000).3.1.1 2D curved truss with 7.32 m heightThis example consists of curved truss with 50.8 m span length with 41 elements as shown inFigure 1. The geometry of this example is taken from [8]. The structure is loaded with a pointload of 222.41 KN on all upper joints in the Z direction. The members of the structure aredivided into 3 groups; first group from element (1-22) and (40, 41), second group from element(26-36) and third group from element (23-25) and (37-39), each group have the same crosssectional area and one design variable. Cross-sectional areas for first group A1=0.04877m2,second group A2=0.009484 m2 and third group A3= 0.01290 m2. The objective function is theweight (or volume) minimized. Maximum tensile stress Mpa895.137t  , maximumcompressive tress Mpa421.103c  and maximum xu and zu displacement all nodes being0.05 m. Material properties are: Young’s modulus, pa199947.96 and material density, 7697 kg/m3.Figure1.  2D curved truss with 7.32 m heightDiscussion of the results: Analysis is done by the FE method coded program and sourceprogram (SAP2000). Table1. Maximum tension stress occurs in elements (16, 26) and maximumcompression stress occurs in elements (1, 10). Table 2 is the result of maximum displacementbefore and after optimization. Maximum displacement occurs in joints (17, 6). The closeagreements between results are seen The GA optimization satisfied displacement constraints.From the results demonstrated in Table 3 it can be observed that the optimum values found forthe final weights. The results got applying the GA for continuous design variable quantities.After 65 iterations, minimum weight design was obtained for continuous design variables. Theweight of the truss is reduced from 521,195 to 238.962 (54 %reduction) for continuous designvariables.Table 1 Comparison of stress for 41 bar 2D curved truss with 7.32 m heightStress (Mpa))Frame NO. Sap2000 Present work1 -44.2000 -43.436912nd International Balkans Conference on Challenges of Civil Engineering, BCCCE, 23-25 May 2013, Epoka University, Tirana, Albania.10 -44.2000 -43.436916 43.3307 43.330726 43.3307 43.3307Table 2 Displacement result before and after optimizationDisplacement (m)Jointno.X-directionU3Y-directionU3BeforeoptimizationAfteroptimizationBeforeoptimizationAfter optimization6 0.0000 0.0000 -0.0200 -0.046917 0.0000 0.0000 -0.0202 -0.0475Table3 Initial and optimum design variables of 2D curved truss with 7.32 m heightDesignvariablesCrosses sectional area (m2) Max. Stress(Mpa)Max. disp.(m)Initial OptimumS10.04877 0.02236103.5695-0.0475S20.009484 0.00408S30.01290 0.006452Weight (kN) 521.195 238.962P.R 54 %3.1.2 3D domeThis example consists of space dome truss with 120-bars and 49 joints, dimensions of dome areshown in Figure 2 [9]. The members are collected into seven different groups, each group havethe same cross sectional area and one design variable. The truss joints are subjected to verticalloading except of supported joints. These are taken as 60.0062 kN at node 1, 30 kN from nodes 2to 14 and 10 kN at rest of the nodes, the dome span and total height shown in the Figure 3. 3Dtruss dome example is analysis and optimized under static loads for determine of maximumstress, displacement and minimum weight. In addition to allowable tensile and compressivestresses, an upper limit for the displacement is taken as -0.005 m. at each node. The allowablecompressive and tensile stresses are 103.4213, 241.3165 Mpa,Used material properties are: density,  8.027172366 kg/m3, Young’s modulusMpa5199947.961E  . Cross-sectional areas for first group A1= 0.00394 m2, second and third12nd International Balkans Conference on Challenges of Civil Engineering, BCCCE, 23-25 May 2013, Epoka University, Tirana, Albania.group A2=A3= 0.00137 m2, forth group A4= 0.00175 m2, fifth group A5= 0.0008 m2, sixth groupA6= 0.00362 m2 and seventh group A7=0.00285 m2.Discussion of the results: Maximum compression stress occurred in second group. Table 4 is theresult of displacements before and after optimization x, y and z directions. Maximumdisplacement occurred in joints (17, 29). The close agreement between results is seen. The GAoptimization satisfied displacement constraints. From the results demonstrated in table 5 it can beseen that the optimum values found for the final weights. The results got applying the GA forcontinuous design variable quantities. After 136 iterations, minimum weight design wasobtained. The weight of the truss is reduced from 118.518 to 57.28 (51.6 %reduction).Figure 2 Top wiev120 element 3D curved truss with 7.0 m height12nd International Balkans Conference on Challenges of Civil Engineering, BCCCE, 23-25 May 2013, Epoka University, Tirana, Albania.Figure 3 Length and height of 3D space trussesTable 4 Comparison of stress for 120 bar 3D curved truss with 7.0 m heightStress MpaGROUP NO. Present Sap200002 -32.54325 -33.2327Table 5 Comparison of displacement for 120 bar 3D curved truss with 7.0 m heightDisplacement (m )JointnoX-direction Y-direction Z-directionBeforeoptimizationAfteroptimizationBeforeoptimizationAfteroptimizationBeforeoptimizationAfteroptimization17 -0.00137 -0.00010 -0.00137 -0.00010 -0.00323 -0.001929 -0.00135 -0.00007 -0.00135 -0.00007 -0.00323 -0.0019Table 6 Initial and optimum design variables of 3D curved roof domeDesignvariablesCrosses sectional area (m2) Max.Stress(Mpa)Max. disp     (m)Initial OptimumS10.00394 0.00189-42.5475 0.00454S20.00137 0.00139S30.00137 0.00063S40.00175 0.00139S50.0008 0.00067S60.00362 0.00073S70.00285 0.00139Weight(kN) 118.518 57.28percentagereduction51.6%12nd International Balkans Conference on Challenges of Civil Engineering, BCCCE, 23-25 May 2013, Epoka University, Tirana, Albania.3.1.3Solid arches under multi point loadThis example involves analysis of arches with circle cross-sections, the geometry and a loadingof arch which has uniform cross-section with 20 m span is considered shown in Figure. 4. Thearches have a radius of curvature R = 11.547 m, the angle ω = 2π/3 (span length l = 24.1383 m),the cross-section area= 0.008968 m2. The following material properties are used: Young’smodulus 610200E kN/m2, material density 9729.76 kN/m3. This example involvesanalyse and optimization of an arch with maximum tensile stress 3t 10120 kN/m3,maximum compressive stress 3c 10120 kN/m3 and maximum xu and zu displacement allnodes is 0.015 m.Figure 5.10 Loading condition of archDiscussion of the results: Analysis is done by source program (SAP2000). Table 7 is the resultof displacements in x and z directions. Maximum displacement occurred in the crown.Table 8 shows the initial and optimal values of design variables and weight. After 42 iterations,minimum weight design was obtained for discrete design variables. The weight of the truss isreduced from 20.618 to 9.698 (53 %reduction).Table 5.13 Displacements of uniform cross-section archesjoint luc / (Disp. in x-direct.) lvc / (Disp. in z-direct.)  /c (Rotation)Crown 0.00000 -0.00264 -0.000705Table 8 Initial and optimum values of design variablesCrosses sectional area (m2)Design variables Initial Optimum12nd International Balkans Conference on Challenges of Civil Engineering, BCCCE, 23-25 May 2013, Epoka University, Tirana, Albania.S1Weight(kN) 20.618 9.698P.R 53 %ConclusionA design methodology of 2D and 3D curved roof trusses and solid arch roofing thatcombines stiffening sizing optimization is an important role in minimizing the amount of steelused in the construction of the structure for economic point of view. The optimization procedureimplemented, combined with accurate FE simulation of steel curved roof and solid arch roofing,resulted in a robust and efficient optimization tool.Optimization algorithm is starts following the implementation of the analysis of thestructure. A FORTRAN program which uses the FEMs based numerical analysis was modified.To achieve size optimization based on GA to perform the analysis and design. The problem ofchoosing the sizes of the bars in order to minimize the weight of the structure while satisfyingstress, displacement, stability.To find the best solution under constrains of allowable displacement and stress GAsearches all the available solution among all available results the best solution is selected. Designvariables were considered corresponding to the sizing of the cross-sectional areas of the bars.For all design variables significant decrease in weight of material with respect to thestress and displacement constraints were get. Finally, it must be emphasized that the algorithmproposed is capable of finding the optimum weight or volume with the least number of groupspossible to make the design practical. Hence, the solution is feasible and the construction of thestructure is easy. The results obtained on these typical problems showed various optimizationexamples were presented for minimizing the weight of the curved roof structures. Crossessection design variables was used. The influence of the number of design variable employed wasalso investigated that the accuracy of the concept presented is more than those of the othermethods.Reductions of (54 %, 51.6 % and in arch case 53%) for the three illustrated examplesrespectively give great encouragement to optimize structures. These reductions are important tosave extra materials in construction projects of curved roof structures consequently serving theeconomical point.REFERENCES[1] Tayşi, N. Göğüş, M.T. and Özakça, M. (2007) Optimization of arches using genetic310968.8  310219.5 12nd International Balkans Conference on Challenges of Civil Engineering, BCCCE, 23-25 May 2013, Epoka University, Tirana, Albania.algorithm. Journal of Mechanics, 41 377–394.[2] Daryl, L. Logan. (2007).  A First Course in the Finite Element Method , Fourth EditionUniversity of Wisconsin–Platteville Publisher: Chris Carson, COPYRIGHT# 2007 byNelson, a division of Thomson Canada Limited[3] Holland, J.H. (1975). Adaptation in Natural and Artificial System. The University ofMichigan Press, Michigan.[4] Gero , M.B.P. Garcia, A.B. and Diaz. J. J. (2005). A modified elitist genetic algorithmapplied to the design optimization of complex steel structures. J. Constr. Steel Res. 61,265–280.[5] Adeli , H. Cheng, N.T. (1994). Augmented Lagrangian genetic algorithm for structuraloptimization. J. Aerosp. Eng., ASCE 7, 104–118.[6] Tan, L.P. Lotfi, A. Lai, E. and Hull, J.B (2004). Soft computing applications in dynamicmodel identification of polymer extrusion process. Appl. Soft Comput. 4, 345–355.[7] Grierson, D.E. and Hajela, P. (1996). Emergent Computing Methods in EngineeringDesign: Applications of Genetic Algorithms and Neural Networks. Springer, New York.[8] Optimized modelling and design of structures using SAP2000CSI Educational Services Computers and Structures, Inc. 1995 University AvenueBerkeley, California 94704 USA[9] Togan, V. and Dalog˘lu, A. T., (2007). An improved genetic algorithm with initialpopulation strategy and self-adaptive member grouping. Computers and Structures, 86,1204–1218.[10] Litewka P. and Rakowski J., (1998). “The exact thick arch finite element”. Computers andStructures, 68, 369-379.

Analysis and Optimum Design of Curved Roof Structures

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Analysis and Optimum Design of Curved Roof Structures

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