optimization of concrete shells using genetic algorithms

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ZAMM · Z. Angew. Math. Mech., 1 – 12 (2013) / DOI 10.1002/zamm.201200215

Optimization of concrete shells using genetic algorithms

Gabriele Bertagnoli1∗, Luca Giordano1, and Simona Mancini2

1 DISEG, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129, Torino, Italy2 DAUIN, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129, Torino, Italy

Received 18 October 2012, revised 14 February 2013, accepted 21 April 2013Published online 17 June 2013

Key words Concrete, genetic algorithms, intensification heuristics, shells, structural design.

In structural design, structures are often modeled using the finite elements method (FEM). One of the most commonelement type is the shell, which is used to model surfaces in a three dimensional space as far as the surface thickness issmaller than the other two dimensions. Many decisions must be taken during the design process, and many physical andloading constraints must be satisfied. Designers are generally interested in providing a solution that respects all the problemconstraints, without trying to further improve it. In fact optimization is not trivial, even if it could yield a huge benefit bothfrom the economic and the construction point of view. Additionally, saving materials is one of the fundamental criteria forthe sustainable approach to the design. In this paper, we address the Skew Reinforcement Design in Reinforced ConcreteTwo Dimensional Elements (SRD2D). It consists of determining the minimum reinforcement required to respect all theconstraints given by the geometric properties and the internal actions working on it. As this problem is strongly nonlinearand non-convex it cannot be easily solved using exact methods, while heuristics and meta-heuristics are suitable to thispurpose. We propose a Genetic Algorithm (GA) and an enhanced version of it, a memetic algorithm, in which we applyan intensification heuristic to the solution obtained by the GA, where each variable is separately optimized by applyinga first improvement based on local search strategy. We report computational results showing both the effectiveness of theproposed method, and the benefit of combining GAs with intensification methods.

c© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In structural design, structures are often modeled using the finite elements method (FEM). One of the most common elementtype is the shell, which is used to model surfaces in three dimensional space as far as the surface thickness is smaller thanthe other two dimensions. Nowadays it is very often necessary to design two dimensional reinforced concrete elementsfor structural reasons in road and railway infrastructures, and in general to satisfy architectural demands. Several concretestructures showing a two-dimensional behaviour are designed by use of shell finite elements, as cores of high rise buildings,tanks, boxed bridges and containment structures for nuclear plants. As a consequence, the use of finite elements analysis(FEA) is constantly rising to model structures and to evaluate internal actions. FEA actually allows for the implementationof a mathematical model of the structure, which can be solved by means of a computer for different load cases. Once theproblem of structural analysis is solved with the evaluation of internal actions for each finite element, a design procedureshould be implemented to verify the concrete stresses and to calculate the necessary steel reinforcement.

The design model suitable to take into account the complete bunch of internal actions may be considered well establishedin the literature [11, 15], at least when layers of orthogonal reinforcement are used. Moreover, this design model has beenrecently improved by the introduction of a new safety criterion for concrete working in biaxial state of stress, based on alarge series of heterogeneous experimental tests [7–9].

We present a Genetic Algorithm (GA) aiming to minimize the required reinforcement under a specific load case. Fur-thermore we propose an enhanced version of it, in which an ad hoc tailored intensification heuristic is applied to the bestsolution found by the GA.

Due to the recent introduction of the problem, exact or heuristic optimization methods to address it are lacking inliterature. A commercial optimization tool as Microsoft Excel is sometimes used in real applications in order to find afeasible solution. For this study, we solved instances taken from a real project, the concrete core of a 170 m high skyscraperin construction in Torino, which is subjected to high concentrated actions. We compared our computational results with thevalues of reinforcement used in the design process and obtained by means of Excel Solver, in order to have a measure of theperformances of the method we developed. We define the treated problem in Sect. 2 and we give a detailed description of theGenetic Algorithm in Sect. 3. The enhanced version of the algorithm and the description of the used heuristic improvementare reported in Sect. 4. Computational results are presented and discussed in Sect. 5, while Sect. 6 is dedicated to conclusionsand future perspectives.

∗ Corresponding author E-mail: [email protected], Phone: +39 011 090 4825, Fax: +39 011 090 4899

c© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

2 G. Bertagnoli et al.: Optimization of concrete shells using genetic algorithms

1 Problem definition

The FEA output given for the shell elements [2] is an octuple of internal actions composed by (see Fig. 1):

• plate components: three membrane forces per unit length (nx, ny , and nxy)• slab components:

– three moments per unit length (mx, my , and mxy)– two out-of-plane shear forces per unit length (vx and vy)

Fig. 1 Shell internal actions.

All the internal actions refer to a unit length of the structure. The ultimate limit state design of such structural elementshas been addressed with some continuity only since the early 80s. If we limit our analysis to the investigations of thelast thirty years, we find a contribution by Bazant and Lin [3], who envisaged a resisting mechanism by which tangentialactions are carried by friction only through the zone compressed by bending moments. Nowadays this mechanism shouldbe viewed as restrictive, since tangential actions can also be transferred through the zone in tension by flexure, providedthat crossed reinforcement is present. A very important observation in Bazant’s report concerns the behaviour of barsarranged in directions which substantially deviate from the stress principal directions, and which therefore display reducedefficiency. In this case, the local damage of concrete may lead to concrete failure by crushing before steel yielding occurs;this phenomenon has actually been observed in several specimens of membrane tested by Vecchio and Collins [39].

In 1985, Brondum-Nielsen [5] proposed a sandwich element model to be used in the ultimate limit state design ofshell elements. Each shell was subdivided into three membranes: the outer ones were meant to carry normal and tangen-tial stresses originated from the six local force components mx, my , mxy , nx, ny , nxy; the inner layer seemed to haveno bearing function, thus the model failed to take into account out-of-plane shear components, vx and vy . Reinforcementdimensioning and concrete verification were carried out according to a lower bound solution within the membranes corre-sponding to the outer layers. Gupta [19] took up Brondum-Nielsen’s proposal and completed it by formulating the set ofequations leading to the ultimate limit state design of the reinforcement and concrete. Also in this case, the shear forcesorthogonal to the shell element plane, vx and vy , were disregarded. The problem was later addressed in a rational andsystematic way through a series of theoretical studies and experimental investigations. Marti and Kong [28] confirmed thenon-conservative nature of the Normal Moment Yield Criterion in presence of significant twisting moments in the directionof reinforcement, i.e. in presence of appreciable differences between the directions of principal stresses and reinforcements.Three years later, Marti [26] presented his improved sandwich model, where the intermediate layer has the task of carryingout-of-plane shear forces, vx e vy , both in presence and in absence of transverse reinforcement; thus, we have a com-plete analogy between a beam consisting of two flanges linked by a web and a slab conceived as a sandwich in which theintermediate layer behaves like a beam web. This approach was further developed by Marti [27]. In 1992, Marti and Mey-boom [29] provided an experimental confirmation of the reduction in bearing capacity arising from the need for extensivecrack reorientation, by comparing the failure behaviour of reinforced concrete vs. prestressed concrete membranes. Fantiand Mancini [15] modified Marti’s model to adapt it to the limit state design philosophy embodied in Model Code 90 [10].They also proposed a guide for reinforcement dimensioning and detailing, taking into account the bearing capacity of thedifferent concrete layers. All the above models were developed within the hypothesis of placing two layers of orthogonalreinforcement within the element. No further developments to the resisting physical model were introduced since 1995,following the approach given by Fanti and Mancini [15]. Two different resisting models should be applied depending onwhether the element is cracked or uncracked. This verification may be performed in agreement with Model Code 2010 [16],by applying the Ottosen criterion [32] to different levels within the element thickness t. If the element results cracked, the

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ZAMM · Z. Angew. Math. Mech. (2013) / www.zamm-journal.org 3

Fig. 2 Plates internal actions.

sandwich model can be used. Two external layers of thicknesses ts and ti and an internal one of thickness tc should beindividuated, as shown in Fig. 2.

The internal layer carries the out-of-plane shear components and should be designed in analogy with the beam approachby using the principal shear v0 as defined in the following:

v0 =√

v2x + v2

y. (1)

Let be vRd1 the shear resistance in absence of specific shear reinforcement according to MC90 [10]:

vRd1 = 0.12(1 +√

200/d)(100ρ fck)(1/3)d (2)

where:

• d is the distance between reinforcement layers and the opposite face of the element;• ρ is the projection in the direction of v0 of the geometrical reinforcement ratios ραr and ρβr as described in the

following paragraphs;• fck is the characteristic cylindrical compressive strength of concrete.

If v0 ≤ vRd1 no shear reinforcement should be provided, on the contrary, if v0 > vRd1, shear reinforcement per unitarea, Asw

s , should be calculated as follows:

Asw

s=

v0

fyd · z · cot θt(3)

where:

• s is the spacing between shear reinforcement in the direction of v0;• fyd is the design yielding strength of steel reinforcement;• z is given in Fig. 2;• θt is the inclination with respect to the element surface of the compression fields in concrete due to shear.

The parameters of Eq. (3) should be chosen in order to respect also expression (4) that explicates the strength of com-pression fields inclined of θt in concrete

v0

sin θt≤ fcd2 · z · cos θt (4)

where fcd2 is given by (5) in agreement to MC90 [10]

fcd2 = 0.6(

1 − fck

250

)fcd (5)

being the cylindrical design strength of concrete, fcd = fck/γc, and γc the partial safety factor for concrete (see MC90 [10]).If a specific shear reinforcement is required and then a truss model is established along the shear principal direction,

the following additional membrane forces coming from the truss model must be taken into account when dealing with theexternal plates:

Δnx =v2

x

v0cot θt, (6)

Δny =v2

y

v0cot θt, (7)

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Δnxy =vxvy

v0cot θt. (8)

The two external layers are subjected to membrane forces due to nx, ny , nxy , eventually increased by the contributionsΔnx, Δny , and Δnxy and mx, my , and mxy . The membrane forces acting on the external layers nxr, nyr, nxyr, withr = s for the top layer and r = i for the bottom one, are given in Eqs. (9), (10), and (11) and are shown in Fig. (2)

nxr = nxz − zr

z± mx

z+

Δnx

2, (9)

nyr = nyz − zr

z± my

z+

Δny

2, (10)

nxyr = nxyz − zr

z± mxy

z+

Δnxy

2(11)

where:

• zr is the distance between the mid-plane of the layer r and the mid-plane of the shell,• the moments contributions should be considered with the plus sign for the top layer (r = s) and with the minus sign

for the bottom layer (r = i),• the contributions Δn should be considered only if a truss model is established in the internal layer according to

Eqs. (6)–(8).

In this way the design of the shell is reconducted to the design of two plates with two orders of skew reinforcement, asshown in the next paragraph.

The stresses σxr, σyr, and τxyr acting on the external layer r can be obtained by dividing the membrane forces nxr, nyr,nxyr by the thickness of the layer tr (see Fig. 3).

Fig. 3 Plates conventions.

Each external layer can then result cracked or uncracked. If concrete is in biaxial tension-compression or tension-tension state and it is cracked, equilibrium can be achieved by providing adequate reinforcement as described below. Thereinforcement amount can be calculated following a pure plastic approach according to a lower bound solution also whenit is oriented in a generic direction. According to this procedure, reinforcement placed across the crack is only subjectedto tensile stresses, and concrete undergoes a uniaxial compression in a direction inclined of an angle θr respect to the xaxis. Therefore no tangential stresses are present along direction θr. Considering a section of the plate element with a planeparallel to the direction of the ultimate compression stress field in concrete (Fig. 4), the equilibrium equations for translationin x and y directions lead to Eqs. (12) and (13):

ραrσsαr =σxr sin θr cosβ − σyr cos θr sinβ + τxyr cos(θr + β)

sin(θr − α) cos(α − β), (12)



ρβrσsβr =σxr sin θr sin α + σyr cos θr cosα + τxyr sin(θr + α)

cos(θr − β) cos(α − β)(13)

where ραr = Asαr/tr and ρβr = Asβr/tr; Asαr and Asβr are the reinforcement areas per unit length in direction α andβ in layer r. Equations (12) and (13) represent the equilibrium of the element portion shown in Fig. 4a, corresponding toτxyr > 0. If τxyr < 0, the same equations can be used considering the absolute value of τxyr and changing α in −α and βin −β as it can be observed by comparing Fig. 4a and Fig. 4b.

(a) (b)

Fig. 4 Equilibrium of the section parallel to the compression field if (a) τxyr > 0 and (b) if τxyr < 0.

(a) (b)

Fig. 5 Equilibrium of the section orthogonal to the compression field if (a) τxyr > 0 and (b) if τxyr < 0.

Now we consider a section of the plate element with a plane orthogonal to the stress field in concrete σcr (Fig. 5). Theequilibrium of the section shown in Fig. 5 allows to evaluate the stress σcr:

σcr =σxr sin α cosβ − σyr cosα sin β + τxyr cos(α + β)

sin(α − θr) cos(β − θr). (14)

If τxyr < 0, Eq. (14) can be used considering the absolute value of τxyr and changing α in -α and β in −β, as donefor Eqs. (12) and (13) (see Fig. 5). In the previous equations, θr should be included in the same quadrant of θer , which isthe angle between the x axis and the principal compression stress at cracking, coincident with the crack inclination. Thesolutions with zero denominator in Eqs. (12)–(14) correspond to cases in which only one order of reinforcement is available(cos(α−β) = 0) or the equilibrium is not possible as no tangential stresses are considered in θr direction (sin(θr −α) = 0and cos(θr − β) = 0). Design stress range for materials may be expressed for reinforcement as:

σskr ≤ fyd (k = α, β; r = s, i). (15)

The reinforcement ratios ραr and ρβr that are the actual unknowns of the problem, can be determined by imposingσskr = fyd. The safety criterion proposed by Carbone et al. [8], may be adopted to verify the concrete stress field. If at



least one order of reinforcement is yielded:

−σcr ≤ fcd2(1 − 0.032 |Δθr|) (16)

and if no order of reinforcement results to be yielded:

−σcr ≤ fcd2

[αcc

fcd

fcd2− σs

fyd

(0.85

fcd

fcd2− 1

)](17)

where:

• Δθr (measured in degrees and not greater than 15◦) is the deviation between the angle of compressive principal stressat first cracking and the plastic compression field in concrete,

• σs is the highest of the two stresses in each reinforcement layer,• fcd is the cylindrical design strength of concrete,• fcd2 is given by (5),• αcc is the coefficient taking account of long term effects on the compressive strength of concrete.

In the presented procedure, the thickness of the layers ts and ti, the inclination of the ultimate stress field in concrete ineach layer θs and θi, and the inclination of the compressed stress field due to principal shear θt are variables that can befreely chosen by the designer according to the plasticity lower bound approach. Every set of the chosen design parametersleads to a solution that is on the safe side.

These parameters become the variables of the following optimization problem: the total volume of the reinforcementper unit area (in [x,y] plane) VTOT is considered as the objective function. The goal of the algorithm is to choose thecombination of variables that allows VTOT to be minimized.

min VTOT =

[∑r

(Asαr + Asβr) +Asw

sh

]· 1 r = i, s. (18)

Restraint conditions are expressed by (15)–(17) and the following variable linear constraints:

Amin ≤ Asαr ≤ Amax (19)

Amin ≤ Asβr ≤ Amax (20)

2Cs ≤ ts ≤ H (21)

2Ci ≤ ti ≤ H (22)

ts + ti ≤ H (23)

0 ≤ θs ≤ π (24)

0 ≤ θi ≤ π (25)

−σcr ≤ fcd2(1 − 0.032 |Δθr|) (26)

1 ≤ cot θt ≤ 2.5 (27)

where H is the total height of the element, Cs and Ci respectively represent the top and bottom cover to the centroid ofthe reinforcement layers, and π is considered equal to 3.141593. Considering Eqs. (12) and (13), and remembering thatσskr = fyd, constraints (19) and (20) can be written as a function of the problem variables, as follows:

Amin ≤ σxr sin θr cosβ − σyr cos θr sin β + τxyr cos(θr + β)fyd · sin(θr − α) cos(α − β)

tr ≤ Amax, (28)

Amin ≤ σxr sin θr sin α + σyr cos θr cosα + τxyr sin(θr + α)fyd · cos(θr − β) cos(α − β)

tr ≤ Amax. (29)

Constraints (21), (22), (24), (25), and (27) indicate the range in which each variable can take values, constraint (23)ensures that the sum of the two layers thicknesses does not exceed the total height of the element, while constraint (26)imposes that the concrete resistance criterion is respected.

Given the values of the variables, VTOT can be computed by means of the procedure presented above.Despite the limited number of variables involved in the optimization problem, its strongly nonlinear and non-convex

nature makes exact methods not suitable to address it, while the use of heuristic or metaheuristic framework is preferable.



2 A genetic algorithm for the Skew Reinforcement Design in Reinforced ConcreteTwo Dimensional Elements

Design optimization methods have been widely applied to reinforced or prestressed concrete structural elements or simpleconcrete structures [4, 21–24, 33], but only in few cases they have been used for real large scale design problems [1, 14].Genetic Algorithms (GAs) are search procedures that employ natural selection mechanics to evolve solutions to problems.Introduced for the first time by Holland [20], they have been successfully applied to analyse several engineering andstructural problems [30], such as material design and identification [40], optimal design of composite structures [31,35,37],structural performance [13, 17] and topology [6]. For an introduction to GAs basic concepts we refer to [36], while for amore deep description of GAs features and hybridization with other heuristic algorithms we refer to [12]. Applications ofGAs on different engineering and science problems are reported by Randy et al. [34] and by Gen and Cheng [18].

In this section, we propose a GA for the Skew Reinforcement Design in Reinforced Concrete Two Dimensional Elements(SRD2D). This algorithm is tailored ad hoc for the problem, in order to take advantage of all the a priori knowledge availableon the search space. In fact, due to the nature of the problem, a variable can assume some critical values that will lead tounfeasible or very bad solutions despite the values of the other variables. These critical values are not a priori known andmay sensibly vary among different instances. In fact, values critical for an element can yield to very good solutions foranother one.

The description of the algorithm will follow these main outlines: individuals representation, initial population, evolutionprocess and operators description.

• Individuals representationEach individual is represented as an array containing the values of the variables ts, ti, θs, θi, and θt. From each com-bination of values it is possible to obtain, as explained in Sect. 1, the values of the requested reinforcement in each oneof the two main directions for each layer, and the shear reinforcement necessary to respect all the model constraints.The sum of these values, the so called VTOT , is considered as the objective function related to the individual.

• Initial populationThe creation of a proper initial population plays a crucial role in the performance of a GA, as stated by LuntalaNsakanda et al. [25] and by Togan and Daloglu [38]. In fact, if individuals belonging to the initial population are toosimilar among each other, also the new generations would continue to be too much homogeneous. Only a small subsetof solution space will then be explored, increasing the probability of remaining trapped in a local minimum. On theother hand, starting with very non homogeneous individuals, comprehensive also of bad solutions, could imply a veryslow convergence of the algorithm to good quality results. For these reasons, diversity and quality must be carefullybalanced in the initial population.The choice of the rules used to create the initial population should then take into account the a priori knowledge wealready have on the behaviour of the objective function when variables assume values within particular intervals. Thiscritical values should not appear in the initial solution, in order to avoid their bad characteristics to be reproduced inthe further generations. For this reason, a randomly generated initial population, which is commonly used in severalapplications, is not suitable for the problem we treat, while the application of an ad hoc developed generation strategyis strongly preferable. The herein proposed procedure can be described as follows.Before creating the initial population, the algorithm separately generates five sets of individuals, each one composedby N individuals, such that, in each set, one variable takes random values within its domain, while the others are keptfixed. More in details, in the first set, ts is randomly chosen for each individual, ti is taken equal to H

2 , while θs andθi are respectively taken equal to θe

s and θei , two values, depending on the internal actions and on H, for which the

probability of finding a feasible solution is maximum. Finally, cot θt is taken equal to 2.5. The second set generationworks exactly as the first one, except that the role of ts and ti are exchanged between each other. The third set iscomposed by individuals having ts = ti = H

2 , θi = θei , cot θt = 2.5, and θs randomly taken within its domain. In the

fourth set, ts = ti = H2 , θs = θe

s , θt = 21.8 and θi is randomly taken within its domain, while in the fifth and last setthe variable which takes a random value is θt, while the other variables assume fixed values (ts = ti = H

2 , θs = θes ,

and θi = θei ). In each set, the objective function, VTOT is calculated for every individual with a feasible solution. If

VTOT <= 1.5 · best, where best is the current best solution value, the individual is kept into consideration, otherwiseit is discarded. If all the individuals in the same set correspond to unfeasible or very bad solutions, the individual withthe best objective function among them is kept into the set, even if it does not respect the feasibility or the quality cri-terion. This procedure allows to a priori prevent variables from taking some values that yield to unfeasible or very badsolutions, regardless of the values taken by other variables. After these operations, the initial population constructionbegins, and each new individual is randomly generated by choosing one individual for each set and taking the value



of the variable ts from the individual chosen from the first set, the value of ti from the one chosen from the secondset, and the values of θs, θi, and θt, respectively, taken from the individual belonging to the third, fourth, and fifth set.This procedure is repeated R times in order to create an initial population composed by R individuals.

• Evolution processThe population of a GA iteratively evolves according to some specific rules. At each iteration, new individuals arecreated, and it has to be decided which ones will enter the population itself. Moreover, individuals must be thrownout of the population in order to avoid a continuous expansion of it. From one iteration to the next, the population canbe totally changed, so that only new individuals are kept in it. This strategy is called generational replacement. Onthe contrary, a steady-state evolution means that only part of the population can be modified between two consecutiveiterations. In our case we use the first technique, that is generational replacement.The choice of the individuals that may reproduce themselves is taken according to a probability distribution given bya fitness function, which is based on the quality of the solution related to the individual. Individuals corresponding tomore promising solutions have a larger probability of reproduction as for species evolution in nature.The fitness function can be defined in several different ways as described in [36]. A preliminary tuning phase of thealgorithm allowed to state that the best way to define the fitness function f(Xi) (combining accuracy of the resultsand small computational time) for our problem is:

f(Xi) =Zi

R∑i=1

Zi

where Zi = 1VT OTi

.

The reproduction probability of an individual is given by its fitness value and it is linearly dependent on Zi. If a childsolution is infeasible it is kept anyway in the population and its fitness is computed following the same rule used forfeasible solutions. We follow this strategy because in our problem a solution could contain good values for a part ofthe variables, and the infeasibility could be due to a single variable. In this case, even a small mutation could transforman infeasible solution in a very promising feasible one. Therefore, it is important to preserve good information even ifit comes from an infeasible solution, because it could become helpful in finding good solution during the reproductionphase. At this point, the evolution process starts and a new generation, composed by R new individuals, is created. Anindividual is selected from the current population with a roulette wheel based on its reproduction probability. Being Xj

the selected individual, if f(Xj) ≥ 1/R, which means that its quality is over the average and its characteristics shouldbe preserved among generations, a mutation operator is applied in order to create a new individual quite similar to it.Otherwise, another individual Xk is randomly chosen following the same probability reproduction, and a crossoveroperator is applied to Xj and Xk in order to generate a new individual. This process is repeated R times to createa completely renewed generation. At this point, reproduction probabilities are recalculated, and the next generationcreation process starts. The whole procedure is repeated NITER times, where NITER is the number of generationsbefore the algorithm terminates and returns the best feasible solution found.

• Operators descriptionWe use two different operators, mutation and crossover. The main difference is that mutation acts on a single indi-vidual, while crossover on a couple of them. Both mutation and crossover group under their name two families ofoperators. Two operators belonging to the same class, (e.i. the crossover family) can have different characteristicsbased on the specific problem addressed and on the individual representation. In the following we describe the op-erators we have developed. Being a generic individual P denoted by an array containing the value associated to thevariables in P = (p1, p2, p3, p4, p5), the mutation operator applied on P , mut(P ), generates a new individual Q,denoted by [q1, q2, q3, q4, q5]. The values qi are randomly drawn, in the interval [ai, bi], being ai = max(ci, pi −λi),bi = min(pi + λi, di), where λi are parameters fixed in advance, while ci and di respectively represent the minimumand the maximum value the ith variable can assume. Each qi can then be different at most of λi from pi. The smaller isthe value of the λi parameters, the smaller is the maximum diversity between P and Q, and consequently the strongeris the intensification of the search around promising solutions. The calibration of this parameter is therefore an issuethat must be carefully addressed. In fact, too large perturbations could yield to a completely different solution, loosingpart of the information coming from the original individual and giving too much power to the randomization compo-nent. On the other hand too small perturbations could yield to an excessive homogenization of the population, witha consequently increased probability of remaining trapped into a local minimum. The crossover operator applied to a



couple of individuals, cross(P, Q), generates a new individual S in which the first k variables assume the same valueas in P , and the others assume the value they have in Q, being k a random variable uniformly distributed in [1,4].

The general philosophy of the evolution process is to preserve promising solutions as much as possible, and to remixbad solutions in order to find new promising ones. A pseudo-code of the algorithm is reported in Algorithm 1.

Algorithm 1 GA for SRD2DCreate an initial populationCalculate fitnessrepeat

repeatCreate an empty new generation populationDraw an individual P from the current population according to a probability function based on the fitnessif f(P ) ≥ 1/R then

apply mutation operator mut(P )else

Draw another individual Q from the current population according to a probability function based on the fitnessapply crossover operator cross(P, Q)

end ifadd the new individual to the new generation population

until new generation population size is equal to Rcalculate new fitness function

until maximum number of generations, NITER, is reached

3 A memetic algorithm for the Skew Reinforcement Design in Reinforced ConcreteTwo Dimensional Elements

Memetic Algorithms are powerful optimization techniques which combine GA algorithm with local search or other heuristicmethods. In this section, we describe a Memetic Algorithm for the Skew Reinforcement Design in Reinforced ConcreteTwo Dimensional Elements, which consists in applying an intensification heuristic on the solution obtained by the GA inorder to improve its performances. This heuristic is a local search based algorithm tailored ad hoc for the problem, whichtries to take advantage from our knowledge about the behaviour of solutions in the search space. When one of the variablesis increasing or decreasing its value while the others are constant, the objective function also monotonically changes itsvalue. The value of the changing variable is then monotonically increased or decreased in order to decrease the objectivefunction until one of the constraints is violated. The method we propose exploits this knowledge in the following way. Theimprovement heuristics, named (IH), consist of addressing one variable at the time. Being P the best solution obtained byGA, and pi the variable we want to analyze, two new individuals are generated starting from P . In the first one, Q, theselected variable qi takes a random value in the range [pi − εi, pi]. In the second individual, S, the i-th variable si takesvalue in [pi, pi + εi]. All the other variables are kept fixed, and equal to the value assumed in P , for both individuals. Thesolutions related to Q and S are then analyzed; if both are infeasible or yield to an objective function greater than the onecorrespondent to P , the search terminates and another variable is addressed. Otherwise, if only one solution is feasible andbetter than P , it is kept as new current best, Y 0. If both are feasible the one with the lower objective function value becomesthe current best. If Y 0 is equal to Q then we follow the decreasing direction, generating a new individual Y t in which yt

i israndomly taken in the range [yt−1

i − εi, yt−1i ], at each iteration t. If Y 0 is equal to S, we go in the opposite directions, i.e.,

yti ∈ [yt−1

i , yt−1i + εi]. This part of the search process terminates when infeasibility is reached. After that, the best solution

obtained becomes the new P , and another variable is analyzed. When all the variables have been addressed, the globalprocedure ends. Since the method is based on the application of micro mutations, the parameter εi should assume a smallervalue than the mutation parameter used in GA, λi. A suggested value is εi = 0.1λi. The pseudo-code of IH is reported inAlgorithm 2. The drawback of this algorithm is that a change in the order used to address the variables may strongly changethe results obtained, thus making the choice of the processing order of the variables crucial for the method performances.Since unfortunately we do not have a priori information on the advantages of considering a variable before another or viceversa, the order choice does not represent a trivial issue. In order to partially overcome this problem, we developed aniterated version of the algorithm, (IIH), in which (IH) is repeated Nrec times. In this way, the order according to whichvariables are processed is less important, because the search for each variable is influenced by the changes made to theother values of the variables in the previous iterations. More in details, we fix randomly the order in which we analyze thevariables, and we proceed applying IH. Once the algorithm is terminated, we restart it again, following the same variable



order. The whole procedure stops after the maximum number of iterations, Nrec is reached, or if at the current step noimprovement has been reached with respect to the previous step.

Algorithm 2 IHstart from an initial solution Prepeat

randomly select a variable i that has still not been analyzedcreate a new solution Q, in which qi is randomly taken in [pi − εi, pi] and the other variables take the same values asin Pcreate a new solution S, in which si is randomly taken in [pi, pi + εi] and the other variables take the same values asin Pif both Q and S are infeasible then

discard the variable i and randomly select a new variable to be optimizedelse if Q is feasible and S is not, or if they are both feasible but Q has a lower objective function than S, then

consider Y 0 = Srepeat

at each iteration t create a new solution Y t, in which yti is randomly taken in [yt−1

i − εi, yt−1i ] and the other

variables take the same values as in Y t−1

until Y t is infeasibleelse if S is feasible and Q is not, or if they are both feasible but S has a lower objective function than Q, then

consider Y 0 = Qrepeat

at each iteration t create a new solution Y t, in which yti is randomly taken in [yt−1

i , yt−1i + εi] and the other

variables take the same values as in Y t−1

until Y t is infeasibleend if

until all the variables have been analyzed

4 Application

A wide number of computational experiments has been effectuated in order to test the performances of the algorithm. Morein details, we have analyzed a structure taken from a real project, and modeled with more than 500 finite elements: theconcrete core of a 170 m high skyscraper in construction in Torino, which is subjected to high concentrated actions. Foreach finite element, 16 octuples of internal actions (nx, ny , nxy , mx, my , mxy , vx, vy) have been taken into account. Eachoctuple corresponds to the maximum and the minimum values that each internal action can assume as a function of theposition of the external loads and the safety coefficients applied to them. For each element, the octuple corresponding to themaximum reinforcement required was taken as leading in the design and its results were used to perform the comparisonbetween different solution criteria. After a tuning phase, effectuated on a small subset of the instances, the following valueshave been assigned to the GA parameters:

• number of individuals in each set used to create the initial population: N = 50;• number of individuals in the population: R = 100;• number of iterations: NITER = 20;• mutation parameter for ts: λ1 = 0.05*H;• mutation parameter for ti:λ2 = 0.05*H;• mutation parameter for θs (expressed in degrees):λ3 = 3;• mutation parameter for θi (expressed in degrees):λ4= 3;• mutation parameter for θt (expressed in degrees):λ5= 1.

We compared the results obtained by GA, GA+IH, and GA+IIH with the ones obtained by Microsoft Excel Solver,which was used in the actual design of the structure. In Table 1 we show, for each octuple, the averaged percentage ofimprovement obtained by GA, GA+IH, and GA+IIH with respect to Microsoft Excel Solver. During the design process,the amount of reinforcement used for each element must be equal to the amount of reinforcement required to satisfy allthe octuples of internal actions, i.e. the amount requested by the octuple that needs the larger amount of reinforcement.This octuple, which may vary among elements, is commonly named “leading combination”. In Table 2 we report, for eachelement, the averaged percentage of improvement obtained on the leading combination by GA, GA+IH, and GA+IIH with



respect to Microsoft Excel Solver. These improvements represent the actual gain in terms of reinforcement amount withrespect to the one used in the real project.

Computational results show that using the proposed GA instead of the Microsoft Excel Solver will reduce the totalamount of reinforcement of 7%, with about the same computational time. This is a relevant result as even a gain of 7%means a huge saving of money when considering large quantities of needed reinforcement. Furthermore, Microsoft ExcelSolver do not find a feasible solution in a consistent number of cases, and its performances are strictly dependent on theinitial solution provided by the user. On the contrary, a change of seed in the GA (where the seed is the parameter that guidesthe random draw) does not imply a significant variation in the method performances, which means that the algorithm isstable. Results also show the benefit of applying IH and IIH, which respectively yield to an averaged improvement of 13.9%and 14.2%. Moreover, GA results are improved in all the cases. Last but not least, the recursive version of the algorithm, IIH,always performs better than the IH. Averaged computational times vary between 1 and 3 seconds for element, dependingon the set to which instances belong, both using our methods and Microsoft Excel Solver.

Table 1 Averaged percentage improvements for each octuple.

SET TYPE GA GA+IH GA+IIH

1 nx max 4.59% 10.54% 10.86%

2 ny max 4.41% 7.69% 7.93%

3 nxy max 5.56% 9.78% 10.36%

4 mx max 6.76% 14.06% 14.35%

5 my max 5.11% 5.13% 5.14%

6 mxy max 8.52% 11.89% 12.20%

7 vx max 9.23% 13.27% 13.80%

8 vy max 1.70% 2.20% 2.35%

9 nx min 5.81% 5.81% 6.25%

10 ny min 7.88% 8.16% 8.16%

11 nxy min 4.44% 4.93% 4.97%

12 mx min 4.64% 4.70% 4.75%

13 my min 8.34% 9.51% 9.68%

14 mxy min 3.83% 4.11% 4.14%

15 vx min 6.77% 9.97% 9.97%

16 vy min 5.19% 6.09% 6.27%

AVG 5.80% 7.99% 8.20%

Table 2 Averaged percentage improvements on leading combination.

Gain on leading GA GA+IH GA+IIH

combination 7.01% 13.88% 14.24%

5 Conclusions

In this paper, we address the Skew Reinforcement Design in Reinforced Concrete Two Dimensional Elements problem,which consists into determining, for each element of a FEM representation of a structure, the minimum reinforcementrequired to respect all the constraints given by the geometric properties of the element and the internal actions workingon it. We propose a Genetic Algorithm (GA) tailored ad hoc for the problem, and an enhanced version of it, a memeticalgorithm, in which GA is combined with an intensification heuristic (IH). In this enhanced version, each variable isseparately analyzed, creating new solutions starting from the best one obtained by GA, and applying micro mutationson the variable value while keeping fixed the other variables. A recursive version of the algorithm (IH), named (IIH), isalso presented. Results show the effectiveness of both the Genetic Algorithm (GA) and its enhanced versions (GA+IHand GA+IIH). Future developments could address three-dimensional finite elements, and problems in which the goal is tominimize the requested reinforcement with respect to all the load cases that can occur on the element. Another interestingissue to be addressed may be the application of IH and IHH to other continuous optimization problems.



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