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2 nd International Conference on Engineering Optimization September 6-9, 2010, Lisbon, Portugal An optimal barrel vault design in the conceptual design stage S. Arnout , G. Lombaert, G. Degrande, G. De Roeck Department of Civil Engineering, K.U.Leuven, Belgium [email protected] Abstract The geometry plays a key role in the structural behaviour of shell structures. Finding the optimal shell geometry is therefore of crucial importance. Structural optimization is well fit to reach this goal. In this paper, structural optimization is used as a design tool during the design of a barrel vault. Based on constructional requirements, a shape for the barrel vault is proposed. Initially, the shape is considered to be fixed and a size optimization is performed to obtain the optimal design. However, if a quadratical variation of the shell thickness is assumed, the volume of the considered barrel vault can be reduced with 7%. Alternatively, if the shell radius is added as a design variable, a volume reduction of 28% is obtained for this example. These results demonstrate that the choice of the design variables and the parametriza- tion strongly influence the resulting optimal design. Moreover, structural optimization gives the design team the opportunity to evaluate a number of design options in terms of material use. This evaluation extends the available information in the conceptual design stage and allows to make a trade-off between aesthetical arguments, constructional requirements and the possibilities for the reduction of material use. Keywords: Size and shape optimization, shell structure. 1. Introduction In contemporary architecture, wide span roofs are often conceived as shell structures. Since the geometry plays a key role in the structural behaviour of such structures an increasing attention is spent on optimal design. Furthermore, recently developed materials such as textile reinforced concrete (TRC) offer the potential to construct new types of light-weight shells easily. This material also allows to reduce the shell thickness since there is no need for a minimal concrete cover to avoid steel corrosion [12, 25]. In this paper, structural optimization is used during the design process of a shell structure to find the optimal geometry. Finding the optimal shape of a shell has received considerable attention in the past. The first ex- perimental method, that identifies the optimal shape as the inverse of a hanging model, was discovered by Hooke, inspiring a lot of later designers such as Wren, Gaudi and Isler. Later, additional physical experiments to find the optimal shape, such as the soap film analogy, are described. The first numerical studies for shape optimization were based on a numerical simulation of these experiments. As an example, a physical hanging model is replaced by a finite element analysis that takes geometrical nonlinearities into account [22]. Alternatively, structural optimization [2, 8, 11] can be used, in which iterative changes in the geometry, proposed by a numerical optimization technique, are evaluated by simulation of the structural behaviour. The goal of structural optimization is to find the best compromise between cost and performance. Due to its very general and flexible formulation [15], structural optimization is now widely used as a powerful design tool [4, 7, 13, 16, 23]. When the aim is to minimize material use, the total volume [16, 17] or the strain energy [7, 15, 18, 24] are often considered in the objective function. The objective function is accompanied by a number of design code based constraints. These constraints limit stresses [16], displace- ments [28] or natural frequencies [21] for multiple load combinations, ensuring an adequate performance of the resulting structure. Traditionally, the design process of structures consists of two stages. A preliminary design is created in the conceptual design stage and subsequently refined in the detailed design stage. The former stage is considered in this paper. It is assumed that an initial concept is available for the topology and the shape of the structure. It has been proposed by an architect, possibly based on morphological indicators [26] or on a design obtained by topology optimization [3]. As a consequence, the topology of the structure is considered to be fixed. Decisions that are taken during the conceptual design stage typically have a large impact on the final design, while they can only be based on a limited amount of information. Structural optimization 1

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Page 1: An optimal barrel vault design in the conceptual design stage · An optimal barrel vault design in the conceptual design stage S. Arnout, G. Lombaert, G. Degrande, G. De Roeck Department

2nd International Conference on Engineering Optimization

September 6-9, 2010, Lisbon, Portugal

An optimal barrel vault design in the conceptual design stage

S. Arnout, G. Lombaert, G. Degrande, G. De Roeck

Department of Civil Engineering, K.U.Leuven, Belgium

[email protected]

Abstract

The geometry plays a key role in the structural behaviour of shell structures. Finding the optimal shellgeometry is therefore of crucial importance. Structural optimization is well fit to reach this goal. Inthis paper, structural optimization is used as a design tool during the design of a barrel vault. Based onconstructional requirements, a shape for the barrel vault is proposed. Initially, the shape is consideredto be fixed and a size optimization is performed to obtain the optimal design. However, if a quadraticalvariation of the shell thickness is assumed, the volume of the considered barrel vault can be reduced with7%. Alternatively, if the shell radius is added as a design variable, a volume reduction of 28% is obtainedfor this example. These results demonstrate that the choice of the design variables and the parametriza-tion strongly influence the resulting optimal design. Moreover, structural optimization gives the designteam the opportunity to evaluate a number of design options in terms of material use. This evaluationextends the available information in the conceptual design stage and allows to make a trade-off betweenaesthetical arguments, constructional requirements and the possibilities for the reduction of material use.Keywords: Size and shape optimization, shell structure.

1. Introduction

In contemporary architecture, wide span roofs are often conceived as shell structures. Since the geometryplays a key role in the structural behaviour of such structures an increasing attention is spent on optimaldesign. Furthermore, recently developed materials such as textile reinforced concrete (TRC) offer thepotential to construct new types of light-weight shells easily. This material also allows to reduce the shellthickness since there is no need for a minimal concrete cover to avoid steel corrosion [12, 25]. In thispaper, structural optimization is used during the design process of a shell structure to find the optimalgeometry.

Finding the optimal shape of a shell has received considerable attention in the past. The first ex-perimental method, that identifies the optimal shape as the inverse of a hanging model, was discoveredby Hooke, inspiring a lot of later designers such as Wren, Gaudi and Isler. Later, additional physicalexperiments to find the optimal shape, such as the soap film analogy, are described. The first numericalstudies for shape optimization were based on a numerical simulation of these experiments. As an example,a physical hanging model is replaced by a finite element analysis that takes geometrical nonlinearitiesinto account [22]. Alternatively, structural optimization [2, 8, 11] can be used, in which iterative changesin the geometry, proposed by a numerical optimization technique, are evaluated by simulation of thestructural behaviour.

The goal of structural optimization is to find the best compromise between cost and performance. Dueto its very general and flexible formulation [15], structural optimization is now widely used as a powerfuldesign tool [4, 7, 13, 16, 23]. When the aim is to minimize material use, the total volume [16, 17] orthe strain energy [7, 15, 18, 24] are often considered in the objective function. The objective function isaccompanied by a number of design code based constraints. These constraints limit stresses [16], displace-ments [28] or natural frequencies [21] for multiple load combinations, ensuring an adequate performanceof the resulting structure.

Traditionally, the design process of structures consists of two stages. A preliminary design is createdin the conceptual design stage and subsequently refined in the detailed design stage. The former stage isconsidered in this paper. It is assumed that an initial concept is available for the topology and the shapeof the structure. It has been proposed by an architect, possibly based on morphological indicators [26]or on a design obtained by topology optimization [3]. As a consequence, the topology of the structure isconsidered to be fixed.

Decisions that are taken during the conceptual design stage typically have a large impact on thefinal design, while they can only be based on a limited amount of information. Structural optimization

1

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allows the evaluation of several options in terms of the objective function by considering different setsof design variables or parametrizations. This additional information gives the opportunity to make atrade-off between aesthetical or construction arguments considered in the initial design concept and thepossibilities to minimize the cost.

In the following section, the structural optimization methodology is presented. Next, the methodol-ogy is applied for the design of a barrel vault. The results for several parameterizations are compared toillustrate the importance of decisions at the conceptual design stage.

2. Structural optimization methodology

Structural optimization can be formulated as a mathematical optimization problem:

minx∈Rn

f(x) with

{

xl 6 x 6 xu

g(x) 6 0(1)

The objective function f(x) is minimized during the optimization. In this paper, the aim is to minimizematerial use, so the total volume of material is considered in the objective function. The vector x containsthe design variables that can be related to the size or the shape of the structure. The optimal set ofvariables has to satisfy a number of constraints. The side constraints define a lower and upper bound xl

and xu on the design variables. Behaviour constraints g(x) enforce limitations on stresses, displacementsor eigenfrequencies in multiple load cases to ensure an adequate performance of the structure.

In structural optimization, three models are involved: the design model, the optimization model andthe analysis model. A strong interaction between the models is essential for successful optimization [27].In this section, these three models are discussed together with their interpretation in the conceptualdesign stage.

2.1. Design modelIn the design model, the geometry of the structure is parameterized with Computer Aided GeometricDesign (CAGD) techniques [10]. In general, all parametrization techniques represent the boundary or thesurface with a linear combination of basis functions [2]. Some of the parameters of the linear combinationsare used as design variables [14]. A good choice for the parametrization is crucial as it fixes the designspace. Particular care is therefore needed when it is difficult to formulate a reasonable guess about theshape of the best solution, as discussed by Bletzinger et al. [6]. In the next section, it will be demonstratedthat the design model strongly influences the resulting barrel vault design.

The influence of the design model can be used purposely to incorporate the preferences of the de-signer as expressed in the initial concept. Moreover, subsequent optimizations of the same structure witha different design model allow evaluating the several design options in terms of the objective function.This information is useful when making the trade-off between reduction of material, constructional re-quirements and aesthetical arguments.

2.2. Analysis modelFor a given value of the design variables, the values of the objective and constraint functions are computedwith an analysis model, constructed with the finite element method (FEM). A simplified analysis modelwith a small number of characteristic static load cases can be sufficient in the conceptual design stagesince no details about the structure are available yet. The load cases are combined in load combinationsusing the safety factors and combination factors according to Eurocode 1 [9].

2.3. Optimization modelThe optimization model consists of the objective and constraint functions and a numerical optimizationalgorithm that drives the optimization.

Two approaches can be distinguished within the objective function and constraints used for theoptimization of shells. The first approach is based on experimental methods for form finding of membranesand minimizes the strain energy [7]. A volume constraint is often added to obtain a light-weight structure.A recent application is described by Kegl and Brank [15]. In the second approach, the volume is minimizedunder a set of constraints that are generally based on the requirements imposed by design codes such asEurocode 1 [9]. Lagaros et al. [16] limit both the von Mises stress in the shell and the bending stressesin the stiffening beams of a cylindrical roof and a storage silo. Lagaros and Papadopoulos [17] impose aminimal buckling load during the optimization of a cylindrical panel. Displacement constraints shouldalso be added to the optimization problem to obtain a design satisfying the requirements of Eurocode 1 [9],

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The first approach is more suitable for membrane structures. Since the thickness of these structuresis determined by the considered material, the volume constraint does not influence the result of theoptimization significantly. Tysmans et al. [24] use the first approach to find the optimal shape of acylindrical shell subjected only to its self weight. The thickness of the shell is determined afterwards tosatisfy all design criteria of the Eurocode. This approach does not guarantee, however, that the optimalcombination of shape and size is found. Lee and Hinton [18] also determine an optimal shell geometryfollowing the first approach and show that the buckling capacity of the optimal shell is not guaranteed. Anadvantage of the second approach is the fact that multiple load cases and design requirements can easilybe accounted for. This ensures that the optimal design meets all relevant design criteria immediatelyafter optimization such that a additional sizing step is avoided. A minimal buckling load can also beimposed easily. For these reasons, the second approach will be used in this paper.

Equation (1) is now elaborated to establish the second approach. Using the volume V (x) as theobjective function, the optimization problem is then stated as follows:

minx

V (x) with

xl 6 x 6 xu

σULS(x) 6 σuSLS(x) 6 u

(2)

The stresses are limited for the load combinations in the ultimate limit state (ULS). The maximaldisplacements are limited for the load combinations in the serviceability limit state (SLS).

If the objective function contains multiple minima, the optimization algorithm should be chosen care-fully. Local gradient-based optimization methods efficiently search for an optimum, but the result is notguaranteed to be the global optimum. Global optimization methods are capable of finding the globaloptimum but then need more calculation time since they explore a large part of the design space randomly.

3. Optimal design of a barrel vault

In this section, structural optimization is used for the design of a barrel vault. The initial design conceptis based on a cylindrical shell with edge beams described by Billington [5]. The design model is presentedfor three different parameterizations. The analysis model and the optimization model are described indetail and the optimization results are discussed.

3.1. Design modelThe cylindrical shell with two edge beams is presented in figure 1a. The width B of the shell is 18 mand the length L is 32 m. The barrel vault is assumed to be carried by 6 columns, modeled by a verticalsupport at each corner and at mid span of the beam. Furthermore, an ideal end bearing wall is added,so that the circular ends of the shell are not allowed to move in the plane of the bearing wall.

The shell is constructed with textile reinforced concrete (TRC) which consists of shotcrete layersalternated with glass fibre fabric. The advantages of fibre fabric compared to steel reinforcement aretwofold. First, the shape of the reinforcement can easily be adopted to the shell geometry. Second, sincethere is no corrosion risk, the requirement for a minimal covering is omitted. Due to these advantages, thetotal shell thickness can be reduced to the minimal thickness required by strength and stiffness demands.Therefore, this new material offers possibilities for the design of efficient shell structures that can beconstructed easily, which will be confirmed by the optimization results in this paper. Up to now, onlylab scale shells have been built using TRC. Hegger and Vos [12] describe the construction of small scalebarrel vault with a minimal thickness of 2.5 cm, constructed by alternating the textile reinforcement withshotcrete layers of 3 to 5 mm. Tysmans et al. [25] manufactured a doubly curved shell spanning 2 m andreport difficulties in obtaining very thin shotcrete layers. Since sufficient layers of reinforcement shouldbe included, these difficulties cause the shell thickness to exceed the necessary thickness. From theseexperiments, it is concluded that the construction, especially the application of the shotcrete layers, stillimposes a minimal shell thickness. In this paper, the minimal value of the thickness is 3 cm. Accordingto Tysmans et al. [25], TRC with a fibre volume of 7% has a Young’s modulus of 20 GPa, a Poisson’sratio of 0.15 and a density of 1900 kg/m3. The ultimate tension strength ftu of this material is 10 MPaand the ultimate compression strength fcu is 35 MPa. The safety factor γck is 1.5 as for normal concreteand γtk is 2 since the fibre reinforcement strength is accounted for.

The edge beam is constructed with concrete of class C30/35 with a Young’s modulus of 33 GPa, aPoisson’s ratio of 0.15 and a density of 2500 kg/m3. Pretensioning of the edge beams is modeled by aload case where a horizontal pressure of 15 kN/mm2 is applied at each beam end. The ultimate tension

3

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(a) BL (b)

Figure 1: The barrel vault: (a) design model and (b) analysis model.

b

h

t2

t1y

H

Figure 2: Possible design variables.

strength ftu is 2.9 MPa and the ultimate compression strength fcu is 30 MPa. For this material, thesafety factors are equal to γck = γtk = 1.5.

Three parametrizations of the barrel vault are presented. The comparison of the optimization resultsfor each alternative will clearly demonstrate the importance of the parametrization. In figure 2, the designvariables are indicated on the structure. In the initial design concept, a height H of 3.73 m is proposedsince this results in a maximal slope of 45 degrees, which is considered as a limit for easy construction.In the first parametrization, the shape of the structure is assumed to be fixed by the initial concept.The shell thickness t is assumed to be constant over the shell surface, so t = t1 = t2. The optimaldimensions, i.e. the shell thickness t and the width b and height h of the edge beam, will be determinedby the optimization considering x = {t b h}. In the second parametrization, the shape of the structureis still assumed as fixed while the shell thickness is allowed to vary in the circumferential direction. Thethickness t is interpolated from the thickness t1 at the side of the beams and t2 at the top of the shell as:

t(y) =4(t1 − t2)

By2 + t2 (3)

with y the horizontal distance to the axis of the cylinder. For this parametrization, the vector of designvariables x is {t1 t2 b h}. In the third parametrization, the height H of the cylindrical shell is consideredas an additional design variable, so that x = {t b h H}.

The results of the three parametrizations are expected to be significantly different and illustrate im-portance of the parametrization. Furthermore, it will also provide additional information to evaluatedesign decisions. For example, the result of the third parametrization reveals to what extent the volumecan be reduced if H is not assumed to be fixed by the initial design concept. This result will indicate if itis advantageous to consider other erection techniques that allow the construction of a shell with a largerslope.

3.2. Analysis modelThe finite element program Ansys [1] is used. For every proposed geometry, a FE model is generatedbased on the Ansys Parametric Design Language (APDL). Both the shell and the edge beams are meshedwith the 8-noded shell element SHELL93. The analysis model is presented on figure 1b.

Five load cases are considered based on Eurocode 1 [9]. The first four load cases are presented infigure 3. In the first load case, the gravity load is considered. The second load case is a service load p

4

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(a)

g

(b)

p

(c)

s

(d)

w1

w2

Figure 3: Load cases: (a) self weight, (b) service load, (c) snow load and (d) wind load.

of 0.5 kN/m2. The third load case is a snow load s of 0.4 kN/m2 on the horizontal projection of theshell surface. In the fourth load case, the wind load is considered. For the estimation of the wind load,the circumference of the shell is divided in four quarters. As can be seen in figure 3c, the two centrequarters are subjected to a pressure w1 of 0.45 kN/m2. The leeward quarter undergoes a wind suctionw2 of 0.2 kN/m2. As described above, the pretensioning of the edge bar is modeled as a fifth load case.A static analysis is performed for each load case.

According to Eurocode 1 [9], the load cases are combined in 9 load combinations. To obtain the 3 ULSload combinations, all load cases are added after multiplication by safety and combination factors to ac-count respectively for uncertainties and the reduced chance that two variable loads reach their maximumvalue at the same time. For these load combinations, the principal stresses are computed and the maximaland minimal value σmax(x) and σmin(x) are determined for the edge beams and the shell. Large stressconcentrations appear at the corner regions of the structure, represented as the white colored regions infigure 1b. Additional local reinforcement will necessarily be added in these zones in the final structure.Therefore, the stresses at these regions are not taken into account during the optimization such that theconcentrations do not influence the overall shell thickness. The 6 SLS load combinations are obtainedusing only safety factors. For these combinations, the maximal horizontal and vertical displacementsumax

hand umax

v are determined.

3.3. Optimization modelAs described in the previous section, the objective function is the volume of the structure. For the threeparameterizations, both the current value of the volume and the derivatives with respect to the designvariables are computed analytically. As an example, the volume for the first parametrization is computedas:

V (b, h, l) = 2bhL +π

2RtL (4)

with R the radius of the cylindrical roof.Side as well as behaviour constraints are defined. Equivalent to equation (2), the problem is now

formulated as:

minx

V (x) with

xl 6 x 6 xu

−fcu/γck 6 σmin(x) 6 σmax(x) 6 ftu/γtk

umax

h6 Ht/200

umaxv 6 B/300

(5)

The only important side constraint is the minimal shell thickness of 3 cm. The behaviour constraintsare related to stresses and displacements. For the ULS load combinations, the maximal and minimalprincipal stress σmax(x) and σmin(x) are computed as described in section 3.2. Since concrete shellsare considered, the stress constraint is based on a Rankine failure criterium. The ultimate compressionand tension strength fcu and ftu are divided by their respective safety factors γck and γtk. The stressconstraint is verified for each material, so for the beam and the shell. For the SLS load combinations,the maximal horizontal displacement umax

his limited to a fraction of the total height Ht/300=0.033 m,

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Table 1: Optimal value of the design variables and objective function value for each parametrization.The design variables are underlined.

t1 t2 b h H V[m] [m] [m] [m] [m] [m3]

Parametrization 1 0.042 0.042 0.109 0.972 3.728 33.61Parametrization 2 0.047 0.030 0.112 1.147 3.728 31.27Parametrization 3 0.030 0.030 0.076 0.791 4.887 24.36

Table 2: The constraint values of the optimum of each parametrization.

Beam Shell Displacementsσmax

1 σmax3 σmax

1 σmax3 umax

humax

v

[MPa] [MPa] [MPa] [MPa] [m] [m]Maximal value 1.93 20.00 5.00 23.00 0.033 0.09Parametrization 1 1.93 15.83 4.39 7.46 0.015 0.018Parametrization 2 1.93 15.95 4.16 7.43 0.016 0.026Parametrization 3 1.93 15.96 3.39 6.95 0.020 0.016

since Ht is assumed to be 10 m. The maximal vertical displacement umaxv is limited to a fraction of the

smallest span B, corresponding to a value of 18 m/200=0.09 m. The derivatives of the constraints withrespect to the design variables are computed by the forward finite difference method.

A sequential quadratic programming method (SQP) [19] is used as the optimization method, as pro-grammed in the Matlab function fmincon. Due to the small number of design variables, it is assumedthat no local minima exist so a local optimization method can be used. This assumption is verified byrepeating the optimization for three initial designs.

3.4. Optimization resultsThe optimal values of the design variables and the objective function for each parametrization are sum-marized in table 1. It is known that optimal material use is most likely obtained for a structure whichdoes not provide any unused capacity and is therefore located near the infeasible region of the designspace. The constraints that have a value close to their maximum are called active. Table 2 shows thatthe maximum tensile strength in the beam is the active constraint in each case.

3.4.1. Parametrization 1The optimization with parametrization 1 is a size optimization problem. The optimal structure, aspresented in table 1, is a very thin shell with slender edge beams and has a volume of 33.61 m3. Dependingon the initial values of the design variables, the optimum is obtained in 30 to 70 iterations with 200 to300 function evaluations. The evolution of the objective function for three initial structures is presentedin figure 4. All three runs lead to in the same volume and the same geometry x, although the initialstructures are quite different.

Now consider the optimization presented as run 1 in figure 4. The initial values of the design variablesare x = {0.10m 0.25m 0.5m} and correspond to a structure with a volume of 71.98 m3 which is not feasiblesince the maximal tensile stress in the beam is 4.07 MPa. The other constraints have rather low valuescompared to their maximal values: the maximal tensile stress in the shell is only 3.14 MPa and themaximal horizontal and vertical displacements are 0.0036 m and 0.0069 m, respectively. During theoptimization, the shell thickness t and the beam height b are reduced and the beam height h is increased.In the final structure, the tensile stress constraint of the beam is satisfied. The capacity of the structureis better exploited since the other constraint values are closer to their maximum. The optimization hasproduced a very useful result: the optimal structure is feasible while the objective function is significantlylower. Such a design is very difficult to find by trial-and-error.

Both in the initial and the optimum structure, the vertical loads at the middle part of the shell aretransferred to the edges by membrane forces. The edge beams, together with the lower parts of the shell,transfer the loads to the supports as a large L-shaped beam. Since the self weight of the structure is a

6

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0 20 40 600

20

40

60

80

IterationV

[m3 ]

Figure 4: Optimization with parametrization 1: evolution of the objective function for three runs startingfrom different initial designs: x = {0.10m 0.25m 0.5m} (solid line), x = {0.06m 0.15m 1.0m} (dashedline) and x = {0.03m 0.05m 0.2m} (dotted line).

(a) (b)

0666667

.133E+07.200E+07

.267E+07.333E+07

.400E+07.467E+07

.533E+07.600E+07

Figure 5: Contour plot of the 1st principal stress in the top plane of the deformed structure (a) beforeand (b) after optimization with parametrization 1, subjected to gravity.

very important load case, the reduction of the weight by the decrease of t and b has a large influence.The moments in the optimal structure are smaller and the membrane behaviour becomes more impor-tant. As presented on figure 5, the maximal principal stress in this load case occurs at the bottom sideof the edge beam. These are longitudinal tensile stresses due to the bending of the edge beam withrespect to its strong axis. By increasing h, the resistance against the bending increases and the maximalstresses are reduced. Since the maximal horizontal displacement umax

h, located at the middle vertical

support of the edge beam, is not close to its maximum, there is no need to increase resistance againstbending with respect to the weak axis. The maximal value of the principal stress in figure 5 is largerthan the allowed maximum, since only the stresses due to the gravity load case are shown. In the loadcombinations, the pretensioning load case is included, which reduces the tensile stresses in the edge beam.

3.4.2. Parametrization 2In the optimal structure of parametrization 1, membrane forces dominate the middle part of the shellwhereas the edges work as a beam. The maximal tensile stress is reached at the edge beams. At theupper part of the shell, however, the maximal allowed stresses are not reached. This reveals a potentialfor further volume reduction by retaining only the necessary material at the top of the shell. For thatpurpose, the size of the shell is parameterized differently. As formulated in equation (3), the thickness tis interpolated from the thickness at the edge beams t1 and the thickness at the top t2.

The optimization algorithm converges for this problem in a similar number of iterations and functionevaluations as for the first parametrization. The optimum, which is presented in table 1, is again verifiedby starting from three initial designs. As expected, the thickness at the top of the shell is decreased to its

7

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lower limit. Compared to the result of the first parametrization, the thickness at the edges and the beamheight increase slightly. The decrease in volume by t2 is not compensated by the increase in volume byt1 and h, so a volume reduction of 7% is obtained.

As can be seen from table 2, the reduction of the stiffness is reflected in the increased value of themaximal displacements. However, the maximal stress values are not influenced since these values arelocated at the shell edges. This is demonstrated in figure 6, showing the maximal principal stress con-tours in the top plane of the shell subjected to the load combination that causes the largest stresses.The maximum tensile stress increases at the top of the shell where the thickness has been reduced. Themaximal allowed value, however, is still situated at the edges of the shell. The optimal design is againdetermined by the maximal tensile stress of the edge beams.

(a) (b)

-410266302

533014799727

.107E+07.133E+07

.160E+07.187E+07

.213E+07.240E+07

Figure 6: Contour plot of the 1st principal stress of in the top plane of the deformed optimal structureof (a) parametrization 1 and (b) parametrization 2 subjected to the worst load combination.

3.4.3. Parametrization 3As membrane forces dominate the upper part of the shell, the curvature of the shell is important. Byconsidering the shell height H as a design variable, the influence of the curvature is studied. The minimalshell height is limited to 0.5 m, which corresponds to an almost flat plate structure. The maximal shellheight is 8.5 m, which is slightly smaller than half the width of the shell (9 m), so that the resultingstructure corresponds to a half cylinder. The higher the shell, the more material is required for thesame shell thickness. Nevertheless, the thickness for a higher shell can be lower as the vertical loads aretransferred more efficiently by the membrane forces. The optimum will be a trade-off between these twoconsequences.

The optimal values of the design variables are included in table 1. Again, the optimization convergesin a similar number of function evaluations. The increase of H makes it possible to reduce the otherdesign variables significantly. The shell thickness t is allowed to be the minimal thickness. Compared tothe result of parametrization 1, the volume is reduced with 28%. Moreover, it is lower than the minimalvolume obtained with parametrization 2. In figure 7, it is shown that the moments around the localX-axes are reduced significantly compared to the result of parametrization 1. As expected, a larger partof the total load is transferred by membrane forces. As demonstrated in table 2, the design is determinedby the maximal tensile stress constraint.

3.4.4. Design optionsBased on the optimization results described above, several design options can be studied and their con-sequences can be evaluated. If the shape is fixed by the initial design concept, the results of parametriza-tion 1 can be used. However, the optimizations with a varying thickness and with variable shell heightresult in a decrease of the objective function value. Changing the height of the shell is the most effectiveoption for volume reduction. The thickness of the shell with the optimal height is the minimal thickness,so a combination of both options is not necessary.

This information can now be incorporated in the design process. First, the design team should considerthe possibilities to construct a shell with the optimal height, since the technological requirement thatdetermined the initial concept has been violated. Additionally, the additional construction costs andaesthetical changes due to the change of the height should be balanced by the advantages of the volumereduction. If the first option is rejected, similar considerations can be made for the varying thickness.

8

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(a) (b)

-1500-1222

-944.444-666.667

-388.889-111.111

166.667444.444

722.2221000

Figure 7: Contour plot of the moments around the element X-axes (parallel to the cylinder axis) of thedeformed optimal structure of (a) parametrization 1 and (b) parametrization 3 subjected to the worstload combination.

Ohsaki et al. [20] have presented an alternative procedure where aesthetical considerations are in-troduced directly in the optimization. For that purpose, an objective function is used that is a linearcombination of the volume and the similarity of the current design to an initial concept. However, it isdifficult to choose the weighting coefficients of these two terms. Moreover, some structures have a largeobjective function value because their distance to the initial design concept is large, but are acceptablebecause they comply with the conceptual idea. For example, changing the height of the barrel vault doesnot violate the initial concept of a cylindrical roof, but introduces more distance to the initial concept.Therefore, it seems more appropriate to compute the minimal volume results for several design optionsand to incorporate this information in a trade-off made by the design team.

4. Conclusions

In this paper, structural optimization is used in the design process of a shell structure. The aim is to obtainan optimal design of shell structures with minimal volume. A small number of load cases is consideredand combined in several load combinations. In these load combinations, stresses and displacements areconstrained to ensure that the resulting structure complies with all relevant requirements of the designcode.

Using this methodology, the optimal design of a barrel vault is determined. Three different param-eterizations are compared. In the first parametrization, only basic size design variables are considered.The second parametrization considers the size design variables with variable shell thickness and the thirdparametrization is an optimization of the radius in combination with the basic size design variables. Thesecond and third parametrization both result in an additional volume reduction compared to the firstparametrization. The optimization results show that parametrization is crucial as it has a large impacton the optimal design. Moreover, these optimization results help the designer to make a trade-off betweenaesthetical arguments and heuristic constructional requirements considered in the initial concept and thepossibilities of material reduction present in the improved structures.

5. Acknowledgements

The first author is a PhD. fellow of the Research Foundation – Flanders (FWO). The financial supportof FWO is gratefully acknowledged.

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