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 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 parametrization 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 experimental 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], displacements [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
allows the evaluation of several options in terms of the objective function by considering different sets
of design variables or parametrizations. This additional information gives the opportunity to make a
trade-off between aesthetical or construction arguments considered in the initial design concept and the
possibilities to minimize the cost.
In the following section, the structural optimization methodology is presented. Next, the methodology is applied for the design of a barrel vault. The results for several parameterizations are compared to
illustrate the importance of decisions at the conceptual design stage.
2. Structural optimization methodology
Structural optimization can be formulated as a mathematical optimization problem:
xl 6 x 6 xu
minn f (x)
with
g(x) 6 0
x∈R
(1)
The objective function f (x) is minimized during the optimization. In this paper, the aim is to minimize
material use, so the total volume of material is considered in the objective function. The vector x contains
the design variables that can be related to the size or the shape of the structure. The optimal set of
variables 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, displacements
or 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 and
the 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 conceptual
design stage.
2.1. Design model
In the design model, the geometry of the structure is parameterized with Computer Aided Geometric
Design (CAGD) techniques [10]. In general, all parametrization techniques represent the boundary or the
surface with a linear combination of basis functions [2]. Some of the parameters of the linear combinations
are used as design variables [14]. A good choice for the parametrization is crucial as it fixes the design
space. Particular care is therefore needed when it is difficult to formulate a reasonable guess about the
shape of the best solution, as discussed by Bletzinger et al. [6]. In the next section, it will be demonstrated
that 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 designer as expressed in the initial concept. Moreover, subsequent optimizations of the same structure with
a 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 requirements and aesthetical arguments.
2.2. Analysis model
For a given value of the design variables, the values of the objective and constraint functions are computed
with an analysis model, constructed with the finite element method (FEM). A simplified analysis model
with a small number of characteristic static load cases can be sufficient in the conceptual design stage
since no details about the structure are available yet. The load cases are combined in load combinations
using the safety factors and combination factors according to Eurocode 1 [9].
2.3. Optimization model
The optimization model consists of the objective and constraint functions and a numerical optimization
algorithm that drives the optimization.
Two approaches can be distinguished within the objective function and constraints used for the
optimization of shells. The first approach is based on experimental methods for form finding of membranes
and 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 minimized
under a set of constraints that are generally based on the requirements imposed by design codes such as
Eurocode 1 [9]. Lagaros et al. [16] limit both the von Mises stress in the shell and the bending stresses
in the stiffening beams of a cylindrical roof and a storage silo. Lagaros and Papadopoulos [17] impose a
minimal buckling load during the optimization of a cylindrical panel. Displacement constraints should
also be added to the optimization problem to obtain a design satisfying the requirements of Eurocode 1 [9],
2
The first approach is more suitable for membrane structures. Since the thickness of these structures
is determined by the considered material, the volume constraint does not influence the result of the
optimization significantly. Tysmans et al. [24] use the first approach to find the optimal shape of a
cylindrical shell subjected only to its self weight. The thickness of the shell is determined afterwards to
satisfy all design criteria of the Eurocode. This approach does not guarantee, however, that the optimal
combination of shape and size is found. Lee and Hinton [18] also determine an optimal shell geometry
following the first approach and show that the buckling capacity of the optimal shell is not guaranteed. An
advantage of the second approach is the fact that multiple load cases and design requirements can easily
be accounted for. This ensures that the optimal design meets all relevant design criteria immediately
after optimization such that a additional sizing step is avoided. A minimal buckling load can also be
imposed 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 the
objective function, the optimization problem is then stated as follows:

6 x 6 xu
 xl
σULS (x)
6
σ
min V (x)
with
(2)
x

uSLS (x)
6
u
The stresses are limited for the load combinations in the ultimate limit state (ULS). The maximal
displacements 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 carefully. Local gradient-based optimization methods efficiently search for an optimum, but the result is not
guaranteed to be the global optimum. Global optimization methods are capable of finding the global
optimum 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 concept
is based on a cylindrical shell with edge beams described by Billington [5]. The design model is presented
for three different parameterizations. The analysis model and the optimization model are described in
detail and the optimization results are discussed.
3.1. Design model
The cylindrical shell with two edge beams is presented in figure 1a. The width B of the shell is 18 m
and the length L is 32 m. The barrel vault is assumed to be carried by 6 columns, modeled by a vertical
support 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 layers
alternated with glass fibre fabric. The advantages of fibre fabric compared to steel reinforcement are
twofold. First, the shape of the reinforcement can easily be adopted to the shell geometry. Second, since
there is no corrosion risk, the requirement for a minimal covering is omitted. Due to these advantages, the
total 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 be
constructed easily, which will be confirmed by the optimization results in this paper. Up to now, only
lab scale shells have been built using TRC. Hegger and Vos [12] describe the construction of small scale
barrel vault with a minimal thickness of 2.5 cm, constructed by alternating the textile reinforcement with
shotcrete layers of 3 to 5 mm. Tysmans et al. [25] manufactured a doubly curved shell spanning 2 m and
report difficulties in obtaining very thin shotcrete layers. Since sufficient layers of reinforcement should
be included, these difficulties cause the shell thickness to exceed the necessary thickness. From these
experiments, it is concluded that the construction, especially the application of the shotcrete layers, still
imposes a minimal shell thickness. In this paper, the minimal value of the thickness is 3 cm. According
to Tysmans et al. [25], TRC with a fibre volume of 7% has a Young’s modulus of 20 GPa, a Poisson’s
ratio of 0.15 and a density of 1900 kg/m3 . The ultimate tension strength ftu of this material is 10 MPa
and the ultimate compression strength fcu is 35 MPa. The safety factor γck is 1.5 as for normal concrete
and 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, a
Poisson’s ratio of 0.15 and a density of 2500 kg/m3 . Pretensioning of the edge beams is modeled by a
load case where a horizontal pressure of 15 kN/mm2 is applied at each beam end. The ultimate tension
3
(a)
L
B
(b)
Figure 1: The barrel vault: (a) design model and (b) analysis model.
t2
H
t1
y
h
b
Figure 2: Possible design variables.
strength ftu is 2.9 MPa and the ultimate compression strength fcu is 30 MPa. For this material, the
safety factors are equal to γck = γtk = 1.5.
Three parametrizations of the barrel vault are presented. The comparison of the optimization results
for each alternative will clearly demonstrate the importance of the parametrization. In figure 2, the design
variables are indicated on the structure. In the initial design concept, a height H of 3.73 m is proposed
since 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 optimal
dimensions, i.e. the shell thickness t and the width b and height h of the edge beam, will be determined
by the optimization considering x = {t b h}. In the second parametrization, the shape of the structure
is still assumed as fixed while the shell thickness is allowed to vary in the circumferential direction. The
thickness 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 ) 2
y + t2
B
(3)
with y the horizontal distance to the axis of the cylinder. For this parametrization, the vector of design
variables x is {t1 t2 b h}. In the third parametrization, the height H of the cylindrical shell is considered
as 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 importance of the parametrization. Furthermore, it will also provide additional information to evaluate
design decisions. For example, the result of the third parametrization reveals to what extent the volume
can be reduced if H is not assumed to be fixed by the initial design concept. This result will indicate if it
is advantageous to consider other erection techniques that allow the construction of a shell with a larger
slope.
3.2. Analysis model
The finite element program Ansys [1] is used. For every proposed geometry, a FE model is generated
based on the Ansys Parametric Design Language (APDL). Both the shell and the edge beams are meshed
with 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 in
figure 3. In the first load case, the gravity load is considered. The second load case is a service load p
4
p
g
(a)
(b)
s
w1
w2
(c)
(d)
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 the
shell 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 centre
quarters are subjected to a pressure w1 of 0.45 kN/m2 . The leeward quarter undergoes a wind suction
w2 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 ULS
load combinations, all load cases are added after multiplication by safety and combination factors to account respectively for uncertainties and the reduced chance that two variable loads reach their maximum
value at the same time. For these load combinations, the principal stresses are computed and the maximal
and minimal value σ max (x) and σ min (x) are determined for the edge beams and the shell. Large stress
concentrations appear at the corner regions of the structure, represented as the white colored regions in
figure 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 the
concentrations do not influence the overall shell thickness. The 6 SLS load combinations are obtained
using only safety factors. For these combinations, the maximal horizontal and vertical displacements
umax
and umax
are determined.
v
h
3.3. Optimization model
As described in the previous section, the objective function is the volume of the structure. For the three
parameterizations, both the current value of the volume and the derivatives with respect to the design
variables are computed analytically. As an example, the volume for the first parametrization is computed
as:
π
V (b, h, l) = 2bhL + RtL
(4)
2
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:

xl 6 x 6 xu



−fcu/γck 6 σ min (x) 6 σ max (x) 6 ftu /γtk
min V (x)
with
(5)
x
umax 6 Ht /200


 hmax
uv 6 B/300
The only important side constraint is the minimal shell thickness of 3 cm. The behaviour constraints
are related to stresses and displacements. For the ULS load combinations, the maximal and minimal
principal stress σ max (x) and σ min (x) are computed as described in section 3.2. Since concrete shells
are considered, the stress constraint is based on a Rankine failure criterium. The ultimate compression
and tension strength fcu and ftu are divided by their respective safety factors γck and γtk . The stress
constraint is verified for each material, so for the beam and the shell. For the SLS load combinations,
the maximal horizontal displacement umax
is limited to a fraction of the total height Ht /300=0.033 m,
h
5
Table 1: Optimal value of the design variables and objective
The design variables are underlined.
t1
t2
b
[m]
[m]
[m]
Parametrization 1 0.042 0.042 0.109
Parametrization 2 0.047 0.030 0.112
Parametrization 3 0.030 0.030 0.076
function value for each parametrization.
h
[m]
0.972
1.147
0.791
H
[m]
3.728
3.728
4.887
V
[m3 ]
33.61
31.27
24.36
Table 2: The constraint values of the optimum of each parametrization.
Maximal value
Parametrization 1
Parametrization 2
Parametrization 3
Beam
σ1max
σ3max
[MPa] [MPa]
1.93
20.00
1.93
15.83
1.93
15.95
1.93
15.96
Shell
σ1max
σ3max
[MPa] [MPa]
5.00
23.00
4.39
7.46
4.16
7.43
3.39
6.95
Displacements
umax
umax
v
h
[m]
[m]
0.033
0.09
0.015 0.018
0.016 0.026
0.020 0.016
since Ht is assumed to be 10 m. The maximal vertical displacement umax
is limited to a fraction of the
v
smallest span B, corresponding to a value of 18 m/200=0.09 m. The derivatives of the constraints with
respect 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 programmed in the Matlab function fmincon. Due to the small number of design variables, it is assumed
that no local minima exist so a local optimization method can be used. This assumption is verified by
repeating the optimization for three initial designs.
3.4. Optimization results
The optimal values of the design variables and the objective function for each parametrization are summarized in table 1. It is known that optimal material use is most likely obtained for a structure which
does not provide any unused capacity and is therefore located near the infeasible region of the design
space. The constraints that have a value close to their maximum are called active. Table 2 shows that
the maximum tensile strength in the beam is the active constraint in each case.
3.4.1. Parametrization 1
The optimization with parametrization 1 is a size optimization problem. The optimal structure, as
presented in table 1, is a very thin shell with slender edge beams and has a volume of 33.61 m3 . Depending
on the initial values of the design variables, the optimum is obtained in 30 to 70 iterations with 200 to
300 function evaluations. The evolution of the objective function for three initial structures is presented
in figure 4. All three runs lead to in the same volume and the same geometry x, although the initial
structures are quite different.
Now consider the optimization presented as run 1 in figure 4. The initial values of the design variables
are x = {0.10m 0.25m 0.5m} and correspond to a structure with a volume of 71.98 m3 which is not feasible
since the maximal tensile stress in the beam is 4.07 MPa. The other constraints have rather low values
compared to their maximal values: the maximal tensile stress in the shell is only 3.14 MPa and the
maximal horizontal and vertical displacements are 0.0036 m and 0.0069 m, respectively. During the
optimization, 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 structure
is better exploited since the other constraint values are closer to their maximum. The optimization has
produced a very useful result: the optimal structure is feasible while the objective function is significantly
lower. 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 are
transferred 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
80
V [m3]
60
40
20
0
0
20
40
Iteration
60
Figure 4: Optimization with parametrization 1: evolution of the objective function for three runs starting
from different initial designs: x = {0.10m 0.25m 0.5m} (solid line), x = {0.06m 0.15m 1.0m} (dashed
line) and x = {0.03m 0.05m 0.2m} (dotted line).
(a)
(b)
0
.133E+07
666667
.267E+07
.200E+07
.400E+07
.333E+07
.533E+07
.467E+07
.600E+07
Figure 5: Contour plot of the 1st principal stress in the top plane of the deformed structure (a) before
and (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 important. As presented on figure 5, the maximal principal stress in this load case occurs at the bottom side
of the edge beam. These are longitudinal tensile stresses due to the bending of the edge beam with
respect to its strong axis. By increasing h, the resistance against the bending increases and the maximal
stresses are reduced. Since the maximal horizontal displacement umax
, located at the middle vertical
h
support of the edge beam, is not close to its maximum, there is no need to increase resistance against
bending with respect to the weak axis. The maximal value of the principal stress in figure 5 is larger
than the allowed maximum, since only the stresses due to the gravity load case are shown. In the load
combinations, the pretensioning load case is included, which reduces the tensile stresses in the edge beam.
3.4.2. Parametrization 2
In the optimal structure of parametrization 1, membrane forces dominate the middle part of the shell
whereas the edges work as a beam. The maximal tensile stress is reached at the edge beams. At the
upper part of the shell, however, the maximal allowed stresses are not reached. This reveals a potential
for further volume reduction by retaining only the necessary material at the top of the shell. For that
purpose, the size of the shell is parameterized differently. As formulated in equation (3), the thickness t
is 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 function
evaluations as for the first parametrization. The optimum, which is presented in table 1, is again verified
by starting from three initial designs. As expected, the thickness at the top of the shell is decreased to its
7
lower limit. Compared to the result of the first parametrization, the thickness at the edges and the beam
height increase slightly. The decrease in volume by t2 is not compensated by the increase in volume by
t1 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 the
maximal displacements. However, the maximal stress values are not influenced since these values are
located at the shell edges. This is demonstrated in figure 6, showing the maximal principal stress contours 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. The
maximal allowed value, however, is still situated at the edges of the shell. The optimal design is again
determined by the maximal tensile stress of the edge beams.
(a)
(b)
-410
533014
266302
.107E+07
799727
.160E+07
.133E+07
.213E+07
.187E+07
.240E+07
Figure 6: Contour plot of the 1st principal stress of in the top plane of the deformed optimal structure
of (a) parametrization 1 and (b) parametrization 2 subjected to the worst load combination.
3.4.3. Parametrization 3
As membrane forces dominate the upper part of the shell, the curvature of the shell is important. By
considering the shell height H as a design variable, the influence of the curvature is studied. The minimal
shell height is limited to 0.5 m, which corresponds to an almost flat plate structure. The maximal shell
height is 8.5 m, which is slightly smaller than half the width of the shell (9 m), so that the resulting
structure corresponds to a half cylinder. The higher the shell, the more material is required for the
same shell thickness. Nevertheless, the thickness for a higher shell can be lower as the vertical loads are
transferred more efficiently by the membrane forces. The optimum will be a trade-off between these two
consequences.
The optimal values of the design variables are included in table 1. Again, the optimization converges
in a similar number of function evaluations. The increase of H makes it possible to reduce the other
design variables significantly. The shell thickness t is allowed to be the minimal thickness. Compared to
the result of parametrization 1, the volume is reduced with 28%. Moreover, it is lower than the minimal
volume obtained with parametrization 2. In figure 7, it is shown that the moments around the local
X-axes are reduced significantly compared to the result of parametrization 1. As expected, a larger part
of the total load is transferred by membrane forces. As demonstrated in table 2, the design is determined
by the maximal tensile stress constraint.
3.4.4. Design options
Based on the optimization results described above, several design options can be studied and their consequences can be evaluated. If the shape is fixed by the initial design concept, the results of parametrization 1 can be used. However, the optimizations with a varying thickness and with variable shell height
result in a decrease of the objective function value. Changing the height of the shell is the most effective
option 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 consider
the possibilities to construct a shell with the optimal height, since the technological requirement that
determined the initial concept has been violated. Additionally, the additional construction costs and
aesthetical changes due to the change of the height should be balanced by the advantages of the volume
reduction. If the first option is rejected, similar considerations can be made for the varying thickness.
8
(a)
(b)
-1500
-944.444
-1222
-388.889
-666.667
166.667
-111.111
722.222
444.444
1000
Figure 7: Contour plot of the moments around the element X-axes (parallel to the cylinder axis) of the
deformed optimal structure of (a) parametrization 1 and (b) parametrization 3 subjected to the worst
load combination.
Ohsaki et al. [20] have presented an alternative procedure where aesthetical considerations are introduced directly in the optimization. For that purpose, an objective function is used that is a linear
combination of the volume and the similarity of the current design to an initial concept. However, it is
difficult to choose the weighting coefficients of these two terms. Moreover, some structures have a large
objective function value because their distance to the initial design concept is large, but are acceptable
because they comply with the conceptual idea. For example, changing the height of the barrel vault does
not 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 options
and 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 obtain
an optimal design of shell structures with minimal volume. A small number of load cases is considered
and combined in several load combinations. In these load combinations, stresses and displacements are
constrained to ensure that the resulting structure complies with all relevant requirements of the design
code.
Using this methodology, the optimal design of a barrel vault is determined. Three different parameterizations 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 third
parametrization is an optimization of the radius in combination with the basic size design variables. The
second and third parametrization both result in an additional volume reduction compared to the first
parametrization. The optimization results show that parametrization is crucial as it has a large impact
on the optimal design. Moreover, these optimization results help the designer to make a trade-off between
aesthetical arguments and heuristic constructional requirements considered in the initial concept and the
possibilities 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 support
of FWO is gratefully acknowledged.
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