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Structural Optimization of Grid Shells: Design Parameters
and Combined Strategies
Downloaded from ascelibrary.org by Tufts University on 10/26/17. Copyright ASCE. For personal use only; all rights reserved.
Ernesto Grande1; Maura Imbimbo2; and Valentina Tomei3
Abstract: The optimization of structures is a tricky process that involves strategies and mathematical algorithms in which the design
parameters—introduced in terms of variables, constraint conditions, optimization functions, penalty conditions, and so forth—play very
important roles. Their selection and introduction in the optimization process could influence the characteristics and the level of optimization of the derived solutions. This paper describes the roles of design parameters used in different optimization strategies. Moreover,
an efficient optimization approach, which combines form-finding (FF), sizing-optimization (SO), and topologic-optimization (TO)
strategies in a multilevel process for which design variables and constraint conditions are opportunely selected, is proposed. Numerical
analyses referring to some canopy case studies derived from the current literature are also presented in the paper. The comparisons
among optimization approaches, featuring FF, SO, and TO optimization strategies performed singularly or in combination throughout a
simple sequence of phases, emphasize the effectiveness of the proposed approach for obtaining light structural solutions for grid shells.
DOI: 10.1061/(ASCE)AE.1943-5568.0000286. © 2017 American Society of Civil Engineers.
Introduction
Nowadays, grid-shell structures are an important structural typology widely chosen in many recent applications thanks to their
capacity to cover large spaces with free form and light solutions.
In the design of grid shells, the shape and the structure cannot be
separated because they are the same thing. For this reason, a grid
shell should have a shape that respects architectural requirements
and is structurally efficient; this requirement means that the architectural aspects are treated together with the structural ones, and
to pursue an optimal design, both aspects should be optimized at
the same time.
Numerous examples of grid-shell structures based on optimized
shapes and involving different structural materials are described in
the current literature (Pone et al. 2013; Williams 2014). In some
cases, they were created through innovative design approaches that
explicitly adopt form-finding (FF) and optimization strategies as efficient support tools for the entire design process. An interesting
and recent example in Italy is the prototype of a timber grid shell
covering an area of 75 m2 built in the Faculty of Architecture courtyard at the University Federico II in Naples (Pone et al. 2013). For
the design of this prototype, a design procedure based on the
particle-spring model, followed by a finite- element analysis,
was developed for the FF phases and structural analysis. Other
fascinating examples of grid-shell structures based on FF or
structural-optimization design processes are present in Europe.
In particular, two examples that are representative of the different
1
Associate Professor, Dept. of Sustainability Engineering, Guglielmo
Marconi Univ., Rome, Italy (corresponding author). E-mail: e.grande@
unicas.it
2
Associate Professor, Dept. of Civil and Mechanical Engineering,
Univ. of Cassino and Southern Lazio, Cassino, Italy.
3
Assistant Professor, Dept. of Civil and Mechanical Engineering,
Univ. of Cassino and Southern Lazio, Cassino, Italy.
Note. This manuscript was submitted on March 28, 2017; approved on
July 26, 2017; published online on October 26, 2017. Discussion period
open until March 26, 2018; separate discussions must be submitted for
individual papers. This paper is part of the Journal of Architectural
Engineering, © ASCE, ISSN 1076-0431.
© ASCE
design strategies and construction techniques are the Mannheim
Multihalle in Germany and the British Museum canopy in
England. Indeed, for the Mannheim Multihalle, which is a standalone multilayered grid system, the designer, Frei Otto, created
the optimal shape by using the physical-hanging model (Linkwitz
and Veenendaal 2014; Williams 2014). In the case of the British
Museum canopy, designed by Foster and Partners, the original
shape was determined from a nonlinear analysis on the basis of
the dynamic relaxation method (Williams 2014).
In the context of function and design, it is then evident that
FF and structural-optimization techniques can have key roles
as efficient supports for the design of grid-shell structures.
Indeed, the growing number of studies concerning grid shells
(Richardson et al. 2013, Bouhaya et al 2014; Adriaenssens
et al. 2014; D’Amico et al. 2014; Dimcic 2011; Dini et al.
2013; Douthe et al. 2006) testify to the interest of the scientific
community, and at the same time, the need to create efficient
and robust design tools involving innovative design strategies.
The present paper has the two-fold aim of examining the influences of design parameters on the optimization process of gridshell structures and an efficient structural-optimization approach
that combines three different strategies: FF, sizing optimization
(SO), and topologic optimization (TO). The main peculiarity of
the proposed approach is that each strategy was considered a
phase of a global-optimization process in which the design parameters and structural requirements were opportunely selected
for each phase and converted into variables and constraint conditions of the optimization problem.
Before presenting the proposed approach, the selected optimization strategies were considered both individually and in combination throughout a simple sequence of optimization phases that were
all characterized by the same set of constraints in terms of displacements and utilization ratios (i.e., the ratio between the stress and
strength of the members). In particular, considering some canopy
case studies derived from the current literature, the optimization
strategies were applied by accounting for different design parameters (boundary restraints, grid densities, ranges of cross-section diameter) and considering specific criteria for introducing these parameters into the optimization process. The results that emerged
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from these analyses are reported in the first part of the paper. The
analyses were finalized to obtain information concerning the influence of the design parameters and structural requirements on the
performance of the selected optimization strategy.
The evidence that emerged from this group of preliminary analyses also accounted for the degree of effectiveness of the proposed
approach. By considering the same canopy case studies, the results
derived by applying the proposed approach, which combines the
optimization strategies created by opportunely selecting the constraint conditions, are presented in detail in the second part of the
paper. Then, the results of these analyses are compared with the
ones derived from the preliminary ones.
In comparison to the approaches in which FF, SO, and TO optimization strategies are taken singularly or combined together
throughout a simple sequence of phases, all characterized by the
same set of design constraints, the proposed approach allows for
lighter structural solutions for grid shells.
The structure of the paper is quite schematic and describes stepby-step the constraints and the objective functions adopted in each
of the analyzed strategies. In fact, after the “Introduction,” the
subsequent paragraphs describe the case studies, the traditional
optimization strategies, and the proposed approach by specifically explaining the underlying roles of the design parameters. In
particular, for both the traditional optimization strategies and the
proposed approach, the constraints and objective functions are
clearly stated. Indeed, the efficiency of the proposed approach
becomes clearer as the paper presents the work because of the
advantage gained through comparisons of it with the basic
strategies.
Case Studies
The case studies analyzed in the paper come from recent research
carried out by Richardson et al. (2013), who presented some interesting results concerning a coupled FF and grid-optimization
approach for single-layer grid shells. In particular, the case studies consist of grid-shell canopies that cover a surface area of
24 24 m, with a geometry of a 6 6, flat, square grid composed
of nodes equally spaced by 4 4 m (Fig. 1).
Three schemes characterized by three different boundary conditions were considered: The first is pin supported on all four sides of
the canopy (R4); the second is pin supported on three sides of the
canopy (R3); the third is pin supported on two opposite sides of the
canopy (R2). The three described canopy schemes present different
values for the peak height (Richardson et al. 2013): 5, 5.85, and
4.728 m for R4, R3, and R2, respectively. In addition, all the
described schemes are characterized by hollow circular-steel cross
sections with a S355-steel-grade material: The yield strength, fyd,
was 355 MPa and Young’s modulus, Es, was 210 GPa. A distributed
gravity load equal to 3 kN/m2 was considered for evaluating the
nodal vertical forces determined during the numerical analyses.
Optimization Strategies and Role of
Design Parameters
The first part of the paper is mainly devoted to showing the analysis
of the influence of some design parameters on the optimization process by FF, SO, and TO strategies. In particular the following
design parameters were considered:
• Boundary constraint schemes R2, R3, and R4;
© ASCE
Fig. 1. Grids described in the paper: (a) 6 6; (b) 3 3; (c) 12 12
Density of the grid composing the canopy with 6 6 [as in
Richardson et al. (2013)], 3 3, and 12 12 grid schemes
(Figure 1);
• Diameters used during the optimization process, including a
single diameter (i.e., same cross section) for all the members
of the canopy (solutions in terms of d1), and three different
diameters used in the canopy (solutions in terms of d3); and
• Criteria for assigning the three diameters among the members
composing the canopy during the optimization process: a criterion based on the orientation of the members, horizontal elements, vertical elements, diagonals (solution denoted d3a); a
criterion based on the utilization ratio U of members, U < 33%,
33% ≤ U ≤ 66%, U > 66% (denoted in the following d3b).
All these parameters were then combined together leading to different structural solutions to consider during the optimization
process.
•
Form Finding
FF was the first strategy considered for the shape derivation of the
described canopies. In particular, to derive an optimal shape for the
canopies, the shape corresponding to the hanging model was determined by considering the three different grid densities for each of
the three different restraint schemes. These shapes were derived
using two different computer codes and selecting different modeling and numerical strategies.
The first-used approach refers to the dynamic relaxation method
(Schek 1974; Day 1965), which is implemented in Kangaroo. In
this case, the optimal shape was set by the level of internal prestress
and on the basis of the boundary supports.
The second approach was performed by using a commercial code
in SAP2000. In this case, the canopies were modeled by using cable
elements instead of truss elements. Second-order incremental analyses were performed by introducing a large-displacements option.
Applications of the different approaches and computer codes
provided the same results in terms of the canopy shapes (Fig. 2).
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Fig. 2. Shapes derived after the FF approach by grid scheme: (a)
3 3; (b) 6 6; (c) 12 12
Nevertheless, more expensive computational time and convergence
drawbacks emerged in the case of the cable-modeling approach.
Furthermore, it was observed that the shapes of the analyzed
canopies were significantly influenced both by the restraint boundary conditions and, moreover, by the grid density.
As emphasized in Tomei (2017) and Grande et al. (2016), the
shape based on the hanging model allows for particular reductions in the structural weight while preserving good levels of
global stiffness. For these reasons, the canopy shapes derived
from this phase were adopted for performing the other optimization strategies.
Sizing Optimization
Taking into account the shapes of canopies derived from the FF, the
SO was performed by considering the different boundary restraint
schemes (R2, R3, and R4), the different grid densities (3 3, 6 6,
and 12 12), and moreover, the range of diameters (d1, d3) and the
criteria for assigning the diameter to canopy members during the
optimization process (d3a, d3b).
The numerical analyses were developed through the commercial software Karamba, a finite-element plug-in developed for
Grasshopper and fully embedded in the three-dimensional (3D)
modeling software, Rhinoceros. Karamba allows for interactive
calculations on the responses of 3D structures while considering
the parametric environment of Grasshopper, and then, it shows all
the potentialities in terms of geometric modeling and structural
optimization. Moreover, it introduces user subroutines for both
the preprocessing and postprocessing phases.
A mono-objective genetic algorithm strategy was used for
developing the SO process according to the following parameters:
population size of 50, crossover rate of 0.8, and mutation rate of
0.1. In particular, constraints in terms of maximum displacement
(Dmax ≤ 0.12 m) and the maximum utilization ratio of members
(Umax ≤ 1, which takes into account both the strength and local
buckling), were both introduced. In addition, the structural
weight, W, was considered the objective function to minimize
during the SO (Fig. 3).
The results derived from the SO strategy are presented in Figs. 4
and 5 in terms of weight and of maximum displacement normalized
with respect to the limit value (Dmax), respectively. The figures
show that the SO provides better results in terms of weight when the
three diameters were considered and assigned according to the criterion that was based on the utilization ratio (solutions in d3b). This
© ASCE
Fig. 3. Flowchart of the optimization strategies
evidence is common to all the restraint schemes and to all the mesh
densities.
All the schemes were characterized by a maximum displacement
value lower than the admissible one. This finding means that the
constraint condition influencing the SO was the utilization ratio of
the canopy members.
The results also reveal that both the mesh density and the
restraint scheme influenced the weight and the stiffness of the solution derived at the end of the SO. In particular, although it was
observed that an increase in the mesh refinement provided a reduction in the weight, the influence of the boundary restraints depended
on both the number of restrained sides of the canopy and the configuration of the restraints. Indeed, the solutions for the restraint configuration R3 (three sides restrained) were characterized by greater
values of structural weight than were the solutions for R2 (two sides
restrained); however, the solutions corresponding to the restraint
configuration R4 (four sides restrained) showed lower values of
structural weight than either the R2 or R3 scheme configurations
did. This evidence is common to both criteria selected for assigning
the diameters to the members.
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Fig. 4. Weight derived from application of only the SO strategy by diameter criteria: (a) d1; (b) d3a; (c) d3b
Sizing-Optimization and Topologic-Optimization
Approach
Considering the previously obtained solutions of the canopies
derived through the FF and the subsequent SO strategy, an additional optimization strategy was analyzed. In particular, a TO based
on the removal of some of the diagonals of the grid shell was performed by considering the same computer code and parameters,
objective function, and constraints of the mono-objective genetic
algorithm as were used for the SO strategy. For this optimization, a
sequence composed of a preliminary FF phase and two subsequent
optimization phases, where both SO and TO were based on the
same objective function and constraint conditions, were used to
derive a final solution that is characterized by a reduced number of
© ASCE
Fig. 5. Displacements derived from application of only the SO strategy by diameter criteria: (a) d1; (b) d3a; (c) d3b
diagonals than either the solution derived from the FF or from the
SO. For the sake of brevity, the proposed strategy that was applied
to the 6 6 mesh for the three restraint schemes is the only one
presented.
The results obtained from the TO approach are reported in Fig.
6 in terms of structural weight: The obtained solutions were compared against the corresponding solutions derived by applying
only the SO. Moreover, Fig. 7 shows the grid configurations
obtained at the end of the optimization process (i.e., after the TO
phase).
From Figs. 6 and 7, it can be observed that, despite a reduction in
the number of diagonals composing the grid, which depended on
both the restraint scheme and the range of diameters and criteria of
assignments used for the SO, the introduction of the TO led to a
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Fig. 7. Grid configurations at the end of the SO followed by the TO
approach: (a) SO(d1)-TO; (b) SO(d3a)-TO; (c) SO(d3b)-TO
Fig. 6. Comparison of solutions derived from the singular SO strategy
and from the SO strategy followed by the TO strategy by specific criteria: (a) same cross-section sizes for all the members; (b) orientation of
members; (c) utilization ratio of members
slight reduction in the structural weight. This effect particularly
depended on the parameters considered for developing the SO: The
lowest reduction of the weight corresponded to the d3b case, for
which the SO depended on a range of three diameters and a criterion
that was based on the utilization ratio of members.
Mixed Sizing-Optimization and Topologic-Optimization
Approach
Because the introduction of the TO did not cause a significant
reduction in the structural weight with respect to the approach that
© ASCE
was based only on the SO, another approach was used that combines the FF, SO, and TO approaches and is referred to as mixed SO
and TO approach; it is composed of four phases (Fig. 3): a preliminary FF phase for deriving the shape; a SO phase based on the same
cross-section size for each member (d1), which differs from the
sizes used in the previous approach; a TO phase; and a final SO
phase developed by considering only the d3a and d3b criteria. In
particular, all phases of the process that were based on the SO and
the TO were developed by considering the same objective function (i.e., the structural weight, W) and the same constraint conditions (i.e., the maximum displacement, Dmax, and the maximum
utilization ratio, Umax) as introduced in the mono-objective
genetic algorithm.
The results derived at the end of the SO phase, introduced after
the TO phase, are reported in Fig. 8 in terms of weight and in comparison to the other analyzed approaches. From the comparisons, it
is clear that, despite some cases in which the mixed SO and TO
approach led to lower structural weights, in some cases, the mixed
SO and TO approach led to solutions of greater structural weight
than the other approaches.
Considerations
At the end of this, the first part of the paper, the roles of some design
parameters in the context of different optimization strategies and
approaches were described. In particular, it was observed that both
the restraint scheme and the grid density affected the hangingmodel shape derived from the FF phase. Moreover, these parameters, together with additional ones introduced in the optimization
process in terms of variables and constraint conditions, also influenced the solutions derived from both the SO and the TO strategies.
Another important aspect that emerged from the first part of the
study was the low level of effectiveness in reducing the structural
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Fig. 8. Comparison of the solutions derived from the SO, SO followed
by the TO, and mixed SO and TO approaches by specific criteria: (a)
orientation of members; (b) utilization ratio of members
weight when a process based on a simple sequence of optimization
strategies was used. This result seems to suggest the use of SO and
TO strategies separately—without combining them—is the better
approach. Nevertheless, it is important to emphasize that the
approaches combining the SO and TO, presented in this first part of
the paper, were based on a simple sequence of SO and TO phases
from which the solution was derived from an optimization phase
that was used for developing the subsequent solution. All these
phases were, indeed, developed by considering the same constraint
conditions.
As shown in the following part of the paper, the peculiarity of
the presented approaches is the main reason for their low effectiveness in reducing the structural weight of the canopies when the SO
and TO strategies are combined together.
Proposed Approach
In remainder of the paper, an optimization approach that draws only
from the combination of different optimization strategies is
proposed.
The proposed approach is similar to the mixed SO and TO
approach presented in the section with of the same title. It is based
on the same sequence of phases but different constraint conditions
that were adopted for each of the optimization phases composing
the entire process.
In particular (Fig. 3):
© ASCE
Fig. 9. Comparison of solutions derived from the SO, the SO followed
by TO, the mixed SO and TO, and the proposed approaches by specific
criteria: (a) orientation of members; (b) utilization ratio of members
Fig. 10. Grid configurations at the end of the proposed approach
1. The FF strategy was preliminary considered for deriving the
shape of the canopy according to the hanging model.
2. The SO strategy was applied according to the solution derived
from the FF: The same cross-section size was used for all the
members (d1); constraint conditions were introduced on both
the displacements and the utilization ratios; and an objective
function, in terms of structural weight, was adopted (i.e., the
same as in the mixed SO and TO approach).
3. The TO strategy was applied by considering the solution
derived from the previous SO strategy solution. Different from
the mixed SO and TO approach, the constraint conditions only
for displacements were introduced. An objective function in
terms of structural weight was adopted.
4. The SO strategy was applied by using the solution derived
from the previous TO strategy. Three different cross-section
sizes and two criteria for assigning them (d3a and d3b) were
considered. Different from the mixed SO and TO approach, the
constraint conditions that concern only the utilization ratios
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Fig. 11. Comparison of solutions derived from the described strategies and approaches
© ASCE
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were introduced, and an objective function in terms of structural weight was used.
Then, the differences with respect to the mixed SO and TO
approach that concern only the type of constraints were introduced
into the two last optimization phases. Nevertheless, despite these
few differences, the results presented in Fig. 9 in terms of structural
weight and in Fig. 10 in terms of grid configuration show that the
proposed approach led to a significant reduction in the number of
diagonals after the TO phase, and it also led to a significant reduction in the structural weight compared to the other approaches considered in the study.
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Conclusions
The optimization of structures involves strategies and mathematical algorithms for which the design parameters, introduced in
terms of variables, constraint conditions, optimization functions,
penalty conditions, and so forth, play a very important role. Their
selection and introduction in the optimization process can influence the characteristics and the level of optimization of the
derived solutions.
In this study, the roles of the design parameters in the context of
different optimization strategies were analyzed. Moreover, an efficient optimization approach that combines the FF, SO, and TO
strategies into a multilevel process, with design variables and constraint conditions opportunely selected, was proposed and assessed
with reference to case studies derived from the literature.
When compared with the usual optimization approaches, in
which the FF, SO and TO optimization strategies are performed singularly or combined together throughout a simple sequence of
phases and all characterized by the same set of design constraints
(stress, displacement, buckling, etc.), the proposed approach is
shown to obtain lighter structural solutions for grid shells (Fig. 11).
Indeed, it was observed that the introduction of constraints that concerned only the displacements into the TO phase led to a substantial
reduction in the number of diagonal members composing the structural scheme. Furthermore, the subsequent SO phase, which used
the solution that emerged from the TO phase and for which constraint conditions were introduced that only concerned the utilization ratio of members, led to a greater reduction of the global structural weight of canopies than the other approaches.
The numerical analyses presented in the paper clearly highlight
the role of the selected design parameters on the structural design
optimization process. For these parameters, the role of the grid density, which influenced both the FF process and the SO and TO optimization solution, was investigated. Indeed, this parameter represented the discretization of the shell surface of the canopy so it
necessarily accounts for the derivation of the optimal shape.
Moreover, it also represents a parameter influencing the length and
slope of the members composing the structural solution better than
the mixed TO and SO process did.
Furthermore, the parameters described in the paper were concerned with the chosen number of diameters as variables of the
problem, and, in particular, the criterion for distributing them during
the SO process. Indeed, it was observed that an increase in the range
of diameters and the selection based on the stress ratio of members
allowed for lighter solutions than were found with use of the other
criteria analyzed in the study.
The paper shows the potential for combining different optimization strategies. Nevertheless, it also revealed the importance in managing the design parameters in terms of variables, constraints, and
objective functions of the mathematical optimization problem, particularly when different optimization strategies are combined.
© ASCE
Indeed, although the solutions derived at the end of each phase by
using genetic algorithms mathematically respect the constraints and
represent a minimum for the objective function, the final solution at
the end of the entire process does not represent the most optimized
structural design solution.
The proposed approach, based on both a combination of different optimization strategies and on a selection of constraints,
improved the level of optimization of canopies in terms of structural
weight with respect to the other approaches.
Finally, although they are not considered in the present research,
the influences of the joints and global buckling, which represent
additional aspects, could play an important role in the optimization
process of grid shells. Indeed, the mechanical behavior, dimensions,
and construction aspects of member-to-member connections can
significantly affect the selection of members and the construction
sequences and the shape of the grid shell. Indeed, some curvatures
are unfeasible because of the rigidity and the limited strength of the
connections. The proneness to global buckling of grid shell solutions derived from only considering the local buckling phenomena
of members requires the introduction of modifications, generally in
terms of the cross-sectional areas of the canopy members.
Aspects of both joints and the global buckling are the subjects of
recent research carried out by the authors [Grande et al. (2017),
“Role of global buckling in the optimization process of grid shells:
design strategies,” submitted to Engineering Structure by Elsevier].
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