Automated optimal design of double- layer latticed domes using

Automated optimal design of doublelayer latticed domes using particle swarm optimization M. Babaei & M. R. Sheidaii

Structural and Multidisciplinary Optimization ISSN 1615-147X Volume 50 Number 2 Struct Multidisc Optim (2014) 50:221-235 DOI 10.1007/s00158-013-1042-2

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Author's personal copy Struct Multidisc Optim (2014) 50:221–235 DOI 10.1007/s00158-013-1042-2

RESEARCH PAPER

Automated optimal design of double-layer latticed domes using particle swarm optimization M. Babaei · M. R. Sheidaii

Received: 1 March 2013 / Revised: 29 September 2013 / Accepted: 21 December 2013 / Published online: 12 March 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract Most of the conventional design methods of large-scale domes need deep engineering insight; furthermore, they hardly give the most economical solutions. Therefore, in this paper, a new practical design algorithm is presented to automate optimal geometry and sizing design of the latticed space domes through the idea of using parametric mathematical functions. Moreover, a simple approach is developed for the optimal sizing design of trusses with outsized number of elements. The robust technique of particle swarm optimization is employed to find the solution of the propounded optimization problem. Some numerical examples on the minimum weight design of several famous domes are provided to demonstrate the efficiency of the proposed design algorithm. Keywords Space structure · Latticed dome · Truss · Automated design · Sizing/geometry optimization · Design algorithm · Particle swarm optimization · Minimum weight

1 Introduction The trend followed by researchers over the past few years shows a significant tendency toward the application of new-born evolutionary algorithms like particle swarm optimization (PSO) and genetic algorithms (GA) in engineering design (Hasac¸ebi et al. 2010; Lagaros et al. 2002; Saka 2007a). These metaheuristic search techniques provide us M. Babaei () · M. R. Sheidaii Department of Civil Engineering, Faculty of Engineering, Urmia University, Urmia, Iran e-mail: [email protected]; mehdi babaei [email protected] M. R. Sheidaii e-mail: [email protected]

with powerful tools for solving complicated multi-variable optimization problems. In this regard, civil engineers are recently to utilize these techniques in order to decrease the constructional cost of large-scale skeletal structures and their weight as well (Dede et al. 2011; Wang et al. 2002). In general, the skeletal weight of structures can be effectively reduced by optimal design of their configuration, considering their strength and serviceability requirements. Therefore, the optimal design of structures leads to the configuration that no better one exists under the given conditions (Lagaros and Papadopoulos 2005). Generally, the optimal design of the pin-jointed structures, as well as the space domes studied in this research, can be classified into three main categories: (i) sizing, (ii) geometry, and (iii) topology optimization. In the sizing optimization, the cross section areas of bars are taken as the design variables of problem. In the geometry optimization, the nodal positions of joints are considered as the design variables and in the topology optimization, optimal connectivity between nodes is to be determined in an optimal way (Deb and Gulati 2001). However, it is too difficult to manually find the global optimum in multi-variable nonlinear problems particularly in the complicated structural design cases. Consequently, engineering researchers are mostly interested in automation of this design process (Bennett 1978; Patnaik and Srivastava 1976). This is also pursued in the current study for designing large space domes. Domes are pin-jointed structural systems that consist of one or two layers of bar elements which are arched in all directions (Makowski 1984). They provide a graceful alternative to cover large and unobstructed areas like stadium and sports arena in which any intermediate obstacle prevents the structure from its proper performance (Ramaswamy et al. 2002). Now that domes are mostly constructed in large scales for covering extended areas, their optimal design

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leads to a vast amount of saving in project cost. Consequently, this field of engineering design has been the subject of many researches for years. Recently, numerous papers have been published on the application of the evolutionary algorithms in the structural design and, in particular pin-jointed structures. Some of them employed GA (Togan and Daloglu 2006; Tang et al. 2005; Balling et al. 2006; Soh and Yang 1994; Cheng 2010; Erbatur et al. 2000; Goldberg 1989), and many others preferred other evolutionary techniques such as simulated annealing, harmony search, and ant colony methods (Sonmez 2011; Degertekin 2012; Lamberti 2008; Hatay and Toklu 2002; Hasancebi and Erbatur 2002; Lee and Geem 2004; Lee et al. 2005) as well as particle swarm optimization (PSO) (Perez and Behdinan 2007; Li et al. 2007, 2009). PSO, which is implemented in this study, is a populationbased paradigm inspired by the behavior of a colony or swarm of living systems (Singiresu 2009). Perez and Behdinan (2007) presented the background and implementation of a particle swarm optimization algorithm suitable for constraint structural optimization tasks. Li et al. (2007) developed a heuristic particle swarm optimizer (HPSO) algorithm for truss structures with discrete variables. Saka (2007b), Kameshki and Saka (2006), Carbas and Saka (2009) mainly investigated the optimal design of the singlelayer latticed domes. Nevertheless, due to stability consideration, single-layer domes are rarely built in practice and most of large-scale domes should be constructed in doublelayer configuration to prevent buckling. The single-layer large-span domes mostly present three distinctive stability problems: (a) overall buckling, (b) snap-through or local buckling and (c) member buckling which is considered in member design (MacGinley 1998). However, in spite of the high practical importance of the double-layer domes, limited researches have been carried out on their optimal design. Consequently, in the current study, this family of domes is taken under consideration. We aim to develop a general design algorithm to automate the geometry and sizing design of these domes. Then, the geometrical layout of the dome is achieved by adopting

a

b

parametrically combined mathematical functions. The sizing design is performed using a straightforward strategy which is introduced to be applied in the optimal design of bar structures with outsized number of elements. Three governing constraints involved in designing the elements are: displacement, stress, and buckling constraints. In the formulation of the problem, the numerical values of these constraints are adopted from AISC code (2005).

2 Configuration of double-layer ribbed and lamella domes The double-layer latticed domes can be constructed in various topologies among them the ribbed and lamella domes are the two most well-known configurations. The ribbed domes consist of ribs or meridional members and some parallel rings or hoops extending around the dome. The ribs and hoops are connected together to constitute trapezoidal panels on the surface of the dome. Shown in Fig. 1 is the plan view of a trimmed double-layer ribbed dome. The lamella dome has a diagonal pattern. Diagonals are extended from the crown down in both clockwise and counter-clockwise. They may have horizontal rings or not. The pattern of a trimmed double-layer Lamella dome is shown in Fig. 2. It is noted that all the rings have equal distance from each other in the plan of the structure. In order to avoid the well-known problem of “overcrowding” of members at the crown of the domes, the upper parts of the domes are cut. Such an operation on the configuration of a dome is referred to as ‘trimming’ and resulting dome is called “trimmed” dome (Nooshin and Disney 2000). Additionally, the domes are usually constructed on circular or regular polygonal bases where apexes touch the circumscribing circle. Other base shapes can also be used (MacGinley 1998). However, the presented algorithm covers various patterns of the base shapes. This advantage is due to the fact that all base shapes will be initially mapped on a unit circle where the geometry functions operate to find the optimal geometry of the domes.

c

d

Fig. 1 The plan view of a trimmed ribbed dome a The bottom layer b The web layer c The top layer d The layers all together

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c

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Fig. 2 The plan view of trimmed lamella dome a The bottom layer b The web layer c The top layer d The layers all together

3 Mathematical formulation

constraint. All the three constraints in their normalized form can be computed as:

A pin-jointed space truss with fixed number of nodes, nn , and members, nm , is to be optimally designed. Its weight function is considered as the objective function of the optimization problem. Since the sizing and geometry optimization problems are not linearly independent (Deb and Gulati 2001), it would be reasonable to perform both operations simultaneously. Thus, in this research, the geometry optimization is carried out together with the sizing one. Design variables incorporate the geometry and sizing variables. In general, the optimal weight design of bar structures can be formulated as: X = {x1 , x2 , . . . , xn } , xil < xi < xiu , i = 1, 2, . . . , n minimize : W (X) =

nm 

(1)

δj,k −1≤0 , δu(j,k) k = 1, 2, 3

j = 1, 2, . . . , nn ,

λi −1≤0 λui

,

i = 1, 2, . . . , nm

(4b)

σi −1≤0 σui

,

i = 1, 2, . . . , nm

(4c)

(1) gj,k =

gi(2) = (3)

gi

=

(1)

(4a)

(2)

(3)

In the above relations, gj,k , gi and gi represent the displacement, slenderness ratio, and stress constraints, respectively. The rest of this section briefly explains the sizing and geometry design approaches employed in this work for modeling the problem.

(2)

ρi Li Ai

3.1 Optimal geometry design

i=1

subjected to: δj,k ≤ δu(j,k) , j = 1, 2, . . . , nn , k = 1, 2, 3

(3a)

λi ≤ λui

,

i = 1, 2, . . . , nm

(3b)

σi ≤ σui

,

i = 1, 2, . . . , nm

(3c)

where W(X) is the weight function and ρi , Ai , Li are the material density, the cross section area, and the length of ith element, respectively. The three last inequalities introduce the problem design constraints which were mentioned before. A solution X is a vector of n decision variables which are allowed to take a value within the lower xil and upper xiu bounds. These bounds constitute a n-dimensional decision space which is called design space. δj,k and δu(j,k) are respectively the computational and allowable deflection of node j in k direction λi is the slenderness ratio of ith member and λui is its upper bound. σi and σui denote the computational and allowable stresses of member i, respectively. Since the cross section areas are selected from a given list of profiles, there is no need to include its bounds as a

Generally, in the geometry optimization problems of trusses, the structural topology and the cross section areas of bars are assumed to be unchanged (Saka 2007a), and the nodal positions of joints are taken as design variables. The same approach can be followed for space domes which are a 3D type of truss structures. However, employing this approach in large domes -having numerous rings- increases the number of design variables,besides, the plurality of these variables is not desirable at all while working with the heuristic algorithms. This deficiency becomes more distinguished while investigating the multi-layer domes with sizable number of nodes. For example, if we gather all nodes of a ring in one group, the number of the geometry variables will be equal to the number of rings. Now, by changing the topology and increasing the number of rings, the dimension of design space proportionally increases and its boundaries will be altered from one problem to another. In these cases, although grouping of the nodes can mitigate the complexity of the problem to some extent,still, it does not completely solve the arisen problem. On the other

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hand, the outline of the common domes usually enjoys a distinguishable symmetry which can be considered in their optimal geometry design. It means that since most of the domes are axially symmetric around their vertical axis passing through their crown, the idea of using math functions possessing this property is found to be useful. This idea was first introduced and tested on a case study by the authors (Babaei and Sheidaii 2013). Now, it seems that applying this approach to the geometry design of the conventional domes will present some significant advantages. First, changing the number of the nodes will not affect the number of the geometry variables any more. Second, there is no need to perform grouping on nodes at all. And third, the design space of dome geometry will be unified for the different design cases. To put this idea into practice, we detected the noteworthy similarities between the layout of several simple math functions and the common domes. Then, numerous functions consistent with the symmetry of the domes were constructed and tested. All these functions should be defined so that they can be molded into any inverse bowl-shape surface. Then, a wide variety of polar functions were checked, and their layout variations were observed while their coefficients were changing. Finally, the following equations were found as the simplest functions which can be utilized for this purpose. These functions have been chosen among many others because they give far better results while testing on a variety of domes with different characteristics: z1 = a1 .r a2

(5)

Exponential function: z2 = a1 .a2r

(6)

Power function:

Polynomial function:

z3 = a1 .r + a2 .r 2 + a3 .r 3

Hyperbolic functions: z4 = a1 . sinh(a2 .r a3 )   z5 = a1 . cosh a2 .r a3

(7) (8) (9)

Although, each of these functions can be individually implemented for searching the optimal geometry of the domes, but it is also found out that their linear combination works even more efficiently:  Z= zi (r, αj ) (10) However, not all the combinations can work with the same efficiency. Accordingly, the following equations are identified as the most appropriate combinations of these functions:   Z1 = a1 r + a2 r 2 + a3 r 3 + a4 cos h a5 r a6  a  + a7 sin h a8 r 9 (11) Z2 = a1 r a2 + a3 a4r + a5 cos h(a6 r a7 )

(12)

Z3 = a1 r + a2 r 2 + a3 r 3 + a4 r a5 + a6 a7r

(13)

where a1 , a2 , ..., a9 are the unknown parameters -either positive or negative- of the problem and the radius r is the mapped polar coordinate of a node onto the inner region of a unit circle. The mapping process enables the functions to operate independently from the plan shape and dimensions of the domes. Therefore, through this mapping, it makes no difference for the plan of the domes to be circular, polygonal, or even elliptical. All of them are initially mapped onto an identical unit circle to be treated the same way. This operation can be simply performed using the following relations: x=

r=

X , R

y=

 x2 + y2,

Y R

(14)

θ = tan−1

y  x

(15)

where R is the radius of the plan; X and Y are the Cartesian coordinates of the nodes; x and y are the mapped coordinates; r and θ are the polar coordinates corresponding to x and y. These components are used in the geometry functions to calculate Z-coordinates of the joints. While the parameters of the functions in an evolutionary process are altering, the functions explore the optimal geometry of the structures. They reshape so that they fit the shape corresponding to the minimum weight. Here, the optimization process has the task of finding the set of the unknown parameters which corresponds to the optimal shape. To have a clear mind to how the geometry of the domes changes through the introduced parameters, several typical surfaces with various values of the parameters are presented in Figs. 3 and 4 for the functions given in (5–6), respectively. In these equations the first parameter mainly concerns the rise of the domes but the second one directly affects the convexity of the domes. In the figures, the value of a1 is assumed to be unchanged while various values for a2 are presented to show different possible layout of these functions. These figures demonstrate the flexibility of these functions in generating various convexities and heights of the domes. The height of the bottom layer is assumed to be a ratio of the height of the upper one: Zbot = k.Zt op

(16)

where Z t op is the shifted surface of the functions Z = Z1 , Z2 , or Z 3 which is moved vertically so that its base ring is located at the level Z = 0. This can be easily achieved through the following: Zt op = Z − Min(Z)

(17)

So, the joint coordinates in the upper layer will be (X,Y,Z t op ) and those of in the bottom layer are (X,Y,Z bot ). Furthermore, a design variable is assigned to the ratio k to be optimally

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225 a1=-4000 , a2=1.5, k=0.75

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z

4000

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1000

0 1

0 1

1 0 y

1 0

0 -1

-1

1 0

0 -1

y

x

-1

y

x

a1=-4000 , a2=3, k=0.75

a1=-4000 , a2=2.5, k=0.75

4000

3000

3000

3000

2000

2000

2000

z

4000

y

0

0 -1

-1

y

x

Fig. 3 Perspective views of the function Z = z1 =

x

1

1

1 0

-1

0 1

0 1

0 1

-1

1000

1000

1000

0

a1=-4000 , a2=3.5, k=0.75

4000

z

z

a1=-4000 , a2=2, k=0.75

4000

z

z

a1=-4000 , a2=1, k=0.75

a1 .r2a

0 -1

-1

x

0 y

0 -1

-1

x

for a1 = −4000, k = 0.75 and various values of parameter a2

determined in the final configuration of the domes. The bottom layers in Figs. 3 and 4 are drawn for a fixed value of k = 0.75. 3.2 Optimal sizing design Although, in theory, it is possible for the evolutionary algorithm to manage problems with any number of variables, large number of these variables reduces the viability of the metaheuristic algorithms. In other words, lengthened string of solution would prevent searching individuals from sufficient flows of information (Togan and Daloglu 2006). In such a condition, the algorithms may fail to find the overall optimum and trap in local optima while exploring large search spaces. Besides, large search spaces require more time and computational efforts to be appropriately explored. To mitigate these difficulties, member grouping is the most known approach propounded in the sizing design problems. Hence, several distinctive patterns are presented to perform this grouping process. Saka (2007b), Kameshki and Saka (2006), Carbas and Saka (2009) usually collect the members having similar positions in one group. They do this task manually before the evolutionary algorithm start working, and the member groups do not change during the design procedure. Afterwards an efficient grouping and regrouping strategy is recently introduced by Krishnamoorthy et al. (2002) and Sudarshan (2000) and then adopted by Togan

and Daloglu (2006). In this grouping strategy, structural members are categorized according to their axial forces. Then, while the geometry and consequently the axial forces of members are changing in the optimal design, regrouping is performed to change the element group according to their current axial forces. In both of the mentioned works, the more the member groups are, the less the structural weight will be. It means that we have to increase the number of member groups to achieve lighter structures even in single-layer sparse domes. Therefore, these strategies were practically inappropriate for dealing with thousands of elements in double layer domes since they require numerous member groups to provide a light weight structure. On the other hand, when members are categorized based on the position or axial force, the element slenderness ratio which plays a dominant role in their strength is neglected. Furthermore, these strategies need to be manually revised and manipulated from one design problem to another. While working on the dense domes, we observed that the previous approaches can be replaced with a simple and efficient one. Alternatively, a modified general approach is proposed in this study to resolve the present deficiencies and automate the sizing design process of the domes. To illustrate, consider we have 30 pipe sections P1 , P2 , . . ., and P30 ; and in addition, it is desired to have just 4 member groups throughout the structure which may have thousands

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a1=-4000 , a2=1.5, k=0.75

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x

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z

2000

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0 1

0 1

0 1

1 0 y

-1

y

x

Fig. 4 Perspective views of the function Z = z2 =

-1

-1

x

1

1 0

0 -1

0

a1=-4000 , a2=1.7, k=0.75

a1=-4000 , a2=1.6, k=0.75

z

z

a1=-4000 , a2=1.4, k=0.75

a1=-4000 , a2=1.3, k=0.75

a1=-4000 , a2=1.2, k=0.75

a1 .a2r

0 -1

-1

x

0 y

0 -1

-1

x

for a1 = −4000, k = 0.75 and various values of parameter a2

of members. Then the structure may be designed using one of the following cross section sets: (P3 , P4 , P15 , P22 ) , (P8 , P10 , P14 , P17 ) , (P1 , P3 , P14 , P27 ) , (P11 , P14 , P17 , P29 ) , . . . , (P1 , P5 , P15 , P22 ) Finally, we will take the set that satisfies all the constraint and at the same time gives the minimum structural weight. In this way the most appropriate set which minimize the weight will be easily found. It is observed that this simple strategy leads to considerably light structures. This idea was the basic motivation of the presented sizing approach. It is clear that creating and evolving the above mentioned sets are the typical task of an evolutionary algorithm. In this regard, PSO is employed to pick up a specified number of discrete cross sections, e.g., 3, 4, 5, or 6 sections, from an introduced list of standard profiles. These selected profiles constitute the input data of a design subprogram which designs the members. It means that, these profiles accompanied with member slenderness ratios and their axial forces will be sent to the design subprogram where all the elements are designed according to the AISC regulations (2005). It is noted that this subprogram uses the smallest sections, among those ones sent from PSO, which fits each member. However, because some of the selected sections by PSO may be left unused in the design subprogram, it may happen to have even less member groups than which

was predicted as its upper bound, i.e., four in the above discussion. This means that the algorithm has the flexibility to decide on the number of sizing groups needed for optimal solution. Therefore, various numbers of sizing groups are checked in a wide variety of possible geometries. Unlike Saka, Togan and Daloglo (Togan and Daloglu 2006; Saka 2007b; Kameshki and Saka 2006; Carbas and Saka 2009), no stand-alone grouping is carried out. Instead, the grouping task will be automatically done while designing process is in progress. In this way, the grouping considerations in Ref. Togan and Daloglu (2006) are also covered in an easy and automated manner. In addition, other determining aspects of the elements like the buckling strength, and slenderness considerations of the members are taken into account. An economical auto-grouping is the first consequence of this approach. Moreover, the number of member groups can be easily fixed on any pre-specified number even in structures with sizable number of members. This method of optimal sizing design can be applied to any other 2D or 3D bar structures such as double-layer grids and barrel vaults.

4 Particle swarm optimization PSO algorithm, originally developed by Kennedy and Eberhart (1995), is a robust global-search technique with emphasis on cooperation. PSO is the engineering model of

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swarm intelligence which is adopted from flock of birds, fish schooling, and swarm of insects (Singiresu 2009). This algorithm has fewer parameters to adjust. It is easy to implement, and it shows a fast convergence speed than the other evolutionary algorithms (Li et al. 2007, 2009). PSO initiates its search constituting a set of randomly created artificial individuals called particles. Each particle is a searching agent of PSO optimizer which flies around a multidimensional search space looking for the best position. It uses its own intelligence and the collective intelligence of the swarm to discover a good path toward food. The rest of the swarm will also be able to track the proper path instantly even if their location is far away in the swarm. In fact, the best position of food is the overall optimum in our engineering problem. The particles of the swarm cooperate with each other to achieve the solution in a search space and they move toward the global optimum by following the current best particle. Each of them has the capability of memorizing its previous personal experience pbest, i.e., its best-so-far position called personal best, and the knowledge of the best neighbor gbest experienced so far by the rest of swarm. Members of a swarm communicate their own information to each other and modify their positions and velocities according to the current positions of pbest and gbest. The mechanism of information transmission in PSO has a one-way flow mechanism (Sivanandam and Deepa 2008). Consider swarm of particles wandering around the parameter space searching for optimum solution. Each particle is characterized by a position vector Xi (t) and a velocity vector Vi (t). Mathematical model of PSO on modification of the particle’s position can be formulated as following:

5 The treatment of the constraints PSO algorithm is an unconstrained optimization technique. Therefore, in constrained problems, it is required to employ one of the known schemes in handling the necessary design constraints. Accordingly, the most famous constraint handling approach of the penalty function is implemented in the current study (Soh and Yang 1996). The penalty function has the task of penalization of violated designs from the determined constraints. Adding the value of the penalty function P(X) to the weight function W(X), the fitness function f (X) of the problem can be written as: f (X) = W (X) + P (X)

(20)

where the value of the penalty function P(X) is calculated as the penalty scheme presented by Rajeev and Krishnamoorthy (1992): P (X) = K.C.W (X)

(21)

where K is penalty multiplier whose value is selected according to the nature of the problem and C is the sum of the normalized violations of the constraints as:  C= ν (n) (22) in which ν (n) is defined as:  (n) (n) g g > 0, n = 1, 2, 3 (n) ν = (n) 0 g ≤0

(23)

and g (n) is the normalized value of the constraint which is computed from (4a–c) depending on the degree of violations for each design.

Vi (t + 1) = wVi (t) + c1 rand1 (pbesti (t) − Xi (t)) + c2 rand2 (gbest (t) − Xi (t)) Xi (t + 1) = Xi (t) + Vi (t + 1)

(18) (19)

where Xi (t) and Xi (t + 1) are the position of agent i at iteration t and t + 1, Vi (t) and Vi (t + 1) are the velocity of agent i at iteration t and t + 1. pbesti , the pbest of the agent i, is the coordinates of the particle (or that set of the design variables) in which the fitness function has had its minimum value up to now. And gbest is that set of the design variables in which the minimum value of the fitness function has so far been observed by all the particles in the swarm. Particle velocities in each direction are bounded to a maximum velocity V max . w is the inertia weight that controls the exploration and exploitation of search space. c1 and c2 , the cognition and social components, respectively, are the acceleration constants which change the velocity of a particle towards the pbest and gbest. Usually, c1 and c2 values are set to 0.5 and 1.5, respectively. Rand1 and Rand2 are vectors with uniformly distributed random numbers between 0 and 1.

6 Proposed problem-solving algorithm In brief, we are to extend a general procedure for the optimal design of the domes so that it can be applied to a wide variety of the domes with different characteristics. In the proposed algorithm we first generate the connectivity data of a desired configuration such as lamella, ribbed, geodesic and so on. It should be noted that the structural topology is assumed to remain unchanged during the optimization process. Then, the nodal coordinates of nodes in the plan of the structure will be determined for the topology under consideration. Now, mapping the nodal coordinates of the joints in the plan of the dome on a unit circle, we can utilize the same operational region for all the geometry functions. So, the bounds of the search space will be established in a way that changing in topology, span, and plan shape of domes with large number of nodes and members has no effect on the search space; furthermore, different domes with any characteristic are designed in the same design space. Next, one of the parametric functions, already introduced for the optimal

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Fig. 5 The basic flowchart of the optimal design of the domes

M. Babaei, M. R. Sheidaii

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Table 1 Design data for examples 1, 2 and 3 Items

Example 1

Example 2

Example 3

Yielding stress (N/m2 )a Density (kg/m3 ) Modulus of elasticity (N/m2 ) Upper and lower bound of area (cm2 ) Max. permissible deflectionsb (mm) Design code Type of sizing variables

4 × 108 7975 2.1 × 1011 20–40 ±10 AISC-ASD Continuous

4 × 108 7975 2.1 × 1011 – ±30 AISC-LRFD Discrete

4 × 108 7975 2.1 × 1011 – ±30 AISC-LRFD Discrete

g = 10 m/s2 in all examples directions

a It is assumed b In all

geometry design, should be sent to PSO algorithm. In this step, choosing a practical and reasonable number of sizing groups will be rather a matter of preference. Subsequently, PSO optimizer is called to find the optimal sizing and geometry related variables. After analyzing the structures created in the preceding steps, elements will be designed according to the analyses output. This process is successively repeated until one of termination criteria stops the algorithm. Then, the optimal design of the dome, which meets all the constraints with minimum weight, will be introduced. A detailed flowchart of the presented design algorithm is shown in Fig. 5.

7 Numerical examples Three design examples of several space domes are investigated using the algorithm presented in this study. First, to provide a comparison between the coded algorithm and the earlier literatures, a single-layer 120-bar dome is designed through the algorithm. This example is merely an academic benchmark rather than a practical design. Then the proposed approach is applied to the two practical examples of doublelayer domes with well-known topologies so-called ribbed and lamella dome. The cognition and social components in PSO are set to 0.5 and 1.5, respectively. The other details

Table 2 Optimum design for 120-bar dome Item

# of member groups in final design Optimum height (m) Number of design variables Size of population Number of generations Max. disp. (mm) Number of design Standard deviation (kg) Weight (kg) The best result

Soh and

Soh and

This study

Yang (1994)

Yang (1996)

Geometry function z1

z2

z3

z4

z5

Z1

Z2

Z3

7

7

7

7

7

7

7

7

7

7

6.985 10

8.050 10

9.96 9

12.01 9

7.796 10

10.51 10

10.47 10

6.840 16

7.21 14

6.94 14

– –

– –

80 4

80 6

80 8

80 11

80 9

80 16

80 16

80 14

– – –

– – –

9.99 4 8.61

9.99 4 16.68

9.72 4 12.84

9.96 4 8.92

10.00 4 1.23

10 5 3.54

9.96 5 10.46

9.94 5 14.5

2659.5

2638.5

2756

2938

2609

2794

2779

2522.0

2536.4

2528.0

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Table 3 Optimum design for 1479-bar ribbed dome Item

Geometry function Z1

Optimum # of member groups in final design # of Repetition in results with the same number of member groups Optimum height (m) # of design variables # of generations Max. disp. (mm) # of design Optimum weight (kg) Standard deviation of results (kg) Weight per unit area of plan (kg/m2 )

Z2

Z3

3 0

4 3

5 1

6 1

3 1

4 3

5 0

6 0

3 3

4 1

5 1

6 0

– 16 – – 5 29939 550 15.247

15.83 16 18 21

14.22 16 12 26.7

16.09 16 12 22

15.30 14 17 28.7 4 29369 943 15.246

13.11 14 11 29.7

– 14 – –

– 14 – –

12.89 14 15 29.7 5 28200 1099 14.362

11.97 14 9 26.8

12.52 14 14 30.01

– 14 – –

needed for each example are given in Table 1. In addition, after applying the algorithm to the numerical examples under consideration with population size of 80 and maximum number of generation set to 20, the results shown in Tables 2, 3 and 4 are obtained. 7.1 Example 1: single-layer 120-bar space dome Figure 6 shows the plan view of 120-bar steel dome, first investigated by Soh and Yang (1994, 1996), Cheng (2010), Erbatur et al. (2000), Goldberg (1989), Sonmez (2011), Degertekin (2012), Lamberti (2008), Hatay and Tokl¨u (2002), Hasanc¸ebi and Erbatur (2002), Lee and Geem (2004), Lee et al. (2005), Perez and Behdinan (2007), Li et al. (2007,

2009), Singiresu (2009), Saka (2007b), Kameshki and Saka (2006), Carbas and Saka (2009), MacGinley (1998), AISC 360-05 (2005), Nooshin and Disney (2000), Babaei and Sheidaii (2013), Krishnamoorthy et al. (2002), Sudarshan (2000), Kennedy and Eberhart (1995), Sivanandam and Deepa (2008), Rajeev and Krishnamoorthy (1992), to find its optimal configuration. Dealing with the sizing design, they categorized the members into seven groups. Accordingly, members with similar position should be collected in one group. Consequently, by taking seven sizing variables, the same grouping will be automatically done through the presented algorithm. This is due to the fact that this structure consists of just seven different types of elements. All the elements in each of these categories have the same length, slenderness ratio, and axial force. Then, if we pick

Table 4 Optimal design for 1488-bar lamella dome Item

Geometry function Z1

Optimum # of member groups in final design # of repetition in results with the same number of member groups Optimum height (m) # of design variables # of generations Max. disp. (mm) # of design Optimum weight (kg) Standard deviation of results (kg) Weight per unit area of plan (kg/m2 )

Z2

Z3

3 0

4 3

5 1

6 0

3 2

4 2

5 1

6 0

3 0

4 0

5 5

6 0

– 16 – – 4 29394 917 14.970

12.87 16 17 26.3

17.11 16 15 19.32

– 16 – –

17.08 14 15 28.36 5 31470 660 16.027

12.46 14 18 22.43

12.63 14 13 29.80

– 14 – –

– 14 – – 5 28818 600 14.676

– 14 – –

12.66 14 16 26.84

– 14 –

Author's personal copy Automated optimal design of double-layer latticed domes

231

Fig. 8 120-bar dome with optimal design Fig. 6 The plan view of 120-bar dome

up seven cross sections from the given profile table, the designing routine will automatically assign each of them to one of these member groups. In addition, to specify the geometry, the proposed approach can be replaced without loss of generality of the problem. In other words, since the height of the rings are just needed to specify the geometry of the dome, it makes no difference to either take them as the direct variables of the problem like Soh and Yang (1996), or use a substitute for this old-fashioned technique. Accordingly, since the main purpose of this example is to confront the new geometry approach with the previous one, we replace their rigidly organized approach with the suggested methodology to compare their efficiency. The span of the plan is 31.78 m. The dome is subjected to gravitational loads of 60 kN at node 1, 30 kN at nodes 2–13, and 10 kN at the rest of the unsupported nodes. The slenderness ratio limit and allowable stresses are calculated according to the AISC-ASD (2005) (see Appendix A). It should be noted that the effective length factor k and the factor of safety FS in these calculation is assumed to be unit. The gyration radii of circular hollow sections are calculated from r = 0.4993A0.6777 (Saka 1990). The optimal geometry of the dome is presented in Figs. 7 and 8 and the obtained results are given in Table 2.

The viability of the geometry functions, Zi and their components, zi , are obvious from the obtained optimal weight. It can be seen that the components by themselves do not yield satisfactorily light structures; however, it is clear from these weights that they all are able to approach the optimal configuration in general. Difference between the weights obtained from the functions zi shows a sort of instability which occurs while we employ these components. This deficiency is completely resolved by combining the components so that the functions Zi work strongly and yield more stable results. It is interesting that the weight of the dome designed with the function z3 is still 1.13 % less than that given in Ref Soh and Yang (1996). Noting that the number of the variables is the same as Ref Soh and Yang (1996), it is concluded that the weight of the dome can be reduced through the presented approach. Although, this occurrence may not be guaranteed for all types of the domes while we are working with each of the component functions; at least it expresses that the proposed idea works properly. Something which is worthy with these functions is that the number of variables remains unchanged, even if the number of rings increases. This facility would not be available if the heights of the rings were taken as the design variables like Soh and Yang (1996). Moreover, the numbers of generations are almost small. It means that use of geometry function results in a fast convergence in geometry design of the domes. 7.2 Example 2: double-layer 1479-bar ribbed dome

Fig. 7 The side view of 120-bar dome

As mentioned before, since both the number of design variables (14∼16) and design space are fixed in the new algorithm for various kind of domes, it allows us to increase number of nodes and members to any desired value without any concern. In other words, the robustness of the algorithm will become clear while we are dealing with the dense domes rather than simple sparse domes like the previously

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investigated 120-bar dome. For this reason, the presented algorithm mainly targets large-scale double-layer domes which have plenty of nodes and members. So, a 1479-bar ribbed dome is considered to be designed in this example. The structure is subjected to vertical gravitational loads of 20000 N at the all unsupported joints of the top layer. This is approximately equivalent to the distributed load of 300 kgf/m2 which is considered to be the maximum value of the combination of the factored loads which can be experienced by the structure during its lifetime under dead and snow loads. The plan view of the considered double-layer ribbed dome is shown in Fig. 1. The objective dome is a 20-segment ribbed dome with 8 rings. It consists of 320 nodes and contains 1479 steel bar elements. Diameter of the plan is 50 m. All layers of the dome are displayed separately in Fig. 9. Tubular steel sections are considered to be used as structural elements. All the sections are selected from an introduced list of standard pipe sections taken from “Steelwork design guide to BS 5950” published by the UK Steel Construction (1990). The material properties and other design details are given in Table 1. Since sizing and geometry optimization are simultaneously carried out, both sizing and geometry-related variables are introduced to PSO to be found at the same time. The allowable tension and compression stresses are computed according to AISC-LRFD (2005) code provision (see Appendix B). The maximum permissible deflections for all nodes are restricted to 30 mm in all three directions. The weight of each element is distributed between its two end nodes. Each design process was repeated several times to reach the results shown in Table 3. The optimal layout of the dome is presented in Fig. 10. The efficiency of the presented approach is clearly distinguished from the results obtained for the optimal weight of the dome.

Fig. 9 The layers of double-layer ribbed dome

M. Babaei, M. R. Sheidaii

Fig. 10 Ribbed dome with optimal design

Considering that the applied load value is almost practical, the structural weight of the final design is obtained to be approximately 15 kg/m2 . This value is much less than the typical weight of these structures which is about 30 to 50 kg/m2 (Kamyab and Salajegheh 2013). It is noted from the results that although the optimum heights varies from 11.97 m to 16.09 m, the integrated geometry and sizing approach could reduce the weight of the dome to 15 kg/m2 . Then, it indicates that the optimal configuration may be found in different possible geometry and its corresponding cross-section set. Moreover, as seen from the results, the final designs are obtained in less than 20 generation. This fast convergence is due to the fact that the math functions rarely generate those geometries which are inappropriately far from the optimal design. Again, this advantage would be more distinctive if we alternatively worked with the heights of the rings as the geometry variables of the design as it was common previously (Soh and Yang 1996). Small value of

Fig. 11 The layers of double-layer lamella dome

Author's personal copy Automated optimal design of double-layer latticed domes

Fig. 12 Lamella dome with optimal design

the standard deviations, which is about 2∼4 %, verifies the stability of the results obtained from the design algorithm. 7.3 Example 3: Double-layer 1488-bar lamella dome The design of double layer Lamella dome shown in Fig. 2 is performed as another example to show the generality of the presented algorithm. A Lamella dome comprising of 1488 steel bars and contains 432 nodes is considered to be designed. It has 24 segments and 6 rings as well. The structure is subjected to vertical gravitational loads of 20000 N at the all unsupported joints of the top layer. This is approximately equivalent to the distributed load of 270 kgf/m2 . Diameter of the plan is 50 m. All layers of the dome are displayed separately in Fig. 11. Tubular hollow steel sections are considered to be used as structural elements. The cross section areas and other assumptions are taken similar to example 2. Moreover, the material properties and some other details are given in Table 1. The final optimal shape of the dome is shown in Fig. 12 and the corresponding results are presented in Table 4. Again, the results in Table 3 verify the previous claims. Something which may be noteworthy with this example is that the optimal weight is obtained very close to the optimal weight of the ribbed dome in the previous example. Therefore, it does not make much difference that whichever configuration is used to cover our 50 m span area.

233

First, in order to test new algorithm, the optimum weight of single layer 120-bar dome is obtained 2522 kg. Compared with what was obtained by Soh and Yang (1996), the proposed algorithm gives approximately a 4.5 % lighter structure. Next, assessment of examples 2 and 3 shows that the number of sizing groups, found for the optimum configuration, has less effect on the weight of the domes and the proposed geometry functions can find a near optimal design even with just three member groups for the studied 1488bar dome. In other words, in all designs repeated for each example, it almost makes no difference whether the optimal number of the member groups is obtained three or six in the final optimal design; however, the geometry functions find the fittest geometry so that their weights all became approximately 30 tones. In summary, the difficulties with the design of the largescale dome structures are discussed in detail and some essential suggestions are made to resolve them. Consequently, in the presented algorithm, all domes are treated in the same way regardless of their topologies, plan dimensions as well as their plan shape, i.e., circular, polygonal or any other shapes. The most significant point with the suggested algorithm is the small dimension of design space required for modeling large-scale problems. It means that in the large-scale problems, the number of the design variables is decreased as much as possible. Less number of design variables leads to a significant level of saving in computational efforts using the evolutionary algorithms. Some other advantages of the algorithm are as follow: •

•

•

• 8 Summary and conclusion A new design algorithm is presented to automate the design process of the large-scale domes. Although this algorithm is tested here on two double layer domes which have almost 1500 members with 320 and 432 nodes, it enjoys the generality of being applied to other domes with even more nodes and members.

•

Use of presented mathematical geometry functions makes it unnecessary to perform node grouping while doing optimal geometry design of large-scale domes. The presented sizing approach automates this process in such a way that there is no need to perform a standalone member grouping. Otherwise, member grouping in a 1488-bar dome would be a very tedious and less practical task while optimizing its sizing design. Regardless of the number of rings, nodes, and members, the total number of design variables is fixed to 14–16, and it does not increase any more. Therefore, increasing the number of nodes and members has no effect on the number of variables. Handling the geometry variables is very easy in the suggested method. The z-coordinates of the nodes are not taken as the geometry variables anymore; therefore, the number of rings does not affect the geometry optimization process of domes any more. The proposed sizing approach is not limited to the dome structures only, and it can be used for other classes of space structures such as barrel vaults and double layer grids.

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Appendix A

References

According to AISC-ASD “American Institute of Steel Construction-Allowable Stress Design”, the permissible slenderness ratio for compression members is considered to be: ki li λi = ≤ 200 (A.1) ri

AISC 360-05 (2005) Specification for structural steel buildings. American Institute of Steel Construction, Chicago Babaei M, Sheidaii M (2013) Optimal design of double layer scallop domes using genetic algorithm. Appl Math Model 37:2127–2138 Balling RG, Briggs RR, Gillman K (2006) Multiple optimum size/shape/topology designs for skeletal structures using a genetic algorithm. J Struct Eng ASCE 132:1158–1165 Bennett JA (1978) Automated design of truss and frame geometry. Comput Struct 8:717–721 Carbas S, Saka MP (2009) Optimum design of single layer network domes using harmony search method. Asian J Civil Eng 10:97– 112 Cheng J (2010) Optimum design of steel truss arch bridges using a hybrid genetic algorithm. J Constr Steel Res 66:1011–1017 Deb K, Gulati S (2001) Design of truss structures for minimum weight using genetic algorithms. Finite Elem Anal Des 37:447–465 Dede T, Bekiro˘glu S, Ayvaz Y (2011) Weight minimization of trusses with genetic algorithm. Appl Soft Comput 11:2565–2575 Degertekin SO (2012) Improved harmony search algorithm method for sizing optimization of truss structures. Comput Struct 92–93:229– 241 Erbatur F, Hasanc¸ebi O, T¨ut¨unc¨u ˙I, Kılıc¸ H (2000) Optimal design of planar and space structures with genetic algorithms. Comput Struct 75:209–224 Goldberg DE (1989) Genetic algorithms in search optimization and machine learning. Addison-Wesley Longman Publishing, Boston Hasanc¸ebi O, Erbatur F (2002) On efficient use of simulated annealing in complex structural optimization problems. Acta Mech 157:27– 50 Hasac¸ebi O, Erdal F, Saka MP (2010) Optimum design of geodesic steel domes under code provisions using metaheuristic techniques. IJEAS 2:88–103 Hatay T, Tokl¨u YC (2002) Optimization of truss using simulated annealing method. In: 5th international congress on advances in civil engineering. Istanbul Technical University, Istanbul, pp 379– 388 Kameshki ES, Saka MP (2006) Optimum geometry design of nonlinear braced domes using genetic algorithm. Comput Struct 85:71– 79 Kamyab R, Salajegheh E (2013) Size optimization of nonlinear scallop domes by an enhanced particle swarm algorithm. Int J Civil Eng 11(3):77–89 Kennedy J, Eberhart R (1995) Particle swarm optimization. In: IEEE international conference on neural networks, vol IV. Piscataway, pp 1942–198 Krishnamoorthy CS, Venkatesh PP, Sudarshan R (2002) Objectoriented framework for genetic algorithms with application to space truss optimization. J Comput Civil Eng 16:66–75 Lagaros ND, Papadopoulos V (2005) Optimal design of shell structure with random geometric, material and thickness imperfections. Int J Solids Struct 43:6948–6964 Lagaros ND, Papadrakakis M, Kokossalakis G (2002) Structural optimization using evolutionary algorithms. Comput Struct 80:571– 589 Lamberti L (2008) An efficient simulated annealing algorithm for design optimization of truss structures. Comput Struct 86:1936– 1953 Lee KS, Geem ZW (2004) A new structural optimization method based on the harmony search algorithm. Comput Struct 82:781– 798 Lee KS, Geem ZW, Lee SH, Bae KW (2005) The harmony search heuristic algorithm for discrete structural optimization. Eng Optim 37:663–684

and for tension members: ki li λi = ≤ 300 ri

(A.2)

Maximum allowable stress for tension members is 0.6Fy and for compression members is obtained from: ⎧

   λ2i   ⎪ 1− Fy ⎪ 3 2 ⎪ 2cc ⎪ 5 3 λi 1 λi ⎪ F = and F S = + − a ⎪ FS 3 8 Cc 8 Cc3 ⎪ ⎪ ⎪  ⎪ ⎪ 2 ⎪ ⎨ f or λ ≤ 2π E i

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Fa = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩



π2E λ2i



FS

and f or

Fy

F S = 23 12  2 λi > 2πFyE (A.3)

Appendix B According to AISC-LRFD the nominal compressive strength is computed as follows: Pn = Fcr Ag

(B.1)

in which Fcr is obtained from: ⎧   ⎨ Fcr = 0.658Fy /Fe Fy f or ⎩ F = 0.877F cr e

f or

 λi ≤ 4.71 FEy  λi > 4.71 FEy

(B.2)

where Ag is the area of cross-section and Pe is the Euler load given by: Pe =

π 2E λ2i

(B.3)

In accordance with AISC-LRFD specification it is required that: Pu ≤ φPn

(B.4)

and for members in tension: Pu ≤ φFy Ag

(B.5)

where Pu is the factored axial load of members and φ is resistance factor which is given as 0.9 for both compression and tension elements.

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