Geometry and topology optimization of geodesic

Struct Multidisc Optim DOI 10.1007/s00158-010-0566-y

RESEARCH PAPER

Geometry and topology optimization of geodesic domes using charged system search Ali Kaveh · Siamak Talatahari

Received: 11 December 2009 / Revised: 12 June 2010 / Accepted: 11 August 2010 c Springer-Verlag 2010

Abstract Dome structures provide cost-effective solutions for covering large areas without intermediate supports. In this article, simple procedures are developed to reach the configuration of the geodesic domes. A new definition of dome optimization problems is given which consists of finding optimal sections for elements (size optimization), optimal height for the crown (geometry optimization) and the optimum number of elements (topology optimization) under determined loading conditions. In order to find the optimum design, the recently developed meta-heuristic algorithm, known as the Charged System Search (CSS), is applied to the optimum design of geodesic domes. The CSS takes into account the nonlinear response of the domes. Using CSS, the optimum design of the geodesic domes is efficiently performed. Keywords Geodesic domes · Optimization · Nonlinear design · Charged system search

early times, sports stadiums, assembly halls, exhibition centers, swimming pools, shopping centers and industrial buildings have been typical examples of structures with large unobstructed areas nowadays (Makowski 1993). Dome structures are the most preferred type of large spanned structures. Domes have been of a special interest in the sense that they enclose large areas without intermediate supports. Although dome structures are economical in terms of consumption of constructional materials compared to the conventional forms of structures (Makowski 1984), a more lightweight design can be obtained using optimization methods. Optimization of an engineering design is an improvement of a proposed design that results in the best properties for minimum cost. Structural optimization can be categorized as: •

1 Introduction Structural systems, which enable the designers to cover large spans, have always been a challenging task for structural engineers. Beginning with the worship places in the • A. Kaveh (B) Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak, Tehran 16, Iran e-mail: [email protected] S. Talatahari Department of Civil Engineering, University of Tabriz, Tabriz, Iran

•

Sizing optimization in which the geometry and topology of the structure remain unchanged but cross sectional properties are optimized (e.g. Rajeev and Krishnamoorthy 1992; Camp et al. 1998; Camp and Bichon 2004; Serra and Venini 2006; Perez and Behdinan 2007; Lamberti 2008; Kaveh and Shojaee 2007; Kaveh et al. 2008a; Kaveh and Talatahari 2009a, b, c, 2010a); Geometry optimization which determines the optimum location of the joints in the structure in addition to the size of members (e.g. Prager and Rozvany 1977; Hasançebi and Erbatur 2002; Saka 2007; Kaveh and Talatahari 2010b); and Topology optimization that involves finding the number of members of the structure and the way in which these members are connected to each other (e.g. Dhingra and Bennage 1995; Rozvany 1998, 2001, 2009; Fourie and Groenwold 2002; Bendsoe and

A. Kaveh, S. Talatahari

Sigmund 2003; Wang et al. 2004; Svanberg and Werme 2005, 2006; Martínez et al. 2007; Kaveh et al. 2008b; Rahami et al. 2008; Huang and Xie 2010). The optimal design of topology and geometry has been investigated far less due to its complexity, despite the fact that the optimization of topology and geometry greatly improves the design (Bendsoe and Mota Soares 1992; Kirsch 1989, 1997; Rozvany et al. 1995; Topping 1983). In general, optimization techniques used in structural problems can be categorized into classical and heuristic search methods. The disadvantages of traditional optimization methods such as complex derivatives and the large amount of required memory have forced researchers to employ heuristic approaches for solving optimization problems (Lee and Geem 2004). Many of the heuristic approaches are inspired by the natural phenomena. These

Fig. 1 Geodesic domes with a three rings, b four rings, c five rings

(a) Nr =3

phenomena include the biological evolutionary process, animal behavior, or the physical processes. Recently, the authors proposed a new optimization approach, so-called Charged System Search (Kaveh and Talatahari 2010c, d), which utilizes a number of Charged Particle (CP). These particles affect each other based on their fitness values and separation distances considering the governing laws of Coulomb and Gauss from electrical physics and the governing laws of motion from the Newtonian mechanics. This paper develops the CSS method to perform an optimum geometrical and topological design of geodesic domes. A simple procedure is developed to calculate the joint coordinates and element constructions in order to determine the configuration of the domes. Here, the joint coordinate equations are formulated for automatic formation of the configuration. The nodal coordinates are calculated using a simple relationship. The joints are

(b) Nr =4

3D view

3D view

Top view

Top view

Side view

Side view

Geometry and topology optimization of geodesic domes using charged system search

divided into two types considering the symmetry, and then the elements are easily determined. The serviceability and strength requirements are considered in the design problems as specified in LRFD-AISC (1991). The algorithm takes into account the nonlinear response of the dome due to effect of axial forces on the flexural stiffness of members. In this paper, unlike the previous studies on the dome optimization (Saka 2007; Kaveh and Talatahari 2010b) which consider only size or geometry optimization of the domes by determining the number of rings and the crosssectional areas of the elements, a method is developed that has the ability to determine the element configurations and

the number of elements automatically during the optimization process, this number being considered as a design variable. A discussion over the effect of the number of rings and the number of elements on the results is presented in Section 5.1. The remaining sections of this paper are organized as follows: Section 2 contains the statement of the dome design problem. Section 3 reviews the CSS, briefly. Section 4 describes the implementation of CSS methodology to optimize geodesic dome structures. The achieved results are explained in Section 5. Finally, Section 6 presents the conclusion of the paper.

2 Geometry and topology optimum design of dome structures

(c) Nr =5

Geometrical optimal design of dome structures consists of finding optimal sections for elements, optimal height for the crown and the optimum number of rings under determined loading conditions. The allowable cross sections are considered as 37 steel pipe sections taken from LRFDAISC (1991) which is also utilized as the code of practice. The mathematical formulation for optimum design of dome structures can be expressed as 3D view

Find

X = [x1 , x2 , ..., xng ], h, Nr xi ∈ {d1 , d2 , ..., d37 } h i ∈ h min , h min + h ∗ , ..., h max

to minimize

W (X) =

nm

ρ · xi · L i

(1)

i=1

Top view

Side view

Fig. 1 (continued)

where X is a vector containing the design variables of the elements; h is the variable of the crown height; Nr is the total number of rings which is taken as 3, 4 or 5 as shown in Fig. 1; d j is the jth allowable discrete value for the design variables (37 pipe sections taken from LRFD-AISC); h min , h max and h ∗ are the permitted minimum, maximum and increased amounts of the crown height, which in this paper are taken as 1.00, 9.00 and 0.25 m, respectively; ng is the number of design variables or the number of size groups; W (X) is the weight of the structure; L i is the length of member i; ρ is the is the material mass density and nm is the total number of elements. Similarly, topological optimization of geodesic dome structures can be considered as finding optimal sections for elements, optimal height for the crown, optimum number of rings and optimum number of the elements. Since the number of elements available in each ring is equal to the number of nodes related to that ring, therefore once the number of


ring nodes as the design variables is determined, the topological optimum design will be achieved. In the other hand, for the geodesic domes the number of nodes in each ring is considered as the number of joints in the first ring (Nn) multiplied by the number associated with that ring. Figure 2 shows the different forms of geodesic domes when Nn is altered. As a result, if we consider Nn as the new design variable, the complete dome topology optimization problem will be defined as Find

X = [x1 , x2 , ..., xng ], h, Nr, N n xi ∈ {d1 , d2 , ..., d37 } h i ∈ h min , h min + h ∗ , ..., h max

to minimize

W (X) =

nm

The design constraints are as follows:

Displacement constraint: |δi | ≤ δimax

i = 1, 2, ..., nn

(3)

Interaction formula constraints: Muy Pu Mux Pu + + ≤ 1 For < 0.2 2φc Pn φb Mnx φb Mny φc Pn Muy Mux Pu Pu 8 ≤ 1 For + + ≥ 0.2 φc Pn 9 φb Mnx φb Mny φc Pn (4)

Shear constraint:

ρ · xi · L i

(2)

Vu ≤ φv Vn

(5)

i=1

Fig. 2 Geodesic domes with different number of elements

(a) Nn =3

(b) Nn =4

3D view

3D view

Top view

Top view

Side view

Side view

Geometry and topology optimization of geodesic domes using charged system search Fig. 2 (continued)

(c) Nn =5

(d) Nn =6

3D view

3D view

Top view

Top view

Side view

Side view

in which δi is the displacement of node i; δimax is the permitted displacement for the ith node; nn is the total number of nodes; Pu is the required strength; Pn is the nominal axial strength; φc is the resistance factor (φc = 0.9 for tension, φc = 0.85 for compression); Mux and Muy are the required flexural strengths in the x and y directions, respectively; Mnx and Mny are the nominal flexural strengths in the x and y directions; φb is the flexural resistance reduction factor (φb = 0.90); Vu is the factored service load for shear; Vn is the nominal strength in shear and φv represents the resistance factor for shear (φv = 0.90).

3 Charged System Search algorithm The Charged System Search (CSS) algorithm is based on the Coulomb and Gauss laws from electrical physics and the

governing laws of motion from the Newtonian mechanics. This algorithm can be considered as a multi-agent approach, where each agent is a Charged Particle (CP). Each CP is considered as a charged sphere with radius a, having a uniform volume charge density and is equal to

qi =

fit(i) − fitworst i = 1, 2, ..., N fitbest − fitworst

(6)

where fitbest and fitworst are the best and the worst fitness of all the particles; fit(i) represents the fitness of the agent i, and N is the total number of CPs. CPs can impose electric forces on the others. The kind of the forces is attractive, and its magnitude for the CP located in the inside of the sphere is proportional to the separation distance between the CPs, and for a CP located

A. Kaveh, S. Talatahari Fig. 2 (continued)

(e) Nn =7

(f) Nn =8

3D view

3D view

Top view

Top view

Side view

Side view

outside the sphere is inversely proportional to the square of the separation distance between the particles qi qi Fj = qj ri j · i 1 + 2 · i 2 pi j (Xi − X j ), a3 ri j i,i= j

j = 1, 2, ..., N i 1 = 1, i 2 = 0 ⇔ ri j < a i 1 = 0, i 2 = 1 ⇔ ri j ≥ a

(7)

where F j is the resultant force acting on the jth CP; ri j is the separation distance between two charged particles which is defined as follows ri j =

||Xi − X j || ||(Xi + X j )/2 − Xbest || + ε

Pi j determines the probability of moving each CP toward the others as ⎧ fit(i)− f itbest ⎪ ⎨1 > rand ∨ fit( j) > fit(i) fit( j)−fit(i) (9) pi j = ⎪ ⎩ 0 otherwise The resultant forces and the motion laws determine the new location of the CPs. At this stage, each CP moves towards its new position under the action of the resultant forces and its previous velocity as X j,new = rand j1 · ka ·

(8)

where Xi and X j are the positions of the ith and jth CPs, respectively; Xbest is the position of the best current CP, and ε is a small positive number. The initial positions of CPs are determined randomly in the search space and the initial velocities of charged particles are assumed to be zero.

V j,new =

Fj · t 2 mj

+ rand j2 · kv · V j,old · t + X j,old

(10)

X j,new − X j,old t

(11)

where ka is the acceleration coefficient; kv is the velocity coefficient to control the influence of the previous velocity; and rand j1 and rand j2 are two random numbers uniformly


(g) Nn =9

3D view

and 1, sets the rate of choosing a value in the new vector from the historic values stored in the CM, and (1-CMCR) sets the rate of randomly choosing one value from the possible range of values. The pitch adjusting process is performed only after a value is chosen from CM. The value (1-PAR), sets the rate of doing nothing. Here, ‘‘w.p.’’ means ‘‘with probability’’. For more details, the reader may refer to Kaveh and Talatahari (2009a, c). In order to have discrete results, a rounding function is utilized which changes the magnitude of a result to the nearest discrete value, as follows

X j,new

Fj = Round rand j1 · ka · · t 2 + rand j2 · kv . mj · V j,old · t + X j,old (13)

The constraint handling approach is the penalty approach and the framework of the CSS algorithm is illustrated in Fig. 3.

Top view

Side view

Fig. 2 (continued)

distributed in the range of (0,1). To save the best design a memory (Charged Memory or CM) is considered. If each CP exits from the allowable search space, its position is corrected using the harmony search-based handling. According to this mechanism, any component of the solution vector violating the variable boundaries can be regenerated from the CM as ⎧ ⎪ w.p. CMCR ⎪ ⎪ ⎪ ==> select a new value for a variable from CM ⎪ ⎪ ⎪ ⎨ ==> w.p. (1 − PAR) do nothing xi, j = ==> w.p. PAR choose a neighboring ⎪ ⎪ value ⎪ ⎪ ⎪ ⎪ w.p. (1 − CMCR) ⎪ ⎩ ==> select a new value randomly (12) where xi, j is the ith component of the CP j. The Charged Memory Considering Rate (CMCR) varying between 0

4 The CSS methodology for deign of optimum dome structures The behaviour of the dome structures is nonlinear due to the change of geometry under external loads. The imperfections arising either from the manufacturing process and/or from the construction of the structure can also be the source of nonlinearity. Inclusion of geometric nonlinearity requires additional considerations in the analysis. Stability check is also necessary during the analysis to ensure that the structure does not lose its load carrying capacity due to instability (Saka 2007) and furthermore, considering the nonlinear behaviour in the design of domes results in lighter structural systems. The elastic instability analysis of domes involves repeated analysis of the structure at progressively increasing load factor. At each increment of the load factor, nonlinear analysis of the structure is carried out. For this, the stiffness matrix for a three-dimensional space member that includes the effect of flexure on axial stiffness and the stiffness against translation is derived. Details of this derivation and related terms of the nonlinear stiffness matrix of a space member are given in Majid (1972) and Ekhande et al. (1989). The stiffness matrix of a stable structure is positive-definite. During the nonlinear analysis iteration, the determinant of the overall stiffness matrix is checked to determine whether at any load increment it becomes negative. This is an indication of a loss of stability in the

A. Kaveh, S. Talatahari Fig. 3 The flowchart of the CSS algorithm

structure and the load factor which causes this case, is identified as the critical load factor. As a result, the geometric nonlinearity is also included in this study as described in Kaveh and Talatahari (2010b). Figure 4 shows the nodes of the geodesic dome, when N n = 10. As it can be seen, the number of nodes in each ring is considered as Nn multiplied by the number associated with that ring. In this paper the total number of rings (Nr) is selected from 3 to 5, and the number of nodes in

the first ring (Nn) can be equal to 3 to 10. The distances between the rings in the dome on the meridian line are generally of equal length. The structural data for the topology of this form of the geodesic dome is a function of the diameter of the dome (D), the total number of rings (Nr), the number of nodes in the first ring (Nn) and the height of the crown (h). The top joint at the crown is numbered as the first joint as shown in Fig. 4a (joint number 1) which is located in the centre of the coordinate system in the x − y plane.


(a)

where n i is the number of ring corresponding to the node i; R = (D 2 + 4h 2 )/(8h) is the radius of the hemisphere as shown in Fig. 4b. The member grouping is determined in a way that members between each consecutive pair of rings belong to one group, and the members on each ring belong to a different group. The diagonal members between the crown and the first ring are group 1, the members between ring 1 and 2 are group 2 and the group number of members on the ring 2 is 3, and so forth. Also, the members on each ring constitute one group. The configuration of elements consists of determining the start and end nodes of each element. For the first group, the start node for all elements is the joint number 1 and the end nodes are those on the first ring. The start and end nodes of elements on each ring are two consecutive nodes on that ring. As an example, the element (4,5) belongs to the first ring’ group and the elements (16,17) and (17,18) belong to the second ring’ group, as illustrated in Fig. 5. The element with the lower and upper numbers on that ring also corresponds to that group. For the other diagonal elements, we have ⎧ n i (n i − 1) ⎪ ⎪ +2 I = Nn × ⎪ ⎪ 2 ⎪ ⎪ ⎪ j −1 ⎪ ⎪ ⎪ ⎨ + n i × Fi x 3 j = 2, 3, ..., 3 × N n n ⎪ n (n + 1) i = 1, 2, ..., Nr − 1 i i ⎪ ⎪ J = Nn × +j ⎪ ⎪ ⎪ 2 ⎪ ⎪ j −1 ⎪ ⎪ ⎩ + n i − 2 × Fi x 3 (15)

(b)

Fig. 4 Nodal numbering and the related coordinate system. a Top view of the geodesic dome. b Section of the geodesic domes

The coordinates of other joints in each ring are obtained as follows: ⎧ D.n i N n × n i (n i − 1) 360 ⎪ ⎪ = i − x cos − 2 i ⎪ ⎪ 2Nr N n · ni 2 ⎪ ⎪ ⎪ ⎪ ⎪ 360 D.n i N n × n i (n i − 1) ⎨ sin −2 yi = i− 2Nr N n · ni 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n i2 D 2 ⎪ ⎪ 2 ⎪ R − − (R − h) ⎩z i = 4Nr 2 (14)

Fig. 5 The dome nodes and their related elements


⎧ ⎪ ni ni − 1 j −1 ⎪ ⎪ I = Nn × + 3 + n i × Fi x ⎪ ⎪ ⎪ 2 2 ni − 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j − 2 n i − 1 × Fi x ( j − 1) 2(n i − 1) − 1 ⎪ ⎪ + Fi x ⎨ 2 ⎪ ⎪ n i (n i + 1) j −1 ⎪ ⎪ + 2 × Fi x ⎪ ⎪ J = Nn × ⎪ 2(n i − 1) ⎪ 2 ⎪ ⎪ j ⎪ ⎪ + Fi x +3 ⎪ ⎪ ⎩ 2

⎧ n n −1 ⎪ ⎨ I = Nn × i i +2 2 n i = 1, 2, ..., Nr − 1 n + 1 n ⎪ i +2 ⎩ J = Nn × i +1 2 (17) where I and J are the start and end nodal numbers of the elements, respectively. We divide the nodes into two types: primary and secondary nodes. The primary nodes are located on the symmetry lines and the number of these nodes is equal to Nn for each ring. Other nodes are considered as the secondary ones. The number of the secondary nodes for the ring n i is equal to Nn × (n i − 1). As an example, in the third ring, the nodes 32, 35, 38, ..., 59 are the primary and the remaining nodes are the secondary ones, when N n = 10 (Fig. 4a). Equations (15), (16) and (17) present the total elements of the diagonal groups. Using (15), the elements with the primary start nodes are determined. As an example, for n i = 1 and j = 4, 5, 6, the second group contains (3,13), (3,14) and (3,15), as shown in Fig. 5. Equation (17) presents only one element which connects the first primary node on the ring n i to the last secondary node on the ring n i + 1 and for the second group, it will be (2,31). The elements related to the secondary nodes are given by (16). For example, using n i = 3 and j = 1, 2, 3, 4 in this equation, the elements (33,63), (33,64), (34,64) and (34,65) are specified. Figure 5 shows the primary and secondary nodes and their related elements when N n = 10 and Nr = 4.

5 Results and discussion In this section, the dome described in the previous section is optimized using the CSS algorithm. The LRFD specification and drift constraints are considered as the constraints for this structure. The modulus of elasticity for the steel is taken as 205 kN/mm2 . The diameter of the dome is selected as 20 m. The limitations imposed on the joint displacements are 28 mm in the all directions for the nodes.

j = 1, 2, ..., 2 × N n × (n i − 1) n i = 2, 3, ..., Nr − 1

(16)

A population of 25 individuals is used for the CPs and the value of constant a is set to one. The acceleration coefficient ka and the velocity coefficient kv are taken as 0.5. Here, CMCR = 0.95 and PAR = 0.10 are used as suggested by Kaveh and Talatahari (2010c). 5.1 Effects of Nr and Nn on the optimum designs Though the number of rings (Nr) and the number of nodes in the first ring (Nn) are defined as the design variables in our program, and the optimum values for these variables can directly be obtained. However, in order to investigate the effect of Nr and Nn on the optimum designs, here we consider all possible conditions for these design variables. The dome is considered to be subjected to equipment loading of 1,000 kN at its crown in this state. Different design variables defined in (2) affect the weight of structures in different manners. Nr and Nn determine the number of elements and the height of dome alters the length of elements and these in turn can change the sum of the element lengths. On the other hand, the design variables xi determines the cross-sectional areas of elements without change in the geometry of the structure. In order to investigate the effect of the variables separately, the relation of weight for the dome structure is modified as W (X) = ρx ·

nm

Li

(18)

i=1

where x is the average cross sectional area of the elements. The sum of the element lengths is a function of the height of the dome. Tables 1, 2 and 3 list the optimal designs for the geodesic dome with different Nn obtained by the CSS algorithm when Nr is equal to 3, 4 and 5, respectively. From these tables it can be observed that a dome with small number of elements (Nn) tends to select the greater height. When Nn increases, almost in the all the tables the height of the dome decreases. For a dome with small Nn, having a large

Geometry and topology optimization of geodesic domes using charged system search Table 1 Optimum design of the geodesic dome with three rings Group number

Optimum sections (designations) Nn = 3

Nn = 4

Nn = 5

Nn = 6

Nn = 7

Nn = 8

Nn = 9

N n = 10

1

PIPESTa (12)

PIPEST (12)

PIPST (12)

PIPST (10)

PIPST (12)

PIPST (12)

PIPST (10)

PIPST (10)

2

PIPST (6)

PIPST (5)

PIPST (4)

PIPST (3.5)

PIPST (3.5)

PIPST (3.5)

PIPST (3)

PIPST (3)

3

PIPST (5)

PIPST (4)

PIPST (3.5)

PIPST (3)

PIPST (3)

PIPST (2.5)

PIPST (2.5)

PIPST (2.5)

4

PIPST (12)

PIPST (8)

PIPST (8)

PIPST (10)

PIPST (6)

PIPST (6)

PIPST (8)

PIPEST (5)

5

PIPST (6)

PIPST (5)

PIPST (4)

PIPST (3.5)

PIPST (3.5)

PIPST (4)

PIPST (3)

PIPST (3)

Height (m)

8.00

7.00

7.25

6.50

5.75

5.00

5.25

5.00

Max. displacement (cm)

2.80

2.80

2.80

2.80

2.80

2.74

2.80

2.80

Max. strength ratio

99.82

86.38

90.19

97.26

91.86

99.98

99.99

97.35

Weight (kg) L i (cm)

7,213

6,312

5,977

5,970

6,105

6,161

6,236

6,454

238.43

258.92

303.65

330.34

353.46

374.38

415.88

445.55

x (cm2 )

38.5

31.1

25.1

23.0

22.0

21.0

19.1

18.5

a PIPST,

PIPEST, and PIPDEST stand for standard weight, extra strong, and double-extra strong, respectively

height helps the dome to prevent instability. In addition, the selected sections for the elements in a dome with a small Nn are stronger than those of a dome with a large value for Nn. This means that although a dome with small Nn has a small value for the sum of the element lengths, however its average cross sectional area is a big value. The lowest weight design is the one which has the smallest values for the average cross sectional area and the sum of the element lengths, simultaneously. From Table 1 related to the domes with three rings, the good designs are obtained when Nn is set to 5, 6 and 7. For the smaller values for Nn, as expected the sections are

very strong and therefore the average cross sectional area becomes a higher value and for the big values of Nn, the sum of the element lengths increases the weight of the dome. Similarly, for the domes with 4 rings (Table 2) when Nn is 7, 8 and 9, the economical designs are obtained. Optimum value of Nn is 7, 8, 9, and 10 for the domes with 5 rings (Table 3). As it can be seen when the number of rings increases, for optimum design, a bigger values should be selected for Nn. For a constant Nn, when Nr increases, the height remains almost constant and the weight reduces. In other words, for a constant Nn, when Nr increases, the height remains almost constant. For small values of Nn,

Table 2 Optimum design of the geodesic dome with four rings Group number


Nn = 5

Nn = 6

Nn = 7

Nn = 8

Nn = 9

N n = 10

1

PIPEST (12)

PIPEST (12)

PIPEST (12)

PIPEST (12)

PIPST (12)

PIPST (12)

PIPST (12)

2

PIPST (6)

PIPEST (3.5)

PIPST (3.5)

PIPST (3.5)

PIPST (4)

PIPST (3)

PIPST (3)

3

PIPST (3)

PIPST (3)

PIPST (2.5)

PIPST (2.5)

PIPST (2.5)

PIPST (2.5)

PIPST (2.5)

4

PIPST (3)

PIPST (2.5)

PIPST (2.5)

PIPEST (2)

PIPST (2)

PIPST (2)

PIPST (2)

5

PIPST (8)

PIPST (6)

PIPST (5)

PIPEST (2)

PIPST (3)

PIPST (2.5)

PIPST (1.25)

6

PIPST (6)

PIPST (5)

PIPST (3.5)

PIPST (5)

PIPST (4)

PIPEST (3.5)

PIPEST (3)

7

PIPST (3.5)

PIPST (3.5)

PIPST (3.5)

PIPST (2.5)

PIPST (3)

PIPST (2.5)

PIPST (3)

Height (m)

7.50

6.50

6.50

5.00

5.00

4.75

4.50


2.80

2.80

2.80

2.80

2.76

2.78

2.80

Max. strength ratio

99.98

99.99

98

95.96

100

93.8

83.57


6,813

6,243

6,112

6,070

5,982

6,085

6,259

371.96

404.65

456.83

466.91

513.79

553.17

591.48

x (cm2 )

23.3

19.7

17.0

16.6

14.8

14.0

13.5

A. Kaveh, S. Talatahari Table 3 Optimum design of the geodesic dome with five rings Group number


Nn = 5

Nn = 6

Nn = 7

Nn = 8

Nn = 9

N n = 10

1

PIPEST (12)

PIPEST (12)

PIPST (12)

PIPST (12)

PIPST (12)

PIPST (12)

PIPST (12)

2

PIPST (8)

PIPST (5)

PIPST (4)

PIPST (4)

PIPST (3.5)

PIPST (3)

PIPST (3)

3

PIPST (3)

PIPST (2.5)

PIPST (3.5)

PIPST (3)

PIPEST (2)

PIPEST (2)

PIPEST (2)

4

PIPST (2.5)

PIPEST (2)

PIPST (2.5)

PIPST (2)

PIPST (2)

PIPST (2)

PIPST (2)

5

PIPST (2.5)

PIPEST (2)

PIPST (2)

PIPST (2)

PIPST (2)

PIPST (2)

PIPST (1.25)

6

PIPST (2.5)

PIPEST (1.5)

PIPST (1.25)

PIPST (1.5)

PIPST (1.25)

PIPST (0.75)

PIPST (0.5)

7

PIPEST (5)

PIPEST (5)

PIPST (5)

PIPST (5)

PIPEST (3.5)

PIPEST (3)

PIPST (4)

8

PIPST (4)

PIPST (3)

PIPST (3)

PIPST (3)

PIPST (3)

PIPST (2.5)

PIPST (2.5)

9

PIPST (3)

PIPST (3)

PIPST (2.5)

PIPST (2)

PIPST (2)

PIPST (2)

PIPST (2)

Height (m)

7.00

6.75

6.50

5.50

5.25

5.25

4.75


2.80

2.74

2.79

2.80

2.80

2.79

2.80

Max. strength ratio

100

99.75

99.9

91.13

98.5

88.57

87.1


6,927

6,332

6,210

5,896

5,848

6,041

6,091

465.16

524.63

582.56

610.68

662.02

722.56

760.23

x (cm2 )

19.0

15.4

13.6

12.3

11.3

10.7

10.2

the weight reduces, while for the large values of Nn the weight increases, as shown in Fig. 6. As an example, for N n = 8, the weight is equal to 6,161, 5,982 and 5,848 kg when Nr = 3, 4 and 5, respectively, while these are equal to 5,977, 6,243 and 6,332 kg when N n = 5. When the number of rings increases, a feasible dome is not found with N n = 3. This means that we must either change the height limits or alter the utilized sections. The dominant constraints of the designs are often the displacement constraint; however changing a section of the reported designs to a lighter one often causes swerving the stress constraint in elements.

Nr = 5 Nr = 4 Nr = 3

7000 6500

Weight

6000 5500 5000 4500 4000

4

5

6

7 Nn

8

9

Fig. 6 The weight compression when Nn and Nr are varied

10

Figure 7a, b show the relation between

nm

L i and the

i=1

height of the geodesic dome when Nn and Nr are changed, respectively. As expected, raising the height of the dome increases the sum of the element lengths; however the rate of the increment is higher when Nn or Nr has a big value. As an example, according to Fig. 7a, when the height of the dome rises from 1 to 9 m, the sum of the element lengths increases 27%, 44% and 53% for N n = 3, 6 and 10, respectively, when Nr is kept constant (Nr = 4). Also according to Fig. 7b, the element lengths increment is 49%, 51% and 51.5% for Nr = 3, 4 and 5, respectively, for N n = 8. From this investigation, two results can be derived. Firstly with a higher probability a small height will be a suitable choice rather than a large one for domes with big Nn. In other words, we expect when the value of the Nn increases, the height of the dome decreases. Secondly, if a height is suitable for a dome with constant Nn, for another dome with a different number of rings, the new height does not differ a great deal. In the other hand, the sum of the element lengths for the geodesic dome with four rings is 1.38 times larger than that of the dome with three rings in average. This value becomes 1.75 times when the domes with five and three rings are compared (Nn is constant). While these differences are smaller when Nn is investigated (Nr remains constant). As an example, the sum of the element lengths for the four, five and six nodes on the first ring is 1.13, 1.26 and 1.39 times larger than that of the dome with three nodes on the first ring, respectively. Therefore, it is expected that the alterations of x must be smaller when the number of nodes

Geometry and topology optimization of geodesic domes using charged system search 4

9

(a)

x 10

8 Sum of the member lengths

Fig. 7 The relation between the sum of lengths and the height of the domes when a Nn is varied (Nr = 4); b Nr is varied (N n = 8)

Nn = 3 Nn = 4 Nn = 5 Nn = 6 Nn = 7 Nn = 8 Nn = 9 Nn = 10

7 6 5 4 3 2

1

2

3

Sum of the member lengths

7

8

9

(b)

4

9

4 5 6 Height of the domes

x 10

8 7 Nr = 3 Nr = 4 Nr = 5

6 5 4 3

1

2

on the first ring is changed compared to it when the number of rings is altered. In addition, the value of x must be decreased as much as possible when Nn or Nr increases. Therefore, for the domes with small Nn and/or Nr, we will have stronger sections. These points are supported by the comparisons of the results made in the three Tables 1, 2 and 3. 5.2 Optimum design of domes under multiple loadings In the process of design, often non-symmetric and/or multiple loading conditions must be taken into account. In order to investigate the efficiency of the present methodology, here two different loading conditions are considered as follows: Case 1: A uniform loading acting on the structural with the sum equal to 1,000 kN. In this Case the equivalent load for each joint is equal to 1,000 kN divided by the number of the unsupported joints.

3

4 5 6 Height of the domes

7

8

9

Case 2: A multiple loading condition containing: the loading of the previous case and a non-symmetric loading condition including loads acting on all the nodes with the sum equal to 800 kN, in addition to loads with the sum of 400 kN acting only on half of the structure, as shown in Fig. 8.

The best weight found for the first case is 2,728 kg which almost is more than 45% economical design than the best one obtained when 1,000 kN acting as a single load on the crown. For Case 2, due to the non-symmetric loading condition, the weight increases. It is 3,186 kg where both symmetric and non-symmetric loads are utilized. Table 4 presents the results of these cases. In all loading cases, the stress constraints are dominant and the displacement is not active. The maximum stress ratio is 97.86% and 99.98% for cases 1 and 2, respectively. The maximum displacement values are very small than its allowable value. For both cases the number of ring is equal four and the value of the elements (Nn) is equal to 6 and 7 respectively which are


(a)

6 Concluding remarks

(b)

Fig. 8 The multiple load condition a a uniform loading; b nonsymmetric loading

found directly by the present methodology. In addition, the height of the dome is reduced in these cases compared to the case with a single loading at the crown. Table 4 Optimum design of the geodesic dome with multiple loadings Group number

Optimum sections (designations) Case 1

Case 2

1

PIPST (2)

PIPST (2.5)

2

PIPST (2.5)

PIPST (2.5)

3

PIPST (2.5)

PIPST (2.5)

4

PIPST (2)

PIPST (2)

5

PIPST (2)

PIPST (2.5)

6

PIPST (2)

PIPST (2.5)

7

PIPST (2.5)

PIPEST (2)

Height (m)

3.25

3.5

Nr

4

4

Nn

6

7


1.14

1.50

Max. strength ratio

98.76

99.98

Weight (kg)

2728

3186

This paper utilizes the Charged System Search algorithm for design of geodesic domes. This algorithm determines the total number of rings, the number of nodes on the first ring, the optimum height and the optimum steel section designations for the members of geodesic domes from the available steel pipe section table and implements the design constraints from LRFD-AISC. The CSS is inspired by the laws from electrostatics and Newtonian mechanics. CSS contains a number of charged particles. Each CP is considered a charged sphere of radius a, which can impose an electric force on other CPs. This force and the laws for the motion determine the new location of the CPs. From optimization point of view, this process provides a good balance between the exploration and the exploitation paradigms of the algorithm which can considerably improve the efficiency of the algorithm. A simple procedure is presented to calculate the joint coordinates and specify the elements to determine the configuration of the geodesic domes. First, the joint coordinates are calculated and divided into the primary and secondary types considering the symmetry of the dome. Then using some simple relationships, the elements are constructed. A complete investigation on the effect of the number of rings and the number of nodes of the first ring on the final optimum design is performed. It is shown that using these parameters as the design variables, it is possible to perform an optimum topological design of dome structures. Semi-actual load conditions containing non-symmetric and multiple loading conditions are also taken into account. Although the results of the CSS algorithm as other heuristic methods should be considered as suboptimal, however the results show that the CSS method is a robust technique that can successfully be utilized in the practical optimum topology design of the domes. Since the height of the dome is considered as a design variable and the wind load is directly determined using the height of the dome, therefore the wind load will change in the optimization process and it will be treated as a function of a design variable. Therefore, the load condition will be changed by altering the design variables and this makes the problem more difficult. However, formulation of such a function gives a more accurate simulation of the real condition. The future work must consider the exact values for the wind load conditions. Since our presented procedure is utilized for different loading conditions, it is expected that such a real condition can also be included in the present methodology. Acknowledgement The first author is grateful to Iran National Science Foundation for the support.


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