Kitipornchai-Factors Affecting the Design of Lamella Dome

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Journal of Constructional Steel Research 61 (2005) 764–785 www.elsevier.com/locate/jcsr Factors affecting the design and construction of Lamella suspen-dome systems S. Kitipornchai a,, Wenjiang Kang a , Heung-Fai Lam a , F. Albermani b a Department of Building and Construction, City University of Hong Kong, Hong Kong b Department of Civil Engineering, University of Queensland, Brisbane QLD 4072, Australia Received 5 August 2004; accepted 13 December 2004 Abstract The suspen-dome system is a new structural form that has become popular in the construction of long-span roof structures. These domes are very slender and lightweight, their configuration is complicated, and hence sequential consideration in the structural design is needed. This paper focuses on these considerations, which include the method for designing cable prestress force, a simplified analysis method, and the estimation of buckling capacity. Buckling is one of the most important problems for dome structures. This paper presents the findings of an intensive buckling study of the Lamella suspen-dome system that takes geometric imperfection, asymmetric loading, rise-to- span ratio, and connection rigidity into consideration. Finally, suggested design and construction guidelines are given in the conclusion of this paper. © 2005 Elsevier Ltd. All rights reserved. Keywords: Suspen-dome; Buckling; Prestress; Geometric imperfection; Asymmetric loading; Connection rigidity Corresponding author. E-mail addresses: [email protected] (S. Kitipornchai), [email protected] (W. Kang), [email protected] (H.-F. Lam), [email protected] (F. Albermani). 0143-974X/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2004.12.007

Transcript of Kitipornchai-Factors Affecting the Design of Lamella Dome

Journal of Constructional Steel Research 61 (2005) 764–785

www.elsevier.com/locate/jcsr

Factors affecting the design and construction ofLamella suspen-dome systems

S. Kitipornchaia,∗, Wenjiang Kanga, Heung-Fai Lama,F. Albermanib

aDepartment of Building and Construction, City University of Hong Kong, Hong KongbDepartment of Civil Engineering, University of Queensland, Brisbane QLD 4072, Australia

Received 5 August 2004; accepted 13 December 2004

Abstract

The suspen-dome system is a new structural form that has become popular in the constructionof long-span roof structures. These domes are very slender and lightweight, their configuration iscomplicated, and hence sequential consideration in the structural design is needed. This paper focuseson these considerations, which include the method for designing cable prestress force, a simplifiedanalysis method, and the estimation of buckling capacity. Buckling is one of the most importantproblems for dome structures. This paper presents the findings of an intensive buckling study ofthe Lamella suspen-dome system that takes geometric imperfection, asymmetric loading, rise-to-span ratio, and connection rigidity into consideration. Finally, suggested design and constructionguidelines are given in the conclusion of this paper.

© 2005 Elsevier Ltd. All rights reserved.

Keywords: Suspen-dome; Buckling; Prestress; Geometric imperfection; Asymmetric loading; Connection rigidity

∗ Corresponding author.E-mail addresses: [email protected] (S. Kitipornchai), [email protected] (W. Kang),

[email protected] (H.-F. Lam), [email protected] (F. Albermani).

0143-974X/$ - see front matter © 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2004.12.007

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Fig. 1. Basic arrangement ofthe suspen-dome system.

1. Introduction

In recent years, space structures have been developing rapidly. Plane lattices, latticedomes, and tensegrity structures are widely used all over the world. The suspen-domesystem, which was developed by Kawaguchi et al. [1–3], is one of the most attractivespace structures due to its excellent structural properties.

The fundamental idea of the suspen-dome system is the stiffening of a single-layerdome with a tensegrity system, as is shown inFig. 1. The topmost single-layer domeprovides rigid support and decreases the flexibility of the bottom tensegrity system, andthus reduces the required prestress force in the cables compared to that of the cable-domesystem [1]. Simultaneously, the bottom tensegrity system reduces the stress in the membersof the topmost single-layer dome. As a result, the buckling capacity of the overall systemis enhanced.

The suspen-dome system has been widely used, with the Hikarigaoka Dome and FureaiDome in Japan [1], and the Kiewitt suspen-dome in Tianjin in China as examples [5].A comprehensive analysis of the modified Kiewitt suspen-dome has been carried out inRefs. [5,9]. Although static [2] and dynamic [4] tests havebeen carried out, many aspectsthat are related to the structural characteristics of the suspen-dome have not been addressedin the literature, such as the design of cable prestress force, and the effects of asymmetricloading, connection rigidity, rise-to-span ratio, and geometric imperfection on the bucklingbehavior of the system. The primary aim of this paper is to comprehensively investigatethe suspen-dome system, and to provide useful guidelines for engineers in the analysis,design, and construction of this type of structure.

Many single-layer dome configurations, such as the Lamella, Kiewitt, and modifiedKiewitt domes (seeFig. 2), can be used as the topmost single-layer dome of thesuspen-dome system. The symmetrical configuration of the Lamella dome makes it very

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(a) Kiewitt dome. (b) Modified Kiewitt dome. (c) Lamella dome.

Fig. 2. Single-layer dome configurations that can be employed in the suspen-dome system.

convenient to be employed as the topmost single-layer dome of the suspen-dome system.Although this paper focuses on the Lamella suspen-dome system, many conclusions in thispaper can be applied to a suspen-dome system with other configuration.

This paper addresses several key issues of Lamella suspen-dome structures. Amethod for the design of the cable prestress force is presented first, followed by acomparison between the Lamella suspen-dome structure and the corresponding single-layer Lamella dome. An efficient analysismethod is then presented for the purposeof studying the effects of cable prestress force and external load on the structuralbehavior of the suspen-dome system. A parametric study of structural characteristics andbuckling capacity follows. Buckling is one ofthe most important problems for domestructures, and there have been cases when domes have suddenly collapsed due to aninsufficient buckling capacity, such as the collapse of a dome with a span of 93.5 m inBucharest, Romania in 1961. An examination of the effect of geometric imperfection,asymmetric loading, rise-to-span ratio, and connection rigidity on the buckling capacityof suspen-domes is carried out, and the results and findings are summarized in thispaper.

2. Cable prestress force design method for the suspen-dome system

2.1. The basic idea of cable prestress force in the suspen-dome system

A suspen-dome system is not a tensegrity system. Tensegrity systems are spatialreticulated systems in a state of self-stress.All tensioned elements (tendons) constitutea continuous set, and strut elements constitute a discontinuous set. Each node receivesonly one strut element [6]. Unlike the tensegrity system, the suspen-dome system is stablebefore the cables are prestressed. Therefore, it does not require form-finding calculations.It is not appropriate to use the analysis methods that are employed for a tensegrity system,such as the traditional force density method [6] and the methods that are presented inRefs. [7,8], to analyze suspen-dome structures.

The tensegrity (cable–strut) system stiffens the suspen-dome system in two differentways. First, the prestressed cables introduce an opposing force to the external gravity load.This idea can be explained through the simply supported beam example inFig. 3. Themoment that is induced by the external load inFig. 3(a) is opposite to the moment that is

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(a) Uniformly distributed load.

(b) Cableprestress force.

(c) Uniformly distributed load and cable prestress force.

Fig. 3. Bending moment diagrams of a simply supported beam under different loading conditions.

induced by the cable–strut system inFig. 3(b). As a result, the maximum bending momentof the combined system inFig. 3(c) is reduced. Another method is to treat the prestressedcables as equivalent pre-tensioned struts. In this way, the suspen-dome system works likea double-layer dome.

It must be pointed out that the applied prestress force of the hoop and radial cables mustbe large enough to prevent cable slack, but not be so large as to buckle the struts. If thecables slack, then the lower end of the corresponding vertical strut will be free, which mayinduce a serviceability problem.

2.2. The cable prestress force design method

According to the aforementioned discussions, theprestress force of the cables must bedesigned to provide enough vertical strut force against the external load, and to preventcable slack. To make this paper self-contained, the cable prestress design method that isproposed in Ref. [5] is briefly outlined here. The interested reader is directed to Ref. [5]for further details of this method.

The basic idea of this method is to stress the hoop cables such that the induced upwardforce in a strut is proportional to the external load that is applied to the corresponding node.The five-ring suspen-dome inFig. 5(a) is employed as an example to illustrate this idea.FromFig. 4(a), the relationship between the axial forces in the hoop cable and the vertical

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(a) Side view of a section.

(b) Plane projection.

Fig. 4. The cable prestress force design method.

strut can be expressed as:

Nvji = 2Nhci cotγicosαi

2

cosβi2

(1)

whereNhcj for j = 1, . . . , 5 is the prestress force in the hoop cables at thej th ring, andNvji is the axial force of the struts at thej th ring, which is induced by the prestress forcesof cables at thei th ring. For a given geometry, Eq. (1) can be simplified as:

Nvji = Ki Nhci (2)

whereKi is given by:

Ki = 2 cotγicosαi

2

cosβi2

(3)

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(a) Five-ring suspen-dome.

(b) Three-ring suspen-dome.

Fig. 5. The finite element models of the two suspen-dome examples with typical node and element numbers.

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whereα j is the inner angle of the neighboring hoop cables at thej th ring, β j is theprojection of the inner angle of the neighboring radial cables at thej th ring on thehorizontal plane, andγi is the angle between the radial cable at thei th ring and the verticalaxis. The resultant upward force in the strutthat is caused by the cable prestress forceis given by K1Nhc1 − K2Nhc2 at both the first and second rings, as is shown inFig. 4.Similarly, the cable prestress force in the hoop cables of the second and third rings inducesan upward forceK2Nhc2 − K3Nhc3 in the strut at the second ring, and so on.

For the five-ring suspen-dome inFig. 5(a), Ki is almost the same fori = 1, . . . , 5.When the dome is loaded with a uniform nodal load, we can ensure that the vertical forcesof all struts are the same if the magnitudes of the prestress forces at the hoop cables atthe first to the fifth rings are in the proportion of 5:4:3:2:1. If the dome is loaded withnodal loads at different rings in a specified proportion (which is always the case when auniformly distributed vertical load is applied to the roof of the dome), then the prestressforces for the hoop cables at different rings must be set so as to produce upward forcesin the struts at different rings in the same proportion as those of the external nodal load.If the model inFig. 5(a) is loaded with a vertical downward uniformly distributed load of1.0 kN/m2, then the equivalent nodal loads for typical nodes two to six are 20.3, 16.3,12.3, 8, and 4.1 kN. According to the proposed cable prestress force design method, theprestress force for hoop cables at rings one to five should be set in the proportion of61(=20.3 + 16.3 + 12.3 + 8 + 4.1):40.7(=16.3 + 12.3 + 8 + 4.1):24.4(=12.3 + 8 +4.1):12.1(=8 + 4.1) : 4.1.

2.3. Realization of the cable prestress force design

Two methods are usually used [15] in the construction of suspen-dome structures. Theseare the two-stage method (TSM) and the multi-stage method (MSM). In the TSM, theprestress force is applied at the first stage,and the external load, which includes both deadand imposed loads, is applied at the secondstage. In the MSM, the prestress force is notapplied to the cables in one single operation, but at several stages. Part of the prestressforce is introduced to the cables, and then part of the external load is applied. This processis repeated until the complete external load has been applied to the system. The TSM issimple and efficient, but the MSM allows engineers to make full use of the cable strength,and to control the deformation of the dome system during construction. Note that the roofof such dome structures usually consists of several layers. A partial external load can beapplied to the system during construction by controlling the construction sequences of thedifferent layers.

If the TSM is employed, then the large cable prestress force may buckle the systembefore it is compensated for by the external load (e.g., the weight of the roof layers).Therefore, the buckling capacity of the system in the absence of an external load mustbe considered in the design. If the designed prestress force is close to the critical bucklingprestress force, then the MSM must be used.

In summary, the following procedures are recommended for the design and applicationof cable prestress force for the suspen-dome system.

1. Calculate the required cable prestress force by the method presented in Ref. [5] (brieflyoutlined in this paper).

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2. Check to ensure that there is no cable slack in any of the design load cases.

3. Check to ensure that the applied cable prestress force will not introduce buckling beforethe application of any external load.

3. Structural characteristics of a Lamella suspen-dome

3.1. The design model

Based on the analytical results in Ref. [5], the contribution of cables and struts at theinner rings in resisting an external load is very small. Therefore, cables and struts at thetwo innermost rings are removed to form a cost-effective system, as is shown inFig. 5(b).The three-ring dome that is shown inFig. 5(b) is employed in this study to investigatethe structural characteristics of the Lamella suspen-dome system. The rise-to-span ratio ofthedome is 0.1, and the span is 48 m. The length of the vertical struts at the first ring is3 m, at the second ring is 2.7 m, and at the third ring is 2.4 m. A uniformly distributedload with a magnitude of 1.0 kN/m2 is applied to the top surface of the dome in thevertically downward direction. This distributed load is converted to a nodal load based onthe corresponding tributary area. As a result, the nodal loads of the typical nodes one toseven inFig. 5are 11.5, 20.3, 16.3, 12.3, 8, 4.1, and 5.3 kN, respectively.

Steel Circular Hollow Sections (CHS) with a diameter of 133 mm and a thickness of6 mm are usedas the principal members of the topmost single-layer dome, which allowsthe slenderness ratio of the dome members to vary between 23.2 and 139.4. Steel CHSwith a diameter of 89 mm and a thickness of 4 mm are used as the vertical struts of thebottom tensegrity system, which allows the slenderness ratio of the strut members to varybetween 79.8 and 99.8. The elastic modulus of the CHS is 210 GPa, and the cross sectionalareas of all of the cables are 4.47 cm2 with an elastic modulus of 180 GPa. An analysis iscarried out by the nonlinear finite element program ANSYS.

Al l of the boundary nodes of the numerical model are pinned and restrained in verticaland tangential directions to the dome boundary. That is, the boundary nodes can moveradially at the boundary. It must be pointed out that a very large horizontal force will beinduced from the dome to the underlying structure if the boundary nodes are restrained inall directions.

The cable prestress force is designed using the proposed method, and hoop cables areonly used at the first three rings. It is noted that the stresses of the cables at rings 1, 2, and3 are in the proportion of 48.9(=20.3 + 16.3 + 12.3):28.6(=16.3+ 12.3):12.3.

3.2. Comparison between the Lamella suspen-dome and the Lamella single-layer dome

Here, the mechanical properties of the Lamella single-layer dome and the Lamellasuspen-dome, which is formed by stiffening the Lamella single-layer dome using atensegrity system, are studied.

The calculated element axial forces for the single-layer dome under an external load andthe suspen-dome with and without an external load are summarized inFig. 6. The figureshows that the axial stresses for the members of the suspen-dome are much smaller thanthose of the corresponding single-layer dome, especially for the members at the outmostring.

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Fig. 6. Axial stresses for elements of the single-layer dome and the suspen-dome systems with and withoutexternalload.

With the help of the bottom tensegrity system, the stress of element 1, as marked inFig. 5,decreases from 278 to 17.3 MPa in the presence of an external load.

In addition to the stress consideration,Fig. 7 shows that the tensegrity system greatlyreduces the nodal displacements. It is clear from the figure that the magnitudes of the dis-placements of the suspen-dome are much smaller than those of the single-layer dome. Ingeneral, the nodal displacements of the suspen-dome are less than 10% of those of the cor-responding single-layer dome. The results of the analysis verify that the Lamella suspen-dome system is an efficient structural form compared with the single-layer dome system.

4. The superposition method for suspen-dome analysis

Unlike the tensegrity system, the suspen-dome system is very rigid, and the nonlineardeformation is very small even under a large external load. As a result, the method ofsuperposition can be applied in the analysis ofthis type of structure without introducing alarge degree of error.

To study the structural behaviors of the suspen-dome, an analysis is carried out in twophases. In Phase I, cable elements are employed in the finite element model to calculatethe system deformation and internal member forces of all of the members under the actionof cable prestress forces without an external load. In Phase II, using the original geometry,all of the cable elements are replaced by truss elements, and the finite element model isused to calculate the system deformation and internal member forces again under differentload cases in the absence of cable prestress force. The overall structural responses of thesuspen-dome system under different load cases can then be obtained by the superpositionof the results from these two phases. With the help of the superposition method, it is easyto identify the contributions of the stress that is due to the cable prestress force and the

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Fig. 7. Nodal displacement-to-span ratio of the single-layer dome and suspen-dome systems with and withoutexternalload.

stress that is due to the external load. Accordingly, engineers can calculate the allowableexternal forces under a given load case without slacking any cable.

The dome example inFig. 5(b) is employed to verify the superposition analysis method.The axial stresses and nodal displacements in Phases I and II, and the superposition ofPhases I and II are given inTables 1and2, together with the calculated results from ageometric nonlinear analysis. It is clear fromTable 1that the percentage errors for thecalculated stresses are generally close to 1%,except for element 4, for which the percentageerror is the largest and is equal to 8.33%. This can be explained by the fact that themagnitude of stress at element 4 is much smaller than it is at other members. Therefore,a small change in the magnitude of stress will result in a relatively large percentageerror. Similarly,Table 2summarizes the calculated displacements for all four cases. Thepercentage errors are generally very small with the largest being 5.36% at typical node 6.It can be concluded that the geometric nonlinear effects of the suspen-dome system undersuch loading conditions are not significant, and the proposed superposition method allowsengineers to calculate the structural responses with acceptable accuracy.

5. Parametric study of buckling capacity and structural characteristics

5.1. Buckling under cable prestress force without an external load

If the prestress force on the cables is toolarge, then the system may buckle in theabsence of an external load. The three-ring model inFig. 5(b) is employed again as anexample. Linear eigenvalue buckling analysis iscarried out, and the calculated eigenvalueis 1.96. Thus, the structure will buckle in the mode that is shown inFig. 8 if a cable

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Table 1Member stresses calculated by the superposition method and the nonlinear analysis method(×106 N/m2)

Element no. Phase I Phase II Phase I+ II Nonlinear analysis Error (%)

1 −125.600 143.120 17.520 17.338 1.052 −83.704 13.384 −70.320 −70.496 0.253 −17.376 −22.936 −40.312 −40.491 0.444 35.701 −40.829 −5.128 −4.7338 8.335 16.384 −43.376 −26.992 −27.325 1.226 1.052 −28.039 −26.987 −27.453 1.707 10.049 −33.075 −23.026 −22.948 0.348 11.646 −32.183 −20.537 −20.547 0.059 7.532 −24.918 −17.386 −17.341 0.2610 1.816 −16.835 −15.019 −15.014 0.0311 −0.435 −9.894 −10.329 −10.287 0.4112 −1.121 −11.719 −12.840 −12.631 1.65

Table 2Nodal displacements calculated by the superposition method and the nonlinear analysis method (mm)

Node no. Phase I Phase II Phase I+ II Nonlinear analysis Error (%)

2 20.5 −44.8 −24.3 −24.5 0.823 44.7 −57.6 −12.9 −13.1 1.534 59.9 −62.2 −2.3 −2.4 4.175 51.1 −59.8 −8.7 −9.0 3.336 44.7 −50.0 −5.3 −5.6 5.367 43.4 −40.3 3.1 3.0 3.33

prestress force that is1.96 timesthe design value is employed. This eigenvalue is low,and a multi-stage construction method is recommended in this case.

5.2. Effects of connection rigidity, asymmetric load, and rise-to-span ratio

The effects of connection rigidity, asymmetric load, and the rise-to-span ratio on thebuckling capacity and structural characteristics of the suspen-dome system are studied.The eigenvalue buckling analysis predicts the theoretical buckling capacity (the bifurcationload) of an ideal linear elastic structure. However, imperfections and nonlinearitiesprevent most real-world structures from achieving the theoretical elastic buckling capacity.Nevertheless, the eigenvalue buckling analysis does provide an upper bound for the criticalload, together with the buckling mode, which gives engineers valuable information aboutthe buckling behavior of the system. Theeigenvalue buckling analysis results of the suspen-dome and the corresponding single-layer domeunder different conditions are summarizedin Table 3.

5.2.1. Connection rigidityRigid connections are always employed in the construction of single-layer domes,

because the buckling capacity of pin-connected single-layer domes is very low.

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Fig. 8. Buckling mode under cable prestress force without external load.

Table 3Buckling load factors of single-layer domes and suspen-domes with different connection conditions, loadingcases, and rise-to-span ratios

Rise-to-spanratioDome 0.06 0.10 0.15 0.2 0.3

Single-layer dome (Rigid/Full-span load) 2.50 6.54 9.84 10.87 9.20Single-layer dome (Rigid/Half-span load) 2.36 6.06 10.22 12.63 11.17Single-layer dome (Pin/Full-span load) 0.15 0.63 1.87 3.84 7.96Single-layer dome (Pin/Half-span load) 0.12 0.54 1.59 3.25 6.78Suspen-dome (Rigid/Full-span load) 4.12 8.98 10.58 10.67 8.27Suspen-dome (Rigid/Half-span load) 4.51 7.94 10.96 12.73 10.16Suspen-dome (Pin/Full-span load) 0.15 0.63 1.87 3.83 7.96Suspen-dome (Pin/Half-span load) 0.13 0.54 1.60 3.26 6.78

However, pin connections are often used in double-layer lattice domes or double-layer plane lattice structures, because the additional layer stiffens the structure.With the help of the tensegrity system, the suspen-dome system performs likea double-layer dome system. Therefore, pin connections can be used in theconstruction of the suspen-dome system. This argument is supported by the eigenvaluebuckling analysis results inTable 3, which show that the buckling capacity of thepin-connected suspen-dome is higher than that of the pin-connected single-layer dome.

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Fig. 9. A suspen-domeunder a half-span load.

Table 3also shows that the buckling capacity ofthe pin-connected suspen-dome is lowercompared with that of the rigidly connected suspen-dome,as is expected. From a designviewpoint, the ball joint is treated as a pin connection, and the welded hollow ballconnection is treated as a rigid connection.As the ball joint is more convenient forconstruction, its employment is recommended when the span is not too large. For example,ball joint connections were employed in the construction of the suspen-dome in Tianjin inChina, which hasa span of35.4 m [5]. However, if the span is large, say 50 m, then weldedhollow ball connections are recommended from the buckling viewpoint.

The buckling modes of the system under rigid and pin connections are shown inFig. 11(a) and (b), respectively. It is clear from the figure that the buckling is localized atthe central region for thepin-connected suspen-dome (seeFig. 11(b)), and is far away fromthe center of thedome in the rigidly connected suspen-dome (seeFig. 11(a)). One possibleway to increase the buckling capacity of a pin-connected suspen-dome system would be tostrengthen the central region with an additional layer, as is suggested inFig. 13.

5.2.2. Asymmetric loadAnother factor to be considered in this study is asymmetric load. One type of

asymmetric load is the half-span load, which is shown inFig. 9. A half-span load canbe resulted during construction, and can also be induced by snow. Another possible causeof asymmetric load is wind load. A wind-load model based on the wind tunnel test thatwas developed by Lin and Albermani [10,11] (seeFig. 10) is used.The anglesφAB andφBC depend on the rise-to-span ratio. When the rise-to-span ratio is small (which is usually

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Fig. 10. Wind-induced pressure over a dome.

the case), the angleφAB is small, and the horizontal component of the wind-induced forceis very small. Therefore, the horizontal effect of asymmetric load on the system can beomitted, and the wind mainly introducesan upward force to the dome system.Table 3shows that the buckling capacities of the Lamella suspen-dome system under a full-spanload and half-span load are very similar. Fig. 12 shows the buckling modes for the rigidlyand pin-connected suspen-dome systems under a half-span load. The buckling modes arevery similar to those with the full-span load inFig. 11.

5.2.3. Rise-to-span ratioApart from the aforementioned factors, the rise-to-span ratio is also an important factor

that affects the buckling capacity of suspen-dome systems. For a pin-connected suspen-dome system,Table 3shows that the higher the rise-to-span ratio, the higher the bucklingcapacity will be. However, a similar trend cannot be found in the rigidly connected suspen-dome system. If the rise-to-span ratio is smaller than 0.2, then an increase in rise-to-spanratio will result in an increase in buckling capacity. If the rise-to-span ratio is larger than0.2, then the buckling capacity will decrease as the rise-to-span ratio increases.

From Table 3, it can be seen that a single-layer dome with a small rise-to-span ratiohas a small buckling capacity. When the rise-to-span ratio is small, say, less than 0.15, thesuspen-dome has the greatest advantage over the single-layer dome.

Apart from buckling capacity concerns, changes in the inner force of suspen-domemembers under different rise-to-span ratios are also studied. To clearly show the effects ofthe rise-to-span ratio under the action of an external load in the absence of a cable prestressforce, only the results of Phase II of the superposition analysis method are shown, and thePhase I results are omitted.

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(a) Rigid connection.

(b) Pin connection.

Fig. 11. Buckling modes of suspen-domes with rigid and pin connections under a full-span load(Rise-to-spanratio = 0.1).

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(a) Rigid connection.

(b) Pin connection.

Fig. 12. Buckling modes of suspen-domes with rigid and pin connectionsunder a half-span load(Rise-to-spanratio = 0.1).

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Fig. 13. Stiffening the central region of a suspen-dome system with an additional layer.

Fig. 14. Stresses of hoop cables at different rings of the suspen-dome system with different rise-to-span ratios.

A series of analyses is carried out using the dome example inFig. 5(b) with differentrises (the span is constant) to study the effect of different rise-to-span ratios. As shown inFig. 14, the stress of the outermost hoop cable will significantly increase when the rise-to-span ratio is reduced. The rate of increase also depends on the rise-to-span ratio. Ingeneral, the rate of increase is larger for smaller rise-to-span ratios. This trend implies thatthe outermost hoop cables attract large forces when the rise-to-span ratio is small.

Fig. 15 shows that the stress of hoop element 1 for the suspen-dome is lower than it isfor the single-layer dome, and that the difference is larger for a small rise-to-span ratio.Therefore, a small rise-to-span ratio is preferred, as it allows engineers to make full use ofthe cable strength. Furthermore, a large rise-to-span ratio implies a long strut length (seeFig. 17(b)). As all of the struts are under compression, a buckling problem may result ifthe struts are too long.

The radial displacements of the boundary nodes of the suspen-dome and single-layerdome with different rise-to-span ratios are calculated and summarized inFig. 16. When

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Fig. 15. Stress of element 1 for suspen-domes and single-layer domes with different rise-to-span ratios.

the rise-to-span ratio decreases, the displacement of the single-layer dome increasesconsiderably. Large radial displacements at the boundary nodes will introduce largehorizontal forces to the underlying structure. This may result in a fabrication problem.FromFig. 16, it can be seen that the increase in displacement that is due to the decrease inthe rise-to-span ratio is relatively gentle for the suspen-dome compared with the single-layer dome. It can be concluded that the tensegrity system is very important for thestiffening of the structure, and thus reduces displacement, especially when the rise-to-spanratio is small.

In conclusion, rise-to-span ratios of less than 0.15 or so are generally recommended inthe design of suspen-dome systems. As the tension forces for cables at the outmost ringare large compared with those at the inner rings, larger diameters for hoop cables at theoutmost ring are preferred.

5.3. Geometric imperfection

Dome structures are very sensitive to geometric imperfection, which is unavoidableduring fabrication. Several methods are available to analyze geometric imperfection.One is the random geometric imperfection method [12], in which randomly generatedimperfection distribution samples are studied. The sample with the smallest bucklingcapacity is identified, and the corresponding buckling capacity is treated as theapproximated critical capacity of the system. Another method is the fundamental modeimperfection method [12], in which the imperfection distribution is assumed to beconsistent with the first buckling mode. Generally, the buckling capacity that is calculated

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Fig. 16. Radial displacement-to-span ratios of the boundary nodes of suspen-domes and single-layer domes withdifferent rise-to-span ratios.

(a) Small rise-to-span ratio (0.125). (b) Large rise-to-span ratio (0.4).

Fig. 17. Examples of suspen-dome structures with different rise-to-span ratios.

by the fundamental mode imperfection method is the lowest of all the other modes, andis therefore the most critical. In this paper, nonlinear elastic analysis is carried out tocompare the buckling capacities of the domes with different magnitudes of maximumnodal imperfections, which are considered using the fundamental mode imperfectionmethod.

Geometric nonlinear analyses are carriedout. The geometric changes that are causedby prestress force and geometric imperfection are all reflected during the analysis. TheNewton–Raphson method and the arc-length method [13,14] are usedto obtain the totalload–displacement equilibrium path.

The topmost single-layer dome of the suspen-dome inFig. 5(b) is used as an exampleto show the importance of geometric imperfection. Rigid connections are used in thissingle-layer dome, but the boundary nodes, at which the displacements are restrained in

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Fig. 18. The load factor–displacement curves of a single-layer dome with different maximum nodal imperfections.

all directions, are pinned. For each geometric imperfection case, the basic nodal loads(defined inSection 3.1) of the typical nodes one to seven inFig. 5(b) are 11.5, 20.3,16.3, 12.3, 8, 4.1, and 5.3 kN, respectively. The buckling load is obtained by scalingthe basic load by the buckling load factor, and the displacement of the center node (nodeseven) is monitored during the analysis. The loadfactor–displacement curves for differentgeometric imperfection conditions are summarized inFig. 18. It is clear from the figurethat the buckling capacity of the single-layer dome is rapidly reduced in the presence ofgeometric imperfections. In a case when the geometric imperfection has a maximum nodalimperfection of 5 cm, which is only 1.0/1000 of the span, a 35% reduction in bucklingcapacity is recorded. As geometric imperfection is unavoidable in construction, thus anadequate safety factor for buckling must be specified.

A nonlinear buckling analysis is carried out for the suspen-dome system with geometricimperfection taken into consideration. The boundary nodes of the dome are pinned, and arerestrained in the vertical direction and the tangential direction of the dome boundary. Theanalysis results for cases of different maximum nodal imperfections are summarized inFig. 19. It is clear from the figure that the buckling capacity of the suspen-dome decreaseswhen the maximum nodal imperfection of the geometric imperfection increases, as isexpected. When the maximum nodal imperfection increases beyond the value of 10 cm(with the imperfection-to-span ratio= 2.1/1000), the ultimate load factor achieves thelimited value of approximately 5.0. Note that the ultimate load factor of the dome withoutgeometric imperfection is 9.4, and therefore, the factor of safety for buckling should be setto 2 in this case.

According to the specifications of the Chinese lattice dome [16], the maximumgeometric imperfection that is caused by construction must not exceed span/300. For the

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Fig. 19. Buckling capacity of suspen-domes with different imperfection-to-span ratios.

model in this study, the span is 48 m, and thus the maximum geometric imperfection canbe controlled below 16 cm. At this value of maximum nodal imperfection, the bucklingcapacity of the system reduces by almost 50%.

6. Conclusions

A comprehensive parametric analysis of the Lamella suspen-dome system is carriedout, and the results show the superior mechanical properties of the Lamella suspen-domesystem over the corresponding single-layer dome. The bottom tensegrity system helps thedome structureto increase the buckling capacity, decrease member stresses, and increasestiffness.

Based on the comprehensive nonlinear buckling analyses, it can be concluded thatgeometric imperfection plays an importantrole in the buckling capacity of the suspen-dome system. According to the numerical study that is presented here, geometricimperfection can reduce the buckling capacity of a suspen-dome system by up to 50%.

Apart from geometric imperfection, the connection rigidity and rise-to-span ratio arealso important factors that affect the buckling capacity. The pin-connected suspen-domesystem has a lower buckling capacity, especially for suspen-dome structures with smallrise-to-span ratios. Although rigid connections are difficult to construct compared with pinconnections, they are highly recommended for suspen-dome structures from a bucklingviewpoint, especially when the span is large.

In this study, it is found that the buckling capacities of the Lamella suspen-dome systemunder the full-span and half-span loads are not evidently different.

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A small rise-to-span ratio, say, less than 0.15 or so, is recommended for suspen-domesystems. For shallow suspen-dome structures, the stresses for cables at the outmost ringrapidly increase under the action of an external load. Therefore, it is necessary to use hoopcables with a relatively largediameter at the outmost ring.

The proposed superposition analysis method is efficient in the study of the suspen-dome system under different load cases. This method allows engineers to calculate thecontributions of member stress from the cableprestress force and the external load, andthus estimate the required cable prestress force that is needed toprevent cables fromslacking under different load cases.

Acknowledgement

The work described in this paper was fully supported by a grant from CityU (ProjectNo. 7001521).

References

[1] Kawaguchi M, Abe M, Tatemichi I. Design, tests andrealization of ‘suspen-dome’ system. Journal of theIASS 1999;40(131):179–92.

[2] Kawaguchi M, Abe M, Hatato T, Tatemichi I, Fujiwara S, Matsufuji H et al. Structural tests on the “suspen-dome” system. In: Proceedings of the IASS symposium. 1994. p. 384–92.

[3] Kawaguchi M, Tatemichi I, Chen PS. Optimum shapesof a cable dome structure. Engineering Structures1999;21(8):719–25.

[4] Tatemichi I, Hatato T, Anma Y, Fujiwara S. Vibration tests on a full-size suspen-dome structure.International Journal of SpaceStructure 1997;12(3–4):217–24.

[5] Kang W, Chen Z, Lam H-F, Zuo C. Analysis and designof the general and outmost-ring stiffened suspen-dome structures. Engineering Structures 2003;25(13):1685–95.

[6] Motro R. Tensegrity, Kogan Page Science. 2003.[7] Hangai Y, Kawaguchi K, Oda K. Self-equilibrated stress system and structural behavior of truss structures

stabilized by cable tension. International Journal of Space Structures 1992;7(2):91–9.[8] Hangai Y, Wu M. Analytical method of structural behaviours of a hybrid structure consisting of cables and

rigid structures. Engineering Structures 1999;21(8):726–36.[9] Kang W, Chen Z. The effect of the Tensegrity system in suspen-domes. Supplement of Industrial

Construction. 2002. p. 401–4 [in Chinese].[10] Lin S, Albermani F. Integrated design system for lattice domes. International Journal of Space Structures

2000;15(1):59–74.[11] Lin S, Albermani F. Lattice dome design using knowledge-based system approach. Journal of Computer-

Aided Civil and Infrastructure Engineering 2001;16(2):158–76.[12] Shizhao S, Xin C. Stability of lattice shell structures. Science Press of China. 1999 [in Chinese].[13] Crisfield MA. A fast incremental/iterative solution procedure that handles snap-through. Computers and

Structures 1981;13(1–3):55–62.[14] Crisfield MA. An arc-length method including line searches and accelerations. International Journal for

Numerical Methods in Engineering 1983;19:1269–89.[15] Lu CL, Yin SM, Liu XL. Modern prestressed steel structures. China Communication Press. 2003 [in

Chinese].[16] Technical specification for latticed shells of China. 2003 [in Chinese].