Lopez-Numerical and Experimental Single Layer Latticed Dome

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Computers and Structures 85 (2007) 360–374

Numerical model and experimental tests on single-layer latticeddomes with semi-rigid joints

Aitziber Lopez *, Inigo Puente, Miguel A. Serna

Institute of Civil Engineering, Tecnun, University of Navarra, Manuel de Lardizabal 13, 20018 San Sebastian, Spain

Received 4 October 2005; accepted 21 November 2006Available online 25 January 2007

Abstract

Geometric non-linearities in single-layer domes can cause the loss of global stability, which is strongly influenced by both geometricparameters and joint rigidity. The rigidity of the joint requires deeper study, since it significantly affects the behaviour of these structures.To this end, a model is proposed for the ORTZ joint. The model is established from the dimensions and properties of the different ele-ments of which the joint is composed and considers the possibility of the material reaching its yield point. After experimental verification,the model is implemented in a computer application. Finally, experimental tests have been conducted on two structures possessing verydifferent features related to geometry and rigidity of joints. In both cases the proposed model has given a good estimation of the exper-imentally observed behaviour of the structures.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Single-layer; Geometric non-linear; Semi-rigid joints; Snap-through; Stability

1. Introduction

Optimising structural design can achieve material sav-ings but can also lead to solutions where members are oftenat the limit of their capabilities. In particular, structures aresometimes required to perform close to instability. This isso with single-layer latticed domes, in which all membersare disposed in only one surface. In return, spaces coveredby single-layer structures offer a different sense oflightness and transparency, when compared to traditionalconstructions.

However, rigidity is much less in single-layer domes thanin double-layer structures. The points defining the surfaceare likely to undergo movements that change the geometryand the global behaviour of the structure. Consequently,analysis has to include geometric non-linear effects. Thisis achieved by means of incremental-iterative procedures.

0045-7949/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2006.11.025

* Corresponding author.E-mail address: [email protected] (A. Lopez).

The one used here is based on the displacement controlmethod [1], although other control methods have been usedto verify the obtained results. In addition, in a large sum-mary on the stability of latticed structures, Gioncu [2]states that material non-linearities, whilst often observedin double-layer structures, are not likely to occur in sin-gle-layer domes (Fig. 1).

The stability of single-layer domes is influenced by thegeometric parameters which define the mesh of elements.Many results can be found in the available literature, fromboth numerical and experimental analyses. In most of themthe structure under consideration is a single-layer diamaticdome. In [3], for instance, it is observed that the bucklingload increases as the span/depth ratio decreases. Sohnet al. [4] consider the reduction of snap-through instabilitywith respect to joint rigidity. In [5], the implication of mem-ber slenderness on global stability is considered. However,the lack of similarity among the types of loads, boundaryconditions and geometric parameters studied in previouspapers makes it difficult to establish a clear trend in thebehaviour of single-layer domes.

mailto:[email protected]

2θ2θ0

P

Fig. 2. Two-member structure.

App

lied

load

Displacement

snap-throughAA'

Fig. 3. Snap-through instability.

geometric nonlinearity

large

materialnonlinearitymediumsmall

single-layer shells

large

medium

small

double-layer shells

double-layer grids

Fig. 1. Different causes of non-linear behaviour affecting latticed struc-tures [2].

A. Lopez et al. / Computers and Structures 85 (2007) 360–374 361

As a result, single-layer latticed domes have often beencalculated by making use of the knowledge accumulatedover the years on continuous shells. Certain criteria canbe used to obtain an equivalent continuous shell for aparticular single-layer structure [6]. Once the latticeddome can be interpreted as a shell, there are formulaethat can then be applied to directly estimate the criticalload of single-layer structures [7–9]. However, in additionto the difficulty of finding the equivalent shell, this socalled equivalent continuum method does not seem tobe appropriate for the study of discrete structures. Forthis reason, the authors have analysed a group of sin-gle-layer domes numerically to achieve a formula thatallows the accurate estimation of these domes directly[10]. This formula maintains the discrete character ofthe structure and takes into account the parameters thatdefine the geometry, the distribution of the loads andthe rigidity of the joints.

Joint rigidity is a particular key factor in the behaviourof single-layer structures, whose flexibility often makesthem unsuitable for spanning large distances. This incom-patibility has sometimes been overcome by a combinationof double and of single-layer structures [11]. Finding asolution to the above has also lead to the development ofnew joints, like the ones shown in [12,13], or the reinforce-ment of existing ones [14]. Advances have also been madein the modelling of joint systems as the use of numericalmethods to foresee real behaviour become more feasible.For instance, Ueki et al. [15] model the joint by means ofan elastic spring with experimentally set rigidity. In thiscase plasticity is considered only likely to occur in the tubu-lar steel member. In contrast, [5,16] assume the tube to beelastic and the properties of the spring are changed with thestress level.

In this paper, the influence of joint rigidity on the globalbehaviour of the structure is analysed numerically. Later, amodel is proposed for the study of the ORTZ joint, whichis widely used, especially in Spain [17]. This model alsotakes into account the possibility of the material reachingits yield point. Finally, the model is checked with twoexperimental tests.

2. The influence of geometric parameters on the behaviour of

single-layer domes

2.1. Study of a basic structure

The study of the influence of geometric parameters onthe behaviour of a single-layer dome begins with the anal-ysis of a simple structure composed of two pinned mem-bers. The structure is represented in Fig. 2 without loads(left) and in final equilibrium under the vertical load P

(right), when the angle between the members and the hor-izontal has changed from h0 to h.

The curve of applied load vs. vertical displacement ofthe node (Fig. 3) is obtained by means of a geometricallynon-linear incremental-iterative method. The use of a dis-placement control method allows plotting the reductionof the applied load that occurs as the members movetowards the horizontal position. This behaviour can alsobe detected in experimental tests when appropriate precau-tions are taken.

In reality, however, the loads acting on a structure donot diminish. Instead, once the applied load has reacheda maximum value (point A in Fig. 3), snap-through instabil-ity appears. As there is no equilibrium position close topoint A, the node moves towards A 0, a new point on thecurve where the value of the applied load equals themaximum value of A. This instantaneous displacement ofthe node implies a dynamic energy which can producethe collapse of the structure. For that reason, the positionof A is critical in relation to the global stability of thestructure.

An exact relation between the applied load and the angleof the members is given by the equation of equilibrium inthe deformed configuration, when the shortening of themembers is expressed as a function of h0 and h:

362 A. Lopez et al. / Computers and Structures 85 (2007) 360–374

P ¼ 2EA 1� cos h0

cos h

� �sin h ð1Þ

In Eq. (1), h0 is the initial angle of the member, E representsYoung’s modulus and A is the section area of the member.The value for the load at A is obtained by maximizing thefunction P(h):

P cr ¼ 2EAð1� cos2=3 h0Þ3=2 ð2Þ

Snap-through instability appears when it occurs beforeEuler load in the members is reached. This conditionimplies that the angle h0 in Fig. 2 satisfies Eq. (3):

cos h0 >k2 � p2

k2

� �3=2

ð3Þ

where k stands for member slenderness. If h0 is larger thanthat in Eq. (3), the Euler load is lower than the criticalvalue for snap-through and the members buckle under

Table 1Results of linear and non-linear analyses

Failure mode

(5,0.3)

x

y P

(0,0) (10,0)

E = 200 GPa A1 = 4.83e-3 m2

A2 = 2.24e-3 m2

Lengths in metres

Snap-thr.

Linear analysis

Euler

Nonlinear analysis

Pcr = 4.2·105 N Pcr = 0.8·105 N

Pcr = 1.9·105 N Pcr = 0.4·105 N

Pcr = 2.0·105 N --

Pcr = 0.4·105 N Pcr = 0.3·105 N

λ1 = 75

λ1 = 75

λ2=120

λ2=120

an

L,

Fig. 4. Diamatic single-layer dome (top) and p

the load given in Eq. (1) when letting h be the angle atwhich Euler load is reached in the members:

P E ¼ 2EAp2

k21� k2 cos h0

k2 � p2

� �2" #1=2

ð4Þ

Some comparative values are shown in Table 1. They havebeen calculated using two structures of the same geometry,but with member slenderness of 75 and 120, respectively.Buckling of members occurs in the structure with moreslender members. The results also demonstrate that thecritical loads are overestimated by the linear analysis.

2.2. Study of single-layer structures

The previous analysis undertaken on a simple structureprovides a better understanding of the behaviour of a dia-matic single-layer dome, like the one shown in Fig. 4 (top).The global dimensions of the structure are the span, D, andthe height, H. Fig. 4 (bottom) also schematises the layoutof the members in ‘a’ rings and ‘n’ meridians lines. Thestructure is completed with diagonal members.

Single-layer structures are often composed of tubularmembers, all of them being of the same section, A, andmoment of inertia, I. The length of a member depends onits position in the mesh. The length, L, of members locatedalong the meridian lines is thus set as the reference length.The non-linear numerical analysis of the structure makes itpossible to study how the parameters defining the structureaffect its stability.

2.3. The effect of member section properties

Numerical non-linear analyses have been carried out onrigidly jointed single-layer domes with six meridians linesand a span/depth ratio of 6. Members’ dimensions wereassigned according with previous linear analysis. The

A, I

D

H

θoθo

α

arameters defining the structure (bottom).


corresponding member properties are listed in Table 2 andthe results are shown in Fig. 5. The structures were loadedwith the same value at every node. Loads in Fig. 5 repre-sent the load applied on each node when the critical pointwas reached.

In Fig. 6, previous results for the critical loads have beendivided by the area of the member section. In this way, itcan be seen how the critical load divided by the corre-sponding section area is the same for all single-layer domeswith the same number of rings. Single-layer domes with pinjoints behave in a similar way. As a result, it can be con-cluded that the critical load for a single-layer dome is pro-portional to the area of the member section, just as it wasfor the simple structure of Fig. 2 (see Eq. (2)).

Table 2Properties of tubular members

Span (m) Dimensionsa Area (cm2) M. inertia (cm4)

20 70 · 2 4.27 24.730 90 · 4 10.8 10040 125 · 5 18.8 34050 200 · 5 30.6 146060 200 · 8 48.3 2230

a Dimensions = outer diameter · thickness (mm2).

0

4

8

12

4 5 6 7number of rings

Cri

tical

load

per

nod

e (1

0 5

N)

50

20 m 30 m 40 m

m 60 m

Fig. 5. Critical loads of single-layer domes obtained numerically (rigidjoints).

0

100

200

300


Cr.

load

/ no

de/

sect

ion

(N/m

m2 )

20 m 30 m 40 m 50 m 60 m

Fig. 6. Critical loads per node and per member section area.

2.4. The effect of the number of rings

The critical loads undergo a significant change as thenumber of rings varies in Fig. 6. The effect of a change inthe number of rings is bigger when the number of rings islow. This is due to the relation between the number of ringsand the angle between members in the structure, since thisangle plays a relevant role as illustrates the simple structureof Fig. 2. In effect, when Eq. (2) is developed for small val-ues of the angle h0, it can be expressed as Eq. (5), where itcan be seen that the critical load is proportional to the thirdpower of the angle between members. Eq. (5) is equal tothat given by [18], where a structure with only one freenode is considered as well.

P cr ¼ 2EAh3

0

3ffiffiffi3p ð5Þ

In the single-layer dome with several rings, the angle be-tween two members located on the same meridian line de-pends on the number of rings ‘a’ and on the span/depthratio D/H:

2h0 ¼ua¼ 1

atan�1 4D=H

ðD=HÞ2 � 4

!ð6Þ

where u is the half-subtended angle of the dome (Fig. 7).Thus, replacing h0 by u divided by 2a and being ‘n’ the

number of meridian lines, which is also the number ofmembers joined at each node, Eq. (5) can be extended forsingle-layer domes so that the critical load is given in Eq.(7):

P cr ¼ ncAEAðu=2aÞ3

3ffiffiffi3p ð7Þ

Coefficient cA has been introduced in Eq. (7) to take intoaccount differences of critical load depending on load dis-tribution. Values of cA proposed by the authors afternumerical analysis of a significant number of single-layerdomes are given in Fig. 8. Eq. (7) is valid for pin-jointedsingle-layer domes but will be extended later for semi-rigidjoints.

Fig. 9 shows reasonable agreement between the valuesgiven by Eq. (7) and those obtained numerically from thenon-linear analysis. The values of Eq. (7) have been joinedin a continuous line in order to make the comparisonclearer.

ϕa

ϕa

ϕa

θ0

θ0

2θ0

Fig. 7. Meridian section of a diamatic dome.

0

8

16

24


Cri

tical

load

per

nod

e (1

0 5 N

) D/H = 6.0

D/H = 4.5

D/H = 3.0

Eq. (7)

Fig. 10. Critical loads for different span/depth ratios (pin-jointed domeunder uniform load).

γA=1

γR=1

γAC

=0.55

γRC

=0.70

γAU

=1.60

γRU

=0.85

Fig. 8. Values of cA and cR according to the load case (C, concentrated; U, uniform).

0

5

10


Cri

tical

load

per

nod

e (1

05 N) 20 m

40 m 60 m Eq. (7)

Fig. 9. Comparison between Eq. (7) and critical loads obtainednumerically.


2.5. The effect of the span/depth ratio D/H

The span/depth ratio affects the critical load of single-layer domes because it affects the angle between members,as has been shown in Eq. (6). Using Eq. (6) in combinationwith Eq. (7), the critical loads can be worked out directlyfor domes with different span/depth ratios. The resultsfor a dome spanning 40 m are shown in Fig. 10 and arecompared with the loads obtained by means of numericalanalysis.

3. The effect of joint rigidity on the behaviour of single-layer

domes

For many years, analyses of single-layer structures haveassumed (conservatively) the use of pin-jointed systems.

However, in order to extend the use of single-layer struc-tures, new designs of semi-rigid joints have been intro-duced. As a result, the behaviour of the structure maydiffer significantly, as is shown in Fig. 11, where theload–displacement curves of two single-layer structuresare compared. Besides the difference in the load values,there is a noticeable variation in the behaviour of the struc-ture. In fact, snap-through instability can disappearbecause of the presence of rigidity in the joints.

The influence of the rigidity of the joint on the criticalload of single-layer domes has been studied in [10], basedon the analysis of a two-member structure with perfectlyrigid joints. The equilibrium equation in the deformed con-figuration for that structure gives Eq. (8):

P ¼ EAðh20 � h2Þhþ 24EI

L2ðh0 � hÞ ð8Þ

The value of the critical load is obtained by maximizing thefunction P(h):

P cr ¼ 2EA1

3h2

0 þ4I

AL2

� �h2

0

3� 8I

AL2

� �1=2

þ 24EI

L2h0 �

h20

3� 8I

AL2

� �1=2" #

ð9Þ

Performing in Eq. (9) the same transformation as for thesimple structure of two pinned members, a new formulafor the direct estimation of the critical load of a single-layerdiamatic dome with semi-rigid joints is obtained in Eq.(10), where coefficients cA and cR multiplies the area andmoment of inertia of the member section, respectively,and have been introduced to take into account the effectsof load distribution. The values of these coefficients forthe two distributions of loads considered here are shownin Fig. 8. When all the nodes are loaded and, consequently,their horizontal movements are limited, the load bearingcapacity is larger.

P cr ¼ ncAEA1

3u=2að Þ2 þ 4cRaI

cAAL2

� �1

3u=2að Þ2 � 8cRaI

cAAL2

� �1=2

þ ncR

12EaI

L2u=2að Þ � 1

3u=2að Þ2 � 8cRaI

cAAL2

� �1=2( )

ð10Þ

In Eq. (10) a is a non-dimensional parameter whose valuesare between 0 and 1, depending on the rotational stiffness k(Eq. (11)), which is the proportionality constant betweenthe rotation at a joint and the applied moment.

0

1

2

3

0.00 0.33 0.67 1.00

Joint rigidity (α)

Cri

tical

load

(10

5 N)

Eq. (10)

numerical

0

1

2

3

0.00 0.33 0.67 1.00

Joint rigidity (α)

Cri

tical

load

per

nod

e (1

05 N

)

Eq. (10)

numerical

Fig. 12. Comparison between Eq. (10) and numerical analysis. Concentrated load (left) and uniform load (right).

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0 0.1 0.2 0.3 0.4

Displacement of the upper node (m)

App

lied

load

(10

5 N

)

Pinned joints

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0 0.1 0.2 0.3 0.4

Displacement of the upper node (m)

App

lied

load

(10

5 N

)

Rigid joints

Fig. 11. Load–displacement curves (six rings and 40 m span).


k ¼ 6EIL

að1� aÞ ð11Þ

Loads predicted with Eq. (10) are compared in Fig. 12 withthose obtained numerically. A very reasonable agreementhas been achieved. Numerical analysis has also verified thatsnap-through instability disappears when the rigidity of thejoint satisfies Eq. (12), which corresponds with imaginaryvalues of Eq. (10):

a >cA

24cR

AL2

Iðu=2aÞ2 ð12Þ

4. Comparison with other proposals found in the literature

The use of single-layer structures to provide interestingarchitectural solutions has prompted the proposal of sev-eral formulations for the rapid estimation of their loadbearing capacity. Most of these formulations are basedon the continuum analogy and are unable to satisfactorilyanticipate the behaviour of a discrete structure. Some ofthe ones applicable to uniform loading are going to becompared here. Eq. (13) was presented by Suzuki et al.[7] and applies to domes with completely rigid joints.

qcr ¼3:6EA

kR2for S < 3:3

qcr ¼38:8EA

k2RLfor S > 3:3

ð13Þ

Saitoh et al. [8] give the corresponding expression forpinned joints in Eq. (14).

qcr ¼ 1:70EAry

LR2ð14Þ

Finally, Eq. (15), taken from [9], is a more complex for-mula which, in certain respects, takes into account the stiff-ness of the dome, but does not truly quantify joint rigidity.When comparing these formulae, Qcr, which representsload per node in Eq. (15), will be converted to qcr, loadper area.

Qcr ¼EA

1þ a2B

8p2

0:47L3

R3þ 3B

IALR

� �ð15Þ

In the previous equations, L is the length of the meridianmembers, A and I are the member section area and mo-ment of inertia, respectively, k stands for member slender-ness and R is the radius of the dome. The value of S in Eq.(13) is given by Eq. (16):

Table 3Equivalent bending stiffness, B [9]

aB 1/32 1/16 1/8 1/4 1/2 1B 0.868 0.873 0.886 0.950 1.176 1.850

aB 2 4 8 16 32 64B 3.15 4.83 6.48 7.35 7.80 7.90


S ¼ffiffiffiffiffiffikLR

rð16Þ

In Eq. (15), B is the equivalent bending stiffness. Its valuesare given in Table 3 as a function of aB, where

aB ¼L2

ryRand ry ¼

ffiffiffiIA

rð17Þ

The equivalent bending stiffness B equals zero for pin-ned joints.

Some calculations have been worked out with five-ringdomes in order to compare the formulae presented in thispaper, with Eqs. (13)–(15) and with the numerical analysis.The diameters of the domes varied from 20 to 60 m and allhad a span/depth ratio, D/H, equal to 6. The results forpinned joints are shown on the left in Fig. 13 and theresults for rigid joints on the right.

For the rigidly jointed domes (Fig. 13, right) only theresults given by Suzuki et al. [7] are close to those obtainednumerically. Besides, applying the formulae available in theliterature to pinned domes (Fig. 13, left) does not provideaccurate values. In addition to the precision which itachieves, the expression proposed by the authors allowsthe introduction of intermediate values of joint rigidity thatare not considered by the continuum analogy-basedformulae.

5. Modelling the joints

Up to this point, the effects of the angle between mem-bers and of the joint rigidity on the load bearing capacityof single-layer structures have been examined only numer-ically. The rigidity of the joint requires deeper study, since

0.0

0.5

1.0

1.5

20 40 60dome span (m)

Cri

t. lo

ad p

er s

q. m

. (10

4 N/m

2 )

Eq. (7) [9]Numerical [8]

Fig. 13. Comparison between existing direct formulae and num

it significantly affects the behaviour of these structures. Tothis end, a model is proposed for the ORTZ joint. Themodel is established from the dimensions and propertiesof the different elements of which the joint is composed.After experimental verification, the model is implementedin a computer application for latticed structural design.

5.1. Proposal of joint model

The ORTZ system consists of a solid steel ball intowhich tubular members are screwed by means of captivebolts. The bolts, which are tightened by two nuts, arethreaded in opposing directions in order to allow an easierassembly of the system. A detailed scheme of the ORTZjoint is shown in Fig. 14.

In the computer application developed for the non-lin-ear numerical analysis of lattice structures, the single-layerdome is considered as being composed of members thatjoin the points of a mesh. Each member consists of a tubewith a joint at either end. The following approximation isproposed for the modelling of the joint–tube–joint group.The stiffness of the ball is considered to be infinite. Thetube is assumed to be elastic and with the same propertiesas the real tube. The bolt is replaced by an elasto-plasticcylinder located between the tube and the balls. As a result,the real joint–tube–joint group shown in Fig. 15 (top) ismodelled by the simplest set in Fig. 15 (bottom).

The stiffness matrix for the new structural members isformed through a procedure of condensation of degreesof freedom (see Appendix A). Finally, the global stiffnessmatrix for the structure is formed by assembling all ofthe members’ stiffness matrices together.

During calculation of the equilibrium curve, the devel-oped software detects the eventual yielding point of thebolts. As soon as plasticity is reached by any of the bolts,the moment of inertia is reduced according to Eq. (18):

Ieq ¼D4

4

uc

8� sinð4ucÞ

32þ 1

3cos3 uc sin uc

� �ð18Þ

where uc is the angle limiting the plastic zone of the boltsection and D is the bolt diameter (see Appendix B). In this

0.0

0.5

1.0

1.5

20 40 60dome span (m)

Cr.

load

per

sq.

m. (

104 N

/m2)

Eq. (10) [9]Numerical [7]

erical analysis. Pinned joints (left) and rigid joints (right).

D L

L L LR

D R

Fig. 14. Detail of ORTZ joint.

L2 L3 L4LlL0

L

Fig. 15. Proposed approximation for the study of structural members.


way, plasticity is taken into account and a distribution ofstress equivalent to that of a linear elastic elementmaintained.

Fig. 16. Experimental testing of joint rigidity.

6. Experimental test for the joint model

The model for the structural element has been testedexperimentally. Fig. 16 shows the test arrangement. A sym-metric layout was chosen in order to avoid torsional effects.Four members were attached to the central node. The otherends were simply supported on the test bench and werereinforced against local buckling.

As a result, the ball was rigidly joined and the loadwas applied at the other ends of the members. Fig. 17shows the vertical load and the bending moment actingat the section which limits the tube and the bolt. As thetube is much longer than the bolt, the largest effect is dueto the bending moment. Therefore, the equivalent lengthof the cylinder model for the real bolt is obtained by keep-ing the same rotational stiffness for a given bendingmoment. This assumption leads to the equivalent lengthof Eq. (19).

LEq ¼IR

IL

LL þ LR ð19Þ

where IR and IL are the moments of inertia correspondingto the diameters DR and DL in Fig. 14.

The experimental curve of applied load versus displace-ment is shown in Fig. 18. The graph also shows the curve

load

(10

3 N)

0

2

4

6

8

0 0.02 0.04 0.06

displacement (m)

experimental

proposed model

Fig. 18. Experimental and numerical results for joint rigidity test.

7 m

disp

lace

men

tm

easu

re

Fig. 19. Overall layout of the basic test structure.

LA

P

P L - L A

L

P

Fig. 17. Actions at the section limiting tube and bolt.

Fig. 20. Overall view of the structure before the experimental test.


predicted by the numerical analysis of the test structuremodelled as explained beforehand. Agreement betweenboth curves was found to be quite satisfactory. Therefore,it has been proved to be a good model for the joint evenafter the initial yielding of the bolts.

7. Testing a basic single-layer structure

In order to check the results achieved with the proposedmodel, an experimental test on a basic single-layer domewas carried out.

7.1. Description of the test structure

The tested single-layer structure consisted of six tubularmembers attached to a central node (Fig. 19). The otherends were joined by six more tubular members formingthe only ring, which was simply supported on the bench.These nodes had a free movement in the radial direction.The structure had a 7-m span and all the tubular memberswere 90 mm in diameter and 3 mm thick.

The height between the ring and the central node has agreat influence on the critical load of the structure, as hasbeen demonstrated by [18]. In his article, Ishikawa andKato [18] show that, for a pin-jointed structure, the criticalload is proportional to the third power of the angle of themembers. In particular, for a pin-jointed structure with thesame geometry as the one in Fig. 19, a 1-mm shortening of

the members would cause a reduction of about 50% of thecritical load. And this angle is relevant when the joints aresemi-rigid as well. In the actual experiment, the expectedvalue for the height of the upper node was 15 cm. However,after the assembly of the members the measured heightturned out to be only 13 cm. Nowadays, in spite of thereduction of fabrication errors, any small deviation intro-duced during the assembly of a shallow single-layer struc-ture can cause changes in the position of the nodes and,consequently, affect its load carrying capacity. Accord-ingly, verification of the node locations after assembly ishighly recommended.

Fig. 20 contains a photograph of the test specimen. Dur-ing the test, the load was applied on the central node andthe vertical displacement was measured. The load wasapplied under control displacement conditions by meansof a specific device located between the node and the actu-ator. A displacement sensor measured the aperture of thering (Fig. 19). In addition, axial strain in all memberswas recorded, as well as bending strain in radial members.


7.2. Experimental test results

The experimental load–displacement curve is shown inFig. 21 together with the numerically derived curves forthe structure when it is assumed to have rigid joints andwhen it is assumed to have pin joints. The experimentalcurve is closer to the pin-jointed curve and also exhibitssnap-through instability. Joint rigidity in the experimentalstructure was low enough to ensure that only certainincreases in the supported loads were introduced that donot significantly change its global behaviour.

7.3. Model to be applied in the analysis of a real structure

In order to take into account imperfections introducedduring the assembly of a real structure, certain variationsneed to be applied to the proposed model. The imperfec-tions are due to lack of adjustment and inadequate tighten-ing of the joints. As has been previously mentioned, smalldifferences in the length of the members lead to relevantchanges in the behaviour of shallow single-layer structures.The fact that defects in the joints increase the flexibility is

-1

0

1

2

0 100 200 300

Displacement of the upper node (mm)

App

lied

load

(10

4 N)

rigid jointsexperimentalpin-jointed

Fig. 21. Experimental load–displacement curve bounded by the numer-ically derived curves.

L Eq

DR

L L L R

DL

Fig. 22. Joint model for the assembled member.

considered by modelling the double-threaded bolt usingits smallest diameter. The resulting model is illustrated inFig. 22.

Unlike the experimental test conducted on a single-nodestructure (Fig. 16), joints in a real structure cannot be con-sidered as perfectly rigid joints. Their greater flexibility ismore accurately approximated by the scheme in Fig. 23,where the actions which appear in the section betweenthe tube and the bolt are shown. There, the shear force ismore relevant than the bending moment. Consequently,the equivalent length for the modelled bolt is obtained inEq. (20) as the length which maintains the same rotationalstiffness for a given shear force.

L2Eq ¼ LLð2LR þ LLÞ þ L2

R

IL

IR

ð20Þ

Considering the element sizes in the actual experimentalstructure and making use of Eq. (20), the joint is modelledby an infinitely rigid ball with a 67-mm long radius and anelasto-plastic cylinder with a diameter of 25.7 mm and alength of 54.6 mm.

The curve obtained from the numerical analysis of themodelled structure is compared in Fig. 24 with the experi-mental results. The reasonable level of agreement achievedproves the capability of the model to reproduce the realbehaviour of the structure for at least this particular case,where joint rigidity is not quite so high.

LA LAP

P2

P2

LA

Fig. 23. Actions at the section limiting tube and bolt.

-1

0

1

2

1000 200 300


App

lied

load

(10

4 N) experimental

proposed model

Fig. 24. Comparison of numerically and experimentally obtained loadcurves.


8. Testing a shallow single-layer dome

One last experiment was conducted on a complete sin-gle-layer dome of the parallel diamatic sort. The purposewas to check whether the proposed model performs wellwith a more complex single-layer structure.

Fig. 26. Hydraulic jack.

8.1. Description of the test

The dome was 7-m-spanned and has four rings. Therewere a total of 156 members and 61 joints. All tubularmembers had a diameter of 40 mm and a thickness of2 mm, except the tubes composing the external ring, whichwere 60 mm in diameter and 3 mm thick. The height of thecentral joint measured before the test was 647 mm.

The external ring rested on a robust support structure.This auxiliary structure allowed the required workspacefor the test equipment and provided the structure with vol-ume enough to undertake configurations quite distinctfrom the initial shape of the dome. The supporting jointswere bolted to the auxiliary structure and were free to movein a radial direction. Small plates of steel and Teflonlocated under the ball-joint ensured that they could slidefreely (Fig. 25). In fact, the numerical analysis revealed thatthe radial displacements were negligible.

The structure was loaded at the central point by meansof a hydraulic jack. A device located between the node andthe actuator guaranteed that the test was performed underdisplacement control conditions (Fig. 26). In addition tothe measurements of the load and the vertical displacementat the central node, displacements in the four nodes encir-

Fig. 25. Detail of the sliding support.

cled in Fig. 27 were registered. Highlighted members inFig. 27 are those for which axial and bending strains weremeasured during the test.

Fig. 27. Location of the instrumented tubular members and nodes.

-4000

-2000

0

0 100 200


Axi

al s

trai

n, f

irst

rin

g ( μ

ε )

buckling

yielding

Fig. 30. Axial deformations of first-ring-members.

Fig. 31. Buckling of first-ring-members.


8.2. Experimental test results

The record of the vertical displacement of the nodes(Fig. 28) shows that initially, whilst the central node wasdescending under the applied load, the adjacent nodeshardly moved. Movement of the central node was, there-fore, not strongly restricted. In fact, it was facilitated inits movement due to the high number of rings and its mark-edly shallow shape. In addition, the experimental load–dis-placement curve in Fig. 29 reveals a linear relationshipbetween load and displacement of the central node forthe first stage which lasted up to 55 mm of the upper nodedisplacement.

The original position of the central members wasinverted when the central node reached a displacement of55 mm. The applied load caused a tensional force in thecentral members whilst the ring members were compressed.As a result, the slope of the load curve in Fig. 29 exhibits asudden increase at 55 mm. A similar change is observed inFig. 30, where the axial strains of the ring members areshown. The behaviour was maintained up to an axial strainof about 1000 le. Beyond this point the steel began to yield.The strain of the yielded members increased notably andthe buckling of three members of the ring occurred. Thesuccessive buckling of the tubes in Fig. 30 correspond tothe peaks observed in the load–displacement curve(Fig. 29). The photograph in Fig. 31 shows the ring withthe buckled members.

In contrast with the structure tested first, the presence ofa constant increase in load in Fig. 29 also leads to the con-

0

100

200

0 100 200Displacement of the upper node (mm)

Dis

plac

emen

t (m

m)

upper nodefirst ringsecond ring

Fig. 28. Displacement of the nodes vs. displacement of the central node.

0.0

0.5

1.0

1.5

2.0

2.5

0 50 100 150 200Displacement of the upper node (mm)

App

lied

load

(10

4 N)

buckling

Fig. 29. Experimental load–displacement curve.

clusion that joint rigidity in this structure was high enough,when compared to the stiffness of the members, to ensurethat the loss of stability due to snap-through was avoided.Two test experiments have, thus, been conducted on struc-tures possessing very different features related to the num-ber of rings and members present in each and to the relativestiffness between their joints and tubes.

8.3. Experimental test results vs. proposed model

The dome of the test experiment has been calculatedwith the non-linear application which includes the model

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 20 40 60 80 100


App

lied

load

(10

4 N) experimental

proposed model

Fig. 32. Comparison of numerically and experimentally obtained loadcurves.


of members proposed beforehand. The joints weremodelled as a 38-mm part of infinite stiffness connectedto an elasto-plastic cylinder with a diameter of 13.6 mmand a length of 32.0 mm, obtained through Eq. (20).

The load–displacement curve arrived at by this analysisis compared in Fig. 32 with the results of the test experi-ment. The graph illustrates how the numerical analysis ofthe modelled structure allows a good estimation of the realbehaviour of the dome.

A B C

Fig. 34. Parts of a member before condensation.

9. Conclusions

The analysis of tubular single-layer structures involvesgeometric non-linear methods that can consider large dis-placements of nodes. In this paper, the influence of variousfactors on the behaviour of single-layer spherical domeshas been studied. These factors are dome geometry, slen-derness of members, joint rigidity and load hypothesis.The angle between members located along the same merid-ian line has been found to have a relevant effect on the loadcarrying capacity of the domes.

When considering structures with rigid or semi-rigidjoints, the influence of the angle between members in thebehaviour of the dome depends on the rigidity with whichthe joint contributes to the global structure. The results ofseveral numerical analyses have allowed the determinationof an expression for the critical load where all the featuresof the dome are taken into account (number of rings andmeridian lines, properties of members, rigidity of the jointsand load distribution).

A model of the members of the structure has been pro-posed in order to obtain a more exact analysis of thedomes. This model establishes a new unique element thatintegrates the joint–tube–joint group. As a result, a stiffnessmatrix has been obtained where all the dimensions andmechanical properties of the real elements are considered,including the plastic behaviour of the bolts of the joint.The model has been tested with satisfactory results.

In addition, two experimental tests have been con-ducted: the first on a simple structure with only one freenode and the second on a dome spanning seven metres.The two test experiments have been conducted on struc-tures possessing very different features related to the num-ber of rings and members present in each and to the relativestiffness between their joints and tubes. In both cases thenumerical analysis of the structure modelled by following

L1L2L0

L

A2 I2

A1 I1

Fig. 33. Lengths and properties of the elem

the proposed model has given a good estimation of theexperimentally observed behaviour of the structures.

Acknowledgements

The authors wish to express their gratitude to theSpanish Government Agency CICYT (Project TAP98-0377-C02-01), and to the Basque Government and theDiputacion Foral de Gipuzkoa. Also, they thank LANIKEngineers for their collaboration with the University ofNavarra.

Appendix A

The stiffness matrix of the member with semi-rigid jointsis presented in this appendix. Cross section properties ofthe different parts of the members are given with referenceto Fig. 33, where Li, Ai and Ii are, respectively, the length,the area and the moment of inertia of each part of themember.

The stiffness matrix for bending of the member inFig. 33 is obtained by the condensation of the stiffnessmatrices of the member in Fig. 34, composed of three parts.The stiffness matrix of the central part is that of a tube (Eq.(21)):

kB ¼EI2

L22

12

L2

6 � 12

L2

6

6 4L2 �6 2L2

� 12

L2

�612

L2

�6

6 2L2 �6 4L2

26666664

37777775

ð21Þ

The stiffness matrices for Part A and C, composed of theball and the bolt, are obtained by imposing a unit displace-ment to each degree of freedom (Fig. 35 shows, as an exam-ple, a unit vertical displacement of the right end of Part A).The forces and moments which appear at the ends are thecolumns of the stiffness matrices (Eqs. (22) and (23)).

L4L3

A3 I3

ents composing a structural member.

FL

ML

MRFR

L0 L1

Fig. 35. Unit vertical displacement of the right end of Part A.


kA ¼EI1

L21

12

L1

6þ 12L0

L1

� 12

L1

6

6þ 12L0

L1

4L1 þ 12L0 1þ L0

L1

� ��6� 12

L0

L1

2L1 þ 6L0

� 12

L1

�6� 12L0

L1

12

L1

�6

6 2L1 þ 6L0 �6 4L1

26666666664

37777777775

ð22Þ

kB ¼EI3

L23

12

L3

6 � 12

L3

6þ 12L4

L3

6 4L3 �6 2L3 þ 6L4

� 12

L3

�612

L3

�6� 12L4

L3

6þ 12L4

L3

2L3 þ 6L4 �6� 12L4

L3

4L3 þ 12L4 1þ L4

L3

� �

26666666664

37777777775

ð23Þ

Finally, the member stiffness matrix is obtained by conden-sation of the three stiffness matrices. For simplicity, in Eq.(24), only the stiffness matrix for a symmetric member isrepresented (L0 ¼ L4; L1 ¼ L3; A1 ¼ A3; I1 ¼ I3). As itcan be noted, torsion rigidity has been neglected.

K ¼

K1;1 0 0 0 0 0 �K1;1 0 0 0 0 0

K2;2 0 0 0 K2;6 0 �K2;2 0 0 0 K2;6

K2;2 0 �K2;6 0 0 0 �K2;2 0 �K2;6 0

0 0 0 0 0 0 0 0 0

K5;5 0 0 0 K2;6 0 K5;11 0

K5;5 0 �K2;2 0 0 0 K5;11

K1;1 0 0 0 0 0

K2;2 0 0 0 �K2;6

K2;2 0 K2;6 0

0 0 0

K5;5 0

S Y M M E T R I C K5;5

2666666666666666666666664

3777777777777777777777775

ð24Þ

where

K1;1 ¼ EA1A2

2L1A2 þ L2A1

ð25Þ

K2;2 ¼12

L1

ED

I1

L1

þ 2I2

L2

� �ð26Þ

K5;5 ¼ 4ED

I1 3þ 3L2

L1

þ 6L0

L1

þ L2

L1

� �2

þ 3L0

L1

L2

L1

þ 3L0

L1

� �2 !"

þI2 9þ 8L1

L2

þ 3L2

L1

þ 6L0

L1

þ 12L0

L2

þ 6L0

L1

L0

L2

� �#ð27Þ

K2;6 ¼�6ED

I1

L1

þ 2I2

L2

� �2þ L2

L1

þ 2L0

L1

� �ð28Þ

K5;11 ¼ 2ED

I1 6þ 6L2

L1

þ 12L0

L1

þ L2

L1

� �2

þ 6L0

L1

L2

L1

þ 6L0

L1

� �2 !"

þ2I2 3þ 4L1

L2

þ 6L0

L1

þ 12L0

L2

þ 6L0

L1

L0

L2

� �#ð29Þ

And D is given in Eq. (30):

D ¼ 4 2L1 þ 3L2 þ 2L2

2

L1

� �þ I1

I2

L2

L1

� �2

L2

þ 4I2

I1

6L1 þ 3L2 þ 4L2

1

L2

� �ð30Þ

Appendix B

The developed software supposes a linear elastic behav-iour of the steel. But stresses in the bolts can be beyond theyielding limit. Therefore, in order to take plasticity intoaccount, the moment of inertia of the bolts is changedwhen yielding is detected.

Fig. 36 shows the stress and strain distributions corre-sponding to a cylindrical section with plastic behaviourfor points located farther than c. The bending momentwhich causes this distribution is given by

M ¼ 2

Z uc

0

ryD4

8csin2 u cos2 uduþ 2

Z p=2

uc

ryD3

4sinu cos2 udu

ð31Þ

M ¼ ryD4

32cuc �

sin 4ucð Þ4

� �þD3

6ry cos3 uc ð32Þ

ϕcDc

M

σy

c

σy

εy

εy

Fig. 36. Stress and strain distributions for a perfect elasto-plasticcylindrical section.

σy

εy

εy

Mc

σy

Fig. 37. Stress and strain distributions for a linear elastic material.


The stress distribution for a linear elastic material withthe same strain distribution is given in Fig. 37. The bendingmoment which corresponds to the linear distribution isgiven by

M ¼ ry

cIeq ð33Þ

Consequently, the moment of inertia, Ieq, used in Eq. (18)to characterize the plastic behaviour with a fictitious linearstress distribution, is obtained by making equal Eq. (32)with Eq. (33) and substituting:

c ¼ D2

sin uc ð34Þ

References

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[2] Gioncu V. Buckling of reticulated shells: state-of-the-art. Int J SpaceStruct 1995;10(1):1–46.

[3] Park SH, Suk CM, Jung HM, Kwon YH. A comparative study onthe buckling characteristics of single-layer and double-layer latticedome according to rise ratio. In: Astudillo, Madrid, editors.

Proceedings of the 40th anniversary congress of IASS, Madrid,1999. p. B2.21–30.

[4] Sohn SD, Kim SD, Kang MM, Lee SG, Song HS. Nonlinearinstability analysis of framed space structures with semi-rigid joints.In: Lightweight structures in civil engineering, proceedings of theIASS international symposium, Warsaw, Poland, 2002. p. 422–7.

[5] Kato S, Mutoh I, Shomura M. Collapse of semi-rigidly jointedreticulated domes with initial geometric imperfections. J ConstructSteel Res 1998;48:145–68.

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[7] Suzuki T, Ogawa T, Ikarashi K. Elastic buckling analysis of rigidlyjointed single-layer reticulated domes with random initial imperfec-tion. Int J Space Struct 1992;7(4):265–73.

[8] Saitoh M, Hangai Y, Toda I, Yamagiwa T, Okuhara T. In: Heki,editor. Buckling loads of reticulated single-layer domes. Proceedingsof the IASS symposium, Osaka, vol. 3, 1986. p. 121–8.

[9] IASS Working Group 8. Analysis, design and realization of spaceframes: a state-of-the-art report. Bull Int Assoc Shells Spatial Struct1984/85;25(1/2).

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[15] Ueki T, Matsushita F, Shibata R, Kato S. Design procedure forkarge single-layer latticed domes. In: Parke, Howard, editor. Spacestructures 4, Guildford conference, 1993. p. 237–46.

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Lopez-Numerical and Experimental Single Layer Latticed Dome

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dierent features

experimentally

layer latticed

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spandepth

layer domes

layer domes