a new approach to geometric nonlinearity of cable structure

A new approach to geometric nonlinearity of cablestructures

A.S.K. Kwan

School of Engineering, University of Wales Cardi, PO Box 917, Cardi CF2 1XH, UK

Received 8 January 1997; received in revised form 1 January 1998

Abstract

The basic structural principles surrounding nonlinear behaviour of cable networks are explained through the

example of a two-link structure. The nonlinear static response to load for this structure is then derived explicitlyusing the proposed simple approach, and results are compared with those obtained from a general two-dimensionalnon-linear bar element (derivation given), and to results quoted in the literature. The proposed approach togeometric nonlinearity is then tested on three three-dimensional cable networks and the results compared with those

obtained by three other techniques, namely geometric stiness matrix, dynamic relaxation and general minimumenergy. The proposed technique has been found to be comparable to established techniques in accuracy, stabilityand speed of solution while at the same time exhibiting the key features of separation of the numerical computation

from the underlying structural mechanics, and the requirement of understanding only the most elementary ofstructural mechanics. The proposed technique is thus also most suitable for introducing cable structures toundergraduate courses. # 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Cable structures; Geometric nonlinearity; Infinitesimal mechanisms; Large displacement analysis; Nonlinear analysis

1. Introduction and purpose of paper

It is clear from the emergence of small cable nets and

fabric structures over doorways to shopping malls,

supermarkets and petrol stations, that such structures

are considered to be elegant and aesthetically pleasing.

There are, however, only a few large-scale examples of

such structures adorning prestigious structures. From a

structural engineering point of view, there can be two

reasons for this. Firstly, cable networks are seen as

highly flexible structures. The problem however, is not

in the flexibility per se, but in the nature of the flexi-

bility, which is not widely expounded in teaching insti-

tutions. The second reason why cable network

examples are few is because they have an entirely non-

linear response to load. Where nonlinear compu-

tational mechanics is taught in undergraduate courses,

the emphasis is on the algorithms, aspects of the com-

putation and computer programming. Such an

approach does not encourage the development of an

intuitive feel for nonlinear behaviour. For these

reasons, it is possible to believe cable networks rather

mystical structures that can only be analysed by some

complex computer programs restricted to a few highly

specialized engineering firms. Such a misconception is

usually a sucient reason to place more conventional

structural forms over cable networks. One of the key

emphases of this paper therefore is the clarity of the

fundamental characteristics of cable networks.

The aim of this paper is to both oer a simple expla-

nation of the nonlinear flexibility inherent in cable net-

works through a straightforward example, and to

propose an analytical technique for the nonlinear static

behavior of cable networks which separates the com-

plexity of a nonlinear numerical algorithm from the

underlying structural principles. However, far from

being a purely demonstrative teaching technique, the

proposed technique has been found to be comparable

to established techniques in speed and accuracy of sol-

utions. A brief review of the main existing techniques

is given in Section 2. Section 3 provides a discussion

on the nature of the flexibility in cablenet structures;

Computers and Structures 67 (1998) 243252

0045-7949/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved.PII: S0045-7949(98 )00052-2

PERGAMON

this forms the basis of the proposed techniquedescribed in Section 4. Some computational aspects are

examined in Section 5 and the technique is illustratedwith comparative examples in Section 6. Concludingremarks are made in Section 7.

2. Review of existing techniques

Several techniques exist for the static (and dynamic)analysis of cable structures and a brief outline is pro-vided in this section. The purpose of this short review

is not to present a detailed comparison but to illustratethe relative complexity previous researchers haveapplied to this problem.

2.1. Dynamic relaxation

Dynamic Relaxation presents itself as an attractive

approach to the design of prestressed cable networksbecause both form-finding and analysis can be carriedout within a single analytical framework. The tech-

nique traces the motion of the structure from the pointof loading to the final steady equilibrium state (mini-mum energy) through use of the DAlembert principle:

pt Md C _d Kd 1where p(t) is the time-dependent vector of externallyapplied load, M is a fictitious mass matrix, C the

matrix of fictitious damping coecients, K is the sti-ness matrix and d is the vector of displacement. Thefirst two terms on the right-hand side of Eq. (1) rep-

resent the residual, or out of balance force, which isiteratively minimized to an acceptably low value, atwhich point the structure reaches steady equilibrium.Iteration is carried out in small time steps using a finite

dierence approach to find values for changes in nodaldisplacements. The speed and stability of the wholeprocess is critically governed by the fictitious mass

assigned to the nodes, the choice of damping coe-cients, as well as the size of the time step. Lewis etal. [9] and Barnes [1] give details on how values can be

assigned to these parameters.Although dynamic relaxation is generally seen as a

specialist technique, it is a popular choice in the designof cable networks and membrane structures. Lewis [10]

reported it to be more stable and ecient for struc-tures with large degrees of freedom than the stinessmatrix approach outlined below.

2.2. Stiness matrix method

The stiness matrix approach, solved iteratively bythe NewtonRaphson method, is fully described byKrishna [6] who also supplied a computer program

listing. For a prestressed cablenet in equilibrium,instead of the usual p= Kd stiness equation for a

geometrically linear structure (where symbols havemeanings defined above), the equivalent relationship isp= KdR(d) where R(d), the vector of residualforces, is a nonlinear function of d. In addition to theNewtonRaphson iteration, Krishna also advocatedincremental loading for situations where error accumu-

lation is too large.

2.3. Minimizing total potential energy

The approach of minimizing total potential energyhas been described by Coyette and Guisset [3], Sufianand Templeman [15], and Stefanou et al. [14]. The

theoretical approach is essentially the same in eachcase, but the choice of the minimizing algorithm dif-fers. Coyette and Guisset favoured a reduced-gradient

algorithm in the MINOS library; Sufian andTempleman chose a quasi-Newton algorithm from theNAG library; and Stefanou et al. used the conjugategradient method.

2.4. Approximate linear methods

Pellegrino [12] proposed an approximate linearapproach to geometric nonlinearity by observing that ageneral load on a cablenet can be decomposed intotwo parts, i.e. one part which causes extensional dis-

placements, and the second which excites inextensionaldisplacements (infinitesimal mechanisms). Geometricnonlinearity arises from the latter which Pellegrino ver-

ified, by experiments, to be small for prestressedcablenets and hence a linear approximation he pro-posed to determine the magnitude of the second type

of displacements was suciently good for practicalpurposes. Vilnay and Rogers [16] also approached theproblem in the same way.

2.5. Finite element method

While the geometric stiness matrix method outlined

above is strictly speaking a form of finite elementmethod (FEM) in which each cable segment is rep-resented by a single element, some researchers haveapplied a more general FEM approach. Gambhir and

de Batchelor [5], for example, developed a curvedmember for shallow cablenets. Their technique, whichensured continuity of slope across nodes, was devel-

oped primarily for dynamic analysis but also producedstatic response to load. Mitsugi [11] formulated a sti-ness matrix for the hypercable (a cable connected to

intermediate pulleys along its length, also known asactive cable after Kwan et al. [7]) element to beincorporated into an FEM solver.

A. Kwan / Computers and Structures 67 (1998) 243252244

3. Inherent characteristics of prestressed mechanisms

While we can instinctively say that cable networks

are more flexible than other structural forms, the

nature of that flexibility and how it can be controlled

are often less intuitive. We shall examine this flexible

behaviour through the simple two-link planar structure

shown in Fig. 1. Links AB and BC shall be treated as

capable of sustaining both tensile and compressive

axial forces, even though cable elements are unable to

carry compressive forces, because the structure shall be

prestressed to a suciently high level such that the lar-

gest compressive axial bar force due to load can be

overcome by the prestress; a compressive bar force is

thus sustained through a reduction in prestress tensile

force.

The well-known but unsophisticated Maxwells rule

for testing statical/kinematic determinacy (3j-b-k,

where j= number of joints, b= number of bars, and

k= number of degrees of freedom removed by sup-

ports) classifies this two-link structure as a statically

determinate structure with no mobility, which it clearly

is not. There can be no vertical component of bar ten-

sion in the two bars to equilibrate a vertical load P

applied at B. In this configuration, the structure is

unable to resist the load; it is a mechanism. The struc-

ture however, undergoes large deformation, and in a

displaced configuration (dotted line in Fig. 1), the

inclined bars AB and BC are capable of equilibrating

the load P. Since the mechanism is stiened by the dis-

placement, it is an infinitesimal mechanism, as

opposed to the more common large deformation,

finite mechanism.

Such structures with simultaneous statical and kin-

ematic indeterminacy prompted Calladine [2] to modify

Maxwells rule to 3j-b-k= s-m (s= number of states

of self-stress, and m= total number of finite and in-

finitesimal mechanisms). For the structure in Fig. 1,

there is a single infinitesimal mechanism (as shown by

the dotted line) and a single state of self-stress which

involves the two collinear bars having equal bar ten-sion. Note that this state of selfstress requires no exter-

nal load for equilibrium and hence the term self-stress.We should therefore clearly attribute the flexibility

of cable networks not to the low axial stiness of the

constituent cables (which is often not true), but to thegeometry of the structure. If point B had been locatedsome distance below the level of A and C, the structure

would indeed be statically determinate with no mech-anism. Since the two-link structure in its original con-figuration is unable to carry the load P, but the

mechanism is stiened by a vertical movement of B,then the PD relationship is clearly nonlinear. Thestiness of the structure to an applied load is depen-dent on its displaced geometry. The nonlinearity how-

ever, is not to be attributed to a nonlinear stressstrainrelationship of the cable material (which is often nottrue), but is due to the geometry of the two-link struc-

ture. The nonlinear behavior arises not so much out ofthe structure undergoing large deformation beyond thelimit of small deflection theory, but precisely because

small deflection theory is not applicable to cable net-works.Having established the fact that cable structures

have a nonlinear response to load because they are in-finitesimal mechanisms in their undeformed configur-ation, and that the nonlinearity is directly dependenton the geometry of the structure, we shall now devise

the nonlinear large deflection PD relationship forthe two-link structure in Fig. 1, which is preloadedwith a state of self-stress of to in the two bars. It will

be seen that the treatment outlined below, followingthe aim of this technique, requires only understandingof elementary structural mechanics. We shall examine

the compatibility (elongationdisplacement), equili-brium (loadtension), and constitutive (elongationten-sion) relationships in turn, and then combine theserelationships to form the PD equation.

3.1. Compatibility

Consider that point B has lowered by D under theload P, so that bar AB now has lengthL= Lo+e= Losecb. The expression for secb can besubstituted by its Taylors series so that

Losecb Lo 1 b2

2! 5b

4

4! . . .

; and that

e Lo Lob2

2! 5Lob

4

4! . . .

Lo

and when we ignore the fourth order term,

Fig. 1. Planar two-link structure composing of bars AB and

BC, both initially pretensioned by to. Joint B is displaced by

D under the vertical load P, and the bar tensions increase byt.

A. Kwan / Computers and Structures 67 (1998) 243252 245

e Lob2

2!; but b1tan b D

Lo

Lo

D2

L2o

2!

D2

2Lo: 2

In this large deflection consideration then, the bar

elongation e is a second order function of the nodaldisplacement.

3.2. Equilibrium

Unlike small deflection theory, the equilibrium ofthe structure is considered at the deformed configur-ation. Vertical equilibrium at B gives:

2t tosinb P; but sin b1b1tanb DLo

and hence

t to PLo2D

: 3

3.3. Constitutive relationship

The relationship between the elongation e (due to

the load alone) and the bar tension is:

e tLoAE PLo

2D to

LoAE

: 4

3.4. Governing PD equation

Eqs. (2)(4) can be combined together to form acubic relationship between the load and the displace-

ment D.

EA

LoD3 2toLoD PL2o 0: 5

The PD relationship in Eq. (5) provides even more in-formation on the behavior of the cablenets than notedabove. If to=0 then the PD curve has zero slope atthe origin, and the two-link structure thus has no re-sistance to load; it is a mechanism. Away from the ori-gin, PD does have a positive slope, and hence themechanism is stiened by displacement; it is thus aninfinitesimal mechanism. When a state of self-stress isapplied, to$0, the PD curve has slope at the originproportional to to and hence the structure does indeedhave initial resistance against load; the infinitesimalmechanism is stiened by self-stress. Note though that

this initial stiness is dependent on the prestress, butnot the axial stiness AE of the bars.

4. Present formulation

We shall now derive the equivalent equations for a

general bar in two-dimensional space and then applythese equations as an example to the two-link structurein Fig. 1. Consider a bar 12, with end coordinates

(x1, y1) and (x2, y2), and initial length L undergoingdeformation so that it ends up in new position 12with a length of L, see Fig. 2a.

4.1. Compatibility

Adopting the short hand notation ( )21=( )2( )1and taking into account joint displacement, the newbar length

L0 x21 dx212 y21 dy212 1=2 L2 1 A

L2

1=2where

A dx221 2x21 dx21 dy221 2y21 dy21

1L 1 12

A

L2

18

A

L2

2 . . .

( )

L dx221

2L dy

221

2L x21 dx21

L y21 dy21

L

x221 dx

221

2L3 y

221 dy

221

2L3 x21y21 dx21 dy21

L3:

Hence

e dx221

2L dy

221

2L x21 dx21

L y21 dy21

L x

221 dx

221

2L3

y221 dy

221

2L3 x21y21 dx21 dy21

L3: 6

We thus have an expression for the bar elongation as a

Fig. 2. (a) Bar 12 of length L in two-dimensional space is

displaced to new position 12; the displaced bar has lengthL. (b) Equilibrium of the bar 12 in displaced position.


second order function of the nodal displacements, hav-ing ignored all higher order terms.

4.2. Equilibrium

We consider equilibrium in the displaced configur-ation, taking into account also the change in the barlength up to first order displacements. Firstly, ex-pressions for sin y and cos y are:

sin y x21 dx21L0

x21 dx21L x21dx21y21dy21L x21 dx21L 1 x21dx21y21dy21

L2

1 x21 dx21

L

1 x21dx21 y21dy21

L2 . . .

1 y21

L y21x21 dx21

L3 y

221 dy21L3

dy21L

and similarly,

cos y1x21L x

221 dx21L3

x21y21 dy21L3

dx21L

:

The equilibrium equations at the two nodes are as

follows.

Px1 t tocos yPy1 t tosin yPx2 t tocos yPy2 t tosin y: 7

4.3. Constitutive relationship

The relationship between the bar elongation and thebar tension, being entirely independent of assumptionsmade in equilibrium and compatibility equations, isstill e= tL/AE.

4.4. Two-link structure

Having derived the governing equations for a gen-

eral two-dimensional bar, we can now apply theequations to the two-link structure in Fig. 1, andexpect the same numerical results as those computed

by Eq. (5). We shall have P= 311.38 N, AE= 564.92kN, Lo=5080 mm and to=4448.2 N because thesevalues represent metric equivalents to imperial values

used by Levy and Spillers [8] analysing the same pro-blem. In the two-link structure, the two equilibriumequations at joint B are:

tAB tBC 0; and tAB 4448:22 dyb5080

tBC 4448:22 dyB

5080

311:38: 8

The two compatibility equations can be combined with

the constitutive relationships, to eliminate barelongations as variables, and the resulting equationsare:

dy2B10160

dxB tAB 5080564:92 103 ; and

dy2B10160

dxB tBC 5080564:92 103 : 9

Eqs. (8) and (9), representing the PD relationshipsfor the two-link structure, are four equations (twoof which are nonlinear) in four unknowns X= (dxB,dyB, tAB, tBC). The nonlinear equations necessitatesome form of iterative solution and a standard Gauss

Newton algorithm (see e.g. Dennis [4]) has proved towork very well. The algorithm requires the equations(linear and nonlinear) to be written as a function F(X)

that returns the residuals for a trial vector X. TheJacobian of F can also be explicitly provided, and inour case this is not dicult, but the algorithm will

work without one by estimating the Jacobian usingfinite dierence. As it turned out, the nonlinearity wehave in our formulation is very well behaved, and the

GaussNewton algorithm converges without diculty,even though the initial starting point for X is alwaysgiven as the null-vector.The GaussNewton algorithm gives the solution to

the current problem as X= (0, 166.449, 303.246,303.246), which is identical to the solution(D= 166.45 mm) obtained from Eq. (5). Since bothanswers are derived from the same theoreticalapproach, this finding is firstly not very surprising, andsecondly provides comforting confirmation that the

GaussNewton algorithm is well-behaved. These twoanswers also compare very well with that provided byLevy [8], dyB=166.54 mm.The formulation for a general three-dimensional

bar, being a straightforward extension of the two-dimensional bar, is not shown but has been used in theillustrative examples in Section 6.

5. Solution techniques

Clearly dierent techniques can be employed tosolve the family of non-linear equations typified byEqs. (8) and (9). Since the form of the nonlinear

equations is known, and the highest order of non-linearity is two, it is possible to develop an algorithmthat specifically exploits these features, but such a


development is not treated in this paper. The presentauthor has used GaussNewton in its most elementary

form because it has been found to be both ecient

and well-behaved, even with complicated examples.

Three simple enhancements though could be supplied

to the GaussNewton approach if they were deemednecessary by the analyst.

Firstly, the initial vector for trial solution is given asthe null-vector, whereas a more reasonable initial esti-

mate for the displacements would be a multiple of the

mechanism vector. A basis for the mechanism vector

can be found from the nullspace of the small-deformation compatibility matrix (see Pellegrino [13]

for more details). Secondly, an explicit Jacobian to the

governing equations could be supplied with little extra

dierential algebraic eort. This would eliminate the

need to estimate the Jacobian by repeated calculations

using finite dierence approximations at every iter-

ation. Thirdly, if convergence proved to be elusive

because of a large amount of nonlinearity for a par-

ticular problem, incremental loading could be used

where the displaced parameters after each load incre-

ment would be used as initial values for the next

Fig. 3. 3 3 flat net in three-dimensional space is loaded bythree equal vertical loads of 150 N.

Fig. 4. Results from the present technique compared to results

given by Lewis [10] (in italics, where available). (a) Bar ten-

sion in N. (b) Nodal displacements in the x-, y- and z-direc-

tions.

Fig. 5. Saddle net with underlying grid showing joint numbers; z-coordinates are given in Table 1.


increment. None of these enhancements, however, havebeen found to be necessary in the examples in the

following section, although they might be useful formore complicated problems.

6. Examples

We shall now examine three three-dimensionalexamples using the governing equations for the three-dimensional bar, whose derivations follow exactly the

pattern described above for the two-dimensional bar.The examples have been chosen to compare resultsfrom the present technique with those quoted in theliterature.

6.1. Flat net

The flat net example consists of a cablenet lying ona 3 3 square grid with cell side lengths of 400 mm,see Fig. 3. The cables have AE= 97.97 kN, and pre-

tension of to=200 N in all cables. The cables arefirmly anchored at the perimeter leaving four freejoints. There are thus 4 3 = 12 equilibriumequations and 12 compatibility-constitutive equations,in 24 unknowns (12 joint displacements and 12 bartensions). Starting with a null vector, the Gauss

Newton algorithm returns nodal displacements andbar tensions shown in Fig. 4, which are exactly theresults obtained by Lewis [10] using both the tech-niques of dynamic relaxation and stiness matrix.

Table 1

Nodal z-coordinates and nodal displacements for selected joints in the saddle net

z Present results Results from Lewis (1987)

Coor dx dy dz dx dy dz

Node (mm) (mm) (mm) (mm) (mm) (mm) (mm)

1 3632 0 0 0

2 2568 0 0 0

3 1808 0 0 0

4 1352 0 0 0

5 1200 0 0 0

10 5000 0 0 0

11 3968 15.55 4.46 81.70 15.5 4.5 82.012 3165 11.50 5.55 61.2213 2592 7.38 4.20 33.3114 2248 5.34 3.11 17.8815 2133 4.11 2.80 11.16 4.1 2.9 11.221 5000 +0 0 0

22 4208 14.43 3.53 97.1423 3592 11.27 4.47 72.9024 3152 7.25 2.97 31.9825 2882 5.67 2.12 10.5426 2800 4.77 0.60 11.3432 5000 0 0 0

33 4352 11.71 1.71 92.4434 3848 9.55 2.11 66.9435 3488 6.30 1.15 20.21 6.2 1.2 19.836 3272 4.92 0.23 14.0537 3200 4.65 0.52 35.7943 5000 0 0 0

44 4400 10.63 0 88.73 10.6 0 88.7

45 3933 8.80 0 62.83

46 3600 5.83 0 13.99

47 3400 4.64 0 22.5248 3333 4.55 0 45.89 4.5 0 46.752 4400 0.92 0 5.86 0.9 0 6.072 3152 3.85 0.78 30.12 3.9 0.7 30.281 2133 4.11 2.80 11.16 4.0 2.9 11.0

85 3968 5.40 1.87 32.17 5.5 1.9 32.6


6.2. Saddle net

Lewis [10] compared the stiness matrix and

dynamic relaxation techniques on five examples ofcable nets of increasing complexity. The most complexexample quoted was the saddle net shown now inFig. 5. Although both techniques converged on similar

solutions (Lewis reported a 0.5% discrepancy) for thefirst four examples, Lewis found the stiness matrixapproach unable to converge on a solution for the sad-

dle net; ill-conditioning of the p= Kd system (whereK= KE+KG, KE is the elastic stiness matrix, and KG

is the geometric stiness matrix) was highlighted as theproblem for such a large system. The dynamic relax-

ation however, converged onto a solution and the pre-sent formulation is thus compared to that technique.The saddle net example is shown in Fig. 5 with an

underlying square grid with superimposed joint num-bers where convenient. The individual square cellshave side lengths of 5.0 m. The net has mirror sym-

metry about both centerlines and the z-coordinates fora quarter of the net is given in Table 1. All cable seg-ments have AE= 44.982 MN, and pretension of

to=60 kN. Some joints, forming a quarter of the netin total, are loaded by 1.0 kN in the x- and z-direc-tions. They are joints 11 4 15, 22 4 26, 334 37,44 4 48, 554 59, 664 70 and 774 81. All perimeterjoints are anchored, giving thus a total of 63 freejoints. These, together with 142 cable segments, pro-vide 331 nonlinear equations in 331 unknowns. The

GaussNewton algorithm, using the 331-dimensionalnull vector as a starting point converges on a sol-ution very similar to that produced by Lewis. Nodal

displacements are compared in Table 1 and cabletensions in Table 2, for specific values given byLewis.

6.3. Hyperbolic paraboloid net

A hyperbolic paraboloid net, shown in Fig. 6, has

been numerically analysed by several researchers aswell as experimentally tested by Lewis [9]. The struc-

Table 2

Cable tension after load on selected cables in the saddle net

Present results Lewis results

Cable Tension (kN) Tension (kN)

1112 53.70 54.14

2324 57.80 58.12

4748 62.51 62.78

6061 59.90 60.18

7273 54.83 55.16

8586 50.23 50.60

111 75.43 75.52

2435 69.30 69.56

2839 48.00 48.28

6273 57.37 57.61

6778 78.69 78.89

8595 63.55 63.66

Fig. 6. Hyperbolic paraboloid net with dimensions in mm. All edge nodes are supported by pin foundations and 10 internal nodes

are applied equal vertical loads.


ture consists of 31 cable segments (AE= 100.72 kN)loaded at some of the joints by a load of 157 N. The

cables were pretensioned to carry 200 N prior to theapplication of load. A table of vertical deflectionsobtained by dierent researchers is shown in Table 3.

The second column consists of experimentallymeasured deflections. Lewis [10] and Sufian et al. [15]analysed this problem by the stiness matrix method,

and a minimum energy method, respectively.

Lewis [9, 10] performed a dynamic relaxation calcu-lation in Fortran on a Prime 750 mainframe computer

and her results are shown in the fifth column. The pre-sent author has also computed the problem withdynamic relaxation on a desktop microcomputer. The

results of this computation, shown in the sixth columnof Table 3, are very similar to those provided byLewis. The final column of Table 3 shows results from

the proposed technique. It can be seen from Table 3

Fig. 7. Displacement of joint 22 of the hyperbolic paraboloid shown in Fig. 6 during three dynamic relaxation computations. As

the fictitious damping value is increased, the final value of displacement is obtained with fewer oscillations.

Table 3

Comparison of displacements (in mm) and computation time (in CPU seconds)

Stiness Minimum Dynamic Dynamic Proposed

Experiment matrix energy relaxation relaxation technique

Joint (Lewis) (Krishna) (Sufian) (Lewis) (present) (present)

5 19.5 19.6 19.3 19.3 19.38 19.52

6 25.3 25.9 25.5 25.3 25.62 25.35

7 22.8 23.7 23.1 23.0 22.95 23.31

10 25.4 25.3 25.8 25.9 25.57 25.86

11 33.6 33.0 34.0 33.8 33.79 34.05

12 28.8 28.2 29.4 29.4 29.32 29.49

15 25.2 25.8 25.7 26.4 25.43 25.79

16 30.6 31.3 31.2 31.7 31.11 31.31

17 21.0 21.4 21.1 21.9 21.28 21.42

20 21.0 22.0 21.1 21.9 21.16 21.48

21 19.8 21.1 19.9 20.5 19.79 20.00

22 14.2 15.7 14.3 14.8 14.29 14.40

CPU 150 120


that there is close agreement in joint displacementsbetween all the dierent methods, and the presently

proposed technique thus compares well with all theestablished techniques.The purpose of reproducing the dynamic relaxation

calculation is to have a comparison of the computationtime required by the dynamic relaxation technique,and that by the present technique. Computation time

in dynamic relaxation is highly dependent on threeparameters: the time increment, the fictitious nodalmasses and the damping coecients at the nodes.

When Lewis [9] original parameters were used, thecomputation time on the present authors desktopcomputer was exceedingly large (>70,000 cpu s).Alternative parameters were thus investigated and

Fig. 7 shows that a stable solution could be attainedafter about 150 cpu s. This value, rather surprisingly,is slightly larger than that obtained for the presently

proposed technique, computing the same problemwithout any of the enhancements discussed in Section5, on the same desktop computer. It would seem there-

fore, that the present technique, although developed in-itially for its simplicity in the treatment of geometricnonlinearity, is also comparable in speed to the more

established technique of dynamic relaxation.

7. Conclusions

A new approach to the statical behavior of geome-trically nonlinear cable structures has been presented.

The key features of this approach are the separation ofthe numerical computation from the underlying mech-anics of the problem and understanding of only the

most elementary of structural mechanics is required.This technique is thus eminently suitable for presen-tation at undergraduate courses on cable structures.On the other hand, in spite of its lack of sophisti-

cation, the proposed technique has been proved to becomparable to more established techniques in its accu-racy of solution, in its speed of calculation and in its

stability of solution, even when optional enhancementsare not used.

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a new approach to geometric nonlinearity of cable structure

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