mechanisms of prestressed reticulate systems with unilateral stiffened components

European Journal of Mechanics A/Solids 27 (2008) 61–68

Mechanisms of prestressed reticulate systems with unilateralstiffened components

Bernard Maurin ∗, Marine Bagneris, René Motro

Laboratoire de Mécanique et Génie Civil, UMR CNRS 5508, Université Montpellier 2, CC48, Montpellier, France

Received 8 December 2005; accepted 10 May 2007

Available online 7 June 2007

Abstract

The stability of space reticulate systems is dependent on the existence of mechanisms. The methods that have been developedto determine them are mainly based on the calculation of a basis of the mechanisms’ vectorial subspace by computing the kernelof the transpose equilibrium matrix of the structure. However, they can only consider a bilateral stiffness of the members, whichapplies to the case for systems composed of bars with traction and compression stiffness. Nevertheless, some classes of reticulatesystems, like tensegrity systems, use unilateral rigidity components such as cables. The objective of this paper is to develop amethod for calculating the mechanisms which can take into consideration the presence of components with unilateral rigidity. Inthis case, specific mechanisms associated with these elements may appear; these are referred to as “unilateral mechanisms”. Anapproach is therefore proposed in order to write a basis of their vectorial subspace. It is included in a methodology devoted to theanalysis of space structures with initial stresses. The process is based on the identification of the possible prestress states and of thebilateral mechanisms and, next, to the characterization of the unilateral mechanisms.© 2007 Elsevier Masson SAS. All rights reserved.

Keywords: Space structure; Mechanism; Unilateral stiffness

1. Objective of the study

This study deals with space reticulate pin-jointed systems. Their static analysis under external loading is generallywithout difficulty, excepted if the structure has one or several mechanism. Moreover, a possible instable behaviour ofthe structure may result if a mechanism is activated by the loads. Mechanically, the system has in this case no rigidityat first order and its global stiffness matrix is generally singular. It is therefore necessary, before any static analysis, tofirstly check the presence of mechanisms and, if existing, to define them. Several methods have been proposed on thatpurpose (Calladine and Pellegrino, 1991). They are not based on a mechanical analysis of the structure with externalloads. It is firstly due to the possible numerical singularities and, more important, to the fact that the designer has noguaranty for the prescribed loading to activate all the existing mechanisms. Theoretically, all the possible loadingsshould be tested, which is quite impossible for large structures. The methods that have been developed are thereforeindependent from external actions and are based on the analysis of the kernel of the equilibrium matrix in order to

* Corresponding author. Fax: +33 4 67 14 45 55.E-mail address: [email protected] (B. Maurin).

0997-7538/$ – see front matter © 2007 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.euromechsol.2007.05.009

62 B. Maurin et al. / European Journal of Mechanics A/Solids 27 (2008) 61–68

determine a basis for the mechanisms. Such approach only considers components with a bilateral stiffness, for instancebars with traction and compression stiffness. Calculations lead thus to the definition of “bilateral mechanisms”.

However, it does not always fit with the reality of some reticulate structures which are composed of elements likecables with unilateral rigidity (tensegrity systems, cable nets or cable beams, etc.). In this case, a slackening of thesecomponents may occur, which means that others specific mechanisms may appear; we will label them “unilateralmechanisms”. The behaviour of the structure is also dependent on the mechanisms and on their activation by theloading. This could be illustrated by the structure represented on Fig. 2 left, composed of two horizontal componentsand one vertical cable 3 connected to one node. If a vertical action is applied to the node downwardly, no mechanismwill occur. On the opposite, an ascendant load activates a unilateral mechanism which leads to the cable 3 slackeningand no structural stiffness at first order.

The objective of this study is therefore to develop a method devoted to the determining of the unilateral mecha-nisms due to the unilaterally stiffened components. The corresponding basis of displacements will be characterizedindependently from the loading.

2. Mechanical background

Reticulate pin-jointed systems comprising m members with bilateral rigidity (bars) or unilateral rigidity (cables)are considered in this work. They are assembled with perfect pins and the structure has N degrees of freedom (dof).

Since such systems can be kinematically and statically indeterminate, they may be characterized by possible mech-anisms and initial prestressed states.

The equilibrium equation, in case of applied external nodal loads f , may be written as (Livesley, 1975):

[A]{q} = {f }. (1)

For the members, we use in this relationship the force density coefficients defined as qj = Tj/ lj for an element j

where Tj represents its resulting axial force and lj its length. A positive value implies tension in the element. Thematrix [A] is the equilibrium matrix of the structure.

If we consider at present the initial non-loaded geometry with possible prestressed forces in some components,the equilibrium equation becomes [A]{q0} = 0 with q0

j = T 0j / l0

j (T 0j being the initial prestressed axial force in an

a element of length l0j ). The values of the coefficients q0

j (i.e. the vectors {q0j }) may be determined by calculating

the kernel of the matrix [A]. It corresponds to the basis of the prestressed states, actually the basis of matrix [A]nullspace. We write {q0

j } ∈ KerA, where KerA is a vectorial subspace of �m (its dimension is associated to thenumber of elements m).

From this initial non-loaded geometry, a virtual displacement {d} of the nodes, compatible with the boundaryconditions, may be considered. The resulting lengths of the components are lj and the corresponding length variationscould be characterized by the parameters:

ej = l0j

(lj − l0

j

). (2)

Such writing is useful, firstly because ej is equal to zero if there is no length variation at first order for an element j

and, secondly, because the vector {e} for the whole system can be determined by using the transpose matrix [A] ofaccording to:

[A]t {d} = {e}. (3)

If the structure has an infinitesimal mechanism, there is no length variation at first order for every component, hence{e} = 0. By writing the associated displacement {dK}, it comes [A]t {dK} = {0}. Hence, the basis of the mechanismscorresponds to the kernel of the matrix [A]t (i.e. the vectorial left nullspace of [A]t ). It implies {dK} ∈ KerAt whereKerAt is a vectorial subspace of the displacement space �N (N dof). We point out that the determining of thesubspace KerAt assumes that all the members have a bilateral rigidity. Hence, we will label these displacements“bilateral mechanisms”.

Moreover, in the displacement space �N , the vectorial subspace ImA (the column space of [A]) is orthogonal andsupplementary to KerAt (Strang, 1980). Such a displacement is written {dI }. Since �N = KerAt + ImA (where “+”represents the direct summation), every displacement {d} can be uniquely splitted in according to {d} = {dK} + {dI }.These subspaces may be represented as illustrated in Fig. 1.

B. Maurin et al. / European Journal of Mechanics A/Solids 27 (2008) 61–68 63

Fig. 1. Vectorial subspaces associated with the application A.

3. Unilateral mechanisms determining

The purpose of this study is to propose a method for the calculation of the unilateral mechanisms. The approach isbased on the following three phases.

(1) Among the system compatible movements (i.e. on the vectorial space �N ), we first identify the system displace-ments that do not modify the length of the elements at first order. It corresponds to the writing of the basis of thebilateral mechanisms (vectorial subspace KerAt ).

(2) Among the remaining movements (on the vectorial subspace ImAt = �N − KerAt , with “−” being the directsubtraction), we secondly identify the displacements that do not modify the length of the prestressed elements.

(3) Among these displacements, we next determine those that shorten the lengths of the non-prestressed cable ele-ments and thus lead to their slackening. It provides the searched basis of the unilateral mechanisms.

The approach may be illustrated with a simple example represented on Fig. 2. It comprises two horizontal cables(number 1 and 2) and one vertical cable (number 3). Their lengths are equal to 1. The node connecting these elementsis free in the plane (dof 1 and 2 according to x and y directions).

The equilibrium matrix of the structure is [A] = [ 1 −1 00 0 −1

]. Its kernel calculation leads to KerAt = φ; hence there

is no bilateral mechanism.Since the mechanisms due to unilaterally stiffened members do not appear in this writing, it means that they can

only belong to the subspace ImA = �N − KerAt .As KerAt is a nil subspace, then ImA = �N and a corresponding basis ImA of is [ImA] = [�N ] = [ 1 0

0 1

].

Next, a basis of ImAt can be defined by considering:

[ImAt

] = [A]t [ImA]; it comes[ImAt

] =[ 1 0

−1 00 −1

]. (4)

Fig. 2. Simple illustrative example (transpose basis matrices are shown on right).


We search at present on the subspace ImA the displacements that do not modify the length of the prestressed elements.It requires the calculation of the basis of KerA in order to determine which elements are prestressed and thereforeinvolved.

In this example [KerA] =[

110

]; which means that only the two horizontal cables 1 and 2 may be pretensioned while

the vertical 3 cannot have an initial force. We split therefore the basis of ImAt according to [ImAt ] =[ [

ImAt]p

. . . . . . .[ImAt

]np

]where a dotted line separates the prestressed (p) and the non-prestressed (np) members according to two sub-matrices

(as also observed on Fig. 2). It comes [ImAt ] =[

1 0−1 0. . . . .0 −1

].

The displacements on ImA that do not change at first order the length of the two prestressed elements will be nowidentified. It could be achieved by calculating the suitable combinations of the displacement vectors on ImA that donot modify these lengths. Such combinations are defined by the combinatory coefficients α which verify:

[ImAt

]p{α} = {0}, here

[1 0

−1 0

]{α1α2

}=

{00

}. (5)

The coefficients α correspond to the kernel Ker([ImAt ]p) of the matrix [ImAt ]p .

In the example, calculation gives [Ker([ImAt ]p)] =[

01

]. The corresponding basis of the displacements is [dp] =

[ImA][Ker([ImAt ]p)], that is to say the vector{

01

}.

Then, it must be checked if this movement shortens the length of the non-prestressed cable 3. By considering thesubmatrix [ImAt ]np, the corresponding length variation is:

[e]np = [ImAt

]np

[Ker

([ImAt

]p

)]. (6)

For the cable 3 it comes [e] = [0 −1][

01

]= −1. It implies that the displacement

{01

}reduces the length of the vertical

cable 3 (slackening) and thus corresponds to the searched unilateral mechanism.

4. First application

This application deals with a 3D system represented on Fig. 3 with one vertical strut (number 1, length equal to 1)and four cable elements (length equal to

√2 for cables 2, 3 and 4 and equal to

√3 for cable 5). The system has three

dof at the top node (dof numbered 1, 2 and 3 according respectively to x, y and z directions, all the black nodes beingfixed).

It comes

[A] =[0 −1 1 0 1

0 0 0 −1 −11 1 1 1 1

]

Fig. 3. 3D example.


and, as a result, KerAt = φ (no bilateral mechanism).Hence, a basis of ImA is

[ImA] = [�N] =

[1 0 00 1 00 0 1

]

and the corresponding basis of ImAt is

[ImAt

] =

⎡⎢⎢⎢⎣

0 0 1−1 0 11 0 10 −1 11 −1 1

⎤⎥⎥⎥⎦ .

The calculation of the kernel KerA leads to the basis of the prestressed states

[KerA] =

⎡⎢⎢⎢⎣

−2 −11 11 00 −10 1

⎤⎥⎥⎥⎦ .

However, we may observe that the second column implies a compression (−1) in the cable number 4 while thecable 5 is tensioned (1). Since such situation is not possible, this vector is eliminated and the first column is the uniquecompatible prestressed state. For this prestress vector, there are only initial forces in the elements 1 to 3; then

[ImAt

] =

⎡⎢⎢⎢⎢⎢⎣

0 0 1−1 0 11 0 1

. . . . . . . . . . . . .0 −1 11 −1 1

⎤⎥⎥⎥⎥⎥⎦ .

The displacements on ImA that do not change the length of these three prestressed elements may calculated with

[ImAt

]p{α} =

[ 0 0 1−1 0 11 0 1

]{α1α2α3

}=

{000

}.

The kernel [Ker([ImAt ]p)] =[

010

]gives the values for the combinatory coefficients α. The associated movement

is [dp] = [ImA][Ker([ImAt ]p)], in this case the vector

{010

}. It must be next determined if this displacement shortens

the length of the non-prestressed cables 4 and 5.With

[e]np = [ImAt

]np

[Ker

([ImAt

]p

)] =[

0 −1 11 −1 1

][010

]=

[−1−1

],

we verify that the lengths of these two element decrease (−1). Therefore, the displacement

{010

}is a unilateral mech-

anism of the system.

5. Second application

This example, represented on Fig. 4, comprises four cable elements with a length equal to 1. The system has fourdof (1 and 2 for the node 1 according respectively to x and y directions; 3 and 4 for the node 2). It has been originallystudied by Tarnai (1989), Kuznetsov (1991) and Vassart et al. (2000) so as to determine the order of its bilateralmechanism.


Fig. 4. Four cable elements example.

Fig. 5. The bilateral and the two unilateral mechanisms.

In this case

[ImA] =⎡⎢⎣

1 −1 0 00 0 1 00 0 0 00 0 −1 1

⎤⎥⎦ .

Hence, it exists a bilateral mechanism

[KerAt

] =⎡⎢⎣

0010

⎤⎥⎦ ;

it is shown on Fig. 5 left.Then, a basis of ImA orthogonal and supplementary to KerAt is

[ImA] =⎡⎢⎣

1 0 00 1 00 0 00 0 1

⎤⎥⎦ .

Moreover, the calculation [KerA] =[

1100

]shows that the system has a prestressed state where the elements 1 and

2 are in pretension while the vertical elements 3 and 4 have not an initial force. It comes the basis

[ImAt

] = [A]t [ImA] =

⎡⎢⎢⎢⎣

1 0 0−1 0 0. . . . . . . . . . . . .0 1 −1

⎤⎥⎥⎥⎦ .

0 0 1


Next, with [ImAt ]p{α} = 0 ⇒[

1 0 0−1 0 0

]{α1α2α3

}=

{000

}, the basis for the coefficients α is then [Ker([ImAt ]p)] =[

0 01 00 1

].

The two corresponding displacements are

[d]p = [ImA][Ker([

ImAt]p

)]=⎡⎢⎣

0 01 00 00 1

⎤⎥⎦ .

The equation [e]np = [ImAt ]np[Ker([ImAt ]p)] provides the length variations for the non-prestressed cables 3 and

4:[

0 1 −10 0 1

][0 01 00 1

]=

[1 −10 1

].

The first column of the resulting matrix shows that, if α1 = 0, α2 = −1 and α3 = 0, then the cable 3 is shortened(−1) while the cable 4 has no length variation (0). The associated displacement is { 0 −1 0 0}t and therefore a unilateralmechanism.

The second column shows that, if α1 = 0, α2 = 0 and α3 = 1 (that is to say the displacement { 0 0 0 1}t ), then thecable 3 is shortened (−1) while the cable 4 is lengthened (1). This movement is not a unilateral mechanism.

However, if the basis [Ker([ImAt ]p)] is rewritten by linear combinations

[0 0

−1 −10 −1

]with the associated movements

[d]p = [ImA][Ker([

ImAt]p

)]=⎡⎢⎣

0 0−1 −10 00 −1

⎤⎥⎦ .

Then

[ImAt

]np

[Ker

([ImAt

]p

)] =[

0 1 −10 0 1

][ 0 0−1 −10 −1

]=

[−1 00 −1

].

In this case, both displacements generate a shortening of the non-prestressed vertical cables. The two unilateralmechanisms are thus { 0 −1 0 0}t (for α1 = 0, α2 = −1 and α3 = 0, as previously determined) and { 0 −1 0 −1}t (forα1 = 0, α2 = −1 and α3 = 1). They are represented on Fig. 5 centre and right.

We note that the movement { 0 0 0 1}t is actually a linear combination of the two unilateral mechanisms. However,its calculation requires the use of a negative combinatory coefficient. It points up that a combination of unilateralmechanisms leads to a unilateral mechanism only, and only if, positive combinatory coefficients are used.

6. Procedure

The sequence of steps to follow in order to determine a basis for the unilateral mechanisms for a prestressedreticulate system with unilaterally stiffened components (N dof and m members) is presented hereafter. They aredivided in three main phases.

Phase 1: – writing of the equilibrium matrix [A](N × m);– basis of the bilateral mechanisms: calculation of [KerAt ](N × b);

Phase 2: – basis of ImA (supplementary and orthogonal to KerAt ). If the solution is not simple, the vectors of[ImA](N × (N − b)) can be calculated by using the Schmidt’s orthonormalization method (see the ap-pendix);

– basis of ImAt with [ImAt ] = [A]t [ImA](m × (N − b));– basis of the selfstress states: calculation of [KerA]; cancellation of the non-compatible selfstress states

(cables in “compression”); identification of the prestressed (p) elements and the non-prestressed (np)

cable elements;– splitting of [ImAt ] in [ImAt ]p(p × (N − b)) and [ImAt ]np(np × (N − b));


Phase 3: – basis of the displacements on ImA that do not modify the lengths of the prestressed elements: calculationof [Ker([ImAt ]p)]((N − b) × dp) and of [dp] = [ImA][Ker([ImAt ]p)](N × dp);

– choice of the displacements on [dp] that shorten the lengths of the non-prestressed cable elements: calcu-lation of [e]np = [ImAt ]np[Ker([ImAt ]p)](np × dp) and cancellation of the displacements characterizedin this matrix columns by at least one non-negative term. The remaining movements define a basis for theunilateral mechanisms.

7. Conclusion

We present in this paper a method devoted to the infinitesimal mechanisms determining for space reticulate sys-tems comprising elements like cables with unilateral rigidity. It completes the existing approaches that only considerbilaterally stiffened components and by leading to the identification of the unilateral mechanisms. It is based on thecalculation of the basis of the prestressed states and of the bilateral mechanisms, followed by the determining of thebasis of the mechanisms associated to the non-prestressed unilaterally stiffened elements.

Appendix. Schmidt’s orthonormalization method

This method allows creating an orthonormed basis (orthogonal and normed vectors) from a given basis. If the initialcomputed basis of KerAt is [KerAt ] = [k1| . . . |kb] (b vectors with N components), a second orthonormed basis of[KerAt ] = [k1| . . . |kb] is generated according to k1 = k1/〈k1, k1〉 (〈, 〉 being the scalar product) and, for i = 2 to b,ki = ki − ∑i−1

j=1〈ki, kj 〉kj followed by ki = ki/〈ki , ki〉.The vectorial subspace ImA is supplementary and orthogonal to KerAt (�N = KerAt + ImA). Hence, if N − b

vectors, orthogonal to the basis defined by [k1| . . . |kb] are identified, they will generate an appropriate basis for ImA.The iterative sequence is maintained for i = b + 1 by selecting kb+1 as one of the N vectors of the canonical basisof �N . In the case where kb+1 = 0, kb+1 belongs to the basis of ImA. In the case where kb+1 = 0, kb+1 dependson [k1| . . . |kb] and another vector kb+1 in the canonical basis is selected. Thereby, N − b non-nil vectors kb+1 to kN ,orthogonal and supplementary to KerAt , are identified for the generation of the search basis [ImA] = [kb+1| . . . |kN ].

References

Calladine, C.R., Pellegrino, S., 1991. First-order infinitesimal mechanisms. Int. J. Solids Structures 27, 505–515.Kuznetsov, E.N., 1991. Systems with infinitesimal mobility. Part I: matrix analysis and first-order mechanisms. J. Appl. Mech. 58, 513–519.Livesley, R.K., 1975. Matrix Methods of Structural Analysis, second ed.. Pergamon Press, Oxford.Strang, G., 1980. Linear Algebra and its Application. Academic Press, New York.Tarnai, T., 1989. Higher order infinitesimal mechanisms. Acta Tech. Acad. Sci. Hung. 102, 363–378.Vassart, N., Laporte, R., Motro, R., 2000. Determination of mechanism’s order for kinematically and statically indetermined systems. Int. J. Solids

Structures 37, 3807–3839.

mechanisms of prestressed reticulate systems with unilateral stiffened components

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