It is convenient to change the vector notation used above to a matrix one. Thus, considering the ith node of the system, the Cartesian coordinates of its position vector can be stored in a column-matrixxi = [xi]3×1. Similarly, components of the external force, acting on the same node, can be stored inFi = [ Fi]3×1. Also, respecting usual conventions of matrix structural analysis, the column-matrix representing the internal force vector, acting on that node, is defined asPi = [− Pi]3×1.
Matrices xi,Fi,Pi, i = 1, . . . , n, can be grouped into three global matrices, respectively, theposition vectorx= [xT1 xT2 . . . xTn]T3n×1, theexternal load vector Fand theinternal load vectorP(with definitions analogous tox).
Nodal displacement can also be grouped in a column-matrix. Storing the com- ponents of the displacement of theith node inui = [ui]3×1, theglobal displacement vectoris written asu= [uT1 uT2 . . . uTn]T3n×1.
The position vector can then be written asx = x0+u, wherex0 is a constant vector which describes an initial configuration. The current geometry of the system
R.M.O. Pauletti
can therefore be defined by eitherxoru. Both vectors can be generically understood asconfiguration parametersof the system.
With the above definitions, the problem of finding the equilibrium configuration of a network of central forces can be posed as
Findu∗such that
g(u∗)=P(u∗)−F(u∗)=0, (3) whereg(u)is theunbalanced load vector, or error vector.
This system can be solved, within some vicinity ofu∗, iterating Newton’s recur- rence formula
ui+1=ui−
∂g
∂u
ui
−1
g(ui)=ui−(Kit)−1g(ui), (4) where thetangent stiffness matrixKit is defined.
It is advantageous to consider the usual sparsity of nodal connections, and the many null interaction loadsNij. So, internal forcesPi are assumed to be imposed by some bars, which connect the nodes of the system. These bars are numbered from 1 tob, and the intensities of the interaction loads are stored in aninternal load vectorN = [N1 N2 . . . Nb]Tb×1, collecting the normal loads developed. Thus, a generic bar, orelement, identified by the indexe, and connecting nodesiandj, is under a normal load Ne = Nij and its space orientation is given by a unit vector
ve= vij, whose corresponding column-matrix isve = [vij]3×1, which also provides the director-cosines of the bar, respect the global coordinate system.
Now, the vector of internal forces can be decomposed as P = CN, where N =N(u)is a vector ofscalar internal loadsandC =C(u)is ageometric oper- ator, collecting the elements’ unit vectorsve. There results, for the tangent stiffness matrix:
Kt =NT∂C
∂u +C∂N
∂u −∂F
∂u =Kg+Kc+Kext, (5) where thegeometric, theconstitutiveand theexternal stiffness matricesare respect- ively defined.
The geometric stiffness matrixKgcorresponds to a reluctance of the network to change its geometry, for a given state of internal loads. It isKgthat most precisely defines the class of taut structures, those that are under tension, and rely essentially on this state to behave properly.
The constitutive stiffness matrixKccorresponds to a reluctance of the network to change its state of internal loads, for a given geometric configuration. Similarly, the external stiffness matrixKextcorresponds to a reluctance of the external force field to change its configuration.
It is remarked that the decompositionP=CNmay be non-unique, and therefore, KgandKcdepend on particular definitions. However, the sumKg+Kcis unique. It may also be convenient to define aninternal stiffness matrix, asKint=Kg+Kc. 120
For conservative problemsKt is symmetric, as well as its components. Besides, ifFis constant,Kext =0. Under geometric linearity,Kg =0. Under material and geometric linearity, and conservative loads,Kt =K0, constant.
It is not computationally convenient to calculate directly the structure’s global stiffness matrix. Instead, the stiffness is calculated for each structural element, then added to the global stiffness matrix. So proceeding, the vector of the nodal displace- ments of thee-th element is written as
ue =Aeu, (6)
whereAeis the order 6×3nBooleanincidence matrixof that element, suchAe1i = Ae2j =I3andAe1k =Ae2k =0, k = i, k =j, where0andI3are, respectively, the null and identity matrices of order three.
It can be readily verified that the same incidence matrixAe appears, transposed, in the relationship between the element and the global internal nodal force vectors
P= b e=1
AeTpe. (7)
Therefore,
Kint= b e=1
AeT ∂pe
∂ueAe = b e=1
AeTkeintAe, (8) where theelement internal tangent stiffness matrixkeintis defined:
keint= ∂pe
∂ue. (9)
Of course, it is not convenient to execute the matrix multiplications presented in equation (8). It is quite more economical to add the element contributions directly to the global stiffness matrix. This text avoids delving into these procedures, direct- ing the reader to the classic literature about matrix structural analysis and the finite element method, as Cook (1989) or Zienkiewicz (1989).
2 Geometrically Exact Truss Element
Direct formulation of the geometrically exact equilibrium of plane trusses was presented for the first time by Turner et al. (1960). Generalization to tri-dimensional trusses is found in several references, for instance, Livesley (1964), Ozdemir (1978) and Pimenta (1988). Different axial strain measures are also discussed by several authors, as Pimenta (1988), Souza-Lima and Brasil (1997) and Volokh (1999). How- ever, seeking a simple explanation, this text considers linear-elastic constitutive rela- tionships and linear strain measurements.
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Fig. 2.A truss element, with local and global nodal indexes.
Referring to Figure 2, nodes indexediandj in the global structural system are indexed as 1 and 2, in thee-th element numeration system. Keeping implicit the element indexe, thedisplacement vectorpand the internal forces vector are defined as
u u1
u2
6×1
and p= −v
v
6×1
N =CN, (10)
where the scalarN=EA(−r)/ris the elementinternal normal load,vis a unit vector directed from node 1 to node 2 andCis a geometric operator. The element is defined in an initial configuration, already under a normal force N0. Thus the reference, zero-stress element length, is given byr =EA0/(EA+N0).
Inserting formula (10) into (9), and proceeding along some straightforward de- rivations, the internal tangent stiffness matrix is obtained:
kint=ke+kg
= EA r
vvT −vvT
−vvT vvT
+N
(I3−vvT) −(I3−vvT)
−(I3−vvT) (I3−vvT)
, (11) wherekestands for alinear elastic constitutive tangent matrix.
Atai and Mioduchowski (1998) have shown that a necessary and sufficient condi- tion for the stability of the equilibrium of elastoplastic cable networks is that normal loads and material tangent modulus are positive everywhere.
In the elastic case, requirements reduce to a field of positive normal loads, which keeps the system taut. Now, since cables and membranes are continuous mechanisms 122
(as their discrete counterparts, for whichkcis not positive-definite), their equilibrium stability relies essentially on this tautness to proper structural behavior.