Modelling of structures made of filiform beams: Development of a curved finite element for wires

Finite Elements in Analysis and Design xxx (xxxx) xxx

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Modelling of structures made of filiform beams: Development of a curved finite element for wires Emanuele Marotta, Lorenzo Massimi, Pietro Salvini * Department of Mechanical Engineering, University of Rome “Tor Vergata”, Via del Politecnico 1, 00133 Rome, Italy

A R T I C L E I N F O

A B S T R A C T

Keywords: Filiform structures Finite element Curved beams Non-linear behaviour

This paper presents a finite element formulation of curved thin beams, useful for modelling structures made of filiform elements. The proposed element is intended to model structures formed by several wires, subjected to very large bending displacements so that their final shapes can be completely different from the original ones. The model is based on the description of the planar wire geometry through the integration of the radius of curvature, which is approximated by means of a cubic polynomial. The solution of an overdetermined system is necessary to compute the coefficients of the polynomial. This approach allows determining the stiffness matrix of the curved wire in closed form, through the application of Castigliano's Theorem. A technique for automatic remeshing during large deformations, based on the curvature change, is also discussed in the paper. To validate the model refined finite element analyses and an experimental test have been carried out. The solution is performed analytically, and it allows to identify the actual stiffness matrix of a curved wire, considering only the degrees of freedom at the ends.

1. Introduction

ranging from
190  C to þ140  C [1,2], deep vacuum, presence of radiations, dynamic stresses in orbit but mainly during launch [3]. At the present time, the space reflectors present almost circular areas with a dimeter as large as 15 m; higher diameters, up to 30 m, are currently being developed [4]. Considering the required size and the other needs, it is mandatory to foresee two configurations, a closed one, packed inside the space rocket, and an opened one, for the effective operation. Of course, the critical feature of the design process is the opening sequence to the working configuration. The most important component of the LDR, crucial to guarantee the proper functionality, is the reflecting surface. Among the alternatives suggested, which ensure the respect of all the requirements, like avoiding Intermodulation of Radio Frequency Signals [5], appropriate reflectivity, accurate reflector's shape [6] and many others, two alternatives have been mainly proposed in literature: knitted metal mesh and Carbon Fiber-Reinforced Silicones (CFRS) [7]. The present study focuses on the knitted mesh. This is composed by single or multiple twisted wires, eventually gold plated, and with different stitch type, primarily Warp type or Weft type [2]. Since the diameter of the wires used is small up to 15 μm, in obtaining the desired shape under load, the knitting procedure needs to be carefully considered [8]. The mechanical characterization is therefore crucial. A two-axes tensile strength-testing machine to analyse the behaviour of the knitted mesh under very low loads and large displacements has been specifically developed [9]. The

In structural mechanics, one of the main research topics is the close bond between load resistance and weight. Many applications require, de facto, the capacity to ensure, optimal design, structural integrity, along with a very lightweight and/or very compact system overall. Many examples can be provided from everything related to freight transport (payload is crucial, especially in aeronautical applications), to biomedical applications, to innovative projects in the field of mechanical and civil engineering. To achieve the best results, the development of both suitable materials and optimal design is critical. Regarding this latter necessity, the use of reticular structures and structural meshes has been a successful practise for many years, thanks to the great flexibility in terms of obtainable geometries, together with essential reduction in size and weight. Over the most recent periods, applications in space explorations brought new challenges. More in details, the present work can be applied to, and it is inspired by, Large Deployable Structures (LDS), and particularly Large Deployable Reflectors (LDR) widely used in space satellite systems. Different primary requirements must be considered carefully. A very efficient reflective surface, with an extremely light structure, and a system highly reliable are the main aims to achieve. The environmental loading conditions are also critical considering: working temperature * Corresponding author. E-mail address: [email protected] (P. Salvini).

https://doi.org/10.1016/j.finel.2019.103349 Received 21 March 2019; Received in revised form 2 July 2019; Accepted 5 November 2019 Available online xxxx 0168-874X/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: E. Marotta et al., Modelling of structures made of filiform beams: Development of a curved finite element for wires, Finite Elements in Analysis and Design, https://doi.org/10.1016/j.finel.2019.103349

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matter of fact, a fully general analytical solution in closed-form appears to be missing. One of the main possible tools for computing stiffness matrices of nonstraight beam, taking advantage of flexibility matrices definition, is the Castigliano's theorem. This instrument, in its fully general application, involves calculating the elastic energy stored in the structure volume. In a one-dimensional problem (like a beam), the integral over the volume becomes a single independent function of the curvilinear abscissa s. Generally, it is very difficult to find a function of s adequately descriptive of the curved beam and analytically integrable. The closed-form solution is so, again, usually possible only in particular trivial cases. In Ref. [28] Tezcan and Ovunc proposed the analysis of plane curved members with constant curvature, using unit load theorem and equilibrium to compute stiffness matrix. Palaninathan and Chandrasekharan [29] have presented a similar approach for the problem. In both cases axial, shear and flexural deformations are considered. Another application of Castigliano's theorem can be found in the studies of Dayyani, Friswell and Flores [30] involving analysis of a super element for curved beam in sandwich structures presenting a remarkable anisotropic behaviour. Already cited studies about exact solutions of beams with quarter or half-elliptical shape are in Ref. [19], parabolic beams are considered by Marquis et Wang [31]. Both last mentioned researches refer to the same approach. Extensions to the non-linear displacements are faced by Mutyalarao et al. [32] considering the large deflections of a cantilever beam, initially straight, end loaded. They adopt a fourth-order Runge-Kutta method for the solution of the non-linear differential equations. Another solution for the same case has been proposed by Shvartsman [33] through a direct numerical method. In Ref. [34] Tari elaborated an approximated model for the large deflection problem of a Euler-Bernoulli cantilever beam with tip point loading, employing a model based on Taylor expansion technique. An analytical solution for the Timoshenko beam under three-point bending, in large deflection, has been proposed by Mohyeddin and Fereidoon [35]. Mainly, the goal of the present study is to develop a non-linear analysis method for slender initially curved beams (or wires). It is capable of computing the stiffness matrix of a general highly curved beam in the plane, considering its geometrical evolution, the primary source of non-linearity. A new curved element, named Wire Element is introduced. Considering structures composed of slender beams or wires, it is not necessary to account of shear effects and changes of the cross section, since wire diameter is small if compared to its axial extension. Other beam theories account of shear deformations considering a constant state of transversal shear strain (Timoshenko beam), or higher orders for the axial displacement (Reddy-Bickford beams) [36]. These more complete beam theories generate problems when slender beams are considered, e.g. locking effects and should account of the effective loading and boundary conditions. The procedure developed is applicable both for small or large deflections. The Castigliano's energy integral is solved in an analytical form after an appropriate substitution for the variable in the integrand. This new variable is the attitude angle, identified by the curve tangent. This change is crucial to eliminate the hurdle due to the square root introduced by the curvilinear abscissa, when working with Cartesian coordinates. If integration is carried out over attitude angle, it is easy to get the actual curvature, curvilinear abscissa and Cartesian coordinates of any point of the wire. This technique to obtain a closed-form solution of the integral for the elastic energy has been used in the already mentioned paper [29], while in the cited study of Lin et al. [24] the same substitution is used to integrate equations for the displacements of particular curved geometries. There are many other papers concerning shear deformation effects, able to minimize or avoid locking or stiffness increase in a static or dynamic analyses, [37,38]; most of them are based on initially straight lines. In the present contest, wires are initially curved and very thin. Thus, it appears obvious to simplify the solution non-considering shear

testing machine is equipped with a strain measurement device, based on the processing of digital images, whose features are described in Ref. [10]. Overall, the mesh is formed by a plane pattern, repeated in two directions, which contains several wire loops in contact among them. Each loop can be structurally modelled as a slender beam with a high value of curvature. Non-linear force-displacement relationship emerges due to the contact evolution and to the behaviour of the wires, which tend to align during the stretching [2]. The main purpose of the present study is to develop a curved wire finite element, so that a huge mesh can be appropriately modelled. This approach of modelling could be useful in many other fields, besides the spatial applications. Elastomeric textile meshes are already used in medical treatments of different diseases, developments of these techniques is the object of several studies. Some specific examples are: the treatment for leg oedema through compression stockings, generating a well-defined level of pressure [11]; a textile mesh positioned around the ventricles, suggested as a therapy for dilated cardiomyopathy [12]; use of synthetic meshes for pelvic organ prolapse and hernia repair [13]. Another possible field of application regards smart fabric and smart clothes, where textile meshes are used as an integral part of different wearable sensors, with possible interest in sport, entertainment and rehabilitation [14,15]. In all above-mentioned applications, an accurate modelling for the mechanical behaviour of the mesh is required, since it has an important influence on the performance of the final product. A crucial step of this study is the analysis of curved beams presenting a high ratio between length and diameter. Since curved elements are present in many mechanical and civil structures, many papers have been proposed on this subject. Kikuchi [16] examined the suitability of modelling circular arches using a series of straight beams in a finite element analysis. Accuracy improves while increasing the number of beams used, but the computational cost also quickly rises. Balasubramanian and Prathap [17] developed a curved element arch beam considering the effect of axial and shear deformations. Other studies consider non-constant curvature, such as parabolic beam element, Marquis et al. [18], or beams with the geometry of a quarter or a half of an ellipse Dahlberg [19]; both last mentioned authors made use of energy theorems. The parabolic arch displacement has been also solved by Gimena et al. [20]. Here the analytical formulation is based on a differential system of equation for a curved beam, extended to variable cross sections and different (concentrated and distributed) loading conditions. The problem of the elastic response of an arch, clamped with a general loading condition have also been considered by Benlemilh and Ferricha [21], who got the solution through a mixed finite element method. An exact analytical solution for non-uniform curved beam, also accounting of shear deformation, in the in-plane static problem have been discussed by Arpaci and Tufecki [22]. The procedure is based on the initial value method but it is a hard task to get the solution in a closed form for any arch shape. An extension of this same analysis for structures with distributed load has been examined by Tufecki et al. [23] for some classic shapes. A closed-form solution for curved beams having particular geometries under pure bending condition (in detail circle, spiral, ellipse, parabola, cycloid, catenary and logarithmic spiral), have been proposed by Lin et al. [24]. A solution for a finite element of a rod spatially curved and twisted, based on a set of governing equations, including equilibrium, constitutive, compatibility and energy equations has been studied by Tabarrok et al. [25]. In Ref. [26], Saje developed a finite element formulation for a planar, generally curved, elastic beam, considering shear deformation, using a variational approach, numerically resolved. In Ref. [27] Koziey and Mirza developed the model of a curved beam, assuming low degree (cubic and quadratic) polynomial function for the approximation of displacements, considering both elastic and elasto-plastic behaviour. However, all the three last-mentioned methods require, when applied in non-trivial cases, numerical integration. All in all, analytical solutions seem to be generally limited to simple cases. As a 2

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deformation for the stiffness element definition. In the present paper, a solution for a more general curved beam is discussed. The curvature radius of each wire is modelled by a polynomial of third order of the slope angle of the wire itself. Therefore, the first point is to interpolate the wire according to a polynomial of the slope angle. This has already been applied by Wu and Yang [39], using the solution of a linear system, prescribing continuity on the slope and on the curvature as boundary conditions, possibly maintaining the arc length. The procedure requires a monotone curvature change to get a closed-form solution. However, the interest focuses on a method suitable for a wider class of curves, with a more general evolution of the tangent angle. It is so necessary to use a piecewise cubic polynomial approximation, with the complete curve divided into a series of pieces with monotonic increase (or decrease) of the tangent-angle. Each approximated portion is described with a different series of cubic coefficients. This approach ensures a C1 continuity at the junctions between different pieces. This procedure allows repeating the interpolation and the stiffness analytical solution at any non-linear step, so that the method accomplishes the modelling of large displacements of wires. 2. Geometric modelling The geometry of any kind of planar wire can be faced through the proposed procedure; if necessary, the curved line can be identified as a sequence of wires attached one after the other. The particular interest in the modelling of thin wires emerges by the consideration that a knitted structure is composed of several patterns that periodically repeats in two directions. Therefore, a knitted structure is structurally formed by a simpler set of matrices that are repeated in the plane. If one computes this submatrix, then it is easy to compose them to form the overall structure by appropriately summing the periodic contributes. In other words, a huge mesh structure takes advantage of a refined loop modelling, although the number of nodes accounted is similar to the loops in the overall model. The discussion of the present approach starts with the modelling of a single piece (or pattern) and its properties. The main application of the present study regards knitted meshes, so that the analysis is restricted to plane problems since structural resistance of fabrics concerns a twodimensional problem. The description of the single piece of wire is based on a cubic polynomial function of the curvature radius ρ, with the tangent angle θ of the curved wire as independent variable: 3

2

ρðθÞ ¼ aθ þ bθ þ cθ þ d

Fig. 1. An example of a cubic curve and its Local Reference System.

A description of the identification of the four coefficients involved in the cubic function is the first step. By hypothesis, we consider curves with the above mentioned limitations, and it is supposed to know the coordinates of a number of points on the curve. The latter can be obtained by some sampling, by an analytical description of the curve or by image analysis processing. When required, if the set of point is poor, interpolation techniques using splines, like B-splines or Hermite splines are suitable. Moreover, each point is associated with its curvilinear length and its slope angle. Note that such quantities can be easily computed with numerical methods starting from Cartesian coordinates. In reverse, it is immediate to express Cartesian coordinates, in the tangent-normal reference, as a function of the curvilinear abscissa:

(1)




being a,b,c,d coefficients. This approach presents two mandatory requirements. Firstly, tangent angle needs to increase or decrease monotonically along the line. Secondly, since the curvature radius is an invariant for rigid motions, it is not possible to store in this equation information about the actual position of the wire with respect to a global reference system. Therefore, a local coordinate system, associated with the initial placing of the curve, needs to be introduced. The reference generally adopted to define a curve is the Frenet–Serret frame. Since the considered problem is plane, a local reference with unit vectors tangent and normal to the starting point is introduced. By means of a rigid rotation and a translation, it is easy to switch between global and local reference system. The class of plane curves that are fruitfully approximated with a single cubic polynomial of the curvature radius is quite wide. These include simple cases (circle, portions of ellipses or parables and many others) and more complex cases (as a curve with one or multiple loops, see also Fig. 1 for a representative example where, for a clearer graphic, the tangent angles are evidenced on the normals rather than on the slopes). Note that this approach does not include straight lines, indeed much easier to model with classical beam element, since their radius of curvature is infinite and cannot be described by Eq. (1).

xðsÞ yðsÞ

8Z <

 ¼

s

0

:Z

9 cosðθðsÞÞds =

s

sinðθðsÞÞds

;

(2)

0

The integration of Eq. (2), in analytical form, is very tricky, since the formulation of ds introduces a square root. ds ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2 þ dy2

(3)

To overcome this difficulty a change of variable, from curvilinear abscissa to tangent angle, is helpful. Substituting the term ds with the product ρ(θ)dθ, Eq. (2) turn into:




xðθÞ yðθÞ

 ¼

8 < :

Zθ

ρðθÞcosðθÞdθ 9 =

0

Zθ

ρðθÞsinðθÞdθ

;

(4)

0

The analytical integration of Eq. (4) is quite easy, being the product of a cubic polynomial and a sine (or cosine) function. The solution for a generic angle θ is 3

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8   > xðθÞ ¼ þ 3θ2 cosðθÞ
6 cosðθÞ þ θ3 sinðθÞ
6θ sinðθÞ þ 6 a > >   > > > þ θ2 sinðθÞ
2 sinðθÞ þ 2θ cosðθÞ b > > > > þ½
1 þ cosðθÞ þ θ sinðθÞc > > < þ½sinðθÞd 



> yðθÞ ¼ þ
6 sinðθÞ þ 6θ cosðθÞ
θ3 cosðθÞ þ 3θ2 sinðθÞ a >   > > > þ
2 þ 2 cosðθÞ
θ2 cosðθÞ þ 2θ sinðθÞ b > > > > þ½sinðθÞ
θ cosðθÞc > > > : þ½1
cosðθÞd

constant their consecutive distance, hereinafter called ds. 3. Stiffness matrix computation Having the coefficients a, b, c, d for every piece of curve, it is possible to compute the stiffness matrix in its tangent-normal reference [40,41]. Again, it is stated that, since the expected application concerns metallic wires with small transverse dimension if compared to total length, shear deformation effects can be discarded, in comparison with bending moment and axial stress effects. External loads, forces and moments, are only applied on the end points of the considered piece; therefore, no effects of distributed loads are accounted. Wires exchange forces each other at their mutual intersections, thus, it is mandatory to introduce endpoints in correspondence of contact points. An example of curved wire element, with uniform cross section is shown in Fig. 2. In the same picture the sequence of the DoFs chosen for nodes i and j is given. The stiffness sub matrix (Kii)is derived after inversion of the flexibility sub matrix (Cii) which is a (3x3) array. This is analytically computed for one end, say i, considering clamped the other one, say j. The sub matrix for the mixed DoFs of the stiffness matrix (Kij) and the other diagonal sub matrix (Kjj) come from equilibrium-based relationships [29]. The Flexibility sub-matrix (Cii) originates by the application of the second Castigliano's theorem, which provides displacements and rotations through partial derivatives of the elastic energy. Bending and axial load contributions are independently computed and superimposed. Considering a generic piece of curve subjected to bending only, the kth generalised displacement is called uk. U is the elastic energy, E is the elastic modulus, I is the section moment of inertia, Pk is the load applied in the same direction of k, M(s) is the internal bending moment as a function of the curvilinear abscissa s, spanning from 0 to sf. If elastic properties and cross-section are supposed to be invariant with s, modified Castigliano's theorem gives

(5)

where x and y are expressed in the tangent-normal local reference. Repeating Eq. (5) at every point allows to get a system of linear equations in the unknowns a,b,c,d. Note that the Cartesian coordinates are the known term of the algebraic system. For each point belonging to the curve, it is possible to write down two equations, x and y coordinates in the tangent-normal reference, using Eq. (5). In matrix form:


xðθÞ yðθÞ



8 9 > >a> > = < Ax ðθÞ b ¼ Ay ðθÞ > > >c> : ; d 

(6)

being Ax, Ay row vectors 1x4. It is important to recall that Cartesian coordinates used must be expressed in the tangent-normal reference. A further equation, related to the total length L of the curve, is considered. This equation can be interpreted as a sort of closing equation, in the form Z L¼

θf 0

1 4

1 3

1 2

ρðθÞdθ ¼ aθ4f þ bθ3f þ cθ2f þ dθf

(7)

Considering now the n points of the curve discretization, being θf the tangent angle to the end point, and including the closing equation on the total length, the linear system turns into an over-determined one. This last can be solved using a Least-Mean-Square approach, generating a pseudo-inverse of the coefficient matrix. 9 3 8 xðθ1 Þ > Ax ðθ1 Þ þ > 8 9 > > > > yðθ1 Þ > > Ay ðθ1 Þ 7 a> 6 > > > > 7 > < xðθ Þ > = 6 = < > 6 7 A ðθ Þ b x 2 7 2 6 ¼6 7 > yðθ2 Þ > A ðθ Þ c > > y 2 > > 7 > ; 6 > : > > 4 ::: 5 > ::: > d > > >

: > ; AL θf L θf

uk ¼

2

∂U 1 ¼ ∂Pk EI

Z 0

sf

  ∂MðsÞ MðsÞ ds ∂ Pk

(8)

In the solution of the system, the last Eq. (7), say closing equation, plays an essential role. In fact, during wire deformation, the total length of the curve is supposed to be invariant (non-compacted curved wires are dominated by flexural displacements and are practically inextensible). Thus, in the computation of the four coefficients, repeated at every nonlinear displacement step, the known term Leeps unchanged, and is not affected by cumulative error. Knowing that, it is reasonable to infer more weight to the last equation in the solution step. This is obtained simply pre-multiplying LHS and RHS of the last Eq. (8) by the same value, bigger than one. An error check of the fitting is necessary. In fact, the cubic interpolation assumed for the curvature could not adequately fits the data. In such a case, the piece of curve tentatively fitted may be bisected, thus generating two different parts, to be independently identified. This bisection procedure can continue until the error is considered acceptable, or a fixed minimum number of points is reached. The fitting provides the coefficients a, b, c, d for every piece in its local Reference; then Eq. (5) applied to every piece needs appropriate rotations and translations to get the global Cartesian coordinates of the whole wire. Although the independent variable for the analytical solution of the integrals is the slope angle, the solution will be always stored in terms of curvilinear abscissa s. In fact, it is possible to have different points in a wire having the same tangent angle. For the sake of simplicity, the sequential data points are interpolated at every step, in order to keep

Fig. 2. Wire element. 4

(9)

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under small displacements. Differently from all the others, the last coefficient c33 is trivial to compute since rotations caused by an end moment results only dependent on wire length. The following matrix relationship is thus formed

as already mentioned, the integration as a function of the curvilinear abscissa is performed exchanging the variable s with the tangent angle θ uk ¼

1 EI

Z

θf

0

  ∂MðθÞ ρðθÞdθ MðθÞ ∂Pk

(10)

8 9 2 c11 < u1 = u2 ¼ 4 c21 : ; u3 c31

The first flexibility submatrix concerns the node i-th as free and node j-th clamped. The resulting generalised displacement are uk (k ¼ 1,2,3). Fig. 3 shows the relative isostatic scheme. Focussing attention, for example, on the displacement u1 of the node i, along x-axis (k ¼ 1) caused by the load F1 one obtains 1 u1 ¼ EI

Z 0

θ* 

½Kii  ¼ ½Cii 
1 2 c11 c12 Kii ¼ 4 c21 c22 c31 c32

(12)

(13)

3 k12 k23 5 k33

(17)

(18)

where (xi yi) and (xj yj) are the coordinates respectively of node i and node j, expressed in the reference tangent to the first point. Considering the relation between displacements and forces on nodes i

Recursion requires to re-use (13) until n ¼ 0. At that stage, the integrals turn trivial, since the integrands are elementary sine or cosine functions. Even if the analytical expression for u1 involves an enormous number of terms, the procedure can be easily carried out using a symbolic solver. The full solution is not represented here for obvious reasons of brevity. It is now possible to reconsider Eq. (11) as follow

u1 ¼ c11 a; b; c; d; θf  F1

k12 k22 k32

8 9 8 9 2 38 9
1 0 0 < F1 = < F1 = < F4 = ¼ 4 0
1 0 5 F2 ¼ ½G  F2 F : ; : 5; : ;
yj
yi xj
xi
1 F6 F3 F3

Z

θ n θn
1 cosðcθÞdθ θn sinðcθÞdθ ¼
cosðcθÞ þ c c Z Z θn n θn
1 sinðcθÞdθ θn cosðcθÞdθ ¼ sinðcθÞ
c c

3
1 2 k11 c12 c23 5 ¼ 4 k21 c33 k31

The two off-diagonal blocks of the stiffness matrix Kij and Kji results from equilibrium relationships. Equilibrium conditions allows to develop a transformation matrix between forces on node i and forces on node j, in the local tangent-normal reference

Integration of Eq. (11) implies the development of several products. However, the recursive application of the hereinafter given formulae helps to find the analytical solution, trivial but very lengthy (251 terms for the considered case) n

(16)

where Cii is the flexibility matrix linking generalised forces and generalised displacements on node i, when node j is clamped. The inversion of Cii generates the stiffness submatrix Kii.

Analytical solution of this integral requires the internal bending moment to be computed as a function of θ; the second equation in Eq. (5) provides it

Z

(15)

fui g ¼ ½Cii fFi g

(11)





M θ ¼ F1  y θ ¼ F1 
a cos θ  θ3 þ 3a sin θ  θ2
b cos θ  θ2 þ2b sin θ  θ þ 6a cos θ  θ
c cos θ  θ
6a sin θ þc sin θ þ 2b cos θ
d cos θ
2b þ dÞ

3 8 9 c13 < F1 = c23 5  F2 : ; c33 F3

Or more synthetically



∂MðθÞ ρðθÞdθ MðθÞ ∂F1

c12 c22 c32

8 9 8 9 < F1 = < u1 = F ¼ ½Kii  u2 : 2; : ; F3 u3

(19)

The stiffness matrix of the mixed term results

(14)

where c11 is simply the result of the integration when F1 ¼ 1. An analogous procedure can be performed applying a force along the y-axis F2, obtaining the coefficient c12, or a concentrated moment F3, obtaining the coefficient c13. In a very similar way, displacement u2 along y-axis and rotations u3 on node i can be found when applying F1, F2, F3, respectively. Indeed, only six integral solutions are required since the flexibility matrix is symmetric

8 9 8 9 8 9 < u1 =  < u1 = < F4 = F u ¼ ½G  ½Kii  u2 ¼ Kji : ; : 2; : 5; F6 u3 u3

(20)

  K ¼ ½G  ½K   ji   T ii Kij ¼ Kji

(21)

Also the Kjj matrix can be computed by the balance relationship (18)

Fig. 3. Isostatic scheme in tangent-normal reference. 5

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8 9 8 9 8 9 8 9 < F4 = < F1 =  < u4 =  < u4 = F u u ¼ ½G F2 ¼ ½G  Kij ¼ Kjj : 5; : 5; : 5; : ; F6 F3 u6 u6

parameters are reduced to two). Finally, an error check (Chk_4) concerns the quality of the fitting obtained. The comparison deals with points having the same curvilinear abscissa. If the distance is too high, so that the fitting does not match enough, the considered piece is divided into two parts. The described procedure is thus dominated by four threshold parameters (one for each check) whose values should be fixed once for all. A reasonable idea is to fix them as a function of the density of the data points within the total length. Generally speaking, a too precise reconstruction is not necessary, since global flexibility of a curved wire between two end points is quite accurate even if the reconstruction is not perfectly superimposed on the data points. At this stage, the curve is divided into a number of pieces, and every piece has its own parametric representation. Fig. 5 shows an example of segmentation on a generic wire that is defined by 500 data points. The original curve is behind in black; blue, red and green represent positive, negative and null curvature portions, respectively. For this example, being np the total number of data points, the check values used are:

(22)

And consequently the Kjj     Kjj ¼ ½G  Kij

(23)

So that the complete stiffness matrix combines all blocks found in Eqs. (17), (21), and (23): " K¼

Kii

KTji

Kji

Kjj

# (24)

For some geometrical shape lines (e.g. thin pipes), the axial compliance gives a non-secondary contribution to the general compliance of the wire, even if the normal elongation can be neglected. Again, Castigliano's theorem applies, considering elastic energy due to the axial load N. uk ¼

1 EA

Z

θf 0

NðθÞ 

∂NðθÞ  ρðθÞdθ ∂ Pk

(25) Step_1: minimum a-dimensional curvature above which the portion is a straight line ¼ 1/(2 np) Check_2: minimum points in a piece ¼ 1/125 np Check_3: minimum angle accepted ¼ 5 degree Check_4: maximum accepted error 5% total length

Keeping linearity in mind, axial contribution provides an additional terms of flexibility formed by a matrix whose elements amn (m,n ¼ 1,2) can be simply added to the previous ones. These terms emerge by the analytical solution of Eq. (25), considering a unitary load applied at the end node; again they depend on a, b, c, d and θf. Below the resulting flexibility matrix for node i is shown. Alike for node j. 8 9 2 c11 þ a11 < u1 = u2 ¼ 4 c21 þ a21 : ; u3 c31

c12 þ a12 c22 þ a22 c32

3 8 9 c13 < F1 = c23 5  F2 : ; c33 F3

When the fitting phase is completed, it is possible to associate a local stiffness matrix at each portion. These last are positioned end-to-end so that it is easy to assemble them to form a global stiffness matrix. However, it is not required to keep all DoF's for a structural analysis, but only the characteristics of the ending nodes. The condensed stiffness matrix is available from Ref. [42]:

(26)

With similar considerations as above, the whole stiffness matrix is formed from flexibility matrices and the general equilibrium of the element. The resulting element matrix, in the local Reference system tangent to the node i, is: 8 9 8 F1 > > > > > > > > > F > > > > > < 2> = < F3 ¼ ½K6x6 F > > 4 > > > > > > > > > > > > : F5 > ; : F6

u1 u2 u3 u4 u5 u6

Kcond ¼ ½Km;m 
½Km;s ½Ks;s 
1 ½Ks;m 

where Kcond is the condensed stiffness matrix to the ending nodes i.e. master nodes whose DoF's are indicated with subscript m; instead, the subscript s identifies slave DoF's. In this manner, a single super-element between the boundary nodes is built. When the full stiffness matrix is computed and the solution step is completed, the finite element solution provides the step displacements and loads on the ending nodes. To obtain the new step configuration of the wire, it is necessary to get the displacements all over in the curvilinear abscissa reference. Here, the change of curvature along the wire is the base information to determine the step displaced configuration of the wire. Change of curvature is defined by the simple beam equation:

9 > > > > > = > > > > > ;

(28)

(27)

It should be added that, if the slope angle θ(s) keeps almost constant with s, the line stiffness is computed with the well-known straight Euler beam. 4. Structural solution for piecewise wires

1

ρðsÞ

The Cartesian coordinates of some consecutive points give the geometric shape of the whole wire. By means of an appropriate interpolation of these points, providing equidistance ds, it is easy to associate the local curvature with the curvilinear abscissa. As above explained straight lines are considered apart, since Eq. (1) is unable to manage them. In Fig. 4 the flow chart of the modelling of a wire, generating an appropriate segmentation, is shown. A first check (Step_1) concerns the identifications of shares with positive, negative or almost null curvature. If the segmentation generates portions with too few data points (Chk_2), the portion is included in the most similar contiguous piece; this avoids that the over determined system in Eq. (8) might provide numerically-ill identification of parameters. A further check (Chk_3) regards the total slope variation of every piece that should be significant enough. The conditioning of the stiffness matrix requires this check; in other words, the matrix should always be semi-positive definite. If this is not the case, the piece can be accounted as a straight line or as a part with a linear-varying curvature (the four




1

ρnew ðsÞ

¼

MðsÞ EI

(29)

where ρ(s) and ρnew(s) are the radii of curvature before and after the applied load step, respectively; M(s) is the step bending along the curve, provided by the end-element loads from the finite element solution. For each data point, the actual bending moment is computed by element equilibrium. As an example, considering the element in Fig. 3, running along the increasing s, local bending is given by MðsÞ ¼ F3
F2 xðsÞ þ F1 yðsÞ

(30)

The new curvature is given by reversing Eq. (29):
1 1 MðsÞ
EI ρðsÞ



ρnew ðsÞ ¼

(31)

Hypothetically, the wire is inextensible, thus the increment ds between consecutive points of the discretization remains constant after the

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Fig. 4. Segmentation algorithm.

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Fig. 5. Example of curve segmentation.

the summation of the loads at every step. For a clearer explanation of the procedure, Fig. 6 shows the flow chart of the algorithm. Contact check is not described within this paper, but it is the subject of a subsequent work. In the incoming development, when contact is achieved between two wires, the two are each divided at the contact point. This is to say that the contact point generates a new node within the wires. The non-linear solution starts setting the parameters of the computation. General parameters are the number of steps and the convergence criteria. The complete wire is divided into pieces according to the criteria given in the previous section, and for each of them the stiffness matrix is computed. After a solution step, the new local curvature is computed according to the nodal loads, and then the new curvature is used to get the new geometrical shape. To achieve this goal, an oversampling of the internal points allows to reduce the drift introduced by curvature discretization; then, the points are re-sampled at their initial number. Whenever the reconstruction does not match with the nodal displacements derived by the condensed stiffness matrix solution, the time step is reduced and the iteration restarts. Otherwise, if the geometrical matching is acceptable, the iteration step continues by computing the residual loads. Furthermore, curved wires tend to change configuration under loading, increasing the tendency to align with the direction of the loads; this may cause a stiffening behaviour. From the point of view of nonlinear solution strategy, some numerical tricks could be necessary [43, 44]. First, as usual in structural analysis, the final loading is divided into some substeps, small enough to guarantee convergence and small displacement (almost linearity between steps). Convergence is achieved when global equilibrium conditions are satisfied with a fixed minimum residual load. A convergence criterion measures how well the solution satisfies equilibrium itself. In general, the criteria are based on some norm of displacement, incremental plasticity, residual forces, or energy. In most cases, residual load check is a suitable choice. Both the solution strategy and the convergence criteria must be tuned according to the forcedisplacement behaviour. Different approaches provide changes in the solution speediness or even non-convergence occurrence. The choice of the appropriate method depends on the kind of structure. This can exhibit a stiffening, softening, bifurcation or returning point behaviour. As already mentioned, in the case of a structure

displacement. It is so possible to compute the new increment dθnew of the tangent angle between consecutive points. dθnew ðsÞ ¼

ds

(32)

ρnew ðsÞ

By a simple summation in the discrete case, the tangent angle accounts also of the initial attitude, given by θ0, in the absolute Reference θTT ðsÞ ¼

X

ds

ρnew ðsÞ

þ θ0

(33)

Given the new curvature and the corresponding tangent angle at each data point of the deformed configuration, it is immediate to get the Cartesian coordinates of all the points, through summation:


xðsÞ yðsÞ




¼

x0 þ y0 þ

X X

ρnew ðsÞcosðθnew ðsÞÞ dθnew ðsÞ ρnew ðsÞsinðθnew ðsÞÞ dθnew ðsÞ

) (34)

Eq. (34) allows computing the new configuration at any curvilinear abscissa in the global reference system. Considering a multi-step loading of the wire, the process requires a new identification of the coefficients a, b, c, d at the end of every step. Thus, the displacement accounts of curvature change ρnew, and a new third order approximation of the curvature radius as a function of the tangent angle is required. The latter may also need a different segmentation of the wire. In the following paragraph, it will be clear that nonlinear analysis requires the identification of the parameters at every iteration. At the end of all the steps, the total solicitation, in terms of bending moment in the wires, can be obtained as a summation of all the bending moments computed at each step with Eq. (30). 5. Geometric non-linear solution Dealing with deformation of thin metallic wires, these could be subjected to large displacements within the elastic response; consequently, a nonlinear analysis follows. The non-linear behaviour of filiform structures requires the subdivision of the computation into small incremental steps. At each step, the solution accounts of the small displacement hypothesis. Therefore, the overall stress and strains on the wires results by 8

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Fig. 6. Flow Chart of the non-linear strategy algorithm.

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composed of knitted wires, the stiffening behaviour is common; as a matter of fact, there is a progressive increasing of the tensile contribution with respect to the bending one. In such a case, the classical NewtonRaphson method can exhibit difficulties in convergence. A possible advantage may arise through quasi-Newton methods, often called secant method [45,46]. This last allows to update the characteristic matrix in a direct way at every iteration, rather than to re-compute it entirely (Newton's method) or leave it unchanged (modified Newton). There exist several methods to obtain the actual secant matrix; here a suitable method for this application is explained. The secant matrix [Ks] should satisfy four requirements [47]: 1. The matrix [Ks] should satisfy the quasi-Newton equation (see hereinafter); 2. The [Ks] matrix should be symmetric; 3. The [Ks] matrix should be positive definite; 4. The matrix [Ks] should be quickly computed and inverted. The quasi-Newton equation prescribes that: h

Table 1 Loop displacement results and comparison with FE solutions.

Fx ¼

1

Fy ¼

1

Mz ¼

10

Displacement [mm]

Curved Beam (1 element)

Multi straight Beams (1000 elements)

Err [%]

Ux Uy Rot z Ux Uy Rot z Ux Uy Rot z

61.0343 11.0563 0.971961 11.0564 7.08100 0.23760 9.71961 2.37560 0.193865

61.0340 11.0560 0.971960 11.0560 7.08080 0.23759 9.71960 2.37590 0.193860

0.00005 0.0027 0.0001 0.0036 0.0028 0.0042 0.0001 0.0126 0.0026

i    

ui
ui
1 ¼ F iInt
F i
1 Int

(35)

where [Kis,n] is the secant matrix at time step n, and iteration i; {ui}, {ui
1} are the displacement vectors at the iteration i and i-1, respectively; {FiInt} and {Fi
1Int} are the internal force vectors at i and i-1 iteration, respectively. For a system with only one degree of freedom, the computation of matrix [Ks] is trivial. On the contrary, for a system with multiple degrees of freedom the determination of [Ks] is more difficult and not unique. Eq. (35) can be modified as:

Fig. 7. Generic loop modelled with a cubic polynomial curvature radius.

Load at Node j [N,Nm]

KiS;n

i  h i i
1 i  γ du ¼ K S;n

(36)

where γ i is the vector that contains the difference of internal load between the iterations i and i-1. The update preserves the requirements above given, in particular positive definiteness, and Eq. (37) shows how to iteratively compute the inverse of the secant matrix:

Fig. 8. Geometric Non-Linear Solution: Comparison between the model proposed and the FE solution. 10

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matrices, this is called Adaptive Descent Technique [48,49]. This method switches to a new characteristic matrix as a tuned combination of the tangent stiffness matrix and the secant one. The tuning is adapted according to actual convergence difficulties; it switches back to the full tangent characteristic when the solution converges, thus optimizing the convergence rate. This approach facilitates the overcoming of critical situations and speeds up the numerical process when convergence becomes easier. The characteristic of the system is tuned by a single parameter ξ:     ½Ki  ¼ ξ KiS þ ð1
ξÞ KiT

(39)

The starting value at every substep of the adaptive parameter ξ is set to 0 and the solution is driven by tangent stiffness matrix [KT]. When the solution shows a divergence propensity, the parameter switches to 1 and the characteristic matrix is the secant one. When the solution comes back to convergence, the parameter is progressively reduced by a factor (typically 4) and the iteration continues. 6. Test cases In this section, a linear and some non-linear examples are discussed. Fig. 7 shows a wire loop-like generated by a cubic function of the curvature radius, modelled with a single wire element. Inside the picture the cubic coefficients are evidenced. This element is fully constrained on node i and active loads are applied on node j. The steel wire has a 1 mm diameter. Table 1 compares the displacements of the wire element versus a multi-beam structure composed of 1000 straight elements. The results regard a linear solution and show a small scatter between the displacements given by the proposed analytical wire element (3 DoF's) and the multi-beam FE model (3000 DoF's). Moreover, the differences tend to decrease if the multi-beam FEM is refined further. The next example regards the extension to a geometric non-linear solution. Fig. 8 represents a quarter of a circle wire. The initial geometry assumes a constant curvature radius (R ¼ 100 mm), which is initially modelled with a single wire element. The wire is constrained at one end and loaded at the other one. The wire diameter is 1 mm, harmonic steel made. The force is directed downwards and its magnitude is such as to require a geometric non-linear computation. The comparison is made with a FE model formed by 500 elements. The displacement scatter at the free end node is less than 6% on the x displacement and 1% for the y displacement, while for the rotation it is almost 2%. Fig. 9 shows the piecewise evolution during the iterations. As already discussed above, the mesh is automatically adapted to the shape evolution of the wire. In fact, for this test case, the wire is initially

Fig. 9. Automatic meshing: elements forming the structure when unloaded or full loaded.

h

KiS;n

i
1



 T 

 T h i
1 i
1  KS;n ½I þ vi wi ¼ ½I þ wi vi

(37)

where: [I] is the identity matrix, [Ki
1S,n] is the previously secant matrix at i-1 iteration (for the first iteration [K1S,n] is given by the initial tangent matrix [KT]), and with:

i v ¼

T

fdui g fγ i g   fdui gT Ki
1 fγ i g S

 1 dui ¼ T i i fdu g fγ g

! 1 2

 i
1  i  i  KS du
γ

i w (38)

An optimum balance is to take advantage of both tangent and secant

Fig. 10. Setup of experimental test: a) unload configuration; b) full load configuration. 11

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Fig. 11. Experimental points and data fitting.

load is recorded during the stroke of the crosshead. The maximum experienced load (0.492 N) is then used for the modelling. In Fig. 11 the experimental points and the fitted data are shown. Fig. 12 presents the result obtained with the proposed model. The figure shows the deformed and undeformed configurations. The displacement values corresponding to the experimental load are shown inside the picture. In the y direction it is equal to 98.24 mm against the 100 mm experimentally imposed. In Fig. 12 the automatic initial and final mesh are also represented. Note that the initial one requires three curved wire elements and a straight beam. Fig. 13 overlaps the displacements obtained numerically with the experimental picture. It appears that the deformed configuration is very close to the experimental one in terms of displacement, rotation and in the overall shape assumed by the wire. The last test case regards a typical knitted mesh structure. This last is made of metallic wires knitted each other to form a repetitive pattern. Each loop is in contact with the nearest one and the two refers to a separate yarn introduced in the longitudinal fabric direction. This manufacturing method, called Warp knitted technology, is majorly used for technical applications such as astromesh reflectors (Fig. 14 a). The mesh modelling is made of a repetitive pattern (Fig. 14 b) composed of 4 elementary curved wires. Each of them is nothing but the element given in Fig. 5, appropriately oriented. The single pattern can be repeated to assemble a fabric-like structure. Indeed, yarns present contacts among them. This aspect is not considered in the present basic example, but contact condition search is already present in the modelling and will be the subject of future reports. Currently, each pattern is jointed to the next one by common nodes. Fig. 15 a) depicts an example made of 3  3 patterns, with loads and constraints. Fig. 15 b) shows the non-linear solution of the whole fabric, lighter is the unloaded, darker the deformed shape. This example is useful to illustrate the potential of the presented modelling procedure, which would have required more than 20000 Finite Beam elements to reach the same accuracy.

modelled with only one element, while at the end of the calculation the same wire is divided into four elements, 2 curved elements (1 with positive and 1 with negative curvature) and 2 straight beams. The next example concerns a single-loop structure, also experimentally tested. The test involves a non-linear computation when large displacements are imposed. The comparison accounts of the proposed wire modelling, Finite Element results and experimental evidences. The steel wire has 1 mm diameter, and an overall geometric shape inside 150 mm  50 mm. The structure is simply supported and the ends are free to rotate on lubricated pins (see Fig. 10). A tensile machine, suitably equipped with a low-capacity load cell, tests the structure. The experiment is conducted imposing a fixed displacement (100 mm) while the

7. Conclusions The paper presents an approach to model structures made of filiform curved beams. The basic element shape is described by means of a cubic polynomial law of the radius of curvature. This allows the modelling of several filiform structures, such as astromesh reflectors, with a limited number of elements. The wire is subdivided into positive, negative or null curvature portions. Each of the identified curved portions is fitted by

Fig. 12. Deformed and Undeformed configuration and mesh of single loop analysis. 12

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The proposed technique should include the contact between the wires to model knitted structures, but this essential aspect requires additional detection and subdivisions that is the subject of future developments. Each piece of wire provides its stiffness matrix through the closed form solution obtained applying the Castigliano's Theorem. The metal wires considered are very slender, so that main stresses are caused by bending; therefore, the hypothesis of inextensibility of the wire is almost verified, and the shear effect can be neglected. The curvilinear element model allows a high reduction of the degrees of freedom required for the analysis. This approach results useful both in small displacement analysis and in non-linear computation of large displacements with elastic slender beams, that is to say small local strains occur. In this latter case, cubic coefficients are recalculated at each iteration, thus providing a selfadapting meshing. The type and the number of portions forming the wire are automatically chosen according to the actual structure shape, considering separately curvilinear or straight parts. The curved model is useful for the modelling of knitted structures such as nets for deployable antennas for space communication or biomedical applications. In such cases the structure is often composed of repetitive patterns that are formed by curved filiform beams. Finite element analyses and an experimental test case have been used to validate the proposed structural model and the adaptive meshing technique. The comparison with numerical model and experimental test shows a limited scatter of the results despite a wide reduction of the required degrees of freedom.

Fig. 13. Comparison of experimental test vs the numerical non-linear results.

means of four coefficients. The quality of the approximation is evaluated through several criteria, the most important of them being the distance from the original structure. If the fitting is not satisfactory according to prefixed criteria, the wire is automatically subdivided into more pieces.

Fig. 14. Example of astromesh a); a single pattern modelled by 4 curved wire elements b).

Fig. 15. A 3  3 pattern structure a); deformed over undeformed shape b). 13

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