Notations
Latin letters:

stiffness matrix of the inner constrained unit cell 

reduced stiffness matrix of the inner constrained unit cell 

chord and diagonal cross section areas 

chords, battens and diagonals second order central moment 

dof’s vector of the girder section


vector of the alternative static quantities on the section i 

state transfer matrix 

girder height 

unit cell stiffness matrices 

length of the unit cell chords, diagonal and battens 

girder span 

bending moment generated by the antisymmetric axial forces on the girder section


resultant of the nodal bending moments and difference between the top and bottom nodal bending moment on the section i 

axial force on the girder section


forces vector of the node


radius of curvature 

state vector of the section i 

axial displacement of the section


axial and transversal displacements of the equivalent beam cross section 

horizontal and vertical displacements of the node


shear force on the girder section i 
Greek letters:

axial stiffness parameters of chords and diagonals 

equivalent bending stiffness 

couple stress bending 

coefficient matrix determinants 

displacements vector of the node


dummy variables 

bending stiffness parameters of chords, diagonals and battens 

slope angle of the girder diagonals 

equivalent axial stiffness 

equivalent shear stiffness 

rotation of the equivalent beam cross section 

rotation of the node


the symmetric and antisymmetric parts of the nodal rotations of the girder section


rotation of the section

1 INTRODUCTION
In the last decades, a growing attention on periodic beamlike structures has been given by researchers and technicians operating in several engineering areas. In fact, this type of structures constitutes an optimal tradeoff between strength and stiffness, on one side, and lightness, economy and manufacturing times, on the other. By these features they are frequently adopted in civil and industrial buildings, naval, aerospace and bridge constructions, material design and biomechanics (^{Salmon et al. (2008}), ^{Cao et al. (2007}), ^{Salehian et al. (2006}), ^{Cheng et al. (2013}), ^{Tej and Tejová (2014}), ^{Fillep et al. (2014}), ^{Zhang et al. (2016}), ^{El Khoury et al. (2011}), ^{Syerko et al. (2013}), ^{Ju et al. (2008})). In addition, the main component of the railway infrastructure, namely the track, has a periodic character and is often designed against the thermal buckling phenomenon assuming that rails, sleepers and fastenings are part of an infinitely long Vierendeel girder constrained to the ground by springs representative of the ballast actions. The importance of this topic is underlined by the fact that several publications have focused the issue, e.g. (^{Kerr and Zarembski (1981}), ^{Pucillo (2016}), ^{De Iorio et al. (2014a}, ^{2014b}, ^{2014c}, ^{2017}).
The response of these types of structures to the service loads is usually analysed in a CAE environment. A f.e. discrete system models the girderlike structures and one of the solution methods for the structural framework problems is then applied to find the unknowns mechanical parameters.
If a considerable number of bays or unit cells composes the girder beam, to avoiding strong calculations charges especially during the preliminary design phase, it may be convenient to approximate its mechanical behaviour by a continuum 1D model, whose properties come from those of the unit cell by a suitable homogenization method. Frequently, this kind of approach also offers the additional advantage of providing analytical closed form solutions for the problem at hand. Moreover, the continuous approximation may be employed as a means of transition to a coarser discrete system with a lower and more tractable set of kinematic and static unknowns.
To analyse correctly the in plane bending of periodic lattice structures, the classical continuum theory does not provide an acceptable approximation. In fact, several micropolar equivalent models have been reported for the analysis of planar lattices (^{Noor (1988}), ^{Bazant and Christensen (1972}), ^{Kumar and McDowell (2004}), ^{Bakhvalov and Panasenko (1989}), ^{Segerstad et al. (2009}), ^{Wang and Strong (1999}), ^{Warren and Byskov (2002}), ^{Onck (2002}), ^{Martinsson and Babuška (2007}), ^{Liu and Su (2009}), ^{Dos Reis and Ganghoffer (2012}), ^{Trovalusci et al. (2015}), ^{Bacigalupo and Gambarotta (2014}), ^{Hasanyan and Waas (2016})). While, the studies on the micropolar models for analysing beam like lattices have not yet achieved the same advances. As far as the authors are aware, only few papers have specifically addressed this topic. In ^{Noor and Nemeth (1980)}, ^{Salehian and Inman (2010}), a rational approach is presented where stiffness parameters of the effective continuum model were obtained using energy equivalence concepts. Nodal displacements of the unit cell were got in an approximated way by a Taylor expansion of the kinematical model of the substitute continuum. Then, the equivalent stiffnesses were derived by equating the potential and kinetic energies of a unit cell of the lattice beam to those of the equivalent continuum. This approach leads to two questionable stiffness couplings between the symmetric and antisymmetric components of the shear stresses and between the bending and couple stress moments, making difficult the solution of the equilibrium equations of the equivalent beam, also for the simplest loading and constraint conditions. In ^{Romanoff and Reddy (2014}) the modified couple stress Timoshenko beam theory (Ma et al. (2008, Reddy (2011) is used to analyse the transversal bending of webcore sandwich panels. The equivalent polar bending stiffness was determined by invoking the spring analogy criterion. According to this rule, along the substitute beam, the ratio of the couple stress moment to the total bending moment is given by the ratio of the chords bending moment to the moment of the couple of axial forces acting in the panels faces. As it is shown in ^{Gesualdo et al (2017b}), this assumption unfortunately leads to an overestimation of the polar bending stiffness.
In this paper, the state transfer matrix eigenanalysis method is applied to evaluate the properties of the micropolar medium substituting a periodic beamlike structure. So far, the transfer matrix methods have been applied mostly for the dynamic analysis of repetitive or periodic structures (^{Mead (1970}), ^{Meirowitz and Engels (1977}), ^{Yong and Lin (1989}), ^{Langley (1996}), i.e.). Just recently, it has also been used for the elastostatic analysis of prismatic, curved and pretwisted repetitive beam like lattices made of pinjointed bars (^{Stephen and Zhang (2004}, ^{2006}, ^{Stephen and Ghosh, 2005}). The main advantage of this method consists in evaluating both the SaintVenant decay rates and the load transmission modes by carrying out an eigenanalysis of the unit cell transmission matrix
Illconditioning, as noted in ^{Zhong and Williams (1995}), arises because the construction of the of the
The present paper introduces a direct technique approach for the homogenization of periodic beamlike lattice structures by the state transfer matrix eigenanalysis. The main advantage of the proposed method is that it operates directly on the subpartitions of the unit cell stiffness matrix and, for this reason, all the drawbacks of the transfer methods till now proposed are avoided, see also ^{Penta et al. (2017}, ^{2018}). For the simpler girder geometries, namely the Pratt and Xbraced girders, closed form solutions for the unit cell force transmission modes are obtained and used to determine the stiffnesses of the equivalent beam. Doing so, neither approximations for the kinematical quantities nor subjective phenomenological assumptions on the inner moments are needed. The polar nature of the substitute beam, which is a Timoshenko couplestress beam, is a direct consequence of the pure bending transmission mode components, since through the unit cell of the analysed girders two kinds of bending moments are transferred: one given by the axial forces, the other one stemmed by nodal moments.
The method can be easily extended also to more complex unit cell geometries composed of two or more bays. In these cases, eigen and principal vectors of
The real capability of the resulting equivalent beams in reproducing the behaviour of real discrete beamlike lattice structures is finally assessed performing some sensitivity analyses by a set of f.e. models. As a consequence, the accuracy of the results associate to the homogenized beams in a wide range of lattice parameters variation, satisfactorily validates the suggested direct technique.
2. EIGENANALYSIS OF THE TRANSFER STATE MATRIX
To depict in a clear and concise manner the homogenization method we propose, some examples of immediate technical and engineering interest are examined in this section. Specifically, the Pratt girder problem is analysed in detail while the main results related to the Xbraced girder are synthetically shown. The Vierendeel girder scheme, equally significative as the previous ones, is not explicitly considered since its solution can be obtained from those of the Pratt and Xbraced girder by simply neglecting the stiffnesses of the diagonal rods. Finally, the problem of the Warren girder is also considered and the peculiar features of the method, making it more convenient when unit cell eigen and principal vectors can only be determined numerically, are highlighted.
2.1 Pratt and Xbraced girders
The unit cell of a Pratt girder is schematically represented in Figure 1. It is made up of two straight parallel chords rigidly connected both to the webs and to the diagonal. All the cell members are BernoulliEuler beams. The top and bottom chords have the same section whose area and second order central moment are denoted
To identify any static or kinematical quantity related to the nodal section i of the girder, the subscript i will be adopted, see Figure 2. To distinguish between the joints or nodes of the same section, the superscripts t or b are used, depending on whether the top or bottom chord is involved. Finally, in a coherent manner, top and bottom nodes of the section i are labelled
In what follows, we denote:
the displacement vectors of the joints
Similarly, the nodal forces applied on the joints
where
In what follows we assume that the positive components of
The Pratt cell stiffness matrix
The static quantities conjugates of the previous kinematic variables are: the axial force
The standard kinematic quantities
being:
Denoting by
where subscript
is the cell stiffness matrix,
The state vector
or equivalently:
In the simplest problems, where a nodal section contains only two nodes, the transfer matrix has size
As a first important consideration we can assert that the force transmission modes of the unit cell are given by the unit principal vectors of the
A state vector is transmitted unchanged or decays through the cell depending on whether its force components constitute a crosssectional force or are selfequilibrating. This is equivalent to a scalar multiplication of the state vector, which leads immediately to an eigenvalue problem. Indeed, by setting:
from eq. (4) the following eigenvalue problem is derived:
The decay eigenvalues occur as three reciprocal pairs depending on whether decay is from left to right, or viceversa. The transmission eigenvalue has unit value and a multiplicity of six, three of which pertain to the rigid body displacements, while the other three are related to the stress resultants of axial and shear forces and bending moment. Expanding the stiffness equation and rearranging the result according to eq. (5) we have:
Since the subpartitions of
Ill conditioning can be avoided either solving the eigenvectors problem in closed form or recasting this problem in an alternative form that is nonpathological from a numerical point of view. Ill conditioning is automatically avoided on adopting the direct approach. Indeed, unit eigen and principal vectors of the transfer matrix
These latter relations follow from the stiffness equation, eq. (5), by imposing the conditions:
Taking the second equation within eq. (3) and adding it to the first one, the vector
where the
in which the subscript e is adopted for the unknowns since they are eigenvector displacement components. By inspection of eq. (9), it is immediately recognized that the unit eigenvectors of
The principal vector
Equivalently, denoting by
attained by substituting the conditions:
in the stiffness matrix equation, eq. (3). By adding term by term the two equations in (10), the successive condition for the displacement vector
where the matrix
Thus, for the known term
Considering also the components of the
Bearing in mind that
Since in both the previous equations the coefficients of the unknown
The principal vector
and also in this case, the displacement subvector is defined up to independent rigid translations along the axial and transversal directions. The antisymmetric part of the nodal rotations is given by:
Instead, the sectional and nodal rotation components
When the displacement subvector
Indeed, on substituting
Shear force transfer properties of the Pratt unit cell are given by the
The shear displacement
The algebraic system generated, whose coefficient matrix is still the
The antisymmetric part of the nodal rotations
The translational components
By this way, the expressions of
where
is the determinant of the coefficients matrix.
The algebraic manipulations to determine the force subvector
The transmission mode of the axial force is finally given by the principal vector
where the symbol
Also for the Xbraced girder, the eigen and principal vectors analysis of the transmission matrix
 bending transmission mode:
 sectional rotational component of the shear transmission mode:
 axial force transmission mode:
The analysis of the components of the
In both kind of examined girders, axial force is transmitted together with antisymmetric selfequilibrated moments applied at the nodes of each cell endsection. In addition, the unit cell of the Pratt girder deforms also with sectional and symmetric nodal rotations. These rotations are instead totally prevented in the Xbraced girder due to symmetry of the unit cell.
2.2 Warren girder
The present approach can be also effectively adopted to analyse unit cells made up by more than one bay. To give an example we consider in this section the case of the Warren girder, whose unit cell is sketched in Figure 1c. To identify nodes and sections of the girder, we adopt a convention very similar to the one of the Pratt girder. The only difference is that here the subscript c labels the kinematical and static quantities of the central or inner nodal section of the cell. Thus, the force and displacement vectors are respectively:
As the Warren unit cell can be obtained by reflecting a Pratt unit cell, the stiffness matrix can be constructed starting from the one reported in Appendix 1 for the Pratt case. Furthermore, being the inner nodal section of the cell free of external load, the cell stiffness equation is:
The second of previous equations allows expressing
When previous result is substituted in the first and third equation of eq. (18), the
The reduced stiffness matrix of eq. (19) has some properties that make very simple the numerical searching of the principal vectors of the
and, for this reason, it is semipositive definite. It exhibits always the following symmetric structure:
When also the dof’s corresponding to the cell rigid longitudinal and transversal translations are constrained, the cell elastic behaviour will be totally defined by the stiffness matrix:
which is positive definite and thus invertible.
In the case of the Pratt and Xbraced unit cells, the algebraic sums of the indirect stiffnesses involving an antisymmetric nodal rotation (i.e. the outdiagonal components in the last column of
must be inverted and this can be performed in closed form, hence avoiding altogether any illconditioning problem.
To compare the direct method with the classical one based on the
The corresponding
Since the stiffness components of this matrix differ at most for two orders of magnitude, its condition number should be of order 10^{2}. In fact, the MATLAB rcond() command, giving an estimate of the reciprocal condition number, for
In Table 1 the rcond() outputs obtained for the displacement and force transfer matrices
From these results, it is clear that when the proposed method is adopted, the force transmission modes of the unit cell are determined by inversion of a matrix of reduced size that is wellconditioned and allows achieving the solutions with greater accuracy.
3. THE EQUIVALENT CONTINUUM
As equivalent continuum, the modified polar Timoshenko beam is adopted (^{Ma et al. (2008}), ^{Reddy (2011})). The displacements
where
the shear strain associated with the directions x and y:
and the curvature:
where
Denoting by
where l is the beam length and A is the area of its cross section,
are the beam axial and shear forces, while:
are the Navier and polar bending moments, respectively. It is worth nothing that the dual shear deformation of
Under the assumption of homogeneous and isotropic linear elastic material, the stressstrain relationships are:
with
where
The beam equilibrium equations can be derived equating the virtual internal work
4. EQUIVALENT STIFFNESSES
The homogenized beam stiffnesses can be determined by averaging over the unit cell length the cell responses under the load conditions defined by the force transmission principal vectors found in sec. 2. Thus, the equivalent axial stiffness of the homogenized beam is:
where
The equivalent Navier bending stiffness
Polar bending stiffness
Hence, the polar and Navier moment of the homogenized beam make work by the same generalized strain, namely the beam curvature
The shear principal vector
which defines the in plane rotation equilibrium of the cell as reported in Figure 5. We recall that the displacement subvector
Bearing in mind the components of the displacement vector
Hence, the equivalent shear stiffness will be:
Axial and bending stiffnesses of the Pratt and Xbraced girders, obtained by eq. (24), (26) and (26) and the results of subsection 2.1, are reported in Table 2.
By inspection of these results it is deduced that Navier bending stiffnesses depend only on the chords axial stiffnesses and that, since bending of the Xbraced unit cell occurs without deformation of the transvers webs, the equivalent polar bending stiffness of this girder is independent of
In addition, axial elongation of the Pratt unit cell is accompanied by rotations both of its joints and end sections. Consequently, its equivalent axial stiffness is dependent also on the bending stiffness of the chords and battens.
Bending stiffness: 


Axial stiffness: 

The eqs. (24)  (27) completely define the elastic behaviour of the equivalent Timoshenko beam. The range of validity of these homogenized equations is analysed in the succeeding section on the basis of the numerical results of a sensitivity analysis.
5. VALIDATION STUDY
The equivalent beam model defined in Section 3, has been validated against a data set including information on the effects of the main geometrical parameters influencing the girder response. This set has been generated by f.e. solution of cantilevered and simply supported girders engendered by assembling BernoulliEuler beams and subjected to a unit vertical load applied respectively at the free end and at the midpoint.
The accuracy of the theoretical predictions has been quantified by the next nondimensional measure of the homogenization error:
where
Furthermore, to have an additional measure of accuracy and to get also direct indications about the influence exerted on the model equilibrium shapes by the couplestress bending stiffness, for each examined girder geometry the maximum displacement
Since, as a first approximation, the main parameter influencing the relative importance of the two bending moments acting on the girder cross section is the height h of the girder, in the first set of f.e. analysis the effects of the changes of this parameter have been considered. Under the assumption that both chords and webs have the same cross section, specifically HEA100, cantilever girder f.e. models having height _{
h=lt=
} 300, 600 and 1200 mm, cell aspect ratios
In Figure 6, as an example, the deformed shapes of f.e. girders having cellaspect ratio
From these results, it can be concluded that for the whole range of considered girder heights, to have accurate estimates of the girder displacements, it is necessary to take into account the bending stiffnesses of chords and webs by means of the couple stress stiffness of the equivalent beam. Furthermore, since small values of the homogenization error have been obtained for all the examined values of the cell shape ratio
A second series of girder models has been prepared to analyse the effects of the changes of diagonal crosssectional area on the equivalent model accuracy, since the girder shear stiffness is strongly influenced by this geometric parameter. For these analysis, more stout girders under three points bending have been considered in order to highlight the shear properties effects in the girder response. For the chords of these models the standard HEA120 section has been chosen. Several back to back angles sections have been considered for the diagonals, while for the battens only the 80 x 8 back to back angle has been used.
α 






e %  
[]  [Nmm]  [Nmm]  [Nmm^{1}]  [mm]  [mm]  [mm]  []  
h=300 mm  0.5  1.969E+13  1.740E+12  3.851E+08  5.840E3  5.823E03  6.356E3  0.214  
β=24  1  “  1.901E+12  2.334E+08  5.810E3  5.791E03  6.381E3  0.261  
2  “  2.017E+12  1.092E+08  5.820E3  5.791E03  6.451E3  0.335  
h=600 mm  0.5  7.876E+13  1.740E+12  2.113E+08  1.610E3  1.579E03  1.648E3  0.261  
β=12  1  “  1.901E+12  1.737E+08  1.610E3  1.583E03  1.663E3  0.620  
2  “  2.017E+12  8.580E+07  1.650E3  1.621E03  1.748E3  1.455  
h=1200 mm  0.5  3.150E+14  1.740E+12  1.701E+08  4.680E4  4.344E04  4.796E4  0.700  
β=6  1  “  1.901E+12  1.594E+08  4.640E4  4.370E04  4.853E4  1.723  
2  “  2.017E+12  8.014E+07  5.170E4  4.805E03  5.746E4  5.003 
α 






e %  
[]  [Nmm]  [Nmm]  [Nmm^{1}]  [mm]  [mm]  [mm]  []  
h=300 mm  0.5  1.969E+13  2.082E+12  5.940E+08  5.557E03  5.725E03  6.343E03  0.193  
β=24  1  “  2.456E+12  4.033E+08  5.643E03  5.634E03  6.355E03  0.212  
2  “  2.726E+12  1.974E+08  5.586E03  5.582E03  6.392E03  0.091  
h=600 mm  0.5  7.876E+13  2.082E+12  3.805E+08  1.574E03  1.557E03  1.618E03  1.050  
β=12  1  “  2.456E+12  3.322E+08  1.554E03  1.553E03  1.623E03  0.157  
2  “  2.726E+12  1.665E+08  1.563E03  1.568E03  1.666E03  0.458  
h=1200 mm  0,5  3.150E+14  2.082E+12  3.298E+08  4.150E04  4.138E04  4.386E04  0.118  
β=6  1  “  2.456E+12  3.150E+08  4.140E04  4.143E04  4.406E04  0.408  
2  “  2.726E+12  1.590E+08  4.280E04  4.359E04  4.855E04  2.350 
The f.e. results and the predictions of the homogenised model are compared in the diagrams of Figure 7, while in Table 6 the homogenization errors and the equivalent stiffnesses are reported. In all the examined cases the model predictions have resulted to be very close to the f.e. outcomes. Thus, the homogenized model is also able to predict the shear dominated girders responses with sufficient accuracy for practical applications.
α 






e %  
[]  [Nmm]  [Nmm]  [Nmm^{1}]  [mm]  [mm]  [mm]  [%]  
h=300 mm  0.5  1.9689E+13  1.7604E+12  3.2517E+08  4.652E02  4.6443E2  5.0640E2  0.177  
β=24  1  1.9474E+12  1.7087E+08  4.614E02  4.6078E2  5.0720E2  0.152  
2  2.0821E+12  6.4477E+07  4.594E02  4.5917E02  5.1000E2  0.056  
h=600 mm  0.5  7.8758E+13  1.7604E+12  1.8875E+08  1.251E02  1.2435E2  1.2790E2  0.565  
β=12  1  1.9474E+12  1.2840E+08  1.245E02  1.2441E2  1.2860E2  0.059  
2  2.0821E+12  5.0387E+07  1.254E02  1.2588E2  1.3210E2  0.478  
h=1200 mm  0.5  3.1503E+14  1.7604E+12  1.5493E+08  3.240E03  3.2335E3  3.3450E3  0.170  
β=6  1  1.9474E+12  1.1781E+08  3.250E03  3.2604E3  3.4040E3  0.497  
2  2.0821E+12  4.6818E+07  3.360E03  3.4410E3  3.7750E3  2.431 
diagonal 






e %  
[]  [Nmm]  [Nmm]  [Nmm^{1}]  [mm]  [mm]  [mm]  [%]  
Pratt girder  80 x 8  9.396E+13  2.702E+12  1.937E+08  9.100E5  8.932E5  1.013E04  1.751 
L=7200 mm  70 x 7  “  2.619E+12  1.500E+08  9.400E5  9.197E05  1.068E04  2.112 
β=12  55 x 6  “  2.548E+12  1.035E+08  9.900E5  9.714E05  1.176E04  1.571 
30 x 6  “  2.505E+12  5.787E+07  1.130E4  1.100E04  1.450E04  2.021  
Xbraced girder  80 x 8  9.396E+13  2.918E+12  3.787E+08  8.500E5  8.481E05  9.226E05  0.763 
L=7200 mm  70 x 7  “  2.744E+12  2.910E+08  8.600E5  8.633E05  9.513E05  0.669 
β=12  55 x 6  “  2.598E+12  1.979E+08  8.900E5  8.923E05  1.009E04  0.668 
30 x 6  “  2.512E+12  1.067E+08  9.700E5  9.668E05  1.165E04  0.424  
Warren girder  80 x 8  9.396E+13  2.708E+12  1.441E+08  6.660E4  6.675E04  7.120E04  0.243 
L=14400 mm  70 x 7  “  2.621E+12  1.187E+08  6.720E4  6.732E04  7.227E04  0.193 
β=12  55 x 6  “  2.548E+12  8.792E+07  6.830E4  6.839E04  7.439E04  0.211 
30 x 6  “  2.505E+12  5.298E+07  7.110E4  7.100E04  7.979E04  0.105 
6. CONCLUSION
A new procedure for homogenizing large repetitive beamlike structures is presented. Such a method is based on the analysis of the eigen and principal vectors of the transfer state matrix of the unit cell. As a substitute medium, a Timoshenko polar beam is adopted. Differently from the approaches until now proposed, the polar character of the equivalent beam is not deduced by kinematical conjectures nor inspired by the microstructure: it is a direct consequence of the pattern of the inner forces acting in the lattice when the pure bending mode of the cell is active.
The main advantage of the presented method is that it allows to operate directly on the subpartitions of the unit cell stiffness matrix. For the simpler unit cells, as those of the Pratt and X braced girders, the method leads to closed form solutions for force transmission modes, that are then used to determine the stiffnesses of the corresponding equivalent beam. When the unit cell instead has a complex geometry and its transmission modes can be determined only numerically, it is shown that the method we propose has a very low computational cost, since the search of the transmission modes, reduces to the inversion of a
The results of a series of finite element simulations are presented for the deformed shapes of some simply supported and cantilever girders. In all the examined cases the predictions obtained with the homogenized models are in close agreement with the numerical f.e.m. outcomes.
The proposed homogenization technique is applicable in several field of structure or mechanical engineering interest. More specifically, it appears to be a serious candidate to analyse the buckling and postbuckling response of periodic beams infinitely long such as the railway track under thermal load (^{Pucillo (2016})) or to analyse the dynamic isolation of fragile goods in tall buildings (i.e. art objects, see ^{Monaco et al. (2014}); ^{Gesualdo et al. (2014}, ^{2017a}). Its range of validity is bounded by the hypothesis of linear elasticity. Further research will thus be needed to extend the proposed method also in the elastoplastic range whereas the response of the unit cell has to be analysed by approximated methods as those reported in ^{Fraldi et al. (2010}, ^{2014}) and ^{Cennamo et al. (2017}).