## 5 Conclusions and Perspectives  
Static and dynamic analyses of grid beams have been addressed in the present paper. To this end, a Timoshenko beam model was formulated in the framework of direct 1D modeling. Its constitutive  
Figure 11: Modal shapes of the equivalent model vs. discrete FE model, for case study I-b and \(n=40\): (**a,b**) First and second flexural mode; (**c,d**) first and second torsional mode. Green dots: discrete FE solution. Continuous blue line: homogenized beam model.  
Figure 10: Modal shapes of the equivalent model vs. discrete FE model, for case study I-a and \(n=100\): (**a,b**) First and second flexural mode; (**c,d**) first and second torsional mode. Green dots: discrete FE solution. Continuous blue line: homogenized beam model.  
Figure 9: Modal shapes of the equivalent model vs. discrete FE model, for case study I-a and \(n=40\): (**a,b**) First and second flexural mode; (**c,d**) first and second torsional mode. Green dots: discrete FE solution. Continuous blue line: homogenized beam model.  
law was determined via a homogenization procedure, based on the energy equivalence between the cell of the periodic grid beam and a segment of the equivalent beam, under the same point-wise displacement fields. The presented model also incorporates corrective factors, analytically determined, which accounts, on average, for the micro-warping effect. Moreover, inertial properties of the coarse model have been determined, under the assumption of lumped masses at the joints.  
The effectiveness and limits of applicability of the Timoshenko beam model have been discussed by numerical analyses, carried out on sample grid beams, taken as case studies. Numerical results, concerning the exact solution of the coarse model in terms of static response and modal properties, have been compared with Finite-Element analyses.  
The following conclusions have been drawn.  
1. An excellent accuracy in terms of translational displacements is given by the Timoshenko model, both in statics and in free dynamics (modal shapes), independently from the number of cells.
2. An excellent agreement between coarse and fine models is also detected, both in statics and in free dynamics (modal shapes), for torsional displacements, where when the aspect ratio of the grid beam cross-section is close to 1, this circumstance reduces the effect of macro-warping. In contrast, when the cross-section is thin, significant errors due to macro-warping appear.
3. Very good agreement is also detected in terms of natural frequencies. In this case, the macro-warping effect also entails worsening of the accuracy in the torsional modes.  
The main perspective of this work relies, therefore, on the development of direct 1D models of beams, and is able to take into account the effects of macro-warping and to investigate the apparent similarity existing between solid and grid beams, when the aspect ratio is far from 1.  
Conceptualization, A.L.; methodology, F.D., M.F. and A.L.; software, F.D. and M.F.; validation, F.D., M.F. and A.L.; data curation, F.D. and M.F.; writing--original draft preparation, F.D., M.F. and A.L.; writing--review and editing, F.D., M.F. and A.L. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The authors declare no conflict of interest.