Spectral solvers are mesh-free, fast and very efficient simulation tools which emerged in the last two decades for solving continuum mechanics problems for heterogenous materials. Applied exclusively on particle-type composites, this paper demonstrates that they can model woven laminated composites with very valuable issues as they are able to capture with high spatial resolution the details of the stress and strain fields in the fabric periodic cell. Assessment of the modelling tool is given within in-plane shear mechanical testing: (i) at the mesoscale, through the comparison of local full-field strain maps obtained experimentally with Digital Image Correlation; (ii) at the macroscale, through the estimation of macroscopic properties from direct computation and comparison to their measured values. As shown in the last part of the paper, this tool provides in-depth knowledge of the strain-stress tensor in the microstructure as it offers an opportunity for precise local 3-D investigation. It should be a matter of high concern in the future as it allows the rapid design of new composite structures.