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Consider the coupling of stretching-bending deformation induced by the arbitrarily laminated composites, and the transverse shear deformation caused by the thick beams. A mathematical model based upon Timoshenko beam theory is established for the general laminated composite beams, and a proper replacement of stiffness is suggested for narrow or wide beam. By using the state-space approach, a general solution is expressed in terms of matrix exponential, which is applicable for any possible loading and boundary conditions. Expand the matrix exponential through Taylor series, an explicit analytical solution is obtained for the arbitrarily laminated composite beams under general loading conditions. With the obtained general solutions, several analytical solutions are derived explicitly such as the Green's functions for an infinite laminated composite thick beam, and the solutions for some typical beam problems. For example, a cantilever beam, a simply supported beam, or a fixed-end beam, subjected to uniformly distributed load, concentrated force/moment, or sinusoidal load, etc. To cover more complicated beam structures, we develop the boundary element method by using the obtained Green's functions. Three numerical examples are implemented to investigate the influences of stacking sequences and beam geometry, and to validate the applicability of boundary element method. [Display omitted] • Derive analytical solutions for arbitrarily laminated composite thick beams. • Incorporate with coupled stretching-bending and transverse shear deformations. • Provide explicit-form solutions for various loading and supported conditions. • Derive Green's function to develop BEM for curved beams with non-uniform thickness. • Discuss the effects of lamination, thickness, width, and non-uniformity. [ABSTRACT FROM AUTHOR]