# Derivation and Verification of Laminated Composite T-Beam Theory
###### Abstract  
This study analyzes the composite laminated T-beams using the composite beam and laminated composite plate theories. The theoretical formula was derived assuming that the composite T-beam has one- and two-dimensional (1D and 2D) structures. The 1D analysis was performed according to the Kirchhoff-Love hypothesis, thereby considering only the axial strain to derive a relationship between the strain and displacement. The 2D analysis was performed considering the T-beam as a combination of two composite sheets. The effective stiffness of the beam was derived from the stress-strain and moment-curvature relationships. Furthermore, the deflection of the beam and the stress of each laminate were calculated. A simple support beam, made of A54/3501-6 carbon/epoxy, was used as a composite laminated T-beam. MSC/NASTRAN finite element software was used for analysis. The accuracy of the theoretical formula and limitations of its use was verified using the finite element analysis. Higher accuracy of the theoretical formula was obtained at a composite beam aspect ratio greater than 15. The formula derived in this study is suitable for thin and long beams. finite element analysis; laminated composite T-beam; bending stress; deflection +
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of the beams under plane stress. [PERSON] [6] analyzed the static bending of symmetrical and nonsymmetrical cross-laminated beams based on the triangular shear deformation theory. The transverse shear and normal strains were considered in the theoretical analysis to achieve higher accuracy. [PERSON] et al. [7] used the plate theory to derive the effect of shear deformation. [PERSON] and [PERSON] [8] conducted theoretical and experimental static analyses of a laminated composite I-beam based on the linear Vlasov theory [9]. The beam was subjected to tip bending and torsional loading. Consequently, the beam was observed to be affected by the lateral shear deformation. The predicted deformation of the beam was compared with the experimental data. [PERSON] and [PERSON] used the Vlasov theory [9] to modify the first-order shear deformation theory for analyzing the dynamic response of an open-section composite beam. In addition, numerical methods have been used to derive the approximate values. [PERSON] and [PERSON] used a finite difference method to analyze the static behavior when a composite blade was subjected to a large deflection. In their study on thin-walled beams, [PERSON] and [PERSON] [10] assumed that the deformation caused by only lateral shear force was considered in the buckling analysis of open-section composite materials. [PERSON] [11] developed a method for analyzing the structural response of laminated composite beams using lamination theory. This analytical method estimated the axial bending stiffness, centroid location, and stress of each layer of the structure. The results were compared with those obtained using the finite element method (FEM). [PERSON] [12] used MATLAB to analyze the normal and shear stresses inside a laminated composite beam and discussed the stress and maximum shear stress in different fiber directions. [PERSON] et al. [13] proposed a solution to analyze the effects of deflection, stress, natural frequency, and critical buckling load of laminated composite beams in static, buckling, and vibration states. [PERSON] [14] proposed a symmetric smoothed particle hydrodynamics (SSPH) code to study the static behavior of laminated composites and sandwich beams at different boundary conditions. The effects of the fiber angle, layup, and aspect ratio on the mid-span displacement and stress were discussed. Moreover, FEM was used to predict and analyze the deformation of composite beams [15; 16; 17; 18; 19]. [PERSON] and [PERSON] [17] used the FEM to analyze the effect of warping on composite beams. [PERSON] [20] used a combination of two different materials, graphite/epoxy, and aramid/rubber, to analyze the neutral plane of the beam after bending, considering the deformation caused by the shear force.