Efficient evaluation of the flexibility of tapered I-beams accounting for shear deformations

The principle of complementary virtual work is used to evaluate numerically the flexibility matrix of tapered I-beams accounting for shear deformations. Equilibrium considerations of the top and bottom fibres reveal that the shear stress is not equal to zero at these locations. To correct for this non-vanishing shear, a statically admissible shear stress field is considered by assuming a parabolic distribution of shear stress which takes non-zero values at the top and bottom fibres such that the global equilibrium is satisfied within the assumed stress profile. The flexibility matrices of the proposed tapered I-beam finite element with different slopes are generated using numerical integration based on Gauss quadrature. The results are compared to full-blown shell finite element models, and stepped beam models constituted by a series of uniform beam elements, to illustrate the effectiveness of the proposed method.