Large deformation analysis of strain-gradient elastic beams

A Corotational Formulation Based on Hamilton’s Principle for Geometrically Nonlinear Thin and Thick Planar Beams and Frames

“…This problem has been solved before in [1,10,14,19]. The solution of James et al [19] failed to converge approximately after P = 4000 kips for n = 0.1 and after P = 3000 kips for n = 0.5 as can be seen in Figures 11 and 12.…”

“…Beheshti [1] compared his results and the results in [19] for loading case n = 0.5. He did not give results for loads behind the load-limits in [19] as shown in Figure 12.…”

“…Consequently, the system equations could be very complicated. This formulation is used by many authors like those of [1][2][3]22]. In the updated Lagrangian formulation [4][5][6], the system equations are defined in terms of a coordinate system which is updated with the last accepted solution.…”

“…The majority of researchers neglected the shear effect in the geometrically nonlinear analysis of beam elements, so they evaluated the stiffness and mass matrices using Euler-Bernoulli beam model. To model the shear effect, Timoshenko beam model was used in [1,[34][35][36] for the evaluation of the stiffness and mass matrices. However, no sufficient examples of beams with small length to thickness ratio have been solved.…”

Large elastic deflections of bars based on nonlocal elasticity

“…Batista [5] has introduced an analytical solution for the deflection of a Reissner's beam under a force acting at the midpoint between two supports is given. Behesti [7] has presented a total Lagrangian finite element formulation for the analysis of large deformation of beams and frames, based on the strain gradient elasticity theory and the Timoshenko beam model. Mohyeddin and Fereidoon [21] have presented an analyze the large deflections of a straight prismatic shear‐deformable beam resting on simple supports at both ends and subjected to a point load at its mid‐span.…”