A new beam element for analysis of planar large deflection

Abstract

In the present study, a simple and efficient finite element approach is presented for large deflection analysis of both the straight and curved Euler-Bernoulli beams in the planar static problems. The linear stress-strain relationship is assumed for the Euler-Bernoulli beam. In some of finite element methods, displacements together with rotations, and in some others, positions are considered as the main fields of interpolation. However, in the present study, the main idea for the interpolation is using the dimensions of the deformed element instead of the displacements. Therefore, the slope angle (like the previous works) and the length of beam centroidal axis (unlike the previous works) are used as the main field parameters. This treatment creates simplicity in the constitutive equations. Next, using the weighted residual method, the constitutive equations are applied to the element. Using the equilibrium equations and kinematic equations, the position coordinates of the nodes are related to the internal forces and the main field parameters. In the present study, three-node element and, consequently, Simpson's 1/3 rule is used for integration. For solving nonlinear equations of the beam, the Newton-Raphson method is used. Finally, several numerical examples are presented and compared with the previous works to illustrate the validity and efficiency of the new element.

Nonlinear dynamics of flexible links in planar parallel robots using a new beam element

“…In a xy global fixed reference frame, as presented by Sharifnia (2018), the proposed beam element No. i is considered with three nodes on the centroid curve.…”

“…A bilinear relation for normal strain italicϵn was presented by Sharifnia (2018) as followsin which parameters a1 and a2 are two unknown parameters. The expression s°(ξ)−S°(ξ)=(a1+2a2ξ) defines the length variation of the centroid curve for the element (see Sharifnia, 2018). Parameters S°(ξ) and s°(ξ) are the length of the centroid curve for the element in the reference and deformed configurations, respectively.…”

“…As presented by Sharifnia (2018), the element has 20 unknowns: two parameters a1 and a2 together with six nodal unknowns for each of the three nodes. For the element, 14 equations are written, and the remaining six equations are provided by six boundary conditions on the kinematic and/or kinetic parameters.…”

“…As discussed in the previous work of the author (Sharifnia, 2018), several subjects can affect the efficiency of the computations in an FEM: (1) selection of a reference frame, (2) selection of field parameters for interpolation, (3) precision of strain–displacement relations, (4) combination and solution of equations, and (5) time integration methods. Sharifnia (2018) presents a beam element for planar static problems with large deflection. The main contributions of the element are around the second and the fourth subject.…”

“…In the present research, a previously presented beam element (Sharifnia, 2018), which has a new formulation, is extended to planar dynamic problems. Some of the equations are written as new forms, and some equations are added.…”