Mathematical Problems in Engineering

Volume 2016, Article ID 8980676, 16 pages

http://dx.doi.org/10.1155/2016/8980676

## A Hybrid Interpolation Method for Geometric Nonlinear Spatial Beam Elements with Explicit Nodal Force

State Key Laboratory of Structural Analysis for Industrial Equipment, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, China

Received 1 November 2015; Revised 20 January 2016; Accepted 4 February 2016

Academic Editor: Chenfeng Li

Copyright © 2016 Huiqing Fang and Zhaohui Qi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on geometrically exact beam theory, a hybrid interpolation is proposed for geometric nonlinear spatial Euler-Bernoulli beam elements. First, the Hermitian interpolation of the beam centerline was used for calculating nodal curvatures for two ends. Then, internal curvatures of the beam were interpolated with a second interpolation. At this point, C1 continuity was satisfied and nodal strain measures could be consistently derived from nodal displacement and rotation parameters. The explicit expression of nodal force without integration, as a function of global parameters, was founded by using the hybrid interpolation. Furthermore, the proposed beam element can be degenerated into linear beam element under the condition of small deformation. Objectivity of strain measures and patch tests are also discussed. Finally, four numerical examples are discussed to prove the validity and effectivity of the proposed beam element.

#### 1. Introduction

In engineering practice, a majority of three-dimensional beams can be considered slender beams and can thus be regarded as Euler-Bernoulli beams. Much research in recent years has been focused on modeling geometrical nonlinearity for 3D Euler-Bernoulli beams subjected to large deformations. Because spatial large rotations are physically nonadditive, an improper discretization of spatial rotations may lead to nonobjectivity of strain measures [1]. Different from the linear beam theory, large rotations of beam cross sections cannot be totally determined by the beam centerline. However, displacement and rotation of the cross section of a slender beam must satisfy the Kirchhoff constraints. Therefore, there is a critical need to develop an efficient interpolation method to solve these problems.

Geometrically exact beam theory, developed by Reissner [2, 3] and Simo [4, 5], has been widely adopted for modeling of geometrical nonlinear beams [6–11]. For a geometrically exact beam, the configuration of a spatial beam can be described by the position vector of the beam centerline and an orthogonal matrix, which specifies rotation of a cross section. A curvature vector is used to describe the rotational strain measure, which is anisotropic and proportional to the stress resultants. Relationship between rotation and rotational strain was found to be consistent with the virtual power principle and is valid for configurations with arbitrary large displacements and rotations. Due to independent interpolations of displacement and rotation, use of geometrically exact beam theory may lead to shear-locking problem for slender beams and nonobjective interpolation of strain measure [1]. In order to avoid these problems, many researchers presented different formulations of geometrically exact beam elements. The core technology in developing these formulations has been on the discretization of rotational degrees of freedom or the incremental rotation [6, 11–13]. In spite of the efforts, a problem still exists with accurate determination of an analytical expression of elastic forces using the displacement-based geometrically exact beam theory.

To avoid the interpolation of rotation parameters, an absolute nodal coordinate formulation (ANCF) was proposed by Shabana [14–18]. Three polynomials were used as the assumed displacement field in the ANCF-based beam element. In ANCF, 12 variables were used for each node of a 3D beam element, which includes the position vector and 9 slopes. Therefore, dimension of governing equations is significantly larger in comparison with traditional beam elements. Based on the interpolation of beam curvatures, a new beam element was proposed by Zupan [7, 19], in which displacement and rotational vector were not interpolated at all. In this formulation, strain measure-interpolation based element was demonstrated to preserve the objectivity, namely, invariant with rigid body motions, and path independence of strain measures. However, use of the beam element based on the curvature interpolation was not convenient when used in calculating nodal displacement and rotation, since they cannot be analytically integrated by strain measures.

In this paper, a hybrid interpolation scheme for geometrically exact Euler-Bernoulli beams is proposed. Global nodal displacement and the rotational vector are the basic unknown variables for the formulation. Interpolation of the beam centerline was used to determine nodal value and its derivatives of the curvatures. These new parameters were also interpolated. Using combined interpolation of the nodal displacement and strain measures, an analytical expression for nodal forces was formulated and objectivity and path independence of strain measures were satisfied.

The outline of this work is listed as follows. In Section 2, basic theory of geometrically exact Euler-Bernoulli beams with large deformation is briefly reviewed. In Section 3, based on the principle of virtual power, weak form of the dynamic equilibrium equations for 3D Euler-Bernoulli beams was deduced using the novel method proposed here, which is concise and convenient for large rotations. Next, hybrid interpolations of nodal displacements and curvatures are introduced in Section 4. In Section 5, governing equations for the discrete finite beam element model are given. Property tests and applications of the beam element are presented in Sections 6 and 7, respectively.

#### 2. General Theory of Geometrically Exact Euler-Bernoulli Beams

In this section, geometrically exact Euler-Bernoulli beam theory is reviewed, including kinematics of large deformation beams, strain measures, stress resultants, and constitutive relations.

##### 2.1. Kinematics and Strain Measures

The initial and deformed configurations of a spatial Euler-Bernoulli beam in the inertial reference frame () are shown in Figure 1. The main assumption of 3D Euler-Bernoulli beam theory is that all cross sections keep rigid and perpendicular to the tangent vector of the centerline during deformation. The configuration can be fully described by the following.