Many widely used beam finite element formulations are based either on Reissner’s classical nonlinear rod theory or the absolute nodal coordinate formulation (ANCF). Advantages of the second method have been pointed out by several authors; among the benefits are the constant mass matrix of ANCF elements, the isoparametric approach and the existence of a consistent displacement field along the whole cross section. Consistency of the displacement field allows simpler, alternative formulations for contact problems or inelastic materials. Despite conceptional differences of the two formulations, the two models are unified in the present paper. In many applications, a nonlinear large deformation beam element with bending, axial and shear deformation properties is needed. In the present paper, linear and quadratic ANCF shear deformable beam finite elements are presented. A new locking-free continuum mechanics based formulation is compared to the classical Simo and Vu-Quoc formulation based on Reissner’s virtual work of internal forces. Additionally, the introduced linear and quadratic ANCF elements are compared to a fully parameterized ANCF element from the literature. The performance of the respective elements is evaluated through analysis of conventional static and dynamic example problems. The investigation shows that the obtained linear and quadratic ANCF elements are advantageous compared to the original fully parameterized ANCF element.

The equations of balance of momentum and energy usually are formulated under the assumption of conservation of mass. However, mass is not conserved when sources of mass are present or when the equations of balance are applied to a non-material volume. Mass then is said to be variable for the system under consideration. It is the scope of the present contribution to review the mechanical equations of balance for variable-mass systems. Our review remains within the framework of the classical, non-relativistic continuum mechanics of solids and fluids. We present general formulations and refer to various fields of applications, such as astronomy, machine dynamics, biomechanics, rocketry, or fluid dynamics. Also discussed are the equations for a single constituent of a multiphase mixture. The present review thus might be of interest to workers in the field of heterogeneous media as well. We first summarize the general balance law and review the Reynolds transport theorem for a non-material volume. Then the latter general formulations are used to derive and to review the equations of balance of mass and linear momentum in the presence of sources of mass in the interior of a material volume. We also discuss the appropriate modeling of such sources of mass. Subsequently, we treat the equations of balance of mass and linear momentum when mass is flowing through the surface of a non-material volume in the absence of sources of mass in the interior, and we point out some analogies to the previously presented relations. A strong emphasis is given in this article to the historical evolution of the balance equations and the physical situations to which they have been applied. The equations of balance of angular momentum and energy for variable-mass systems will be treated separately, in a second part of the review, to be published later. This review article cites 96 references.

This article deals with the non-linear modelling of beams that are bent, sheared and stretched by external forces and moments. In the following, we restrict to plane-deformations and static conditions. Our task is to present a continuum mechanics-based interpretation of the celebrated large displacement finite deformation structural mechanics theory, which was presented by Eric Reissner [On one-dimensional finite-strain beam theory: the plane problem, J. Appl. Math. Phys. 23 (1972), pp. 795–804]. The latter formulation was restricted to the notions of structural mechanics and thus did not use the notions of stress and strain, which are fundamental for continuum mechanics. Thus, the common continuum mechanics-based constitutive modelling at the stress–strain level cannot be utilized in connection with Reissner's original theory. Instead, Reissner suggested that constitutive relations between certain generalized strains (bending, shear and axial force strains) and generalized static entities (bending moments, shear and normal forces) should be evaluated from physical experiments. This means that the beam to be studied must be first built up, and the experiments must be performed for the real beam as a whole. Although such physical experiments are indeed to be performed in practice for safety reasons in sensible cases, for example, bridge decks or aircraft wings, it is nevertheless felt to be a drawback that the results of simple standardized stress–strain experiments concerning the constitutive behaviour of the materials, from which the beam is built up, cannot be used. Moreover, relying only on physical experiments on the whole beam means that computations (virtual experiments) can be made only after the beam has been built up. To overcome this problem, we subsequently present a continuum mechanics-based interpretation of Reissner's structural mechanics modelling, by attaching a proper continuum mechanics-based meaning to both the generalized static entities and the generalized strains in Reissner's theory [E. Reissner, On one-dimensional finite-strain beam theory: the plane problem, J. Appl. Math. Phys. 23 (1972), pp. 795–804]. Consequently, these generalized static entities can be related to the generalized strains on the basis of a constitutive modelling on the stress–strain level. We show this in some detail in this contribution for a hyperelastic material proposed by Simo and Hughes [Computational Inelasticity, Springer, New York, 1998]. An illustrative numerical example is given which shows the results of large bending and axial deformation behaviour for different constitutive relations. This article represents an extended version of a preliminary work published in [H. Irschik and J. Gerstmayr, A hyperelastic Reissner-type model for non-linear shear deformable beams, Proceedings of the Mathmod 09 Vienna, I. Troch and F. Breitenecker, eds., 2009, pp. 1–7].

In extension to a former work, a detailed comparison of the absolute nodal coordinate formulation (ANCF) and the floating frame of reference formulation (FFRF) is performed for standard static and dynamic problems, both in the small and large deformation regimes. Special emphasis is laid on converged solutions and on a comparison to analytical and numerical solutions from the literature. In addition to the work of previous authors, the computational performance of both formulations is studied for the dynamic case, where detailed information is provided, concerning the different effects influencing the single parts of the computation time. In case of the ANCF finite element, a planar formulation based on the Bernoulli–Euler theory is utilized, consisting of two position and two slope coordinates in each node only. In the FFRF beam finite element, the displacements are described by the rigid body motion and a small superimposed transverse deflection. The latter is described by means of two static modes for the rotation at the boundary and a user-defined number of eigenmodes of the clamped-clamped beam. In numerical studies, the accuracy and computational costs of the two formulations are compared for a cantilever beam, a pendulum, and a slider-crank mechanism. It turns out that both formulations have comparable performance and that the choice of the optimal formulation depends on the problem configuration. Recent claims in literature that the ANCF would have deficiencies compared with the FFRF thus can be refuted.