Planar Beam Element Research Articles

A TUTORIAL PRESENTATION OF ALTERNATIVE SOLUTIONS TO THE FLEXIBLE BEAM ON RIGID CART PROBLEM

A classical planar problem in forward flexible multibody dynamics is thoroughly investigated. The system consists of a damped flexible beam cantilevered to a rigid translating cart. The problem is solved using three distinctly different conventional approaches presented in roughly the chronological order in which they have been applied to flexible dynamic systems. First, a modal superposition formulation based on Bernoulli-Euler beam theory is developed. Second, an alternative solution is developed drawing exclusively on methods for rigid body dynamics combined with a knowledge of the theoretical modal behaviour of continuous beams. Third, a formulation based on the conventional finite element method using four-degree-of-freedom planar beam elements is adapted to include the rigid body motion of the cart. The relative merits of the three formulations are discussed and numerical simulation results generated using each of the three formulations are compared with each other and with a solution from a general-purpose flexible multibody dynamics formulation that is briefly outlined. The relative accuracy and efficiency of the methods and the challenges associated with generalizing each formulation are discussed.

Application of the Absolute Nodal Coordinate Formulation to Large Rotation and Large Deformation Problems

There are three basic finite element formulations which are used in multibody dynamics. These are the floating frame of reference approach, the incremental method and the large rotation vector approach. In the floating frame of reference and incremental formulations, the slopes are assumed small in order to define infinitesimal rotations that can be treated and transformed as vectors. This description, however, limits the use of some important elements such as beams and plates in a wide range of large displacement applications. As demonstrated in some recent publications, if infinitesimal rotations are used as nodal coordinates, the use of the finite element incremental formulation in the large reference displacement analysis does not lead to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. In this paper, a simple non-incremental finite element procedure that employs the mathematical definition of the slope and uses it to define the element coordinates instead of the infinitesimal and finite rotations is developed for large rotation and deformation problems. By using this description and by defining the element coordinates in the global system, not only the need for performing coordinate transformation is avoided, but also a simple expression for the inertia forces is obtained. The resulting mass matrix is constant and it is the same matrix that appears in linear structural dynamics. It is demonstrated in this paper that this coordinate description leads to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. Nonetheless, the stiffness matrix becomes nonlinear function even in the case of small displacements. The method presented in this paper differs from previous large rotation vector formulations in the sense that the inertia forces, the kinetic energy, and the strain energy are not expressed in terms of any orientation coordinates, and therefore, the method does not require interpolation of finite rotations. While the use of the formulation is demonstrated using a simple planar beam element, the generalization of the method to other element types and to the three dimensional case is straightforward. Using the finite element procedure presented in this paper, beams and plates can be treated as isoparametric elements.

An energy‐conserving planar finite beam element for dynamics of flexible mechanisms

Using examples of flexible mechanisms, demonstrates that while the Newmark method is unstable for nonlinear dynamics, time step refinement could in some cases lead to even earlier onset of instability in the form of a blown‐up response. As a remedy, develops a plane finite beam element based on the Simo‐Vu Quoc formulation for dynamics and integrates it with an energy‐conserving midpoint time‐stepping rule for solving problems in nonlinear dynamics. Shows that this combination produces a consistently stable and accurate dynamic analysis method even for large time steps.

3D stress/deflection analysis of tubesheet and U-tubes

Service records of many shell-and-tube type heat-exchangers demonstrate the vulnerability of tube penetration joints to structural failures, especially due to the combined action of corrosion and thermomechanical stresses. For a reasonably complete assessment of such failures and related issues (e.g., remedial measures) it is often essential to know the stresses and relative importance of various factors causing these stresses. The main objective of this paper (and its companion) is to present and discuss the results of stress analyses for a specific U-tube type heat-exchanger typical of a pressurized water reactor (PWR) design. First, the deformations imposed by the overall structure - consisting of the thick tubesheet and its shell attachments - on the tube-to-tubesheet junctions are estimated for normal operating conditions of pressure and temperature; the restraining influence of divider plate (between the hot and cold legs) on the tubesheet deformation is explicitly included. The thermal-structural analysis was done by means of a 3-dimensional finite element model using eight-noded isoparametric brick elements under elasto-static conditions. The significant results are presented as deformation patterns for the in-plane and out-of-plane displacements of the tubesheet; also, the maximum rotation of the tubesheet and the tube-end displacements relative to the first support (baffle) plate holes are estimated. The above deformations produce axial loads and sectional bending moments on the tube-to-tubesheet junctions; these loads and moments are also impacted by the thermal differential expansion of the hot and cold legs of a typical U-tube. Therefore, using the results of above model to define the end conditions, the load-deflection response of three typical sizes of U-tubes was investigated; the full length tubes were modeled by planar beam elements with nodal temperatures defined by the heat transfer analysis. Relatively high stiffness of the tubesheet and flexibility of the U-tubes are confirmed by the analysis. These results are discussed in relation to the expected stress influence of the various factors investigated in this work.

A Graphical, User-Driven Newton-Raphson Technique for Use in the Analysis and Design of Compliant Mechanisms

The analysis and design of compliant mechanisms, undergoing large (geometrically nonlinear) deflections, have been assisted by the Newton-Raphson method to find the load which satisfy a prescribed set of force and displacement boundary conditions. This paper introduces a graphical, user-driven Newton-Raphson technique that allows easy access to good initial design variable estimates, and subsequently accurate and expeditious solutions. These design variables may include loads as well as material and geometric properties of the beam segments composing the mechanism. A line search step-restriction technique is included to enhance the stability of the method. The method uses six-degree-of-freedom planar beam elements in a chain calculation that cumulatively evaluates the large deflections corresponding to a given load set.

Improved method of free vibration analysis of frame structures

The objective of this paper is to develop more accurate procedures in the generation of consistent mass matrices of structural elements which can then be embedded in a usual finite element program. This is accomplished by the use of exact displacement functions for the elements, rather than approximate ones, obtained from the solution of the differential equations governing the free vibration behavior of structural components. At the present time, two types of elements are considered, namely, straight beam and curved (circular arc) beam elements, both of which have a uniform cross section. However, the general concepts developed in this study are equally applicable to other types of elements, such as plate elements. Obviously, when several types of elements are used to model a structure, due care must be exercised in enforcing compatibility between two different classes of elements. The implementation of this procedure in connection with planar straight beam elements has been accomplished leading to the generation of the consistent mass matrix of the element. This matrix is obtained in explicit form resulting in improved computational efficiency. The generation of the mass matrix for the curved beam element is in progress. Based on the improved inertial properties and discrete element techniques, a computer program has been developed for the free vibration analysis of frame type structures. Results have been obtained for various type of structures, including free-free, clamped-free (cantilever) and continuous beams, and simple portal frames. Comparison with exact solutions in these cases indicate that with a relatively small number of elements high accuracy can be achieved in computing the natural frequencies and modes of vibration of these systems. In addition, contrary to other approximate methods, such as the Rayleigh-Ritz method, the present procedure yields very good approximations to higher frequencies.

Finite element approach to mathematical modeling of high-speed elastic linkages

A general approach is described for deriving the equations of motion of planar linkages in high-speed machinery. Based on the work of several current authors, well-known displacement finite element method is used to develop the mass and stiffness properties of an elastic linkage. To demonstrate the various steps in the analysis, a 4-bar linkage is utilized; however the method is readily extendible to other planar multi-loop linkages. Starting with a typical elastic planar beam element, the nodal displacement and acceleration expressions are derived including the terms coupling the elastic and rigid-body motions. The linkage is modeled as beam elements and its equations of motion are stated in matrix form. Methods are described for systematic assembly of all elements, resulting in the undamped equations of motion of the total system. Conventional forms of structural damping are reviewed and appended to this paper for inclusion in the equations of motion. This paper also includes assumptions made in order to simplify the analyses here as well as facilitate numerical solutions.