Geometrically non‐linear formulation for three dimensional curved beam elements with large rotations

Abstract

SUMMARYThis paper presents a geometrically non-linear formulation (GNL) for the three dimensional curved beam elements using the total Lagrangian approach. The element geometry is constructed using co-ordinates of the nodes on the centroidal or reference axis and the orthogonal nodal vectors representing the principal bending directions. The element displacement field is described using three translations at the element nodes and three rotations about the local axest. The GNL three dimensional beam element formulations based on these element approximations are restricted to small nodal rotations between two successive load increments. The element formulation presented here removes such restrictions. This is accomplished by retaining non-linear nodal terms in the definition of the element displacement field, and the consistent derivation of the element properties. The formulation presented here is very general and yet can be made specific by selecting proper non-linear functions representing the effects of nodal rotations. The details of the element properties are presented and discussed. Numerical examples are also presented to demonstrate the behaviour and the accuracy of the elements. A comparison of the results obtained from the present formulation with those available in the literature using a linearized element approximation clearly demonstrate the superiority of the formulation in terms of large load steps, large rotations between two load steps and extremely good convergence characteristics during equilibrium iterations. The displacement approximation of these elements is fully compatible with the isoparametric curved shell elements (with large rotations), and since the elements possess offset capability, these elements can also serve as stiffeners for the curved shells.

Eulerian Formulation for Large‐Displacement Analysis of Space Frames

“…With 2 elements, the quartic formulation gives identical results to those in Ref. (Surana and Sorem 1989), as shown in figure (8). Whereas 1 quartic element still provides a very good comparison even though the bend curvature corresponds to high imperfection levels, thus compromising the assumption of small deformation relative to the element chord.…”

Beam element verification for 3D elastic steel frame analysis

“…17, first analysed by Bathe & Bolourchi [53], is perhaps the most popular example for verifying 3D nonlinear beam elements. It is recommended as a benchmark problem by NAFEMS (National Agency for Finite Element Methods and Standards, UK), and has been analysed as such by many computational mechanics researchers [54][55][56][57][58].…”

Conceptual issues in geometrically nonlinear analysis of 3D framed structures

“…It is noted that the above nonlinear equilibrium conditions (6)(7)(8) are normally obtained, in an identical form, using the virtual work method, where the virtual displacement modes are those associated with infinitesimal changes of individual freedoms u i . However, the principle of stationary total potential energy is utilised here simply to facilitate the exposition of conceptual issues which have been a source of previous confusion.…”

“…Observing the equilibrium conditions (6)(7)(8), it is now clear that the first expression for G in (1) can be thought of as representing the out-of-balance between R and e P which are work conjugate with u. If such a definition is adopted for G, the tangent stiffness matrix defined in (3) can be expressed as:…”

“…The geometrically nonlinear analysis of 3D framed structures has received considerable attention by numerous researchers [1][2][3][4][5][6][7][8][9], particularly focussing on the treatment of the difficulties associated with finite nodal rotations in 3D space. These difficulties arise mainly from the non-commutativity of finite rotations about fixed axes and the dual issue of nonconservative moments about fixed axes.…”