Michail Samofalov* and Rimantas Kacianauskas Department of Strength of Materials
Vilnius Gediminas Technical University, Vilnius, Lithuania e−mail: ms@fm.vtu.lt
ABSTRACT
Summary The semi-analytical finite elements (SAFE) for approximation of three-dimensional state variables into cross-sectional direction of thin-walled beams are proposed. The method is able to describe distortion of cross-section by additional DOF. Attention is focussed on simulation of buckling. The versatility of the SAFE approach is demonstrated by the solution of practical examples for U-section beam.
1. Introduction
The engineering theory of thin-walled beams describes bending, non-uniform torsion and out-of-plane warping, but restricts in-plane distortion of the cross-section. It is based on an analytical approximation of displacements and stresses within cross-section. As an alternative to the existing classical models the semi-analytical finite element approximation [1] of cross-sectional distribution of state variables is further developed in this report. The proposed approach, extended here to distortion, considers a thin-walled section as an assemblage of finite elements, where linear warping is described by membrane deformations of individual elements, while linear distortion of cross-section by bending of elements.
2. Conception of Semi-Analytical Finite Elements
Generally, three-dimensional distribution of the displacement field of the beam u(x, y, z) is reduced to one-dimensional one by an approximation in the form of
(
x y z) [
f( )
y z] ( )
U xu , , = , , (1)
where [ f(y, z) ] is the cross-sectional approximation matrix and U(x) is the vector of generalised displacements. The stresses may be approximated independently in the same manner.
Let us consider the beam as a thin-walled cylinder. The contour of the cylinder build up of straight segments and shaping cross-section is considered as domain of one-dimensional finite elements. These elements are termed here as semi-analytical finite elements. The flat segment of cross-section, which is defined by single SAFE, is termed here as subelement. Actually, the construction of global displacement approximation (1) for the entire cross-section focuses on local approximation within a single semi-analytical element se along perimetric co-ordinate p:
( )
x p[ ]
se( )
p se( )
xse f U
u , = . (2)
The second task − derivation of conventional finite elements − follows a standard well-defined path. Longitudinal distribution of generalised displacements U se(x) within subelement se may be expressed in a standard manner, while three-dimensional approximation of displacement is
( )
se( )
se( )
seHere, U se is the vector of nodal displacements, [ N se(x) ] is the longitudinal displacement approximation matrix.
Now, the global approximations (1) is element-wise approximations, which may be constructed by assembling local approximations (2-3). Indeed, once semi-analytical elements se and corresponding shape [ f se(p) ] and [ N se(x) ] functions are determined, they will be used to establish alternative energy expressions or governing equations of the beam in terms of new generalised variables.
3. Finite Element Discretisation
As stated above thin-walled subelement couples membrane behaviour and distortion. The membrane behaviour is described by simplified two-dimensional membrane element (Fig. 1).
Here, the displacement field contains two components u(x, p) = { ux(x, p), up(x, p) }T. Finally, after semi-analytical discretisation, displacements are defined by three independent nodal variables Use
( )
x ={Usexk(x), Usexl(x), Usep(x)}T. The longitudinal components Usexk(x) and Usexl(x) attached to lines l and k are usually approximated by the simplest Lagrangian polynomials. For the transversal component Usep(x) the higher-order interpolation polynomials providing a link between longitudinal and transverse variables are required.x
Fig. 1. Membrane subelement with shear
Distortion as local out-of-plane bending is illustrated in Fig. 2. The cubic distribution of transverse displacement un(x, p) along co-ordinate p is assumed for the distortion model. The vector of generalised distortion displacements Udse
( )
x ={Usenk(x), ϕsexk(x), Usenl(x), ϕsexl(x)}T in approximation contains two translations Usenk(x) and Usenl(x) and two rotations ϕsexk(x) and ϕsexl(x), while approximation matrix [ fnse( )
p ] contains Hermitian polynomials. The same polynomials are used for construction of [ N se(x) ].The membrane and distortion DOF of subelement are stuck together by selection of complex thin-walled elements. The example of U-section beam element composed of three subelements is presented in Fig. 3.
x
Fig. 2. Distortion DOF of subelement and transversal displacement distribution
x
Fig. 3. Finite element of U-section beam and its global DOF 4. Distortion in Buckling Problem
Buckling and especially local buckling is chosen here to illustrate distortion phenomenon of a thin-walled beam. A critical condition, at which buckling of a structure impends, is obtained considering the second variation of the total potential energy of a continuum, expressed in terms the internal strain energy Eint stored in the volume V of a continuum and the external work Wext
done. This condition provides buckling problem as a linear eigenvalue problem. By separating membrane and distortion behaviour the strain energy may be expressed in terms of linear and non-linear membrane energies Elin m and Enon m as well as distortion energy Elin d, where subscript m indicates membrane and d indicates distortion variables.
Linear membrane energy Elin m produces membrane stiffness matrix [ Klin m ] the general expression for which is obtained using mixed approach [1]. The distortion part of strain energy Elin d may be presented in terms of curvatures κx(x, p) and κp(x, p) and bending moments Mx(x, p) and Mp(x, p) using simplified model of bending plate. It produces distortion stiffness matrix [Klin d].
The non-linear strain energy Enon m expressed in terms of non-linear strain εnon x(x, p) produces geometric stiffness matrix. Generally, the non-linear strain associated with transverse displacements Um produces membrane geometric stiffness matrix [ Kg mm ] while strain
associated with distortion displacement Ud produces distortion geometric stiffness matrix [ Kg md ]. Finally, the buckling problem now may be presented as eigenvalue problem
Usually, by solution of buckling problem of beams or plates, matrix [ Kg mm ] is neglected but in thin-walled beams its role is significant.
5. Numerical Example and Conclusions
Various different examples of U-section beams under longitudinal loading have been tested to verify the proposed SAFE distortion approach. The example of compression loads F1 > F2 is presented for illustration (Fig. 4).
a) b) c) d)
Fig. 4. Beam and local buckling mode obtained by different discretisation of DOF occurs
As expected by variation of relative wall thickness, the change of buckling modes occurs. In the range of thin section local instability occur (Fig. 4b-c). The results show a rather good agreement between semi-analytical and ‘exact’ shell approaches (Fig. 4d). In the range of moderate thickness global buckling is observed. This may be explained as predominating of shear effects, which is insufficiently incorporate into our model.
The SAFE has some advantages in comparison to the classical thin-walled beam theory: a)the proposed on-dimensional semi-analytical model of thin-walled beam is much more simple comparing it with expensive shell finite element model; b) higher-order deformations as well as buckling modes may be described by introducing additional DOF; c) new types of local loads and supports my be introduced;
REFERENCES
[1] J. Argyris and R. Kacianauskas. Semi-analytical finite elements in the higher-order theory of beams. Comp. Meth. Appl. Mech. Eng., 138, 19-72, (1996).