Dejan Zupan

The three-dimensional beam theory: Finite element formulation based on curvature

Abstract The article introduces a new finite element formulation of the three-dimensional ‘geometrically exact finite-strain beam theory’. The formulation employs the generalized virtual work principle with the pseudo-curvature vector as the only unknown function. The solution of the governing equations is obtained by using a combined Galerkin-collocation algorithm. The collocation ensures that the equilibrium and the constitutive internal force and moment vectors are equal at a set of chosen discrete points. In Newton’s iteration special update procedures for the pseudo-curvature and rotational vectors have to be employed because of the non-linearity of the configuration space. The accuracy and the efficiency of the derived numerical algorithm are demonstrated by several examples.

THE LATERAL BUCKLING OF TIMBER ARCHES

This paper presents the stability analyses of glulam arches subjected to distributed vertical loading. The present analysis employs a strain-based formulation of a nonlinear geometrically exact three-dimensional beam theory. The influence of the relative height of the arch on the lateral buckling load is studied. The buckling load is determined by bisection method with observing the sign of the determinant of the tangent stiffness matrix. The post-critical load deflection path is traced by a modified arc–length method. Such influences are shown for arches with a constant cross-section or constant volume. After determining the most favorable height of the arch, the influence of the number and position of lateral supports is shown. We also compare the deflections, bending, and radial stresses at the lateral buckling states to the limit values which are recommended by European standards.

The Rattling of Euler's Disk

The motion of a disk spinning on a horizontal surface has drawn a great deal of interest recently. The objectives of the researches are to find out what produces an increasing rattling sound and why the spinning ends so abruptly. In order to understand the behaviour of the spinning disk better, we derived a mathematical model of the rolling/sliding motion of a thin, rigid disk on a rigid, rough horizontal plane, and found the numerical solution of the related initial value problem. Then we studied the motion of the commercially available Tangent Toy disk [3]. The results show that the normal contact force becomes very large whenever the inclination of the disk becomes small. As the inclination of the disk oscillates with time, the time-graph of the normal contact force exhibits periodical peaks, which correlate well with the peaks in the recorded sound response. They could well be responsible for the rattling sound heard during the motion.

Buckling of Asymmetrically Delaminated Three-Dimensional Twisted Composite Beam: Exact Solution

The analytical solution of a buckling force of a pretwisted delaminated composite column with a proper consideration of the extensional and bending stiffness coupling and transverse shear effect is presented. The system of homogenous linearized differential equations with nonconstant coefficients obtained for this problem is solved with the help of the mathematical theory of analytic differential systems. The parametric studies are presented considering the effect of slenderness, the length of delamination, the asymmetrical position of delamination, and the transverse shear on the critical buckling force.