We present the finite-element formulation of the geometrically exact spatial beam formulation with constant strains. Its main advantages are that the equations can be analytically integrated and that the convergence of discrete solution to the exact solution is proven. This work is also important for industrial applications to be implemented in a cooperation with the Fraunhofer Institute for Industrial Mathematics ITWM in 2013.
COBISS.SI-ID: 5825121
The classical concept of parametrizing the rotation matrix by the rotational vector is completely abandoned and subtituted by the rotational quaternion representation for both rotations and rotational strains. Because the quaternions are the elements of a four dimensional linear space, their use has advantages compared to the elements of the special orthogonal group. The linearity of quaternions enables, e.g. to interpolate the rotational quaternions in a standard additive way and to apply standard Runge-Kutta time integration methods. The benefit is that we can apply a family of theoretically well-based adaptive time-step algorithms and avoid the analytical linearization of the discrete equations.
COBISS.SI-ID: 5751393
The paper presents a new mathematical model and its analytical solution for the analysis of the stress-strain state of a linear elastic beam cracked in flexure and strengthened with plates on its lateral sides. Both the longitudinal and the transversal, yet linear interactions at the side plate/beam interface are considered. The method is based upon the linearised planar beam theory of Reissner. The weakening of the beam induced by the flexural crack is modelled as a rotational spring. The suitability of the theory is demonstrated in a case presentation involving the comparison between analytical results of the present 1D beam model, the experiments and the numerical results of a full 3D solid model created in the LUSAS FEA software. An excellent agreement between the results is observed and the proposed formulation is found to be accurate and reliable. Finally, the solution is employed in an engineering analysis, discussing the effects of the material and the geometric properties of selected characteristic cases of the observed beams on the static and kinematic quantities, including the boundary conditions of the side plates, the longitudinal and the transversal stiffness of the connection, the size of the cracks, the span of the beam, and the length and the stiffness of the sideplates. For the cracked cantilever beam, a substantial effect of any of these parameters is found. In contrast, for the cracked two-span continuous beam, only the effect of the stiffness of the side plates and the effect of the length of the beam spans are noticeable.
COBISS.SI-ID: 6031457
The paper presents a consistent model of a 3D delaminated composite column with a proper consideration of the extensional and bending stiffness coupling and transverse shear effect to determine the axial buckling load. The exact analytical solution of the buckling force is obtained using the linearized stability theory. The three dimensional model allows us to consider a rather general set of delaminations including those that are not necessarilly perpendicular to the symmetry axis of the cross-section or/and have non-symmetrical surfaces. The parametric studies are presented showing the effects of shear, the delamination position, the angle of rotation of the delamination and the ratio of elastic to shear moduli.
COBISS.SI-ID: 5913697
Different mathematical models are proposed and their analytical solutions derived for the analysis of linear elastic Reissner's multilayer beams. The models take into account different combinations of contact plane conditions, different material properties of individual layers, different transverse shear deformations of each layer, and different boundary conditions of the layers. The analytical studies are carried out to evaluate the influence of different contact conditions on the static and kinematic quantities. A considerable difference of the results between the models is obtained.
COBISS.SI-ID: 6058849