This talk concerns the analysis of the effects of high flexibility on buckling and post-buckling of beams and beam-like systems. Preliminary, the Feodosiev system [1], consisting of a planar inverted pendulum, axially soft, is analysed in the nonlinear field. It is shown that, when the axial stiffness of the rod is lowered, the bifurcated paths detach from the non-trivial fundamental path, thus explaining the disappearance of buckling [2,3].

Successively, the lateral buckling of a highly flexible fixed-free beam, subjected to bending in a principal inertia plane, is investigated [4]. A 1D exact polar continuum model, internally constrained, is used. The nonlinear, nontrivial path is determined, in which the beam undergoes large in-plane displacements. Then, the model is linearized in the out-of-plane and twist displacements, to capture the critical points at which flexural-torsional buckling manifests. Notably, the relevant linear boundary value problem is ingeniously tackled through an analytical-numerical approach, strategically exploiting linearity. The study unveils the crucial role of precritical deformations, showcasing their propensity to augment critical loads compared to classic theory assumptions, thereby positively influencing the buckling process. These findings resonate with the existing literature concerning axially soft compressed beams, albeit via a distinct eigenvalue divergence mechanism.

The implications of the results extend beyond mere validation, suggesting further explorations concerning loading condition interactions, nonlinear analysis, imperfection sensitivity, constitutive law modelling, and innovative buckling suppression mechanisms. To this end, a paradigmatic reverse pendulum, embedded in a 3D space and grounded with a Cardanic joint, could be used from preliminary investigations.

This research paves the way for a deeper understanding of bifurcations from non-trivial path in the presence of large precritical deformations, offering a rich landscape for future works.

[1] Feodosiev, V. I. (2005). Advanced stress and stability analysis: worked examples. Springer Science & Business Media.

[2] Ferretti, M., Di Nino, S., Luongo, A. (2021). A paradigmatic system for non-classic interactive buckling. International Journal of Non-Linear Mechanics, 134, 103735.

[3] Luongo, A., Ferretti, M., Di Nino, S. (2023). Stability and bifurcation of structures: statical and dynamical systems. Springer Nature.

[4] Luongo, A., Ferretti, M. (2024). Beneficial effects of the precritical nonlinearities on the lateral buckling of extremely flexible beams. International Journal of Non-Linear Mechanics, 159, 104593.

Stiffness is a key term of structural mechanics. The same applies to the adjectives stiffening and softening in the context of structures subjected to proportional loading. In the course of the loading process, originally stiffening (softening) structures may become softening (stiffening) structures. This occurs at the unknown load level at which the stiffness of the structure concerned attains a maximum (minimum) value. Extreme values of the stiffness of proportionally loaded structures are points of inflection of their mechanical behavior. Therefore, it is astonishing that analytical criteria for stiffness maxima and minima of such structures do not exist. In recent publications it has been claimed that points of inflection of eigenvalue functions of a special linear eigenvalue problem in the framework of the Finite Element Method (FEM) mark extreme values of the stiffness of proportionally loaded structures. The task of this lecture is to present the scientific foundation of this assertion in the form of criteria for stiffness maxima and minima based on variational calculus. This amounts to the solution of an inverse problem, with the mentioned numerical results as the observed effect and the sought variational criteria as the unknown cause. In general, analytical solutions for extreme values of the stiffness of proportionally loaded structures are inaccessible. Nevertheless, knowledge of their scientific basis is not only a fundamental scientific value in its own right, but also enhances the understanding of the intricacies of FE analysis for numerical determination of the load level of stiffness maxima and minima.

The investigations were made on thin plates made of laminate with a layer arrangement where extension-bending mechanical coupling exists. The plates under analysis were subjected to in-plane compressive harmonic load. The equations of motions describing the plate's deflection in time were derived analytically considering classical plate theory employing the Galerkin method. The damping was also included. To check the correctness of obtained results using the proposed analytical-numerical the finite element method was employed. The numerical model was prepared, and the obtained from both methods results were compared. Different parameters describing harmonic load, i.e., the mean value and amplitude, were assumed. In all analyzed cases, the mean value of harmonic load had the compressive character and was in the range from 0 to critical static buckling load. The amplitudes were assumed in such a way that the maximal load could be even higher than the buckling static load or/and the minimal load value could have the tension character. The plate's behavior was analyzed based on phase portraits and Poincare maps, assessing if they have periodic or chaotic behavior. The obtained results show different behavior of such a plate depending on the amplitude and mean value of harmonic excitation load. It could mean that such structures with proper dimensions could be used in microelectromechanical systems (MEMS) as sensors that generate energy and give different signals depending on excitation load parameters.

This work introduces a shear deformable beam model designed for the nonlinear stability analysis of laminated beam-type structures. Each wall of the cross-section is symmetric and balanced angle-ply laminate. The incremental equilibrium equations for a straight thin-walled beam element are derived within the framework of updated Lagrangian formulation and the nonlinear displacement field of cross-sections, which accounts for the restrained warping and the large rotations effects. Throughout the analysis, Hooke’s law is assumed to be valid. The shear deformable beam theory incorporated in this model addresses the flexural-torsional response of a composite beam. It also considers the coupling between bending and non-uniform torsion, particularly when dealing with non-symmetric cross-sections. Cross-section properties are calculated based on the reference modulus, allowing for the modelling of various angle-ply laminates. The accuracy and reliability of the proposed numerical model were validated through verification on benchmark examples. The results obtained affirm that the model is free from shear locking issues, indicating its effectiveness in capturing the behaviour of laminated beam-type structures under different types of loads and support conditions.