## 2 General Approach of the Elastic Stability Issue in the Case of Beam-Type Elements  
The property of elastic equilibrium of deformable bodies is considered their capability of recurrence to their initial state after small perturbations appear in the neighborhood of their equilibrium position. Under significant amounts of loads, beam-type elements can change their behavior. Behavior changes take place through the increase or decrease of their stiffness.  
In some situations, the intensity of loads may reach critical values for which the system becomes unstable and may lose its load-bearing capacity for any slight increase in load intensity. At a critical state, if a lateral force is applied then a small deflection is produced. In an elastic analysis, the resulting deflection disappears when the lateral force is removed, thus the element returns to its initial form [20]. In the case of beam-type elements, significant compressive forces and bending moments may cause the loss of their elastic equilibrium configuration.  
To evaluate on a quantitative and qualitative scale the effects of critical loads, the elastic equilibrium conditions must be formulated based on the final deformed shape of the structure, typical for stability issues and second-order theory [20]. The problem of elastic equilibrium of structures may be defined from three different perspectives, static methods, energy methods, and dynamic methods.  
The definition of stability, through the dynamic approach, is fundamental and has an overview of all structural stability issues. Dynamic stability analysis is employed when structures are subjected to nonconservative loads, such as wind or pulsating forces. Throughout dynamic stability analysis, false equilibrium states can be identified, which can be caused by accelerated motions or vibrations with increasing amplitude. In modern structural engineering, the dynamic approach to stability is very important [20].  
The static approach can be applied in the case of conservative structural systems, for which the critical loads can be computed, but cannot give a complete overview of the stability issue. Statics can only answer the question if the equilibrium state is stable or unstable. For simple structures, with no initial curvatures, it proves to be an effective method, but in the case of structures with imperfections, the energy approach is used to decide the stability of the system. Compared to the dynamic approach, it brings a great simplification [20].  
The stability analysis of discretized elastic systems throughout the energy method is based on the Lagrange-Dirichlet theorem, which checks the positive definiteness of the energy function of the structure-load system. The energy criterion offers a comprehensive answer to the question of stability, which is given by the dynamic definition of stability for conservative systems [20].  
The energy method can be applied for discrete and discretized systems, and for continuous structures. Continuous structures such as beam-type elements, which do not represent a discrete system, may usually be approximated by a discrete system, characterized by generalized displacements, which are kinematic variables [20]. [PERSON] and [PERSON] used a discrete model in order to capture the onset of dynamic instability of statically determined columns [3].