This paper proposes a method for structural damage detection through the sensitivity analysis of modal shapes in the calculation of modal STRAIN ENERGY (MSE). For this purpose, sensitivity equations were solved to determine the STRAIN ENERGY based on dynamic data (i.e., modal shapes). An objective function was then presented through the sensitivity-based MSE to detect structural damage. Due to the nonlinearity of sensitivity equations, the objective function of the proposed formulation can be minimized through the shuffled shepherd optimization algorithm (SSOA). The first few modes were employed for damage detection in solving the inverse problem. The proposed formulation was evaluated in a few numerical examples under different conditions. The numerical results indicated that the proposed formulation was efficient and effective in solving the inverse problem of damage detection. The proposed method not only minimized sensitivity to measurement errors but also effectively identified the location and severity of structural damage.

The paper discusses the theory of critical STRAIN ENERGY levels for structures with lumped parameters. The theoretical assumptions and profs for common case are presented. The idea of external actions and STRAIN ENERGY field separation leads to the minimum STRAIN ENERGY principle. It has the self-stress of the structure physical sense. In the general case, a structure's extremal values of parameters are determined from an eigenvalue problem. The critical levels criterion means the self-stress state change. The STRAIN ENERGY consists of two parts: STRAIN ENERGY, which equilibrates the action work, and residual STRAIN ENERGY, which does not allow a deformable body to collapse. This allows for the total and residual STRAIN ENERGY to be calculated. The traditional problem formulation does not give us that option. The proposed theory is illustrated on a rod system, which explains the change in the self-stress state of the structure in a simple manner. The static matrix and stiffness matrix are obtained for the three-bar structure. The eigenvalue problem allows us to obtain the principal values of the nodal reactions and displacements of the structure. New formulations of structural design and structural analysis tasks are given. The results are compared with classical methods of solution. The formulations of weak link problems and progressive limit state problems are given. A structure's residual load capacity is evaluated by the residual STRAIN ENERGY.

A new model is constructed to validate a non-linear elastic response of rubber-like materials. The proposed model is phenomenological and is the combination of Rivlin expression truncated in invariant I1�, 1 with a term in I2�, 2 coming from the Hart-Smith model and which has been modified by Gornet et al. This model has five rheological parameters adjusted with a genetic algorithm. The genetic algorithm method is used in binary code and two sets of experimental data (Treloar data and Nunes and Moreira data) from the literature to compare the proposed model. With Treloar rubber data the simple tension suffices for the determination of the parameter values. Thus, the fit of experimental data is shown to provide an accurate response and predictive description in biaxial tension and pure shear with the same sets of material parameters. Unlike, with Moreira data, the proposed model has given direct material parameters in simple shear with good correlation. On the other hand, a comparison is made between the proposed model and the Beda model (2007), Gornet et al. model and hybrid integral approach (HIA) model. The proposed model is the most efficient among three other classical models because the proposed model emerges from very small relative errors with Treloar's experimental data: 0. 42 , ±,  , 0. 08 in simple tension, 0. 44 , ±,  , 0. 07 in biaxial tension and 0. 42 , ±,  , 0. 05 in pure shear. This model also incorporates all the criteria for model validation described in the literature.