It is well known that the structural failure now is considered as a consequence of accumulated structural damage exceeding a certain, in general, unknown limit. One of the aims of this paper is to investigate how such a limit can be defined. The present paper atempts to lay the foundations for a rigorous evaluation of structural damage description during numerical simulations, observed and registered on the level of external variables of the structure. First, the authors consider the nonlinear equation of motion taking into account the nonlinear structural response. This response is evaluated by incremental-iterative techniques based on tangential operators. The fundamental equation is derived by decomposing the external nodal kinematics and external loads of a certain unknown neighboring structural state into the corresponding variables of a known fundamental state and increments reaching from the fundamental to the neighboring position. This allows to obtain tangential equation of motion, and then to apply standard finite element techniques for the evaluation of unknown variables, including constitutive laws containing damage components. In the framework of a quasi-static process, the authors generate a set of linear algebraic equations related to the tangential stiffness matrix. Then, a homogenization of material damage description is carried out in order to identify the effects of diffferent material damage phenomena on structural level. The physical modeling of such phenomena is incorporated in known constitutive laws, and the homogenization technique is addressed as multi-level simulation strategy. By this all damage constituents modeled in the constitutive laws are transformed to the structural level with identical algorithms in the stress state. The tangent stiffness equation is integrated over life time intervals with successively changing stiffness and internal nodal force properties. Material damage is understood as a stiffness-softening phenomenon caused by material deterioration due to certain origin. Then, the authors introduce new damage indicators on the base of positive eigenvalues of tangential stiffness matrix. Taking into account structural eigenfrequencies of the tangential stiffness matrix and the global mass matrix of the structure, the authors state the basic generalized eigenvalue problem. Finally, the tangential matrix is decomposed into an upper triangular matrix with unit values on the main diagonal, and a pure diagonal matrix, the decomposition which is later used in a numerical example. The numerical examples are devoted to (i) damage development in a reinforced concrete slab highway bridge under condition of constant dead load and quasi-statically incremented traffic load, and (ii) damage evolution in a large cooling tower shell with reinforcements under environmental loads, namely due to dead weight, quasi-static wind loading, and winter service temperature effects.