Studies of natural frequency degradation and the nature of destruction of reinforced concrete frames in case of an earthquake

. The article presents the results of a numerical study of the impact of local damage to load-bearing structures on the degradation of natural frequencies and the integrity of the structure during an earthquake. The research was carried out using LS-DYNA software complex, which implemented Continuous Surface Cap Model (CSCM), the nonlinear model of concrete. The design model of a reinforced concrete building is modeled using volume finite elements for concrete, and rod finite elements for reinforcement. A two-storey building with a frame structural diagram was chosen as an object of modelling. Studies have shown a significant (up to 22%) decrease in the frequencies of natural oscillations during an earthquake of 8 points intensity. This suggests that damage appears in the structure and overall stiffness of the building decreases. Attempting to estimate damage using the percentage of failed items shows that structural damage exceeds 7%. Analysis of the results shows that the most intense decrease in the frequencies of natural oscillations occurs during the active phase of the earthquake. When the active phase ends, the frequencies stabilize and stop decreasing. The LS-DYNA software package allows to perform studies of the impact of local damage to load-bearing structures on the dynamic properties during an earthquake when concrete is reinforced with reinforcement bars, and can be used in calculations and design of buildings and structures.


Introduction
Since the end of the 20th century, dynamic methods for studying the state of buildings and structures have been actively developed.The essence of the methods is to assess the state of building structures by the features of dynamic (frequency) properties, including those after seismic impact.The frequencies and forms of modal analysis play the role of properties of the stiffness of buildings and structures.Methods using frequency analysis make it possible to determine the presence of defects in load-bearing structures, as well as the degree of their danger for further operation.These methods are based on field experimental studies, including the use of FE software systems [1].Full-scale use of FE analysis requires methods that allow modal analysis at certain points of time, taking into account damage in load-bearing structures.At the same time, it is necessary to select adequate deformation diagrams and verify material models.In such software packages as ANSYS, LS-DYNA, etc., the diagrams presented in [2] can be used and joint operation of concrete and reinforcement using rod (for reinforcement) and solid (for concrete) finite elements can be modelled [3].
The purpose of this work is to study the effect of local damage to reinforced concrete buildings during seismic impact on the degradation of properties of natural frequencies and assess the nature of the destruction of the structure during an earthquake by conducting numerical studies.

Materials and methods
The study used LS-DYNA finite element software complex which implements nonlinear, static and dynamic methods.
Solving problems in the nonlinear formulation of the effect, it is more rational to use methods that implement explicit schemes for integrating the equations of motion of the system.To simulate the process of earthquake impact on buildings, this study uses explicit methods for integrating equations of motion.Explicit methods use recurrent relationships that express displacements, velocities, and accelerations at a given stage through their values at previous stages.LS-DYNA uses the method of central differences with explicit integration.To determine displacements, the expression with a delay in time is used [4]: (1) Implicit methods of integrating equations of motion are used for modal analysis.LS-DYNA implements the ability to perform a transient calculation, during which an earthquake calculation is performed by explicit methods, with periodic calculation of eigenvalues by implicit methods [5].In this case, changes affecting the eigenvalues are considered: geometry, stresses and forces, contact conditions.This should make it possible to assess the degradation of frequency parameters during an earthquake, and the technical state of buildings.
The design scheme of the building uses direct reinforcement of load-bearing elements.LS-DYNA implements a function that allows to consider direct reinforcement of concrete using solid (for concrete) and rod (for reinforcement) finite elements [6].
For concrete modeling, the Continuous Surface Cap Model (CSCM) [7][8] is used for solid finite elements.The concrete model is a "cap" model with a smooth continuous interface between the fracture surface and the reinforcing "cap".This model was verified and used in the studies [9][10][11][12][13][14][15][16].The general shape of the yield surface in the space of the main stresses is shown in Fig. 1.The damage formula is based on study [7]: Here d is the scalar parameter of a damage, which converts the stress tensor without damage, denoted by σ vp , as the stress tensor with damage, denoted by σ d .Damage parameter d varies from zero (no damage) to 1 (total damage).
To simulate the reinforcement, an ideally elastic-plastic Prandle model with an initial modulus of elasticity of E= 2.1•10 5 MPa is used, the yield strength is taken equal to σ t =245 MPa, the ultimate plastic deformations ε plastic = 0.1.
The object of study is a two-storey building of the frame structural arrangement (Fig. 2).Overall dimensions in the plan are 6 x 6 x 6.6 (h) m.Slabs are cast-in-situ reinforced concrete beams.Thickness of slabs is 20 cm, crossbars and columns are 40x40 cm in section.
Longitudinal The calculation was made on a stiffness basis, taking into account physical, geometric and constructive nonlinearities.Seismic impact is set in the form of 2 component accelerograms, normalized by 8 points on the MSK-64 scale [12].The accelerogram is shown in Fig. 3

Results
On fig.5-12 shows the results of the calculation in the LS-DYNA software package.Tables 1 show the frequencies of the three main forms of natural oscillations at given time points, which were determined during and after the earthquake.
Analysis of the results shows that during an earthquake with an intensity of 8 points there is a significant (up to 20%) decrease in the frequencies of natural oscillations (Table 1), there is a destruction of concrete at the base of the columns and at the joints of the columns with the crossbars (Fig. 6).At the base of the columns, there is an increase in plastic deformations and failure of the longitudinal reinforcement.Fig. 7 shows a graph of the growth in the number of failed elements as a percentage over time.Fig. 7 shows a graph of the change in the first 3 frequencies of the natural oscillations of the building during an 8-point earthquake.Fig. 8 shows a graph of the damage accumulation of the construction, and fig.8 shows a graph of the damage accumulation in one finite element of the column.Fig. 9 shows graphs of the roof displacement of the building.Fig. 11 shows the stress at the base of the column.When designing the building for a 9-point seismic impact, the floor slab collapses.The nature of the destruction is shown in Fig. 12. Fig. 7. Graph of natural frequencies change during an 8-point earthquake.

Discussion
Analysis of the results shows that during an intense earthquake, significant damage occurs in the load-bearing elements of the building, which can be assessed through numerical or field studies.The proposed approach can be used in assessing the consequences of earthquakes, including for an informed decision on the possibility of further operation of damaged buildings and structures.
5. The approach to assess the degree of damage proposed in the article makes it possible to evaluate the state of the load-bearing elements of buildings and structures by analyzing the dynamic properties in the process of a nonlinear dynamic calculation.
6.The LS-DYNA software package allows to study the impact of local damage to loadbearing structures on the dynamic properties during an earthquake, considering the direct reinforcement of concrete with reinforcement bars, and can be used in research, calculations and design of load-bearing reinforced concrete elements of buildings and structures in non-linear dynamic model.This work was financially supported by the Ministry of Science and Higher Education of Russian Federation (grant # 075-15-2021-686).Tests were carried out using research equipment of The Head Regional Shared Research Facilities of the Moscow State University of Civil Engineering

Fig. 1 .
Fig. 1.The general shape of the concrete model yield surface in the principal stress space.

Fig. 6 .
Fig. 6.The intensity of plastic deformations at different times (a -5 sec, b -6.5sec, c -11.3sec) (1 on the scale corresponds to the complete exhaustion of the bearing capacity of the element).

Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .
Fig. 9.The graph of damage accumulation in the element of the upper part of the column during an 8point earthquake.(1 on the scale corresponds to the complete exhaustion of the bearing capacity of the element)