Mechanical safety of eccentrically compressed RC-column in the event of emergency impact

. The safety analysis of the outermost column of the 8 m high building stylobate was performed under eccentric compression. The bending moments act in two main planes. Modeling of the emergency based on physical and geometrical nonlinear calculation in dynamic statement is proposed. The dynamic overload is modeled simultaneously by longitudinal force and bending moments. It is assumed that such an emergency impact is caused by an impacting body falling from a certain height onto the deformable slab in the vicinity of the slab-column connection zone. An impulse load modeled the load, the magnitude and duration of which were approximated based on the conservation of momentum law. As a result, an area of structural strength, limited by the boundary surface under dynamic loads, was plotted. The mechanical safety consisted in assessing the risk of material damage associated with the collapse of a part of the building structure when the column collapsed.


Introduction
Shock dynamic overloading of the outermost columns occurs during new construction, resumption of work on unfinished structures, and in emergency situations.For facilities under construction, as shown in Figure 1, dynamic overloading of columns can occur due to local settlement of the foundation soil under the adjacent more heavily loaded columns.In addition, during construction, parts of structures or products may have fallen from a great height onto the stylobate.Man-made accidents related to mechanical damage are not uncommon nowadays.The cause of such damage can be a collision of machines with the columns or the effect of an explosion.In general, dynamic impacts are studied in the scientific literature [1,2].Dynamic impacts, including overloads transmitted through adjacent structures as well as directional impacts [3][4][5], can cause in columns, in addition to the destruction of the structural material, the phenomenon of loss of stability.This phenomenon is most commonly observed in flexible reinforced concrete columns.In addition to mechanical damage, columns are deformed by other dangerous influences related to corrosion, high-temperature heating, ground subsidence, horizontal shocks, etc.In order to prevent the failure of the entire support system, researchers are developing various optimization and safety improvement methods [6][7][8][9].At the same time, it is possible to study the reliability of a structure with damage [10].In general, the risk of progressive failure of a structural system or its individual elements is prevented [11][12][13][14].Increasing the resistance of newly designed structures to accidents [15,16], as well as structures with accumulated damage [17][18][19], optimization of structural, technological and organizational solutions [20][21][22], taking into account the behavior of columns under combined impacts [23] -all these studies are aimed at improving the mechanical safety of buildings.Therefore, the topic of this work can be considered relevant.

Methods
Based on the implicit integration of the following system of equations describing the motion, the problem of dynamics is solved by the finite element method: . (3) Here   K is the designation of the tangent matrix of finite element stiffness,   M ,

 
С are nodal mass matrix, and damping matrix.The indices at the matrices have notations: r -rebars,  -for the matrix of small deformations, G -for the geometric matrix.The values () yt , () yt , () yt , are, respectively, displacements, accelerations, and velocities; () Ft this is the vector of the time-varying nodal external load,  is the constructional damping coefficient.
A model of the column with initial material parameters is shown in Fig. 2. For concrete, 8-node elements were used, and their deformations are described according to the DP model, which considers concrete dilatation at a relative stress of 0.3, as well as a compressive strain unloading function, which has four characteristic points shown in Fig. 3.The kinematic scheme of the column constraints assumes a rigid pinch in the foundation and an articulated connection to the slab at the top.This connection is chosen to provide a safety margin because the concrete of the articulation node often collapses under dynamic overloads and the bending moment is not transmitted to the column.The spatial rebars deforming in a elastic-plastic diagram.The connection between concrete and reinforcement is considered to be ensured under both static and dynamic loading.The loading in time is carried out using the coefficient of dynamic overloading, which characterizes the dynamic effect associated with the change in the structural system under the emergency impact.It is conventionally considered that the bending moment is transmitted to the column with some delay in relation to the longitudinal force.
The numerical research program includes: -construction of a conditional volume bounded by the surface of ultimate strength, determination of ranges of values of maximum crash loads; -risk assessment of the accident consequences on a concrete example.
The limiting surface will be built along the axes ult N , , where these values denote the limiting dynamic longitudinal force and the limiting dynamic bending moments, which can be resisted by the column.In order to achieve the objective, we will carry out the following series of calculations, implying the following steps of the search: 1) Longitudinal force 0, 0 2) Bending in one plane: Stiffnesses of concrete and reinforcement were corrected by coefficients taking into account the duration of loading according to SP 63.13330 by introducing coefficients to the modulus of elasticity of these materials.The transition to the static equivalent of the dynamic load, additionally loading the structure as a result of an accidental impact, was carried out on the basis of the classical definition of the force momentum.An impact is considered inelastic if most of the kinetic energy is converted into the internal energy of the deformed system: 21 () where M is the mass of the impact indenting body, V its velocity at the moment of impact, t  is the time of dynamical overloading, 21 , PP are levels of force loading after the impact and before the impact, respectively.
In the dynamic formulation, the physical nonlinearity was considered by the Newton-Raphson method, where, for each moment of time, an internal cycle of accounting for the elastic-plastic behavior of materials was performed.The maximum number of iterations of this cycle was taken to be 30, with a force-weighted discrepancy of less than 0.001 to obtain a solution.
The dynamic loading sequence was carried out according to the graphs shown in Fig. 4. The process of safety assessment is carried out in the following stages: -construction of the limit surface limiting the strength area of the column under dynamic loading and under the selected material parameters; -replication of operating conditions of the column; -estimation of failure probability of the column on the basis of hypothesis about normal distribution of random values of mechanical characteristics of materials and parameters of emergency impact.

Results
The evaluation of the mechanical reliability of the column is performed, the model of which is shown in Fig. 2. The mechanical characteristics of the concrete correspond to the class B20 according to SP 63.13330, and the characteristics of the reinforcement -design resistance (yield stress) .
The results of ultimate surface calculation are given in table 1.In table 1, the lines corresponding to the characteristic points of the boundary (ultimate) surface are highlighted in green, the lines corresponding to the auxiliary points, which allowed to complete the search, are highlighted in orange.In order to simulate the load in the normal operation mode, the static calculation of a simplified scheme (Fig. 5, a) with a grid of 6x6 columns and column height of 8m, with the full design load on the road surface of 60kN/m2 (taking into account the presence of cars on it) was performed.The calculation showed the typical internal forces for the part of the column marked with a circle (Fig. 5, b).

Fig. 5. Simulation of the normal operation mode of the column
To illustrate the calculation results, some data on the stress-strain state of the column in central and eccentric compression are given.Figure 6 shows diagrams of stresses in reinforcement and concrete versus time.The horizontal axis shows the number of integration steps.One step corresponds to a time of 0.02 seconds.From Fig. 6, it can be seen that for the selected column structure, at the maximum dynamic overload, the stresses The stress-strain state of the column elements under biaxial eccentric compression at the time moment corresponding to the peak level of loading is shown in Fig. 7.The load on the column corresponds to pos.18 1) the risk R, will be different, at the same time by its value it is possible to judge the degree of danger of the emergency impact.From this calculation it can be seen that for this mode of operation there is a risk of material loss in an emergency situation.In order to minimize this risk, it is possible to propose the use of a square angle-shaped pylon instead of a square corner column.Such a design will significantly reduce the risk of emergencies outside the contour shown in Fig. 1 in case of a fall of the mass from the height h v onto the floor slab, and it will also allow to extend the range of permissible load weights.However, such a solution also requires economic justification or cost optimization.

Discussions
The proposed approach can be used for safety assessment both for key elements and for structural systems as a whole.At the same time, it is possible to model different scenarios of emergencies.Formation of the basis of ultimate strength surfaces under dynamic influences for the most common shapes of column cross-sections and a set of standardized dimensions of their cross-sections will allow to avoid unacceptable risks of emergency situations.1.The method for quantitative assessment of emergency risks is proposed, which redefines the mechanical safety of the structure in case of failure of a reinforced concrete building column.Combined dynamic overloading by longitudinal force and bending moment is considered as an emergency.
2. The search method for the construction of an ultimatum surface has been developed based on data obtained during the calculation of a volume finite element model in the dynamic formulation, taking into account physical and geometric nonlinearity.The formation of the strength area with the help of such a surface allows to identify the degree of danger of the emergency and to determine the limit values of the emergency impact parameters in the space of permissible risks.
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 object of unfinished construction with 8m long columns as part of the frames

Fig. 2 .Fig. 3 .
Fig. 2. Initial data: design diagram of the column (a); fragment of the bearing system with the area S of the impact, bounded by the curve mnk (b); volume model of the column (c)

Fig. 4 .
Fig. 4. Dynamic loading mode of the column: for a longitudinal force of 10 6 N, (a) and for a bending moment of any value (b)

Fig. 6 .
Fig. 6.Changes of stresses in time for center-compressed column: Mises stresses (a) for reinforcing steel; minimum principal stresses (b) for concrete

Fig. 7 .
Fig. 7. Stresses in the elements of the column under biaxial eccentric compression: equivalent stresses in the reinforcement cage (a), (b), the principal compressive stresses in the concrete (c), (d) The geometric interpretation of the ultimate surface is shown in Fig. 8.In Fig. 9, the point  ( ;; ху N ММ ) denotes one of the possible stress states of the column in the operational mode (when the full value of the load on the slab equal to 60 kN/m is realized).The point   (

Fig. 9
Fig.9 The boundary (ultimate) surface for column dynamic strength If the point is above the boundary surface or exceeds its projection on the horizontal axes, the strength condition of the column is not fulfilled.Therefore, the force margin of safety can be determined as follows:,

Table 1 .
Finding of limiting surface for estimation of column strength under dynamic loads from free falling body with mass of 0.5 tons in Table 1 (t=1.87c). .5, (at different combination of the variables h v , V, (table