Introduction. According to the reliability theory, the level of design loads is determined by limiting the structure failure probability. If the permissible probability is set, then the structures are equally reliable, regardless of the damage caused by transition to the ultimate limit state. It is obvious, that the reliability of structures, the failure of which leads to significant damage, should be higher. The authors propose to limit not the probability of failure, but the risks themselves. To do this, it is necessary to develop an equation that will allow one to determine the acceptable risk depending on the degree of seismic structure strengthening and on the probability of f seismic action in a given area.

Aim. The purpose of the study is to evaluate the design seismic accelerations that cause the same expected damage (risk) to designed structures.

Materials and methods. This paper estimates the magnitude of design peak accelerations for assessing the seismic resistance of buildings and structures. For this purpose, a condition for limiting seismic risk is written down, from which the desired design acceleration is obtained. To calculate the risk, it is necessary to define a vulnerability function and probability density function of seismic action. The vulnerability function is presented basing on literature data, depending on the action intensity and the degree of structure strengthening. To determine the probability density function of a seismic action, it is necessary to construct a density function of peak accelerations at the construction site. This function was constructed using a well-known result of the distribution of peak accelerations within a single intensity according to the Weibull law. Further, summation over integer intensity numbers was replaced by corresponding integration over continuous intensity.

Results. A density function of peak accelerations for the construction site was obtained. The risk value was obtained as an integral over continuous action intensity. An example of calculating the density function was performed for the Ust-Kamchatsk region. The calculation results demonstrate the acceptability of the proposed approach.

Conclusions. The calculated accelerations depend significantly on the acceptable risk and make the results of designing quite clear.

Introduction. To ensure the functioning of earthquake-affected areas, it is very important to ensure the earthquake resistance of bridges and other transport infrastructure facilities. One of the characteristic damages of bridges during an earthquake is the dumping of superstructures from their supports. The discharge of superstructures is caused by different ground conditions under the supports along the length of the structure, which in turn leads to different perturbations of the supports during an earthquake.

Materials and methods. The assessment of the course of the supports of a girder split bridge in Uzbekistan is considered. The ground conditions under the supports of the bridge vary along the length of the structure. It is necessary to calculate the motion of the bearing of bridge in order to exclude the dumping of the superstructure. The authors propose to evaluate the course of the supporting part of the bridge in the development of the linear spectral theory of structures calculating for earthquake resistance, using the mathematical apparatus of the theory of random functions. Usually, the correlation coefficients e are determined under the assumption of stationary processes. In this case, each process is defined by its own spectral density. Various cases of correlation were considered: the correlation between fluctuations of the points of the daytime surface under the supports, the correlation between fluctuations of the top of neighboring supports and the correlation of fluctuations of the base and top of the supports. For all cases, the corresponding correlation coefficients were calculated, which were further taken into account when calculating the motion of the supports.

Results. For the considered three-span bridge crossing with different structural solutions of supports and different soil conditions at their base, the following displacements of the base, relative displacements of the top of the supports, and total displacements were calculated. An assessment of the mutual offsets from above has been performed. Displacements of the top of the channel supports, taking into account the heterogeneity of the acceleration field under the supports. The mutual displacements of the base points in the absence of correlation of vibrations, the mutual displacements of the top of the support in the absence of correlation of vibrations are determined, and the mutual displacements of the base points and the mutual displacements of the top of the support are calculated taking into account the correlation for the case of white noise.

Discussion. An analysis of the results showed that the displacements of the top of the supports are quite significant. Even estimating the displacements as the root of the sum of the squares gives the motion of the supporting part more than 50 cm. Based on the results obtained, recommendations were made for designers, firstly, on the need to develop the support head, and, secondly, on additional reinforcement of the support to compensate for the moment from the vertical reaction of the superstructure with a large motion of the support part, and, thirdly, on the expediency of installing stoppers to eliminate excessive displacements.

Conclusions. The calculation of the motion of the supporting part is a necessary element in assessing the seismic resistance of bridges. The motion of the support part depends on the relative displacements of the top of the supports and the heterogeneity of the acceleration field along the length of the bridge. When designing bridges with different supports and with different ground conditions under the supports, it is advisable to combine the spans into a chain, for example, by using rubber support parts.

Introduction. The paper considers an example of calculating a structure for which taking into account the method of summing up forces by vibration modes becomes essential. The structure under consideration is a building with a rigid structural scheme and a relatively flexible upper floor divided into two parts. The foundation of the structure is not rocky. The calculation takes into account the base soil flexibility and damping heterogeneity.

Aim. To evaluate the impact of different methods of accounting for the summation of forces by vibration modes.

Materials and methods. To estimate the design force, four summation methods are considered: the sum of the modules, the root of the sum of the squares, the Gupta formula, and taking into account the correlation, using the A.A. Petrov formula.

Results. The calculation results differ significantly, and the normative method, the root of the sum of the squares, gives a non-conservative force estimate. The normative calculated forces turn out to be 20 % less than similar forces calculated taking into account the correlation of the modes, using the A.A. Petrov formula.

Conclusions. The method of accounting for the correlation of vibration modes significantly affects the calculation results. It is recommended to include the requirement to take into account the correlation of the vibration modes in the normative calculations.

Introduction. The paper considers engineering methods of calculating adaptive seismic isolation systems. Yet, such methods are absent and this hinders the use of such systems in earthquake engineering. Materials and methods. A seismically isolated system with double support types is considered, including relatively rigid supports with limited bearing capacity and flexible seismic isolating supports. It is believed, that at the moment of the rigid connections switch-off their potential energy is converted into kinetic energy of the superstructure. The displacement of seismic isolation supports is estimated under the assumption that their maximum deformation energy is equal to the obtained kinetic energy. A more accurate calculation is considered, taking into account additional kinematic excitation from an earthquake. Results. Calculation formulas for selecting the parameters of double support of adaptive seismic isolation and formulas for estimating the forces and displacements in the elements of seismic isolating supports have been obtained. An example of calculating a road bridge in a highly seismic region of Dagestan is given. Discussion. Although the seismic isolation system under consideration is essentially nonlinear, its calculation can be performed quite simply, without using complex software packages. The authors of the paper used the main laws of classical mechanics and MathCad or MatLab tools. The work was carried out at the St. Petersburg University of Railway Transport and the Limited Liability Company Sroykompleks-5

Introduction. The effectiveness of investments in antiseismic reinforcement of a building is largely determined by seismic risk, while social risk accounting is an important part of seismic hazard assessment and is subject to serious analysis.

Aim. The article provides an assessment of the seismic risk and economic efficiency of anti-seismic strengthening of residential buildings taking into account human losses and situational seismicity at the construction site.

Materials and methods. The risk assessment methodology proposed by L.V. Kantorovich was used to conduct the research. The risk assessment was performed using known in the literature payoff matrices and data on the recurrence of earthquakes in regions with different situational seismicity. The study developed a method for assessing human losses based on the economic characteristics of residential development.

Results. The ratio of social and commercial seismic risk was estimated. In the conditions of the Russian Federation, social risk is about 5-10% of the economic risk. It is shown that the required degree of strengthening of buildings and structures for the conditions of the Russian Federation is largely determined by economic losses.

Concusions. With an increase in insurance payments for loss of life to 50 million rubles, social losses begin to determine the degree of anti-seismic strengthening.

Introduction. Recently, there has been an increase in the intensity of extreme weather conditions, which leads to an increased probability of flood risk. At the same time, according to the estimate of the Ministry of Emergency Situations, the annual damage from floods in the Russian Federation averages approximately 50 billion rubles. In order to reduce the possible degree of vulnerability of buildings from these natural disasters, various methods of protection are used in world practice, one of which is the use of pontoon foundations and telescopic piles.

Materials and methods. To assess the effectiveness of building construction on pontoons in areas of possible waterlogging of territories, the risk theory in the formulation of Academician                      L.V. Kantorovich, who considers risk as a mathematical expectation of damage, was used. Statistical material on flood damages in the Far East in the Amur River basin was used for risk assessment. The payment matrix of damages was made on the basis of these data. Repeatability was assessed by using the available material on floods in the same region. It is noted that the dependence of investments in floating buildings on the possible height of their surfacing is significantly nonlinear.

Results. The paper presents the results of evaluation of the effectiveness of the proposed solution. It is shown that the device of floating bases for residential buildings with a floating height of 4–6 m is an effective means for construction in waterlogging zones. It is noted that in spite of rather large initial investments the further increase of the surfacing height is relatively small. The economic benefit of the adopted solution is revealed.

The question of setting the seismic design input on high rise buildings is considered. The existing approaches to accounting for increased responsibility of high rise buildings in Russia are described. The proposal to reduce the probability of an acceptable building failure in proportion to the number of floors and Guideline proposals to increase the reliability factor and using maps of general seismic zoning are analyzed. The main disadvantages of methods described are indicated. It is shown that the current regulatory documents in the field of earthquake engineering do not provide the same reliability of designed structures in general and high-rise buildings in particular. The influence of seismic dangers in according with seismic zoning maps on the reliability of the designed objects is noted. An approach to generating the design input based on the permissible probability of its exceeding is considered using the example of five five-storey buildings and one 25-storey buildings. The probability of the admissible damage value included in the normative calculations is estimated. An estimate of the allowable failure probability on the value of acceptable damage (risk) is proposed under the assumption of a normal distribution of damage caused by earthquake. It is shown that the allowable failure probability decreases with decreasing acceptable damage only in the area of small damages. An approach to the assignment of seismic action based on an assessment of seismic risk has been formulated. The system of design coefficients used to calculate seismic loads on high-rise buildings is analyzed. It is noted that along with an increase in the design level of seismic acceleration, it is necessary to increase the coefficient, taking into account the low damping of high-rise buildings oscillations. At the same time, it is possible to significantly reduce the reduction coefficient by regulating the strains between the building floors.

The application of the response spectra method for extended damped systems with a point support on the base is considered. The support excitations can be different. External and internal damping is taken into account, it being assumed proportional i.e. admitting an exact or approximate decomposition of motion according to the undamped system modes. It is noted that the presence of external damping leads to additional system perturbation, i.e. to additional members appearance in the right part of motion equations. It is shown that well-known formulas of the response spectra method for systems on a rigid platform are kept intact, but the vector of load projections on the directions of generalized coordinates is modified taking into account the influence of support displacements on generalized loads in the system.

The authors present their view of the problem, with the frame of the discussion devoted to equations of seismic oscillations taking into account damping. The oscillation equations taking into account external and internal damping are written down. The cases of homogeneous and inhomogeneous acceleration field of the day surface along the length of the structure are considered.

The main problem of modeling statistical seismic vibrations is correct input accelerogram setting. The analysis of the known seismic input models showed the erroneousness of using them in analyzing seismically isolated systems. These statistical models allow one to obtain either reliable accelerations or reliable displacements. However, complicated input models do not quite correspond to real earthquakes. Energy characteristics were not considered at all in the problems of accelerogram statistical modeling. A new model of seismic input, including a random pulse, has been considered. Three parameters has been added to the input model: the magnitude Мw , the epicentral distance R, and the moment when the pulse appears. Varying these parameters within the set limits allows one to adjust additional input characteristics. An example of the proposed process is given.

Summary:  At present, calculation of structures using earthquake accelerograms is becoming more and more important. However, the problem of modeling design accelerograms causes debates and contains various suggestions based on contradictory assumptions. Therefore, the value of complicated and time-consuming dynamic calculations can be rather dubious. The paper deals with some parameters of modeling design accelerograms. It is shown that the widely used method of input generation, developed by A.A. Dolgaya, gives rather rough approximations of synthetic accelerograms to real 
ones due to the insufficient number of model parameters.

Article is devoted to the 150 anniversary of  «Buildings» chair and the 120 anniversary of «Theoretical Mechanics chair» of FGBOU VPO PGUPS Imperator of Alexander I. The main scientific achievements in the field of aseismic construction executed at university are presented.

Results of researches are analyzed in a chronological order. The assessment of settlement and theoretical researches in the field of seismic stability of buildings and various transport constructions, including bridges is given. It is featured questions of studying of seismoisolation. The contribution of scientists of university to development of aseismic construction is shown.

Practical value of the presented analysis consists in possibility of its use by the beginning young scientists, engineers dealing with issues of construction of buildings and constructions in regions of high seismic activity.

The reduction factor K1 included in the formulas of Guidelines for calculating seismic loads for bridge piers has been evaluated. The dependence of the reduction factor on the ratio of pier reinforcement has been obtained. To this aim the non-linear finite element calculation of the pier under monotonous loading has been carried out and the elastic limit and damaging displacement has been obtained.

The comments consider the possibility of using the response-spectra method (RSM) taking damping into account in seismic calculations. The results of the paper under consideration are fair if the influence of damping on the shape modes has not been taken into consideration. At present there exists a variant of RSM, which takes into account the damping influence on shape modes. This variant of RSM has been described in the comments. In this case complex eigenvalues and eigenvectors of the damped system are calculated. The imaginary part of the eigenvalues determines the modal frequency, and the real part of the eigenvalues determines the modal damping. Seismic forces are calculated taking into account the amendments caused by modal damping.

The paper describes a statistical method taking into account the velocity pulse and the energy characteristics of the reference impact. An example of generation of an impact is also given, which includes the impulse component.

The problem of calculating harmonic oscillations of a linear damped system with inhomogeneous damping is considered. Along with the solution available in literature, which requires the inversion of the matrix that determines the eigenvalues of the undamped system, the authors propose a new solution that does not require the inversion of the above mentioned matrix, but requires the inversion of the system damping matrix. The cases of hysteretic, viscous and mixed damping are considered. It is shown that the known solution gives an error near the resonance. At the resonance point, the result is not defined at all, and near the resonance it may be incorrect. An example of building the amplitude-frequency characteristic of a system with two mass tuned dynamic dampers and three peaks in the amplitude-frequency characteristic is given. The proposed formulas for calculating displacements are convenient for constructing the amplitude-frequency characteristics of damped systems with viscous, hysteretic and mixed types of damping.

Seismic isolation of a road bridge in the seismic region of Uzbekistan with a design seismicity of 9 degrees on the MSK-scale is considered. A special feature of the bridge is the large mass of the spans, which is almost 50 times greater than the reduced pier mass. At a first glance, this rules out tuning the isolation to mass dynamic regime. The rigidity of seismic isolation is determined basing on the condition of limiting the mutual displacement of spans. Even at this limitation, seismic loads were reduced by approximately two times, i.e. by one intensity number. A more effective isolation method has been proposed, that is, when one span is rigidly fixed to the pier and the other is seismically isolated, and then the isolation can be tuned, ensuring the operation of the isolated span as the mass damper. In this case, the load on one of the piers is reduced by more than two times and that on the adjacent pier by more than four times.

The aim of the research is to assess the influence of the method accounting damping on the mass damper efficiency and selecting their optimal parameters for seismic impact. Particular attention is paid to the behavior of mass dampers of large mass. The steady-state harmonic oscillations of a damped system with a mass damper are considered. The design model is represented by two masses connected by springs. The mass damper optimal parameters are determined by minimizing the peak displacements on the amplitude-frequency characteristic. Three methods of damping accounting are considered: hysteresis in the structure and damper spring, equivalent viscous in the structure and in the damper spring, as well as hysteresis in the structure and in the damper spring and additional viscous in the damper spring. It has been established that when the mass of the damper is less than 10% of the mass of the protected structure, the method of accounting for energy losses in the structure and in the mass damper spring is not essential. With a damping mass of more than 50% of the structure weight, the method of damping accounting becomes fundamental important. It is noted that for highly damped systems, the efficiency of mass damper is noticeably reduced. Apparently, the ambiguity and inconsistency of recommendations on the damping selection for mass dampers available in the literature is associated with a significant influence of the way system damping is taken into account. It is recommended to consider the damping in the damper spring and in the structure separately. In a structure, damping does not usually exceed 15% of the critical one and the way it is taken into account is not important, but special dampers are used in the damper spring and their consideration must correspond to the actual damper loading diagram.

Three characteristics are proposed to describe the excitation differences under supports for calculating multi-supported structures. These characteristics include the non-synchronicity indicator t0, the incoherence indicator r and the non-uniformly scaled indicator z. The first characteristic shows the time shift between the excitations, the second characteristic shows the difference in the excitation frequency composition, and the third one shows the difference in the action amplitudes. To estimate the nonsynchrony index, two approaches have been proposed – maximizing the correlation function and minimizing the norm of the excitation difference. The proposed parameters fully characterize the acceleration inhomogeneity field at the construction site. However, they affect the system response in different ways. Small deviations in the index of excitation amplitudes z weakly reduce the structure response. As for the excitation incoherence, this effect can bring the system closer or away from resonance.