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. 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

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.

It is shown that in many cases while calculating bridge spans it is necessary to take into account the soil-structure interaction. If the pier is higher than 5 m, energy dissipation for the main oscillation mode is determined substantially by energy losses, caused by wave propagation in the soil base, thereby the dynamic ratio drops significantly. Furthermore, if a bridge span is shorter than 55 m and the base modulus of deformation Eo < 30 MPa, the force in the piers can be determined by the second mode, for which the superstructure and piers oscillate in antiphase. For bridges with spans over 80 m long at Eo > 30 MPa, seismic forces for spans and piers are determined by the first oscillation mode. 

An analytical solution of the motion equation of nonlinear seismically isolated systems on the phase plane is obtained. The cases of systems with increasing and decreasing stiffness are considered. A typical example of a system with increasing rigidity is a system with restrictors. Systems with decreasing stiffness include well-known types of seismic isolation with kinematic supports of A.V. Kurzanov and Yu.D. Cherepinsky. For systems of the first type, the motion trajectories on the phase plane are always limited, but with strong displacement restrictions, an increase in accelerations takes place. The proposed solution makes it possible to analyze the dependence of the acceleration growth on the displacement limitation. Solutions of the second type can be unstable, which is clearly seen on the phase trajectories. This requires an analysis of their stability when designing seismic isolation.