Influence of the Vertical Earthquake Component on the Shear Vibration of Buildings on Sliding Foundations

The paper studied the vibrations of the buildings on the sliding foundation with dry friction under the action of real earthquakes at the intensity of 8 and 9 on the MSK-64 scale. It was developed a unique algorithm for calculating the displacements, velocities, accelerations, and shear forces resulting from the simultaneous action of the horizontal and vertical components of the seismogram record. It was studied four-story and nine-story buildings under the set of the three earthquake records. It was shown that the use of a sliding foundation does not always lead to a significant reduction in the shear force on the building floors and that the vertical component of the seismic effect has a significant influence on the shear vibration of the building.


Introduction
In recent decades, various measures have been taken in countries worldwide to protect the buildings and structures from the unfortunate effects of the strong earthquakes [1][2][3][4][5][6]. One effective method of the seismic isolation is the sliding foundation using a layer of fluoroplastic [2,3].
In [4], a study of seismic isolation in the pipeline system in the form of Teflon interlayer was carried out. In the numerical studies, the seismic isolation was modeled as dry friction, but the dry friction model is replaced by an "Ideal elastoplastic body" model during implementation.
In [5], a study of seismic vibrations of the multi-story building with sliding support is given. The condition of the joint motion used the force of inertia of the whole building. In [6], the results of studies of the high-rise building with an application of the seismic isolation in the form of the steel-laminated rubber bearings are given.
In [11], the results of measured processing accelerations during the earthquakes of the three buildings constructed in Almaty in 1989 are described. These three buildings had the same above-ground foundation (9-storey large-panel in 158-series houses) but the different foundations: conventional strip, kinematic, and supports with fluoroplastic layers. The contact surfaces with fluoroplastic have inclined planes. On August 16, 2014, an earthquake was recorded with its epicentre located 41 km east of Almaty. Underground shocks were felt in Almaty at 4 -5 points on MSK-64 scale. The maximum values of spectral coefficient β for seismically isolated buildings at the 9th-floor level are less than those for the analogue building: for buildings with fluoroplastic layers -on 11%, for buildings with kinematic foundations -on 63%. When selecting the optimal value of dry friction coefficient to reduce the earthquake effect on buildings with sliding foundations, when the building is modeled as a mass with a spring, the mass of the grillage is not considered. In [7][8][9], the results of studies of the spatial structures of buildings on the action of real earthquakes using the LS-DYNA software package are given. In many studies, when studying the motion of the material point on a rough plane, it is assumed that the material point is always subjected to the dry friction force against the motion, with multiple stops and slips not considered.
In nonlinear problems of seismodynamics of the underground structures, the model of the dry friction in the pipeline-soil interaction is used [12]. In [13], a finite-difference approximation of the equation of the motion of the rod with external dry friction was built, and an algorithm of the solution was constructed; later, this algorithm was used in [14][15][16].
In [17], vertical vibrations of the buildings with distributed and concentrated parameters are considered. In [18][19][20] seismodynamics of the underground pipelines under the harmonic and real impacts are numerically studied.
In this work, an algorithm similar to the one described in [13] will be used to calculate the buildings with sliding foundations under the action of the seismic waves of real earthquakes.

Materials and methods of solution
Let the horizontal and vertical motions of the building foundation be given as a seismogram of the real earthquake. Let us assume that the lower part of the building foundation acquires the same motions, and the upper part of the foundation or the grillage is separated by a twolayer fluoroplastic [2,3]. As a model of the interaction of the two fluoroplastic layers in the horizontal direction, we will take the Coulomb dry friction model; we will assume that they are absolutely rigidly connected in the vertical direction. In this formulation of the problem, the ground deformation is not considered; it is considered a rigid fixed of the lower foundation with the base soil.
The calculation scheme of the multi-storey building considering seismic isolation with dry friction according to the normative document takes the following form (figure 1). The building is represented by a one-dimensional shear model with concentrated masses and inertialess elastic bonds M is the diagonal matrix of masses, the masses are located at the floor levels,  In the joint motion, the displacement 0 u is determined by equality (2), and the equation of the motion of the mass 1 M takes the following form In this case 1 true solution is the solution in the first formulation because the applied dry friction force makes a move in different directions and hence the unknown force is smaller than the limiting value of the dry friction force, i.e., the masses of the lower and upper foundations move together without sliding at this step in time. If relative velocities in the second and third problem statements have the same signs, then the true solution is the solution of the problem in the statement in which the relative velocity in absolute value is the smallest because the dry friction force is directed against the relative motion. All three problems are solved by the Newmark method [1]; the digitized earthquake seismogram is approximated by a linear function in the recording step interval when the time step of the approximation is smaller than the earthquake recording step.

Results and Discussion
Let us discuss the results of the calculations using the following examples. Let the characteristics of the 4 and 9 storey buildings are given and the seismograms of the following earthquakes: 1.  Hz. In the numerical solution with dry friction problems, regardless of the choice of explicit or implicit finite-difference scheme, the time step must be chosen to ensure sufficient accuracy. In our example calculations, the time step was 0.0001 s.
The results are shown below in the form of displacements and shear forces figures for the four-story and nine-story buildings. Figure 2 shows the results of the calculations of the changing of displacements in time of the upper and lower parts of the foundation of the four-story building under the earthquake №1, taking into account the horizontal impact (a) and simultaneously the horizontal and vertical impacts (b) of the real earthquake records. The slip times from the beginning of the seismic wave impact process are equal to: 3.67 s (figure 2, a) and 3.01 s ( figure 2, b). The occurrence of the first slip is associated with a change in the direction of the motion of the lower foundation. The transition from sliding with dry friction to comotion and vice versa occurs many times, depending on the conditions discussed above. By the end of the process, the residual shear equals 0.0028 m for horizontal impact and 0.0019 m if vertical motion is considered.   (figure 3, b). The transition from slip with dry friction to co-motion and vice versa occurs many times, depending on the conditions discussed above. By the end of the process, the residual displacement is 0.0011 m for horizontal impact and 0.0014 m if vertical motion is considered. Figures 2 and 3 show the strong influence of the vertical impact on the horizontal oscillation process of the buildings. In [3], based on experimental results, it is written that fluoroplastic can reduce the load by up to four times. A computational experiment has shown that in some cases, fluoroplastic in the sliding foundation can reduce the maximum load from an earthquake by up to 9 times. Increasing the dry friction coefficient reduces the efficiency of the sliding foundation. For weak earthquakes, buildings do not feel the presence of the sliding foundation [11]. The vibrations of the buildings in earthquakes №2 and №3 are also strongly influenced by the vertical component of seismic impact.
In earthquake №2 decreases of the maximum shear force value by the factor of 8.9 and 1.1 times with respect to the case without sliding foundation, while accounting for the vertical motion decreased by the factor of 4.2 and 1.6 times respectively four-story and the nine-story buildings. In earthquake №3, the maximum shear value force decreases by a factor of 3.2 and 3.8 concerning the case of no sliding foundation, while accounting for vertical motion decreased by 3.0 and 3.5 times, respectively four-story and the nine-story buildings.

Conclusions
An algorithm for the numerical solution of the problem of the building's vibration with sliding foundations using the Coulomb dry friction model with consideration of slip stops under the simultaneous effect of horizontal and vertical components of the seismogram of the real earthquakes is presented. It is shown using the example of the four-story and the nine-story buildings based on the recordings of the three earthquakes that the use of a sliding foundation does not always lead to multiple reductions in the shear force the consideration of the vertical component of the seismogram significantly influences the shear vibration process of the building. When using the sliding foundation to reduce the effect of an earthquake, it is necessary, based on the construction site, to select seismogram records that are close in terms of the prevailing frequencies and to carry out calculations using the algorithm described above while simultaneously affecting the horizontal and vertical components of the seismogram.