A Method of Determination of Dominant Vibration Periods Values for Nonhomogeneous Multilayer Ground Sites

This paper is presenting development of a method for determination of dominating periods (frequencies) of vibrations of heterogeneous, multilayer foundations. By the method of wave mechanics precise complex transcendent equations of frequencies have been obtained for a foundation of two, three and four layers. To obtain such equations for a foundation having more than four layers is practically impossible. To avoid obtaining such complex frequency equations and to solve such equations using special computer programs a simplified method has been developed in this paper. The main point of the presented method is in successive reduction of a multilayer foundation to two-layer one for which depending upon ratios of amplitudes and periods of vibrations of layers, were computed roots of transcendent equations for more than 5000 combinations of these ratios have been tabulated, extrapolation formula has been derived. The procedure designed for determining values of T01 periods of vibrations for any thickness of n layers is performed in the following sequence. First on the basis of data obtained during well-boring or by other geotechnical methods, according to values of Hk thickness, shear waves vsk velocities, shear moduli Gk, and densities ρk of layers are calculated all periods of each layer (as of homogeneous) by the formula Tk=4Hk/vsk. Next it is regarded that the foundation consists of only two upper layers with their H1, H2 and T1 and T2. By a special table and a formula the vibration period 2 1 01 T − is calculated. Then two upper layers are regarded as one layer of 2 1 H H + thickness and 2 1 01 T − period, and it considered with the third layer of H3 thickness and T3 period. In similar manner by ratio values 2 1 H H + and H3 and 2 1 01 T − and T3 a the period 3 1 01 T − of conditional sandwich thickness. This process proceeds for all layers. The obtained at the last stage the value of the period n 1 01 T − is assumed as a sought value of dominated period of the entire thickness of n layers. The method is illustrated be examples and estimation of its error is given.

T − is calculated. Then two upper layers are regarded as one layer of 2 1 H H + thickness and 2 1 01 T − period, and it considered with the third layer of H 3 thickness and T 3 period. In similar manner by ratio values 2 1 H H + and H 3 and 2 1 01 T − and T 3 a the period 3 1 01 T − of conditional sandwich thickness. This process proceeds for all layers. The obtained at the last stage the value of the period

Introduction
Fundamental characteristic of compound vibratory motion of ground in earthquakes is a spectrum of their frequencies (periods) depending upon local geological conditions of a building site. They usually have complex geological structure involving conglomerate of silt layers, sand, and gravel comparatively late period (alluvial-deluvial deposits) over more young stratifications. Layers of upper alluvial grounds are much more loose and complex than underlying ground. More dense lower ground is called "ground foundation", and upper layers of sedimentations -"superficial layer" or "near-surface thickness". Thus, the basic singularity of alluvial-deluvial grounds is the presence there of complex structures composed of materials having different mechanical properties. In earthquakes this structure has a great influence on behavior of vibration motion of the upper layer of thickness, on which the foundation of a structure rests. It is known that seismic waves in reaching the Earth surface generate in it surface seismic waves. Surface waves propagation velocities in upper layers are less than in lower deep layers. Therefore, according to refraction and reflection rules seismic transversal waves will fall on layers of near-surface thickness of the Earth at almost right angle and inside of the thickness will take place their multiple refraction and reflection. As a result of these phenomena in the Earth's upper layer a more continuous vibration process of definite vibration periods is generated conditioned physical-and-mechanical characteristics and thicknesses all layers of stratum. These periods of vibrations are assumed to refer to as dominant periods of ground vibrations.
Seismic load on structures is of dynamic behavior, hence maximum level of the effect occurs when values of ground vibration and surface structures' free vibrations periods coincide (resonance) or differ slightly. In addition the analysis of earthquake consequences shows that the character and peculiarities of damages of buildings and 58 A Method of Determination of Dominant Vibration Periods Values for Nonhomogeneous Multilayer Ground Sites structures at the time of earthquakes differ depending on ground conditions. On loose grounds during earthquakes considerable soil settlement, inclination and even overturning of structures are observed and on bedrock ground -cracks in most components of structures, large relative displacements and relative deformations through the entire body of the structure. Thus, ground conditions not only have an influence on kinematic characteristics of seismic load but also essentially change the character of the damage of structures. According to the building norms of earthquake-proof structures building sites in all countries are divided into a number (4-6) of categories -from the most dense to the most loose soils. We believe that in the capacity of integral characteristic while soils are divided into categories according to seismic properties along with other parameters it is reasonable to assume a value of dominant period T 01 , since physical-and-mechanical characteristics of all layers' thicknesses affect on its formation, as is presented below. A version of such classification of grounds is presented in Table 1 where data have been taken from Earthquake-proof Building Code of the Republic of Armenia [1].
According to [1] in order to avoid resonance phenomena it is necessary to meet the following conditions: T 01 >1.5T 1 or 1.5T 1 <T 01 where T 1 is the period of the first mode of free vibrations of surface structures, T 01 -is the first dominant period of ground thickness.
Therefore, while designing new buildings and structures it is necessary reliably forecast not only maximum value of the ground acceleration, but also the value of dominant period of the ground vibration during earthquake.
Reliable data on dominant periods in earthquake can be obtained by registration of acceleration during severe earthquake through its spectral (Fourier) analysis.
Taking into consideration a wide variety of litho logic columns' structures in building sites when designing new building and structures it is difficult to find similar site on which at least one moderate earthquake was registered. Today the most widespread ways for defining values of dominant periods of vibrations are:  by recording microvibrations on the upper surface of the thickness (building site excavation) and its spectral analysis,  by solving wave equations of multilayer continuum with respective thicknesses, elasticity and density characteristics established as a consequence of engineering-and-geological survey (well boring) and laboratory research. This paper concerns development of precise and simplified methods designed for defining dominant periods of multilayer foundation vibration by solving wave equations and comparative analysis of results obtained by applying different methods.

Determination of Free Vibrations Periods of Nonhomogeneous Foundations
Near-surface no homogeneous strata is considered as continuum composed of n layers with their physical-and-mechanical characteristics: densities ρ k , shear module G k, and thicknesses H k (Fig.1.) We hold that no homogeneous surface layer of the total thickness H rests on the "ground of foundation". When vertical propagation of transverse wave there is only one horizontal component u(x,t), which inside each layer u k (x,t) must satisfy wave equation. Taking into consideration the fact that viscous properties of the medium have little influence on the periods of free vibrations, they can be ignored and as a wave equation we get [2]  For further simplification of the problem, let us numerate the layers starting from the top layer, take the origin of coordinates on the earth's surface and assume designations: where р i is sought-for angular frequency i-th mode of free vibrations of the surface non-homogeneous multilayer bedding , u k (x,t) is shear strain of k-th layer of i-th mode. Substituting (3) in (2) for a partial solution of ( ) where k k 2 i 2 ki G p ρ = λ (5) Solution of (4) is sought-for in the form To determine 2n А ki and В ki factors and angular frequency р we get the following two boundary conditions and 2n-2 equality conditions for shear strains and transverse forces on levels of the layers division planes, that is k = 1, 2,..., n-1 Substituting (6) in boundary conditions (7) and condition of compatibility (8) for 2n unknown coefficients А ki and В ki , we have Since the set of equations (9) relative to 2n unknown coefficients А ki and В ki is homogeneous, then their unique non-trivial solution takes place only when the determinant of 2n order, built up of unknown coefficients from the set of equations (9), is equal to zero. This exactly is the desired complex transcendent partial solution for determination of unknown frequancy р i . In case of double-layer foundation it will be a determinant of 4-th order, in case of a sandwich -6-th order, and in case of a six-layer -12-th. They having been developed, we get When n=2 To reduce computational load and make use of frequency (10), (11), and (12) when there is a good many layers it is possible the whole system in advance to reduce to equivalent two-three-or-four-layer systems of reduced (average) shear waves characteristics velocities sk v , densities k ρ , and multilayer continuum k H and make use of respective transcendental (11), (12). It is recommended to calculate the average parameters by formulas [3] , As is seen from (14), characteristics of k-th layer of the reduced system, calculated by that equation, comprises real characteristics of alternating one another j-m number of layers from the sequential number j to the number m. The alternating j-m layers are chosen in such a way, that their real characteristics be closely different one from another. It means that different reduced layers can comprise different quantity of real layers.

Simplified Technique for Determining Periods of Multilayer Foundation
As is seen from the foregoing, determination of free vibrations periods is impossible without using special computer programs. Below a simplified method is stated for determining the value of the basic period T o1 of the multilayer bedding free vibration without expansion and use of a 2n determinant of the system (9). We have made an attempt to solve this problem by sequential reducing of a multilayer system (Fig.1) to a double-layer one (Fig.2).
where T 1 and T 2 are periods of the fundamental tone of free vibrations of the first and second layers, respectively, on the assumption of their independent existence as one-layer foundation, v sl and v s2 are shear wave velocities.
Taking these designations into consideration (10) can be written in the following form where Т 0i is the period of free vibrations of the entire double-layer system. Designating To cover all practicaly possible cases of the double-layer bedding roots of (18) (for the first mode of vibrations) for various α are shown in Fig.3.
Values of constants a and b for various α are presented in Table 2. Аt that for T 1 /T 2 <1 it is necessary to take values a 1 и b 1 from the table, for 1<T 1 /T 2 <1 -values a 2 и b 2 , and for  Table 1 by linear interpretation. We assume a=1, b=0 for T 1 /T 2 >25. The procedure used for determining values of fundamental period of vibration for any width composed of n layers is performed in the following sequence. At first, by boring data or other geotechnical techniques, on the basis of Н k layers thicknesses, velocities v sk of shear waves or shear moduli G k and densities ρ k, periods of free vibrations of each layer (as a homogeneous medium) are determined by a formula Next, at first it is considered that double-layer bedding consists of only two top layers with their H 1 and H 2, T 1 and T 2 . According to the ratio T 1 /T 2 for the given ratio H 1 /H 2 according to Table 2 values of parameters a and b are determined, and by the formula (20) the period of free vibrations of the conventional double-layer bedding is calculated (from the first two layers). This period is denoted by T 01 1-2 . Then, the two top layers are considered as one equivalent layer of H 1 +H 2 width and by already calculated period T 01 1-2 and it, then, is considered with the third layer of H 3 thickness and T 3 period.
Similarly according to the ratio T 01 1-2 /T 3 and (H 1 +H 2 )/H 3 from the same Table 2 new values of a and b parameters are determined, and by the same formula (20) ) the period of free vibrations of the conventional three-layer bedding is calculated and denoted T 01 1-3 . This process is proceeded for all layers. The value T 01 1-n of the period obtained at the last stage is the desired value of fundamental period of free vibrations of the entire thickness of n layers. In an another form (with the help of scaling graphs) such an approach was employed in [4]. If at any k-th stage it turns out that T 01 1-k /T k > 25, then in the capacity of T 01 1-k the following is assumed If amongst layers there is a very thin one having a sequential number k, it can be combined with the next k+1 layer of the reduced thickness k H and velocity sk v In this case the number of layers decreases by a unit.

Examples
Let us consider two typical examples to illustrate the method. As the first example let us determine the first period of a six-layer section for grounds of Gyumri City, Armenia, with the following parameters [5,6].   Comparative analysis of fundamental period Т 01 values for seven different sections obtained by transcendental equations and suggested method are presented in Table 3.
The above tabulated data prove that the suggested simplified technique provides satisfactory accuracy in calculation of the first period of a nonhomogeneous multilayer foundation, eliminating expansion of high order determinants, for deriving complicated transcendental equations and their solutions using special computer programs and computer.

High Order Periods of Anon homogeneous Foundation
In homogeneous near-surface layer the nature of high mode vibrations differs from the fundamental mode by odd numbers 3, 5, 7 … 2n-1. А natural question can be raisedwhat is the range of these ratios variation in case of no homogeneous sections. In the general case to give a reasonable answer is quite difficult. Therefore, we confine the study of this problem theto double-layer section with a large range of characteristic layers' variation (Fig.2). We assume that the obtained results somehow can also be used for multilayer sections.
Besides, we have limited our research within Т 02 и Т 03 values of periods of vibrations second and third modes. As an initial transcendental equation for determining Т 01 , Т 02 , and Т 03 (18) was used for large variations of  Table 4 respectively when shear waves velocity v S2 of the second-lower layer is constant and equal to 800m/s, and velocities v S1 of the first-higher layer varies within 50m/s to 700m/s in different ratios of the first layer H 1 thickness to the total H thickness. Similar results, when v S1 is constant and equal to 800m/s, and v S2 varies from 50m/s to 700m/s are shown in Table 5. In both Tables real ratios of periods for variant close to 3  and 5 are separated by gray tone. It is seen from Table 4 that the maximum difference оf Т 01 /Т 02 and Т 01 /Т 03 , respectively from 3 and 5, in increasing direction occurs when under the first heavy stiff layer of v S1 =800m/s loose layer of v S2 =50÷150m/s and small thickness. The foregoing suggests that in such sections values of periods of the second and third modes of vibrations will be considerably smaller than periods of vibration's fundamental mode, therefore, their effect, as a very high-frequency component of influence, can be neglected.
And vice versa, from Table 4 it is clear that the maximum difference Т 01 /Т 02 and Т 01 /Т 03 from 3 and 5 to the decrease occurs when on a stiff heavy layer of v S2 =800m/s loose layer is located of v S1 =50÷150m/s and small thickness (Н 1 /Н=0.1÷0.15). In this case Т 02 and Т 03 will have comparable with Т 01 values and, therefore, their effect in the total influence cannot be neglected.
As for the value of the period Т 01 of the fundamental mode, then for its calculation respective correcting coefficients k 1 and k 2 have been introduced for H=30 meter foundation, which in many countries have been included in seismic building code. These values are tabulated in Tables 6 and 7, k 1 corresponding to the case when v S2 =800m/s, and v S1 varies within 25÷700m/s, and k 2 corresponding to the case when v S1 =800m/s, and v S2 varies within 25÷700m/s. For hatched variants in Table 6 the influence on T01 exerted by the top layer can be neglected, and for variants in Table 6 the influence on T 01 exerted by the bottom layer can be neglected.
Computations have shown that in total thickness of multilayer foundation Н=30m, values of periods at v S2 =800m/s and v S2 =1000m/s for Н 1 /Н>0.4 slightly differ from each other, therefore, the value v S =1000m/s can be counted as a lower limit for bed rock in determining category of no homogeneous foundation on the basis of seismic properties and using Т 01 values [1].
Values of Т 01 ,Т 02 ,Т 03 have been experimentally found for a number of Tokyo City area sections [8]. According to Okomoto data central part of Tokyo City territory has been divided by a one-kilometer-net-shaped map. On that map dominant frequencies are indicated at points of lines intersections, established experimentally.