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Investigation of block foundations resting on soil–rock and rock–rock media under coupled vibrations
Accepted Manuscript Investigation of block foundations under coupled vibrations resting on soil-rock and rock-rock medium Ms. Renuka Darshyamkar, M.Tech. Student, Mr. Ankesh Kumar, Ph.D. Student, Dr. Bappaditya Manna, Assistant Professor PII:
S1674-7755(16)30271-2
DOI:
10.1016/j.jrmge.2016.09.006
Reference:
JRMGE 311
To appear in:
Journal of Rock Mechanics and Geotechnical Engineering
Received Date: 16 June 2016 Revised Date:
25 September 2016
Accepted Date: 29 September 2016
Please cite this article as: Darshyamkar R, Kumar A, Manna B, Investigation of block foundations under coupled vibrations resting on soil-rock and rock-rock medium, Journal of Rock Mechanics and Geotechnical Engineering (2017), doi: 10.1016/j.jrmge.2016.09.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Title of the Manuscript: Investigation of block foundations under coupled vibrations resting on soil-rock and rock-rock medium.
Author 1:
Address : Department of Civil Engineering Indian Institute of Technology, Delhi Hauz Khas, New Delhi – 110016, India
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E-mail: [email protected]
Author 2: Name : Mr. Ankesh Kumar* Affiliation : Ph.D. Student
Address : Department of Civil Engineering
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Indian Institute of Technology, Delhi
Hauz Khas, New Delhi – 110016, India
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E-mail: [email protected]
Author 3:
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Name : Dr. Bappaditya Manna
Affiliation : Assistant Professor Address : Department of Civil Engineering Indian Institute of Technology, Delhi
Hauz Khas, New Delhi – 110016, India
E-mail: [email protected] Phone: +91-11-26591211
*
Corresponding Author
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Affiliation : M.Tech. Student
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Name : Ms. Renuka Darshyamkar
ACCEPTED MANUSCRIPT Abstract In the present study, the dynamic response of block foundations of different equivalent radius to mass (Ro/m) ratios under coupled mode of vibration is investigated for various homogeneous and
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layered systems. The frequency dependent stiffness and damping of foundation resting on homogeneous soil and rocks are determined using the half-space theory (Velestos et. al., 1971). The dynamic response characteristics of foundation resting on the layered system considering rock-rock
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combination is evaluated using the theory proposed by Kausel (1974) with the help of finite element program with transmitting boundaries. Frequencies versus amplitude responses of block
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foundation are obtained for both translational and rotational motion. A new methodology is proposed for determination of dynamic response of block foundations resting on soil-rock and weathered rock –rock system in the form of equations and graphs. The variation of dimensionless
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natural frequency and dimensionless resonant amplitude with shear wave velocity ratio are investigated for different thickness of top soil/weathered rock layer. The dynamic behaviour of block foundations are also analysed for different rock-rock systems by considering sandstone, shale
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and limestone underlain by basalt rock. The variation of the stiffness, damping and amplitudes of
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block foundations with frequency are shown in this study for various rock-rock combinations. In the analysis two resonant peaks are observed at two different frequencies for both translational and rotational motion. It is observed that dimensionless resonant amplitudes decrease and natural frequencies increase with increase in shear wave velocity ratio. Finally the parametric study is performed for block foundations of size 4m × 3m × 2m and 8m × 5m × 2m by using generalised graphs. The variation of natural frequency and peak displacement amplitude are also studied for different top layer thickness and eccentric moments.
ACCEPTED MANUSCRIPT Key words : Rock-Rock system; block foundation; coupled vibration; homogeneous medium; equivalent radius to mass ratio; half-space theory
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1. INTRODUCTION The geology of earth consists of all types of soil, rocks and minerals belonging to different geological period. Due to its diversity in nature, geotechnical engineers and geologist are
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exposed to completely new set of problems. They encounter the problem related to nonhomogeneity because of the presence of discontinuities such as bedding planes of varying strength,
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fissures, joints and faults. Due to the presence of such geological structures, the studies related to the foundations constructed on such surfaces become most important. Normally block foundations are required to support machines, machine tools and heavy equipments on such non homogeneous
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surfaces. These types of machine foundations are provided in heavy industries like nuclear power plant, hydro power plant, petrochemical industry etc. to ensure the satisfactory operation of the machines which have wide range of operating speeds, dynamic loads and operating conditions. To
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design the machine foundation successfully it becomes crucial to carry out careful engineering
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analysis and the response of block foundation to the dynamic loads. The block foundations are subjected to either static load, or combination of static and dynamic load repetitively over a long period of time. This load leads to the generation of dynamic forces such as balanced and unbalanced forces due to the operation of rotating type or reciprocating type machinery. In case of the coupled vibrations, horizontal and rocking motions are generated simultaneously under the action of unbalanced horizontal dynamic forces of rotating type of machines.
ACCEPTED MANUSCRIPT In the Indian scenario most of the heavy construction is under progress on the Deccan trap rocks. These rocks are mostly consists of weathered basalt underlain by fresh basalt. Apart from this, the rocks which are slightly weathered at present may become weaker in future due to the
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environmental action. Many researchers (Xiao et al., 2014a, Xiao et al., 2014b, Xiao et al., 2015, Xiao and Desai, 2016, and Xiao et al., 2016) investigated the characteristics of the weathered rock and granular soils related to their dilatancy, particle breakage and transitional behavior. It is found
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from the research that the density and pressure have great influences on the strength and stressstrain behaviors of weathered rocks and granular soils. These materials exhibits strain hardening,
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postpeak strain softening, volumetric contraction, and expansion with a range of densities and pressures. Hence the shear wave velocity, stiffness and damping values are greatly influenced by the state of the material which varies from hard rock to highly weathered rock or granular soils.
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Therefore the foundations for heavy machinery and structures on such type of ground conditions must be studied thoroughly for the future safety. In general these layered combinations can be classified as (i) soil-rock / weathered rock - rock system and (ii) rock-rock system. The dynamic
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analysis of block foundations resting on these layered systems is very complex. A very few studies
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are available in the literature regarding the methodology of determination of the natural frequencies and resonant amplitudes of block foundation under coupled horizontal and rocking vibration. Hence in the present study, an attempt is made to analyze the effect of various soil-rock and rockrock foundation system on dynamic response of block foundations of different mass and equivalent radius under coupled mode of vibrations. The dynamic response characteristics of foundations are studied by using the theory proposed by Velestos and Wei (1971) and Kausel (1974) for homogenous and layered system respectively. The variations of natural frequency and resonant amplitude with shear wave velocity are investigated for different top layer thickness. The dynamic
ACCEPTED MANUSCRIPT behaviour of block foundations is analyzed for different rock-rock systems by considering sandstone, shale and limestone underlain by basalt rock. The variation of the stiffness, damping and amplitudes of foundation with frequency are shown in this study for various rock-rock
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combinations. The variation of natural frequencies and peak displacement amplitudes are also studied for different top layer thickness and eccentric moments. Finally the parametric study is carried out to determine the natural frequency and resonant amplitude for block foundations of size
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2. THEORETICAL INVESTIGATION
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4 m × 3 m × 2 m and 8 m × 5 m × 2 m using generalized relations proposed in the study.
The classical work on vibratory response of foundations was carried out by Lamb (1904). Subsequently Reissner (1936) developed an analytical solution for periodic vertical displacement at
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the centre of the circular loaded area with elastic half-space mathematical model. Arnold et al. (1955) extended the elastic half-space theory to include other modes of vibrations using rigid base type contact stress distribution and weighted average displacement condition. Hsieh (1962)
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obtained the expressions for the frequency dependent stiffness and geometrical damping in terms of
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displacement functions. Lysmer (1965) and Lysmer and Richart (1966) showed that vertical vibration of a rigid circular footing can be represented by a simplified analog consisting of a simple damped oscillator with physical parameter that gives good agreement with the response curve obtained from the elastic half-space theory. It is well established now that the dynamic response of foundations depends on several factors namely, size and shape of the foundation, depth of embedment, soil profile, soil properties, frequency of loading, and mode of vibration (Richart et al.1970 and Gazetas and Stokoe, 1991). The finite element approach has been used to calculate the stiffness and damping of embedded foundations by a number of investigators (Urlich and
ACCEPTED MANUSCRIPT Kuhlemeyer, 1973; Kausel and Ushijima, 1979). Beredugo and Novak (1972) and Novak and Sachs (1973) presented the stiffness and damping for the coupled and torsional mode of vibration of embedded footings and it was found that both the stiffness and damping values are increased due to
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the embedment of footing. Veletsos and Verbic (1973) and Veletsos and Nair (1974) proposed stiffness and damping parameters of embedded footing for different modes of vibrations. Further, Warburton (1957), Hadjian and Luco (1977) and Kagawa and Kraft (1981) studied the dynamic
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response of foundations considering the heterogeneities of the soil.
The dynamic impedance of block foundation resting on homogenous deposit considered as halfspace
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is evaluated using the theory proposed by Veletsos and Wei (1971). Furthermore, the theory proposed by Kausel (1974) are used for the layered system analysis in which top layer is treated as stratum and bottom layer is considered as halfspace.
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2.1 Homogenous Medium
In the present study an approximate solution proposed by Veletsos and Wei (1971) is used to study the steady state response of rigid circular disk of infinitesimal thickness supported at the surface of a
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non dissipative, homogeneous, linear elastic halfspace. The weight of the disk is assumed to be
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negligible. The exciting forces considered includes a harmonically varying lateral force directed along the x-axis, and harmonically varying moment acting about the y-axis. The system has following assumptions: (i) the normal component of the contact pressure is assumed to be zero due to horizontal force (ii) the horizontal or shearing components of the interface pressure are assumed to be zero due to overturning moment. Thus, it permits the horizontal translational motion of the disk to be evaluated independently. The surrounding region of disk is assumed to be stress free.
ACCEPTED MANUSCRIPT In order to compute the coupled displacements, the amplitudes of the steady-state horizontal displacement and rotation of the disk, u and φ, stated as u f11 + ig11 φ r = f + ig o 21 21
f12 + ig12 u st f 22 + ig 22 φst ro
(1)
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where f’s and g’s are dimensionless flexibility coefficients; ust = horizontal displacement by static
2−v P 8 G ro
φ st =
3(1 − v ) M 8 G ro 3
(2)
(3)
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u st =
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horizontal force, P; φst = rotation of disk due to static moment, M; where,
in which ν = Poisson’s ratio for material of half-space; G =shear modulus of elasticity; ro = equivalent radius. The flexibility coefficients, f and g are functions of ν and of the dimensionless frequency parameter (ao), where ao = ω ro/Vs, ω = circular frequency of the excitation, Vs= shear
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waves velocity in the halfspace.
The force amplitudes, P and M, corresponding to the displacement amplitudes, u and φ, may be expressed as:
k 2 1 + ia o c 2 1 u 2−v ( k 2 2 + ia o c 2 2 ) φ ro 3(1 − v )
(4)
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P k 1 1 + ia o c 1 1 M = K x k 2 1 + ia o c 2 1 r o
in which the k’s and c’s are dimensionless coefficients depending on ao and ro, i =√-1; where ;Kx = P/ust and Kφ = M/φst . Specifically, Kx represents the horizontal static force necessary to produce a unit horizontal displacement of the disk and Kφ represents the static moment necessary to rotate the disk by a unit amount with no restriction on the value of the horizontal displacement. The quantities k will be referred to as the stiffness coefficients, and the quantities c will be referred to as the damping coefficients.
ACCEPTED MANUSCRIPT If the off-diagonal terms of the flexibility matrix are neglected, f12, f21, g12, and g21 are assumed to be zero, the force-displacement relationship for the disk is obtained from the following equation: P ( k 1 + ia o c 1 ) K x M = 0
0 ( k 2 + ia o c 2 ) K φ
u φ
(5)
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The stiffness function for horizontal and rocking motion are calculated from equation (5). The above methodology is used to investigate the dynamic response of block foundation resting on
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homogeneous medium by using the computer program DYNA 5 (Novak, 1972). 2.2 Layered Medium
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The analysis is carried out using FEM based formulation and computer program developed by Kausel and Ushijima (1979). A three-dimensional axisymmetric finite element model with transmitting boundaries is used to model a rigid circular foundation resting on a homogeneous rock
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stratum of finite depth resting upon a much stiffer rock-like material. The transmitting boundaries used were developed by Kausel (1974) to the three dimensional case with axisymmetric geometry. Kausel and Ushijima (1979) presented a numerical method for the dynamic analysis of
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axisymmetric foundation resting on viscoelastic soil layers over rock of infinite horizontal extent.
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The solution technique developed by Waas and Lysmer (1965) was extended for the analysis of axisymmetric systems subjected to arbitrary non axisymmetric loading using Fourier expansion method.
For the plane symmetric displacement modes, the forced horizontal displacement is referred as swaying and the forced rotation about horizontal axis is referred as rocking. The stiffness functions can be given as K s = K so (k1 + iao c1 )(1 + 2i β h1 )
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Kψ = Kψ o (k 2 + iao c2 )(1 + 2i β h 2 )
(7)
ACCEPTED MANUSCRIPT Where Kso , KΨo are real parts of the stiffness functions in the static case, ao is dimensionless frequency, k1, k2, c1, c2 are the stiffness and damping coefficients and β is internal damping coefficient.
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Using the methodology proposed by different researchers the analyses are carried out to determine the dynamic response of block foundation in soil-rock system, weathered rock-rock system and rock-rock system. The methodology involved in this study is incorporated in the computer program
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DYNA 5 (Novak, 1972). This program is used to present the dynamic behaviour of block
constants for layered system.
3. RESULTS AND DISCUSSIONS
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foundation in terms of frequency response curves for displacement, stiffness, and damping
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The analyses are carried out for three eccentric moments i.e. mee = 0.45, 0.366, 0.278 N-m for both homogeneous and layered medium. The results are then plotted in dimensionless form using
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following equations,
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Dimensionless translational amplitude,
Dimensionless rotational amplitude,
Dimensionless frequency,
Ax =
x.m me e
Aψ =
Iψ
ao =
me eZ e
ωro vs
(8)
(9)
(10)
where, x is the amplitude of system, m is the total mass of the footing and machine, IΨ is moment of inertia, me is the mass of eccentric rotating part of oscillator, e is eccentricity of rotating part of
ACCEPTED MANUSCRIPT oscillator, ψo is rotational moment, ω is the angular operating frequency, ro is the radius of foundation and vs is the shear wave velocity. Two different ratios of equivalent radius with mass of block foundation i.e. (Ro/m)1= 2.85×10-5 and
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(Ro/m)1= 1.74×10-5 are considered to study the effect of mass and size of foundation on the dynamic response of block foundation, where Ro is equivalent radius of footing in meter and m is total mass of the footing and machine in kN. The properties of soil and rocks considered for the
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analysis is shown in Table 1 (Zhao, 2010).
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3.1 Homogeneous Medium
In case of homogenous medium, the theory proposed by Velestos and Wei (1971) has been used for the analysis block foundation resting on homogeneous soil and four type of homogeneous rocks i.e.
k h ,k r = k o =
c k Soil ( ao = 0 )
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(12)
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ch ,cr = co =
k k Soil ( ao = 0 )
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sandstone, shale, limestone and basalt. The stiffness and damping constant are normalized as
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where ko is normalized stiffness constant (where ko is kh, and kr for horizontal stiffness and rocking stiffness respectively), co is normalized damping constant (where co is ch, and cr for horizontal damping and rocking damping respectively), k and c are stiffness and damping constant of layered system and ksoil(ao= 0) is stiffness of soil at ao= 0. The variation of normalised horizontal stiffness (kh) with dimensionless frequency (ao) for all type of homogeneous soil and rocks are shown in Fig 1 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively. The variation of normalized rocking stiffness (kr) with dimensionless natural frequency (ao) are shown in
ACCEPTED MANUSCRIPT Fig 2 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively. It has been observed that both horizontal and rocking normalized stiffness (kh) are found maximum for the basalt among all geological materials considered here i.e. limestone, shale, sandstone and soil. Hence, foundation resting on basalt has
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higher resistance for both horizontal and rocking displacement as compared to other rocks and soil. The variation of normalized horizontal damping (ch) with dimensionless frequency (ao) are shown in Fig 3 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively. The variation of normalized rocking damping
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(cr) with dimensionless frequency (ao) also shown in Fig 4 (a) and (b) for (Ro/m)1 and (Ro/m)2
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respectively. It is noted that the normalized damping increases significantly as dimensionless frequency decreases at low frequency condition for both horizontal and rocking motion because of the conversion of frequency-independent material damping (β) to the equivalent viscous damping coefficient (c) as c = 2β/ω (where ω = circular frequency). It is also observed that the normalized
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horizontal and rocking damping for soil is much lower than the homogeneous rock. The variation of dimensionless translational amplitude (Ax) and dimensionless rotational amplitude (Aψ) with dimensionless frequency (ao) are shown in Fig 5 (a) and 5 (b) respectively for both (Ro/m)1
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and (Ro/m)2. It is observed from these figures that there are two dimensionless amplitude peaks at
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two different dimensionless frequencies. For (Ro/m)1, the maximum values of dimensionless frequency is lower and maximum dimensionless amplitude is higher than the (Ro/m)2 for both horizontal and rocking motion. The maximum value of dimensionless amplitude is higher for soil as compared to the rocks for both translational and rotational case. It can also be seen that the dimensionless frequency (ao) corresponding to maximum value of dimensionless amplitude for soil is lower than the rocks for both translational and rotational motion.
ACCEPTED MANUSCRIPT 3.2 Layered Medium By using the theory proposed by Kausel (1974), the analyses are carried out to study the dynamic response of block foundations resting on layered soil-rock and rock-rock system. In the case of
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layered medium, the dimensionless parameter which appreciably influences the dynamic response of block foundation is the H/B ratio, where H is depth of top layer and B is width of foundation. In the present study three different H/B ratios, i.e. 0.5, 1.0 and 1.5 are considered. The results of dynamic
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impedance of foundations, pertaining to all possible translational and rotational mode of vibration are plotted in the form of dimensionless parameters of stiffness, damping and amplitude. The ratio of
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stiffness and damping of the layered system with the stiffness of homogeneous igneous rock is taken to obtain the normalized stiffness and damping of layered system k1 k Basalt ( ao = 0 )
c h ,c r = c o =
c1 k Basalt ( ao = 0 )
(13)
(14)
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k h ,k r = k o =
where, ko and co are normalized parameters for stiffness and damping of layered system, kl and cl are
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stiffness and damping constant of layered system, kbasalt (ao= 0) is the stiffness of basalt rock at the
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zero dimensionless frequency, kh and kr are normalized stiffness parameters for horizontal and rocking motion respectively and ch and cr are normalized damping parameter for horizontal and rocking motion respectively.
3.2.1 Soil-Rock and Weathered Rock-Rock Systems It is well established that the shear wave velocity values is greatly influenced by the state of the material which varies from hard rock to highly weathered rock or granular soils. Therefore in case of soil-rock and weathered rock-rock systems, analyses are carried out by considering shear wave
ACCEPTED MANUSCRIPT velocity as a important parameter. In the layered system, normally the soil or weathered rock layer is underlain by hard rock and hence the layered medium can be defined by two shear wave velocities i.e. Vs1 and Vs2 (where Vs1 is the shear wave velocity of top layer, and Vs2 is the shear
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wave velocity of bottom halfspace). Therefore in present study the dynamic behaviour of block foundation on layered medium is investigated in the form of shear wave velocity ratio (Vs1/Vs2). Three different shear wave velocity ratios i.e. 0.8, 0.6 and 0.3 are considered in the analyses to
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represent limestone-basalt, shale-basalt and sandstone-basalt system respectively. Then the variation
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of natural frequency and resonant amplitude of translational and rotational mode of vibration are obtained for H/B = 0.5, 1.0 and 1.5. The maximum dimensionless amplitude with its subsequent dimensionless natural frequency is obtained for all shear wave velocity ratios. The trend lines are plotted for two cases (i) for natural frequency and shear wave velocity (ii) for resonant amplitude and
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shear wave velocity ratios. The shear wave velocity ratio (Vs1/Vs2 ) ≤ 0.5 is normally considered to represent the soil – rock and weathered rock – rock systems (Gupta and Rao, 1998). Therefore the
line.
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extrapolation is done for shear wave velocity ratio (Vs1/Vs2 ) up to 0.5 by using the equation of trend
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The variation of dimensionless natural frequency for first peak with shear wave velocity ratio is shown in Fig. 6 (a) for both (Ro/m)1 and (Ro/m)2. Similarly, the variation of dimensionless natural frequency for second peak with shear wave velocity ratio is shown in Fig.6 (b) for both (Ro/m)1 and (Ro/m)2. The trend lines shown in the graph are the variation of first and second natural frequency with shear wave velocity ratios for the coupled vibration. It is observed from the figures that the trend lines shows decrement with the decrease in shear wave velocity ratio. The variation of dimensionless translational amplitude (Ax) with shear wave velocity ratio is shown in Fig 7 (a) and
ACCEPTED MANUSCRIPT (b) for (Ro/m)1 and (Ro/m)2 respectively. Similarly, the variation of dimensionless rotational amplitude (Aψ) with shear wave velocity ratio is shown in Fig 8 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively. It can be seen from these figures that the value of dimensionless natural frequencies are
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higher and dimensionless translational and rotational amplitudes are lower for (Ro/m)2 as compared to (Ro/m)1. It can be seen from the Fig. 7 and 8 that as H/B ratios increase the dimensionless rotational and translational amplitudes decrease. A converging trend is found for the trend lines representing
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resonance frequency and amplitudes for various H/B ratios as the values of Vs1/Vs2 increase.
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Therefore the effect of H/B ratio reduces as the shear wave velocity ratio increases. It is observed that with increase in shear wave velocity ratio the dimensional natural frequencies increase and dimensionless resonant amplitudes decrease. Therefore, the dimensionless natural frequencies are found lower and dimensionless resonant amplitudes are found higher in case of soil-
3.2.2 Rock-Rock System
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rock system than weathered rock-rock system.
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In the present study, the rock-rock combinations considered are limestone-basalt, shale-basalt and sandstone-basalt with shear wave velocity ratio of 0.8, 0.6 and 0.3. The variation of dimensionless
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frequency, dimensionless translational and rotational amplitudes are studied for various H/B ratios and shear wave velocity ratios. The variation of normalized horizontal stiffness (kh) with dimensionless frequency (ao) is shown in Fig 9 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively for sandstone-basalt, shale-basalt and limestone-basalt systems. Similarly the variation of normalized rocking stiffness (kr) with dimensionless frequency (ao) is shown in Fig 10 (a) and (b). It can be seen that normalized horizontal and rocking stiffness values are higher for limestone basalt system than other two rock-rock system
ACCEPTED MANUSCRIPT because of higher shear wave velocity ratio of limestone-basalt system. Also the normalized horizontal and rocking stiffness value of shale-basalt system is more than the sandstone basalt system. It is also observed that the normalized stiffness values decrease with increase in H/B ratios
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approximately up to dimensionless frequency (ao) of 1.5 for both horizontal and rocking motion. The variation of normalized horizontal damping (ch) with dimensionless frequency (ao) is shown in Fig 11 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively for all rock-rock systems. Similarly the
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variation of normalized rocking damping (cr) with dimensionless natural frequency (ao) is shown in
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Fig 12 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively. It is found that both normalized horizontal and rocking damping (ch and cr) values increase as shear wave velocity ratio increases. It is also noted that normalized damping is very low for sandstone-basalt system as compared to other rockrock systems. It can also be seen that the variation of normalized damping (ch and cr) with H/B ratio
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is very less for lower dimensionless frequency, ao < 0.2. It is also observed that the normalized damping increases with increase in the thickness of top layer approximately up to dimensionless frequency (ao) of 1.5 for both horizontal and rocking motion.
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The variations of dimensionless translational amplitude (Ax) and rotational amplitude (Aψ) with
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dimensionless frequency (ao) for sandstone-basalt system are shown in Fig.13 (a) and (b) respectively for both (Ro/m)1 and (Ro/m)2. Similarly, the variations of dimensionless translational and rotational amplitude with dimensionless frequency are shown in Fig 14 (a) and (b) respectively for shale-basalt system. Further, the variations of dimensionless translational and rotational amplitude with dimensionless frequency are shown in Fig 15 (a) and (b) respectively for limestone-basalt system. It is found that in most of the cases the response is dominated by the first resonant peak and the second resonant peak is entirely suppressed for both translational and rotational motion. It is also
ACCEPTED MANUSCRIPT found that the dimensionless natural frequencies increase and the dimensionless resonant amplitudes decrease with increase in shear wave velocity ratios. It is observed that the maximum values of dimensionless frequency is lower and maximum dimensionless amplitude is higher for (Ro/m)1 than
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the (Ro/m)2 for all rock-rock system in case of both horizontal and rocking motion.
4. PARAMETRIC STUDIES
The parametric study is carried out for two block foundations of size 4 m × 3 m × 2 m and 8 m × 5 m
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× 2 m, subjected to coupled horizontal and rocking vibrations. The values of horizontal and
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rotational amplitude corresponding to their natural frequencies are shown in Table 2 and Table 3 for block foundations of size 4 m × 3 m × 2 m and 8 m × 5 m × 2 m respectively. The frequency and amplitude values are presented for two different eccentric moments (mee = 0.028 and 0.045 N-m). The values shown in the tables are obtained with the help of generalized graphs proposed in the
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present study.
The natural frequencies for soil-rock and weathered rock-rock system are found by using Fig 6 (a)
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and (b). Similarly, Fig 7 (a) and 7 (b) are used for determining the values for translational amplitudes and Fig 8 (a) and 8 (b) are used for rotational amplitudes for soil-rock and weathered rock-rock
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system. It is observed from the tables that values of both the natural frequency and maximum amplitude varies with mass and size of block foundations. For soil-rock system, the resonant amplitudes are found higher and the natural frequencies are found lower as compared to other rockrock systems. It is also observed that the values of natural frequencies are higher and resonant amplitudes are lower for limestone-basalt system as compared to all other systems. Therefore the limestone-basalt system can be considered as best foundation medium for block foundation among
ACCEPTED MANUSCRIPT all other systems considered in this study because the limestone-basalt system is suitable for machines with high operating frequencies with low amplitude of vibration.
5. CONCLUSIONS
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In the present study, an effort is made to study the dynamic response of block foundations resting on homogeneous and layered medium under coupled horizontal and rocking vibration. Two different Ro/m ratios are considered to investigate the effect of size and mass of block foundation.
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Analyses are carried out to study the variation of stiffness, damping and amplitude with frequency
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of block foundation for three different medium: (i) homogeneous soil and rocks, (ii) soil/weathered rock-rock system and (iii) rock-rock system. For all the cases, the response is dominated by the first resonant peak and the second peak is entirely suppressed in case of translational motion. However the reverse trends are found in case of rotational motion.
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In the case of homogeneous medium, it is found that the normalized horizontal and rocking stiffness for block foundations on soil are lower than the block foundations on rocks. It is also
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observed that the normalized horizontal and rocking damping for soil is much lower than the homogeneous rock. Therefore, the dimensionless translational and rotational amplitudes for soil are
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found higher than the rocks and the dimensionless natural frequencies for soil are found lower than the rocks.
In the case of layered medium i.e. soil - rock and weathered rock – rock systems, different equations are proposed to calculate the dimensionless natural frequency and resonant amplitudes for both translational and rotational motion in terms of H/B ratios and shear wave velocity ratio. It is observed that the dimensional natural frequencies increase and dimensionless resonant amplitudes decrease with increase in shear waive velocity ratio. Therefore, the dimensionless
ACCEPTED MANUSCRIPT natural frequencies are found higher in case weathered rock-rock system than soil-rock system and the dimensionless resonant amplitudes are found higher in case soil-rock system than the weathered rock-rock system. These parameters are presented in the form of shear wave velocity, as the testing
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for strength parameters of these weathered rocks is very difficult in laboratory. The graphs presented can directly be used by the practicing engineers for estimation of natural frequency and resonant amplitude of block type machine foundations resting on different layered combinations
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like soil-rock and weathered rock-rock systems.
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For rock–rock systems, it is observed that both the normalized stiffness decreases and damping increases with increase in the thickness of top layer approximately up to dimensionless frequency of 1.5 for both horizontal and rocking motion. It is found that both normalized stiffness and damping increases with increase in shear wave velocity ratio. Therefore, the dimensionless resonant
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amplitudes decrease and natural frequencies increase with increase in shear wave velocity ratio for both translational and rotational motion. It is also found that both the dimensionless resonant amplitudes and natural frequencies decrease with increase in H/B ratio.
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The effect (Ro/m) ratio on the frequency – amplitude response is also investigated for homogeneous
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and layered systems. It is observed that the maximum values of dimensionless frequency is lower and maximum dimensionless amplitude is higher for (Ro/m)1 than the (Ro/m)2 for both horizontal and rocking motion.
The parametric study is also carried out for the case of block foundation of size 4 m × 3 m × 2 m and 8 m × 5 m × 2 m, using normalized graphs proposed in the present work. The values of translational and rotational amplitude corresponding to their natural frequencies are presented. It is observed from the tables that the natural frequencies are lower and resonant amplitudes are higher
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values. Therefore, the limestone-basalt system can be considered as good foundation medium for the block foundation because of its high value of natural frequencies and low value of resonant
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amplitudes.
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References 1. Arnold, R. N., Bycroft, G. N., and Warburton, G. B. (1955), Forced Vibrations of a Body on an infinite Elastic Solids, Journal of Applied Mechanics Trans., ASME, Vol. 77, pp.
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391-401.
2. Beredugo,Y.O. and Novak, M. (1972), Coupled Horizontal and Rocking Vibration of Embedded Footings, Canadian geotechnical Journal, Vol. 9, No. 4, pp. 477-97.
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3. Gazetas, G., and Stokoe, K. H. (1991), Free Vibration of Embedded Foundations: Theory versus Experiment, Journal of Geotechnical Engineering, ASCE, Vol. 119, No. 9, pp.
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1382-1401.
4. Gupta, A. S. and Rao, K. S. (1998), Index properties of weathered rocks: interrelationships and applicability, Bull Eng Geol Env, Vol. 57, pp. 161–172. 5. Hadijan, A. H., and Luco, J. E. (1977), On the Importance of Layering on Impedance
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Functions, Proc. 6th WCEE, New Delhi.
6. Hsieh, T. K. (1962), Foundation Vibrations, Proc., Institution of Civil Engineers (U.K.), pp. 211-226.
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7. Kausel, E. (1974), Forced Vibrations of Circular Foundations on Layered Media, PhD Thesis, Massachusetts Institute of Technology.
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8. Kausel, E. and Ushijima, R. (1979). Vertical and Torsional Stiffness of cylindrical Footing, Civil Engineering Department Report R79-6, MIT, Cambridge, Massachusetts.
9. Kagawa, T., and Kraft, L. M. (1981), Machine Foundations on Layered Soil Deposits, Proc., 10th International Conference on Soil Mechanics and Foundation Engineering, Stockholm, Vol. 3, pp. 249-252.
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10. Lamb, H. (1904), On the Propagation of Tremors over the Surface of an Elastic Solid, Philosphical Transactions of the Royal Society, London, Ser. A, Vol. 203, pp. 1-42. 11. Lysmer, J. (1965), Vertical Motion of Rigid Footings, U.S. Army Engg. Waterways
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Experiment Station, Report No. 3, 115, Vicksburg, Misissippi.
12. Lysmer T, and Waas G (1972), Shear Waves in Plane Infinite Structure, J Eng Mech Div ASCE, Vol. 98, pp.85–105
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13. Lysmer, J. and Richart, F. E. Jr. (1966), Dynamic Response of Footings to Vertical Loading, Journal of Soil Mechanics and Foundation Engineering Division, ASCE, Vol.
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92, SM 1, pp. 65-91.
14. Novak, M. and Beredugo, Y.O. (1972), Vertical Vibration of Embedded Footings Journal Soil Mechanics and Foundation Division, ASCE, SM12, pp.1291-1310. 15. Novak, M. and Sachs, K. (1973), Torsional and Coupled Vibrations of Embedded
No.11, p.33.
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Footings International Journal of Earthquake Engineering and Structural Dynamics, Vol.2
16. Richart, F. E., Hall, J. R., and Woods, R. D. (1970), Vibrations of Soils and Foundations,
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Prentice-Hall, Inc., Englewood Cliffs, N.J., pp. 414. 17. Reissner, E. (1936), Stationare, Axialsymmetrische Durch Eine Schut-telnde Masse
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erregto Schwingungen eines Homogenen Elastischen Halbraumes. Ingenieur archive, Berlin, Germany, Vol. 7, No. 6, pp. 381-396.
18. Urlich, C. M. and Kuhlemeyer, R. L. (1973), Coupled Rocking and Lateral Vibrations of Embedded Footings, Canadian Geotechnical Journal, Vol. 10, No. 2, pp. 145-160.
19. Veletsos, A.S. and Wei, Y.T. (1971), Lateral and Rocking Vibration of Footings. Journal Soil Mechanics and Foundation Division, ASCE, SM9, pp. 1227- 1248.
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20. Veletsos, A.S. and Verbic, B. (1973), Vibration of Viscoelastic Foundation, Journal of Earthquake Engineering and Structural Vol. 2, pp. 87-102. 21. Veletsos, A.S. and Nair, V. V. D. (1974), Torsional Vibration of Viscoelastic
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Foundaation. Journal of Geotechnical Division, ASCE, Vol. 100, No. GT3, pp. 225-246. 22. Warburton, G. B. (1957), Forced Vibration of a Body upon an Elastic Stratum, Journal of Applied Mechanics, ASME, Vol. 24, pp. 55-58.
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23. Xiao, Y. and Desai, C. S. (2016), General Stress-Dilatancy Relation for Granular Soils. J. Geotech. Geoenviron. Eng. Vol. 142, No. 4, 02816001.
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24. Xiao, Y., Liu, H., Chen, Y. and Jiang, J. (2014a), Bounding surface model for rockfill materials dependent on density and pressure under triaxial stress conditions. J. Eng. Mech Vol. 140, No. 4, 04014002.
25. Xiao, Y., Liu, H., Chen, Y. and Jiang, J. (2014b), Bounding surface plasticity model
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incorporating the state pressure index for rockfill materials. J. Eng. Mech 140, No. 11, 04014087.
26. Xiao, Y., Liu, H., Ding, X., Chen, Y., Jiang, J. and Zhang, W. (2016), Influence of
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Particle Breakage on Critical State Line of Rockfill Material. Int. J. Geomech., Vol. 16, No. 1, 04015031.
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27. Xiao, Y., Liu, H., Sun, Y., Liu, H. and Chen, Y. (2015), Stress-dilatancy behaviors of coarse granular soils in three-dimensional stress space. Eng. Geol., Vol. 195, pp. 104-110.
28. Zhao, J (2010), Handbook on rock mechanics for civil engineers, Civil and Environmental Engineering, Swiss Federal Institute of Technology, Lausanne.
ACCEPTED MANUSCRIPT FIGURE CAPTION Fig.1. Variation of Normalized Horizontal Stiffness (kh) With Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio and (b) (Ro/m)2 ratio. Fig.2. Variation of Normalized Rocking Stiffness (kr) With Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio and (b) (Ro/m)2 ratio.
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Fig.3. Variation of Normalized Horizontal Damping (ch) With Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio and (b) (Ro/m)2 ratio. Fig.4. Variation of Normalized Rocking Damping (cr) With Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio and (b) (Ro/m)2 ratio.
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Fig.5. Variation of Dimensionless Amplitudes With Dimensionless Frequency (ao) for: (Ro/m)1 and (Ro/m)2 (a) Translational, Ax and (b) Rotational, Aψ
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Fig.6. Variation of Dimensionless Natural Frequency With Shear Wave Velocity Ratio (Vs1/Vs2) for: (Ro/m)1 and (Ro/m)2 (a) First Peak, ao1 and (b) Second Peak, ao2 Fig.7. Variation of Dimensionless Translational Amplitude (Ax) With Shear Wave Velocity Ratio (Vs1/Vs2) for: (a) (Ro/m)1 ratio and (b) (Ro/m)2 ratio. Fig.8. Variation of Dimensionless Rotational Amplitude (Aψ) With Shear Wave Velocity Ratio (Vs1/Vs2) for: (a) (Ro/m)1 ratio and (b) (Ro/m)2 ratio.
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Fig.9. Variation of Normalized Horizontal Stiffness (kh) with Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio, and (b) (Ro/m)2 ratio. Fig.10. Variation of Normalized Rocking Stiffness (kr) with Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio, and (b) (Ro/m)2 ratio.
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Fig.11. Variation of Normalized Horizontal Damping (ch) with Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio, and (b) (Ro/m)2 ratio.
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Fig.12. Variation of Normalized Rocking Damping (cr) with Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio, and (b) (Ro/m)2 ratio. Fig.13. Variation of Dimensionless Amplitude with Dimensionless Frequency (ao) for Sandstone-Basalt System for (Ro/m)1 and (Ro/m)2, for: (a) Translational Amplitude (Ax) and (b) Rotational Amplitude (Aψ). Fig.14. Variation of Dimensionless Amplitude with Dimensionless Frequency (ao) for ShaleBasalt System for (Ro/m)1 and (Ro/m)2, for: (a) Translational Amplitude (Ax) and (b) Rotational Amplitude (Aψ). Fig.15. Variation of Dimensionless Amplitude with Dimensionless Frequency (ao) for Limestone-Basalt System for (Ro/m)1 and (Ro/m)2, for: (a) Translational Amplitude (Ax) and (b) Rotational Amplitude (Aψ).
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Unit weight (N/m3)
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Soil Weathered rock Sandstone Shale Limestone Basalt
Shear wave velocity (m/s) 185 1680 1110 2220 2960 3700
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Table.1. Properties of soil and rocks (Zhao, 2010)
16000 20810 22000 23000 25000 26000
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Table.2. Observed Natural Frequency and Resonant Amplitude for Different Eccentric Moment for Foundation of size 4 m × 3 m × 2 m. H/B ratio
mee = 0.028 N- m fn1 (Hz)
AH1-res (mm)
mee = 0.045 N- m AH2-res (mm)
fn2 (Hz)
Ψr1-res (Rad)
Ψr2-res (mm)
fn1 (Hz)
AH1-res (mm)
Ψr1-res (Rad)
fn2 (Hz)
AH2-res (mm)
Ψr2-res (mm)
15.7 32.1 19.3 163 229 189
0.0147 0.00782 0.00625 0.00755 0.00639 0.00509
0.0593 0.00144 0.00155 0.0308 0.00147 0.00131
34.6 33 31.4 300 293 287
0.0462 0.000961 0.000961 0.034 0.000943 0.000943
0.00295 0.00111 0.00111 0.00155 0.00101 0.00948
85.9 71.6 70 159 146 140 210 200 195
0.77652 0.48049 0.37044 0.46887 0.36024 0.30734 0.36882 0.32918 0.29376
0.28421 0.16096 0.14003 0.18667 0.14197 0.12775 0.01412 0.12612 0.11562
155 195 170 324 388 356 472 534 496
1.0432 0.05281 0.06285 0.07396 0.05018 0.05643 0.06588 0.054 0.06166
0.17311 0.07169 0.0797 0.09934 0.07156 0.07544 0.0788 0.0704 0.07144
Soil-rock/weathered rock-rock systems 15.7 32.1 19.3 163 229 189
0.0092 0.00487 0.00389 0.0047 0.00398 0.00317
0.0369 0.00099 0.00097 0.0192 0.00092 0.00082
34.6 33 31.4 300 293 287
0.0288 0.000598 0.000598 0.00212 0.000587 0.000587
0.00184 0.000692 0.000598 0.000969 0.000634 0.00059
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Weathered rock-basalt (Vs1/Vs2 = 0.45)
0.5 1 1.5 0.5 1 1.5
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Soil-basalt (Vs1/Vs2 = 0.05)
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Type of rock and soil
Rock-rock system
Shale-basalt (Vs1/Vs2 = 0.6) Limestonebasalt (Vs1/Vs2 = 0.8)
0.5 1 1.5 0.5 1 1.5 0.5 1 1.5
85.9 71.6 70 159 146 140 210 200 195
0.00719 0.004449 0.00343 0.004653 0.003575 0.00305 0.003415 0.003048 0.00272
0.0022 0.001246 0.001084 0.001445 0.001099 0.0009889 0.001093 0.0009763 0.000895
155 195 170 324 388 356 472 534 496
0.00966 0.000489 0.000582 0.000734 0.000498 0.00056 0.00061 0.0005 0.000571
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Sandstonebasalt (Vs1/Vs2 = 0.3)
0.00134 0.000555 0.000617 0.000769 0.000554 0.000584 0.00061 0.000545 0.000553
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fn1, fn2 = First and second resonant frequencies, AH1-res, AH2-res = First and second resonant amplitudes for horizontal motion, ψr1-res, ψr2-res = First and second resonant amplitudes for rocking motion
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Table. 3. Natural Frequency and Resonant Amplitude for Different Eccentric Moment for Foundation of size 8 m × 5 m × 2 m. H/B ratio
mee = 0.028 N- m fn1 (Hz)
AH1-res (mm)
Ψr1-res (Rad)
fn2 (Hz)
mee = 0.045 N- m AH2-res (mm)
Ψr2-res (mm)
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Type of rock and soil
fn1 (Hz)
AH1-res (mm)
Ψr1-res (Rad)
fn2 (Hz)
AH2-res (mm)
Ψr2-res (mm)
15.6 11.6 10.4 126 111 107
0.00148 0.0000927 0.0000477 0.000801 0.0000696 0.0000471
0.000125 0.000152 0.0000733 0.000104 0.000103 0.0001
9.97 8.89 9.75 111 106 104
0.000467 0.000161 0.000322 0.00037 0.00018 0.000338
0.000712 0.000142 0.000126 0.000273 0.000131 0.000121
70 66.8 66.8 152 133 152 195 190 195
0.27604 0.16264 0.10983 0.08521 0.12927 0.10756 0.13089 0.11437 0.10756
0.06697 0.09349 0.04903 0.04538 0.06816 0.05124 0.055159 0.058133 0.051419
108 152 122 203 292 248 310 381 339
0.08715 0.0486 0.05702 0.0703 0.05216 0.05896 0.06577 0.05508 0.06058
0.22522 0.06612 0.06926 0.07283 0.0645 0.06595 0.07062 0.06223 0.06272
Soil-rock/weathered rock-rock systems 15.6 11.6 10.4 126 111 107
0.000925 0.0000578 0.0000298 0.000499 0.0000434 0.000294
0.0000781 0.0000953 0.0000603 0.0000648 0.0000647 0.0000628
9.97 8.89 9.75 111 106 104
0.000291 0.000101 0.000201 0.000231 0.000112 0.000211
0.000444 0.000089 0.000079 0.000171 0.000082 0.000076
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Weathered rockbasalt (Vs1/Vs2 = 0.45)
0.5 1 1.5 0.5 1 1.5
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Soil-basalt (Vs1/Vs2 = 0.05)
Rock-rock system
Shale-basalt (Vs1/Vs2 = 0.6) Limestonebasalt (Vs1/Vs2 = 0.8)
0.5 1 1.5 0.5 1 1.5 0.5 1 1.5
70 66.8 66.8 152 133 152 195 190 195
0.000852 0.000502 0.000339 0.000263 0.000399 0.000332 0.000404 0.000353 0.000332
0.0000788 0.00011 0.0000577 0.0000534 0.0000802 0.0000603 0.0000649 0.0000684 0.0000605
108 152 122 203 292 248 310 381 339
0.000269 0.00015 0.000176 0.000217 0.000161 0.000182 0.000203 0.00017 0.000187
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Sandstonebasalt (Vs1/Vs2 = 0.3)
0.000265 0.0000778 0.0000815 0.0000857 0.0000759 0.0000776 0.0000831 0.0000732 0.0000738
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fn1, fn2 = First and second resonant frequencies, AH1-res, AH2-res = First and second resonant amplitudes for horizontal motion, ψr1-res, ψr2-res = First and second resonant amplitudes for rocking motion
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S1674-7755(16)30271-2
DOI:
10.1016/j.jrmge.2016.09.006
Reference:
JRMGE 311
To appear in:
Journal of Rock Mechanics and Geotechnical Engineering
Received Date: 16 June 2016 Revised Date:
25 September 2016
Accepted Date: 29 September 2016
Please cite this article as: Darshyamkar R, Kumar A, Manna B, Investigation of block foundations under coupled vibrations resting on soil-rock and rock-rock medium, Journal of Rock Mechanics and Geotechnical Engineering (2017), doi: 10.1016/j.jrmge.2016.09.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Title of the Manuscript: Investigation of block foundations under coupled vibrations resting on soil-rock and rock-rock medium.
Author 1:
Address : Department of Civil Engineering Indian Institute of Technology, Delhi Hauz Khas, New Delhi – 110016, India
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E-mail: [email protected]
Author 2: Name : Mr. Ankesh Kumar* Affiliation : Ph.D. Student
Address : Department of Civil Engineering
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Indian Institute of Technology, Delhi
Hauz Khas, New Delhi – 110016, India
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E-mail: [email protected]
Author 3:
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Name : Dr. Bappaditya Manna
Affiliation : Assistant Professor Address : Department of Civil Engineering Indian Institute of Technology, Delhi
Hauz Khas, New Delhi – 110016, India
E-mail: [email protected] Phone: +91-11-26591211
*
Corresponding Author
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Affiliation : M.Tech. Student
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Name : Ms. Renuka Darshyamkar
ACCEPTED MANUSCRIPT Abstract In the present study, the dynamic response of block foundations of different equivalent radius to mass (Ro/m) ratios under coupled mode of vibration is investigated for various homogeneous and
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layered systems. The frequency dependent stiffness and damping of foundation resting on homogeneous soil and rocks are determined using the half-space theory (Velestos et. al., 1971). The dynamic response characteristics of foundation resting on the layered system considering rock-rock
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combination is evaluated using the theory proposed by Kausel (1974) with the help of finite element program with transmitting boundaries. Frequencies versus amplitude responses of block
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foundation are obtained for both translational and rotational motion. A new methodology is proposed for determination of dynamic response of block foundations resting on soil-rock and weathered rock –rock system in the form of equations and graphs. The variation of dimensionless
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natural frequency and dimensionless resonant amplitude with shear wave velocity ratio are investigated for different thickness of top soil/weathered rock layer. The dynamic behaviour of block foundations are also analysed for different rock-rock systems by considering sandstone, shale
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and limestone underlain by basalt rock. The variation of the stiffness, damping and amplitudes of
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block foundations with frequency are shown in this study for various rock-rock combinations. In the analysis two resonant peaks are observed at two different frequencies for both translational and rotational motion. It is observed that dimensionless resonant amplitudes decrease and natural frequencies increase with increase in shear wave velocity ratio. Finally the parametric study is performed for block foundations of size 4m × 3m × 2m and 8m × 5m × 2m by using generalised graphs. The variation of natural frequency and peak displacement amplitude are also studied for different top layer thickness and eccentric moments.
ACCEPTED MANUSCRIPT Key words : Rock-Rock system; block foundation; coupled vibration; homogeneous medium; equivalent radius to mass ratio; half-space theory
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1. INTRODUCTION The geology of earth consists of all types of soil, rocks and minerals belonging to different geological period. Due to its diversity in nature, geotechnical engineers and geologist are
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exposed to completely new set of problems. They encounter the problem related to nonhomogeneity because of the presence of discontinuities such as bedding planes of varying strength,
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fissures, joints and faults. Due to the presence of such geological structures, the studies related to the foundations constructed on such surfaces become most important. Normally block foundations are required to support machines, machine tools and heavy equipments on such non homogeneous
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surfaces. These types of machine foundations are provided in heavy industries like nuclear power plant, hydro power plant, petrochemical industry etc. to ensure the satisfactory operation of the machines which have wide range of operating speeds, dynamic loads and operating conditions. To
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design the machine foundation successfully it becomes crucial to carry out careful engineering
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analysis and the response of block foundation to the dynamic loads. The block foundations are subjected to either static load, or combination of static and dynamic load repetitively over a long period of time. This load leads to the generation of dynamic forces such as balanced and unbalanced forces due to the operation of rotating type or reciprocating type machinery. In case of the coupled vibrations, horizontal and rocking motions are generated simultaneously under the action of unbalanced horizontal dynamic forces of rotating type of machines.
ACCEPTED MANUSCRIPT In the Indian scenario most of the heavy construction is under progress on the Deccan trap rocks. These rocks are mostly consists of weathered basalt underlain by fresh basalt. Apart from this, the rocks which are slightly weathered at present may become weaker in future due to the
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environmental action. Many researchers (Xiao et al., 2014a, Xiao et al., 2014b, Xiao et al., 2015, Xiao and Desai, 2016, and Xiao et al., 2016) investigated the characteristics of the weathered rock and granular soils related to their dilatancy, particle breakage and transitional behavior. It is found
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from the research that the density and pressure have great influences on the strength and stressstrain behaviors of weathered rocks and granular soils. These materials exhibits strain hardening,
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postpeak strain softening, volumetric contraction, and expansion with a range of densities and pressures. Hence the shear wave velocity, stiffness and damping values are greatly influenced by the state of the material which varies from hard rock to highly weathered rock or granular soils.
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Therefore the foundations for heavy machinery and structures on such type of ground conditions must be studied thoroughly for the future safety. In general these layered combinations can be classified as (i) soil-rock / weathered rock - rock system and (ii) rock-rock system. The dynamic
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analysis of block foundations resting on these layered systems is very complex. A very few studies
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are available in the literature regarding the methodology of determination of the natural frequencies and resonant amplitudes of block foundation under coupled horizontal and rocking vibration. Hence in the present study, an attempt is made to analyze the effect of various soil-rock and rockrock foundation system on dynamic response of block foundations of different mass and equivalent radius under coupled mode of vibrations. The dynamic response characteristics of foundations are studied by using the theory proposed by Velestos and Wei (1971) and Kausel (1974) for homogenous and layered system respectively. The variations of natural frequency and resonant amplitude with shear wave velocity are investigated for different top layer thickness. The dynamic
ACCEPTED MANUSCRIPT behaviour of block foundations is analyzed for different rock-rock systems by considering sandstone, shale and limestone underlain by basalt rock. The variation of the stiffness, damping and amplitudes of foundation with frequency are shown in this study for various rock-rock
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combinations. The variation of natural frequencies and peak displacement amplitudes are also studied for different top layer thickness and eccentric moments. Finally the parametric study is carried out to determine the natural frequency and resonant amplitude for block foundations of size
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2. THEORETICAL INVESTIGATION
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4 m × 3 m × 2 m and 8 m × 5 m × 2 m using generalized relations proposed in the study.
The classical work on vibratory response of foundations was carried out by Lamb (1904). Subsequently Reissner (1936) developed an analytical solution for periodic vertical displacement at
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the centre of the circular loaded area with elastic half-space mathematical model. Arnold et al. (1955) extended the elastic half-space theory to include other modes of vibrations using rigid base type contact stress distribution and weighted average displacement condition. Hsieh (1962)
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obtained the expressions for the frequency dependent stiffness and geometrical damping in terms of
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displacement functions. Lysmer (1965) and Lysmer and Richart (1966) showed that vertical vibration of a rigid circular footing can be represented by a simplified analog consisting of a simple damped oscillator with physical parameter that gives good agreement with the response curve obtained from the elastic half-space theory. It is well established now that the dynamic response of foundations depends on several factors namely, size and shape of the foundation, depth of embedment, soil profile, soil properties, frequency of loading, and mode of vibration (Richart et al.1970 and Gazetas and Stokoe, 1991). The finite element approach has been used to calculate the stiffness and damping of embedded foundations by a number of investigators (Urlich and
ACCEPTED MANUSCRIPT Kuhlemeyer, 1973; Kausel and Ushijima, 1979). Beredugo and Novak (1972) and Novak and Sachs (1973) presented the stiffness and damping for the coupled and torsional mode of vibration of embedded footings and it was found that both the stiffness and damping values are increased due to
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the embedment of footing. Veletsos and Verbic (1973) and Veletsos and Nair (1974) proposed stiffness and damping parameters of embedded footing for different modes of vibrations. Further, Warburton (1957), Hadjian and Luco (1977) and Kagawa and Kraft (1981) studied the dynamic
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response of foundations considering the heterogeneities of the soil.
The dynamic impedance of block foundation resting on homogenous deposit considered as halfspace
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is evaluated using the theory proposed by Veletsos and Wei (1971). Furthermore, the theory proposed by Kausel (1974) are used for the layered system analysis in which top layer is treated as stratum and bottom layer is considered as halfspace.
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2.1 Homogenous Medium
In the present study an approximate solution proposed by Veletsos and Wei (1971) is used to study the steady state response of rigid circular disk of infinitesimal thickness supported at the surface of a
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non dissipative, homogeneous, linear elastic halfspace. The weight of the disk is assumed to be
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negligible. The exciting forces considered includes a harmonically varying lateral force directed along the x-axis, and harmonically varying moment acting about the y-axis. The system has following assumptions: (i) the normal component of the contact pressure is assumed to be zero due to horizontal force (ii) the horizontal or shearing components of the interface pressure are assumed to be zero due to overturning moment. Thus, it permits the horizontal translational motion of the disk to be evaluated independently. The surrounding region of disk is assumed to be stress free.
ACCEPTED MANUSCRIPT In order to compute the coupled displacements, the amplitudes of the steady-state horizontal displacement and rotation of the disk, u and φ, stated as u f11 + ig11 φ r = f + ig o 21 21
f12 + ig12 u st f 22 + ig 22 φst ro
(1)
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where f’s and g’s are dimensionless flexibility coefficients; ust = horizontal displacement by static
2−v P 8 G ro
φ st =
3(1 − v ) M 8 G ro 3
(2)
(3)
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u st =
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horizontal force, P; φst = rotation of disk due to static moment, M; where,
in which ν = Poisson’s ratio for material of half-space; G =shear modulus of elasticity; ro = equivalent radius. The flexibility coefficients, f and g are functions of ν and of the dimensionless frequency parameter (ao), where ao = ω ro/Vs, ω = circular frequency of the excitation, Vs= shear
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waves velocity in the halfspace.
The force amplitudes, P and M, corresponding to the displacement amplitudes, u and φ, may be expressed as:
k 2 1 + ia o c 2 1 u 2−v ( k 2 2 + ia o c 2 2 ) φ ro 3(1 − v )
(4)
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P k 1 1 + ia o c 1 1 M = K x k 2 1 + ia o c 2 1 r o
in which the k’s and c’s are dimensionless coefficients depending on ao and ro, i =√-1; where ;Kx = P/ust and Kφ = M/φst . Specifically, Kx represents the horizontal static force necessary to produce a unit horizontal displacement of the disk and Kφ represents the static moment necessary to rotate the disk by a unit amount with no restriction on the value of the horizontal displacement. The quantities k will be referred to as the stiffness coefficients, and the quantities c will be referred to as the damping coefficients.
ACCEPTED MANUSCRIPT If the off-diagonal terms of the flexibility matrix are neglected, f12, f21, g12, and g21 are assumed to be zero, the force-displacement relationship for the disk is obtained from the following equation: P ( k 1 + ia o c 1 ) K x M = 0
0 ( k 2 + ia o c 2 ) K φ
u φ
(5)
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The stiffness function for horizontal and rocking motion are calculated from equation (5). The above methodology is used to investigate the dynamic response of block foundation resting on
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homogeneous medium by using the computer program DYNA 5 (Novak, 1972). 2.2 Layered Medium
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The analysis is carried out using FEM based formulation and computer program developed by Kausel and Ushijima (1979). A three-dimensional axisymmetric finite element model with transmitting boundaries is used to model a rigid circular foundation resting on a homogeneous rock
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stratum of finite depth resting upon a much stiffer rock-like material. The transmitting boundaries used were developed by Kausel (1974) to the three dimensional case with axisymmetric geometry. Kausel and Ushijima (1979) presented a numerical method for the dynamic analysis of
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axisymmetric foundation resting on viscoelastic soil layers over rock of infinite horizontal extent.
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The solution technique developed by Waas and Lysmer (1965) was extended for the analysis of axisymmetric systems subjected to arbitrary non axisymmetric loading using Fourier expansion method.
For the plane symmetric displacement modes, the forced horizontal displacement is referred as swaying and the forced rotation about horizontal axis is referred as rocking. The stiffness functions can be given as K s = K so (k1 + iao c1 )(1 + 2i β h1 )
(6)
Kψ = Kψ o (k 2 + iao c2 )(1 + 2i β h 2 )
(7)
ACCEPTED MANUSCRIPT Where Kso , KΨo are real parts of the stiffness functions in the static case, ao is dimensionless frequency, k1, k2, c1, c2 are the stiffness and damping coefficients and β is internal damping coefficient.
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Using the methodology proposed by different researchers the analyses are carried out to determine the dynamic response of block foundation in soil-rock system, weathered rock-rock system and rock-rock system. The methodology involved in this study is incorporated in the computer program
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DYNA 5 (Novak, 1972). This program is used to present the dynamic behaviour of block
constants for layered system.
3. RESULTS AND DISCUSSIONS
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foundation in terms of frequency response curves for displacement, stiffness, and damping
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The analyses are carried out for three eccentric moments i.e. mee = 0.45, 0.366, 0.278 N-m for both homogeneous and layered medium. The results are then plotted in dimensionless form using
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following equations,
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Dimensionless translational amplitude,
Dimensionless rotational amplitude,
Dimensionless frequency,
Ax =
x.m me e
Aψ =
Iψ
ao =
me eZ e
ωro vs
(8)
(9)
(10)
where, x is the amplitude of system, m is the total mass of the footing and machine, IΨ is moment of inertia, me is the mass of eccentric rotating part of oscillator, e is eccentricity of rotating part of
ACCEPTED MANUSCRIPT oscillator, ψo is rotational moment, ω is the angular operating frequency, ro is the radius of foundation and vs is the shear wave velocity. Two different ratios of equivalent radius with mass of block foundation i.e. (Ro/m)1= 2.85×10-5 and
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(Ro/m)1= 1.74×10-5 are considered to study the effect of mass and size of foundation on the dynamic response of block foundation, where Ro is equivalent radius of footing in meter and m is total mass of the footing and machine in kN. The properties of soil and rocks considered for the
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analysis is shown in Table 1 (Zhao, 2010).
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3.1 Homogeneous Medium
In case of homogenous medium, the theory proposed by Velestos and Wei (1971) has been used for the analysis block foundation resting on homogeneous soil and four type of homogeneous rocks i.e.
k h ,k r = k o =
c k Soil ( ao = 0 )
(11)
(12)
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ch ,cr = co =
k k Soil ( ao = 0 )
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sandstone, shale, limestone and basalt. The stiffness and damping constant are normalized as
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where ko is normalized stiffness constant (where ko is kh, and kr for horizontal stiffness and rocking stiffness respectively), co is normalized damping constant (where co is ch, and cr for horizontal damping and rocking damping respectively), k and c are stiffness and damping constant of layered system and ksoil(ao= 0) is stiffness of soil at ao= 0. The variation of normalised horizontal stiffness (kh) with dimensionless frequency (ao) for all type of homogeneous soil and rocks are shown in Fig 1 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively. The variation of normalized rocking stiffness (kr) with dimensionless natural frequency (ao) are shown in
ACCEPTED MANUSCRIPT Fig 2 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively. It has been observed that both horizontal and rocking normalized stiffness (kh) are found maximum for the basalt among all geological materials considered here i.e. limestone, shale, sandstone and soil. Hence, foundation resting on basalt has
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higher resistance for both horizontal and rocking displacement as compared to other rocks and soil. The variation of normalized horizontal damping (ch) with dimensionless frequency (ao) are shown in Fig 3 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively. The variation of normalized rocking damping
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(cr) with dimensionless frequency (ao) also shown in Fig 4 (a) and (b) for (Ro/m)1 and (Ro/m)2
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respectively. It is noted that the normalized damping increases significantly as dimensionless frequency decreases at low frequency condition for both horizontal and rocking motion because of the conversion of frequency-independent material damping (β) to the equivalent viscous damping coefficient (c) as c = 2β/ω (where ω = circular frequency). It is also observed that the normalized
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horizontal and rocking damping for soil is much lower than the homogeneous rock. The variation of dimensionless translational amplitude (Ax) and dimensionless rotational amplitude (Aψ) with dimensionless frequency (ao) are shown in Fig 5 (a) and 5 (b) respectively for both (Ro/m)1
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and (Ro/m)2. It is observed from these figures that there are two dimensionless amplitude peaks at
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two different dimensionless frequencies. For (Ro/m)1, the maximum values of dimensionless frequency is lower and maximum dimensionless amplitude is higher than the (Ro/m)2 for both horizontal and rocking motion. The maximum value of dimensionless amplitude is higher for soil as compared to the rocks for both translational and rotational case. It can also be seen that the dimensionless frequency (ao) corresponding to maximum value of dimensionless amplitude for soil is lower than the rocks for both translational and rotational motion.
ACCEPTED MANUSCRIPT 3.2 Layered Medium By using the theory proposed by Kausel (1974), the analyses are carried out to study the dynamic response of block foundations resting on layered soil-rock and rock-rock system. In the case of
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layered medium, the dimensionless parameter which appreciably influences the dynamic response of block foundation is the H/B ratio, where H is depth of top layer and B is width of foundation. In the present study three different H/B ratios, i.e. 0.5, 1.0 and 1.5 are considered. The results of dynamic
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impedance of foundations, pertaining to all possible translational and rotational mode of vibration are plotted in the form of dimensionless parameters of stiffness, damping and amplitude. The ratio of
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stiffness and damping of the layered system with the stiffness of homogeneous igneous rock is taken to obtain the normalized stiffness and damping of layered system k1 k Basalt ( ao = 0 )
c h ,c r = c o =
c1 k Basalt ( ao = 0 )
(13)
(14)
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k h ,k r = k o =
where, ko and co are normalized parameters for stiffness and damping of layered system, kl and cl are
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stiffness and damping constant of layered system, kbasalt (ao= 0) is the stiffness of basalt rock at the
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zero dimensionless frequency, kh and kr are normalized stiffness parameters for horizontal and rocking motion respectively and ch and cr are normalized damping parameter for horizontal and rocking motion respectively.
3.2.1 Soil-Rock and Weathered Rock-Rock Systems It is well established that the shear wave velocity values is greatly influenced by the state of the material which varies from hard rock to highly weathered rock or granular soils. Therefore in case of soil-rock and weathered rock-rock systems, analyses are carried out by considering shear wave
ACCEPTED MANUSCRIPT velocity as a important parameter. In the layered system, normally the soil or weathered rock layer is underlain by hard rock and hence the layered medium can be defined by two shear wave velocities i.e. Vs1 and Vs2 (where Vs1 is the shear wave velocity of top layer, and Vs2 is the shear
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wave velocity of bottom halfspace). Therefore in present study the dynamic behaviour of block foundation on layered medium is investigated in the form of shear wave velocity ratio (Vs1/Vs2). Three different shear wave velocity ratios i.e. 0.8, 0.6 and 0.3 are considered in the analyses to
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represent limestone-basalt, shale-basalt and sandstone-basalt system respectively. Then the variation
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of natural frequency and resonant amplitude of translational and rotational mode of vibration are obtained for H/B = 0.5, 1.0 and 1.5. The maximum dimensionless amplitude with its subsequent dimensionless natural frequency is obtained for all shear wave velocity ratios. The trend lines are plotted for two cases (i) for natural frequency and shear wave velocity (ii) for resonant amplitude and
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shear wave velocity ratios. The shear wave velocity ratio (Vs1/Vs2 ) ≤ 0.5 is normally considered to represent the soil – rock and weathered rock – rock systems (Gupta and Rao, 1998). Therefore the
line.
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extrapolation is done for shear wave velocity ratio (Vs1/Vs2 ) up to 0.5 by using the equation of trend
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The variation of dimensionless natural frequency for first peak with shear wave velocity ratio is shown in Fig. 6 (a) for both (Ro/m)1 and (Ro/m)2. Similarly, the variation of dimensionless natural frequency for second peak with shear wave velocity ratio is shown in Fig.6 (b) for both (Ro/m)1 and (Ro/m)2. The trend lines shown in the graph are the variation of first and second natural frequency with shear wave velocity ratios for the coupled vibration. It is observed from the figures that the trend lines shows decrement with the decrease in shear wave velocity ratio. The variation of dimensionless translational amplitude (Ax) with shear wave velocity ratio is shown in Fig 7 (a) and
ACCEPTED MANUSCRIPT (b) for (Ro/m)1 and (Ro/m)2 respectively. Similarly, the variation of dimensionless rotational amplitude (Aψ) with shear wave velocity ratio is shown in Fig 8 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively. It can be seen from these figures that the value of dimensionless natural frequencies are
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higher and dimensionless translational and rotational amplitudes are lower for (Ro/m)2 as compared to (Ro/m)1. It can be seen from the Fig. 7 and 8 that as H/B ratios increase the dimensionless rotational and translational amplitudes decrease. A converging trend is found for the trend lines representing
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resonance frequency and amplitudes for various H/B ratios as the values of Vs1/Vs2 increase.
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Therefore the effect of H/B ratio reduces as the shear wave velocity ratio increases. It is observed that with increase in shear wave velocity ratio the dimensional natural frequencies increase and dimensionless resonant amplitudes decrease. Therefore, the dimensionless natural frequencies are found lower and dimensionless resonant amplitudes are found higher in case of soil-
3.2.2 Rock-Rock System
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rock system than weathered rock-rock system.
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In the present study, the rock-rock combinations considered are limestone-basalt, shale-basalt and sandstone-basalt with shear wave velocity ratio of 0.8, 0.6 and 0.3. The variation of dimensionless
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frequency, dimensionless translational and rotational amplitudes are studied for various H/B ratios and shear wave velocity ratios. The variation of normalized horizontal stiffness (kh) with dimensionless frequency (ao) is shown in Fig 9 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively for sandstone-basalt, shale-basalt and limestone-basalt systems. Similarly the variation of normalized rocking stiffness (kr) with dimensionless frequency (ao) is shown in Fig 10 (a) and (b). It can be seen that normalized horizontal and rocking stiffness values are higher for limestone basalt system than other two rock-rock system
ACCEPTED MANUSCRIPT because of higher shear wave velocity ratio of limestone-basalt system. Also the normalized horizontal and rocking stiffness value of shale-basalt system is more than the sandstone basalt system. It is also observed that the normalized stiffness values decrease with increase in H/B ratios
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approximately up to dimensionless frequency (ao) of 1.5 for both horizontal and rocking motion. The variation of normalized horizontal damping (ch) with dimensionless frequency (ao) is shown in Fig 11 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively for all rock-rock systems. Similarly the
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variation of normalized rocking damping (cr) with dimensionless natural frequency (ao) is shown in
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Fig 12 (a) and (b) for (Ro/m)1 and (Ro/m)2 respectively. It is found that both normalized horizontal and rocking damping (ch and cr) values increase as shear wave velocity ratio increases. It is also noted that normalized damping is very low for sandstone-basalt system as compared to other rockrock systems. It can also be seen that the variation of normalized damping (ch and cr) with H/B ratio
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is very less for lower dimensionless frequency, ao < 0.2. It is also observed that the normalized damping increases with increase in the thickness of top layer approximately up to dimensionless frequency (ao) of 1.5 for both horizontal and rocking motion.
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The variations of dimensionless translational amplitude (Ax) and rotational amplitude (Aψ) with
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dimensionless frequency (ao) for sandstone-basalt system are shown in Fig.13 (a) and (b) respectively for both (Ro/m)1 and (Ro/m)2. Similarly, the variations of dimensionless translational and rotational amplitude with dimensionless frequency are shown in Fig 14 (a) and (b) respectively for shale-basalt system. Further, the variations of dimensionless translational and rotational amplitude with dimensionless frequency are shown in Fig 15 (a) and (b) respectively for limestone-basalt system. It is found that in most of the cases the response is dominated by the first resonant peak and the second resonant peak is entirely suppressed for both translational and rotational motion. It is also
ACCEPTED MANUSCRIPT found that the dimensionless natural frequencies increase and the dimensionless resonant amplitudes decrease with increase in shear wave velocity ratios. It is observed that the maximum values of dimensionless frequency is lower and maximum dimensionless amplitude is higher for (Ro/m)1 than
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the (Ro/m)2 for all rock-rock system in case of both horizontal and rocking motion.
4. PARAMETRIC STUDIES
The parametric study is carried out for two block foundations of size 4 m × 3 m × 2 m and 8 m × 5 m
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× 2 m, subjected to coupled horizontal and rocking vibrations. The values of horizontal and
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rotational amplitude corresponding to their natural frequencies are shown in Table 2 and Table 3 for block foundations of size 4 m × 3 m × 2 m and 8 m × 5 m × 2 m respectively. The frequency and amplitude values are presented for two different eccentric moments (mee = 0.028 and 0.045 N-m). The values shown in the tables are obtained with the help of generalized graphs proposed in the
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present study.
The natural frequencies for soil-rock and weathered rock-rock system are found by using Fig 6 (a)
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and (b). Similarly, Fig 7 (a) and 7 (b) are used for determining the values for translational amplitudes and Fig 8 (a) and 8 (b) are used for rotational amplitudes for soil-rock and weathered rock-rock
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system. It is observed from the tables that values of both the natural frequency and maximum amplitude varies with mass and size of block foundations. For soil-rock system, the resonant amplitudes are found higher and the natural frequencies are found lower as compared to other rockrock systems. It is also observed that the values of natural frequencies are higher and resonant amplitudes are lower for limestone-basalt system as compared to all other systems. Therefore the limestone-basalt system can be considered as best foundation medium for block foundation among
ACCEPTED MANUSCRIPT all other systems considered in this study because the limestone-basalt system is suitable for machines with high operating frequencies with low amplitude of vibration.
5. CONCLUSIONS
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In the present study, an effort is made to study the dynamic response of block foundations resting on homogeneous and layered medium under coupled horizontal and rocking vibration. Two different Ro/m ratios are considered to investigate the effect of size and mass of block foundation.
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Analyses are carried out to study the variation of stiffness, damping and amplitude with frequency
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of block foundation for three different medium: (i) homogeneous soil and rocks, (ii) soil/weathered rock-rock system and (iii) rock-rock system. For all the cases, the response is dominated by the first resonant peak and the second peak is entirely suppressed in case of translational motion. However the reverse trends are found in case of rotational motion.
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In the case of homogeneous medium, it is found that the normalized horizontal and rocking stiffness for block foundations on soil are lower than the block foundations on rocks. It is also
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observed that the normalized horizontal and rocking damping for soil is much lower than the homogeneous rock. Therefore, the dimensionless translational and rotational amplitudes for soil are
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found higher than the rocks and the dimensionless natural frequencies for soil are found lower than the rocks.
In the case of layered medium i.e. soil - rock and weathered rock – rock systems, different equations are proposed to calculate the dimensionless natural frequency and resonant amplitudes for both translational and rotational motion in terms of H/B ratios and shear wave velocity ratio. It is observed that the dimensional natural frequencies increase and dimensionless resonant amplitudes decrease with increase in shear waive velocity ratio. Therefore, the dimensionless
ACCEPTED MANUSCRIPT natural frequencies are found higher in case weathered rock-rock system than soil-rock system and the dimensionless resonant amplitudes are found higher in case soil-rock system than the weathered rock-rock system. These parameters are presented in the form of shear wave velocity, as the testing
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for strength parameters of these weathered rocks is very difficult in laboratory. The graphs presented can directly be used by the practicing engineers for estimation of natural frequency and resonant amplitude of block type machine foundations resting on different layered combinations
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like soil-rock and weathered rock-rock systems.
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For rock–rock systems, it is observed that both the normalized stiffness decreases and damping increases with increase in the thickness of top layer approximately up to dimensionless frequency of 1.5 for both horizontal and rocking motion. It is found that both normalized stiffness and damping increases with increase in shear wave velocity ratio. Therefore, the dimensionless resonant
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amplitudes decrease and natural frequencies increase with increase in shear wave velocity ratio for both translational and rotational motion. It is also found that both the dimensionless resonant amplitudes and natural frequencies decrease with increase in H/B ratio.
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The effect (Ro/m) ratio on the frequency – amplitude response is also investigated for homogeneous
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and layered systems. It is observed that the maximum values of dimensionless frequency is lower and maximum dimensionless amplitude is higher for (Ro/m)1 than the (Ro/m)2 for both horizontal and rocking motion.
The parametric study is also carried out for the case of block foundation of size 4 m × 3 m × 2 m and 8 m × 5 m × 2 m, using normalized graphs proposed in the present work. The values of translational and rotational amplitude corresponding to their natural frequencies are presented. It is observed from the tables that the natural frequencies are lower and resonant amplitudes are higher
ACCEPTED MANUSCRIPT for soil-basalt system as compared to all other rock-basalt system because of the low shear wave velocity ratio of soil-basalt system. The best possible option to be considered as foundation medium for block foundation is the one having high natural frequency and low resonant amplitudes
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values. Therefore, the limestone-basalt system can be considered as good foundation medium for the block foundation because of its high value of natural frequencies and low value of resonant
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amplitudes.
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References 1. Arnold, R. N., Bycroft, G. N., and Warburton, G. B. (1955), Forced Vibrations of a Body on an infinite Elastic Solids, Journal of Applied Mechanics Trans., ASME, Vol. 77, pp.
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391-401.
2. Beredugo,Y.O. and Novak, M. (1972), Coupled Horizontal and Rocking Vibration of Embedded Footings, Canadian geotechnical Journal, Vol. 9, No. 4, pp. 477-97.
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3. Gazetas, G., and Stokoe, K. H. (1991), Free Vibration of Embedded Foundations: Theory versus Experiment, Journal of Geotechnical Engineering, ASCE, Vol. 119, No. 9, pp.
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1382-1401.
4. Gupta, A. S. and Rao, K. S. (1998), Index properties of weathered rocks: interrelationships and applicability, Bull Eng Geol Env, Vol. 57, pp. 161–172. 5. Hadijan, A. H., and Luco, J. E. (1977), On the Importance of Layering on Impedance
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Functions, Proc. 6th WCEE, New Delhi.
6. Hsieh, T. K. (1962), Foundation Vibrations, Proc., Institution of Civil Engineers (U.K.), pp. 211-226.
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7. Kausel, E. (1974), Forced Vibrations of Circular Foundations on Layered Media, PhD Thesis, Massachusetts Institute of Technology.
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8. Kausel, E. and Ushijima, R. (1979). Vertical and Torsional Stiffness of cylindrical Footing, Civil Engineering Department Report R79-6, MIT, Cambridge, Massachusetts.
9. Kagawa, T., and Kraft, L. M. (1981), Machine Foundations on Layered Soil Deposits, Proc., 10th International Conference on Soil Mechanics and Foundation Engineering, Stockholm, Vol. 3, pp. 249-252.
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10. Lamb, H. (1904), On the Propagation of Tremors over the Surface of an Elastic Solid, Philosphical Transactions of the Royal Society, London, Ser. A, Vol. 203, pp. 1-42. 11. Lysmer, J. (1965), Vertical Motion of Rigid Footings, U.S. Army Engg. Waterways
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Experiment Station, Report No. 3, 115, Vicksburg, Misissippi.
12. Lysmer T, and Waas G (1972), Shear Waves in Plane Infinite Structure, J Eng Mech Div ASCE, Vol. 98, pp.85–105
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13. Lysmer, J. and Richart, F. E. Jr. (1966), Dynamic Response of Footings to Vertical Loading, Journal of Soil Mechanics and Foundation Engineering Division, ASCE, Vol.
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92, SM 1, pp. 65-91.
14. Novak, M. and Beredugo, Y.O. (1972), Vertical Vibration of Embedded Footings Journal Soil Mechanics and Foundation Division, ASCE, SM12, pp.1291-1310. 15. Novak, M. and Sachs, K. (1973), Torsional and Coupled Vibrations of Embedded
No.11, p.33.
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Footings International Journal of Earthquake Engineering and Structural Dynamics, Vol.2
16. Richart, F. E., Hall, J. R., and Woods, R. D. (1970), Vibrations of Soils and Foundations,
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Prentice-Hall, Inc., Englewood Cliffs, N.J., pp. 414. 17. Reissner, E. (1936), Stationare, Axialsymmetrische Durch Eine Schut-telnde Masse
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erregto Schwingungen eines Homogenen Elastischen Halbraumes. Ingenieur archive, Berlin, Germany, Vol. 7, No. 6, pp. 381-396.
18. Urlich, C. M. and Kuhlemeyer, R. L. (1973), Coupled Rocking and Lateral Vibrations of Embedded Footings, Canadian Geotechnical Journal, Vol. 10, No. 2, pp. 145-160.
19. Veletsos, A.S. and Wei, Y.T. (1971), Lateral and Rocking Vibration of Footings. Journal Soil Mechanics and Foundation Division, ASCE, SM9, pp. 1227- 1248.
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20. Veletsos, A.S. and Verbic, B. (1973), Vibration of Viscoelastic Foundation, Journal of Earthquake Engineering and Structural Vol. 2, pp. 87-102. 21. Veletsos, A.S. and Nair, V. V. D. (1974), Torsional Vibration of Viscoelastic
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Foundaation. Journal of Geotechnical Division, ASCE, Vol. 100, No. GT3, pp. 225-246. 22. Warburton, G. B. (1957), Forced Vibration of a Body upon an Elastic Stratum, Journal of Applied Mechanics, ASME, Vol. 24, pp. 55-58.
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23. Xiao, Y. and Desai, C. S. (2016), General Stress-Dilatancy Relation for Granular Soils. J. Geotech. Geoenviron. Eng. Vol. 142, No. 4, 02816001.
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24. Xiao, Y., Liu, H., Chen, Y. and Jiang, J. (2014a), Bounding surface model for rockfill materials dependent on density and pressure under triaxial stress conditions. J. Eng. Mech Vol. 140, No. 4, 04014002.
25. Xiao, Y., Liu, H., Chen, Y. and Jiang, J. (2014b), Bounding surface plasticity model
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incorporating the state pressure index for rockfill materials. J. Eng. Mech 140, No. 11, 04014087.
26. Xiao, Y., Liu, H., Ding, X., Chen, Y., Jiang, J. and Zhang, W. (2016), Influence of
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Particle Breakage on Critical State Line of Rockfill Material. Int. J. Geomech., Vol. 16, No. 1, 04015031.
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27. Xiao, Y., Liu, H., Sun, Y., Liu, H. and Chen, Y. (2015), Stress-dilatancy behaviors of coarse granular soils in three-dimensional stress space. Eng. Geol., Vol. 195, pp. 104-110.
28. Zhao, J (2010), Handbook on rock mechanics for civil engineers, Civil and Environmental Engineering, Swiss Federal Institute of Technology, Lausanne.
ACCEPTED MANUSCRIPT FIGURE CAPTION Fig.1. Variation of Normalized Horizontal Stiffness (kh) With Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio and (b) (Ro/m)2 ratio. Fig.2. Variation of Normalized Rocking Stiffness (kr) With Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio and (b) (Ro/m)2 ratio.
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Fig.3. Variation of Normalized Horizontal Damping (ch) With Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio and (b) (Ro/m)2 ratio. Fig.4. Variation of Normalized Rocking Damping (cr) With Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio and (b) (Ro/m)2 ratio.
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Fig.5. Variation of Dimensionless Amplitudes With Dimensionless Frequency (ao) for: (Ro/m)1 and (Ro/m)2 (a) Translational, Ax and (b) Rotational, Aψ
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Fig.6. Variation of Dimensionless Natural Frequency With Shear Wave Velocity Ratio (Vs1/Vs2) for: (Ro/m)1 and (Ro/m)2 (a) First Peak, ao1 and (b) Second Peak, ao2 Fig.7. Variation of Dimensionless Translational Amplitude (Ax) With Shear Wave Velocity Ratio (Vs1/Vs2) for: (a) (Ro/m)1 ratio and (b) (Ro/m)2 ratio. Fig.8. Variation of Dimensionless Rotational Amplitude (Aψ) With Shear Wave Velocity Ratio (Vs1/Vs2) for: (a) (Ro/m)1 ratio and (b) (Ro/m)2 ratio.
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Fig.9. Variation of Normalized Horizontal Stiffness (kh) with Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio, and (b) (Ro/m)2 ratio. Fig.10. Variation of Normalized Rocking Stiffness (kr) with Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio, and (b) (Ro/m)2 ratio.
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Fig.11. Variation of Normalized Horizontal Damping (ch) with Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio, and (b) (Ro/m)2 ratio.
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Fig.12. Variation of Normalized Rocking Damping (cr) with Dimensionless Frequency (ao) for: (a) (Ro/m)1 ratio, and (b) (Ro/m)2 ratio. Fig.13. Variation of Dimensionless Amplitude with Dimensionless Frequency (ao) for Sandstone-Basalt System for (Ro/m)1 and (Ro/m)2, for: (a) Translational Amplitude (Ax) and (b) Rotational Amplitude (Aψ). Fig.14. Variation of Dimensionless Amplitude with Dimensionless Frequency (ao) for ShaleBasalt System for (Ro/m)1 and (Ro/m)2, for: (a) Translational Amplitude (Ax) and (b) Rotational Amplitude (Aψ). Fig.15. Variation of Dimensionless Amplitude with Dimensionless Frequency (ao) for Limestone-Basalt System for (Ro/m)1 and (Ro/m)2, for: (a) Translational Amplitude (Ax) and (b) Rotational Amplitude (Aψ).
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Unit weight (N/m3)
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Soil Weathered rock Sandstone Shale Limestone Basalt
Shear wave velocity (m/s) 185 1680 1110 2220 2960 3700
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Table.1. Properties of soil and rocks (Zhao, 2010)
16000 20810 22000 23000 25000 26000
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Table.2. Observed Natural Frequency and Resonant Amplitude for Different Eccentric Moment for Foundation of size 4 m × 3 m × 2 m. H/B ratio
mee = 0.028 N- m fn1 (Hz)
AH1-res (mm)
mee = 0.045 N- m AH2-res (mm)
fn2 (Hz)
Ψr1-res (Rad)
Ψr2-res (mm)
fn1 (Hz)
AH1-res (mm)
Ψr1-res (Rad)
fn2 (Hz)
AH2-res (mm)
Ψr2-res (mm)
15.7 32.1 19.3 163 229 189
0.0147 0.00782 0.00625 0.00755 0.00639 0.00509
0.0593 0.00144 0.00155 0.0308 0.00147 0.00131
34.6 33 31.4 300 293 287
0.0462 0.000961 0.000961 0.034 0.000943 0.000943
0.00295 0.00111 0.00111 0.00155 0.00101 0.00948
85.9 71.6 70 159 146 140 210 200 195
0.77652 0.48049 0.37044 0.46887 0.36024 0.30734 0.36882 0.32918 0.29376
0.28421 0.16096 0.14003 0.18667 0.14197 0.12775 0.01412 0.12612 0.11562
155 195 170 324 388 356 472 534 496
1.0432 0.05281 0.06285 0.07396 0.05018 0.05643 0.06588 0.054 0.06166
0.17311 0.07169 0.0797 0.09934 0.07156 0.07544 0.0788 0.0704 0.07144
Soil-rock/weathered rock-rock systems 15.7 32.1 19.3 163 229 189
0.0092 0.00487 0.00389 0.0047 0.00398 0.00317
0.0369 0.00099 0.00097 0.0192 0.00092 0.00082
34.6 33 31.4 300 293 287
0.0288 0.000598 0.000598 0.00212 0.000587 0.000587
0.00184 0.000692 0.000598 0.000969 0.000634 0.00059
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Weathered rock-basalt (Vs1/Vs2 = 0.45)
0.5 1 1.5 0.5 1 1.5
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Soil-basalt (Vs1/Vs2 = 0.05)
RI PT
Type of rock and soil
Rock-rock system
Shale-basalt (Vs1/Vs2 = 0.6) Limestonebasalt (Vs1/Vs2 = 0.8)
0.5 1 1.5 0.5 1 1.5 0.5 1 1.5
85.9 71.6 70 159 146 140 210 200 195
0.00719 0.004449 0.00343 0.004653 0.003575 0.00305 0.003415 0.003048 0.00272
0.0022 0.001246 0.001084 0.001445 0.001099 0.0009889 0.001093 0.0009763 0.000895
155 195 170 324 388 356 472 534 496
0.00966 0.000489 0.000582 0.000734 0.000498 0.00056 0.00061 0.0005 0.000571
TE D
Sandstonebasalt (Vs1/Vs2 = 0.3)
0.00134 0.000555 0.000617 0.000769 0.000554 0.000584 0.00061 0.000545 0.000553
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Table. 3. Natural Frequency and Resonant Amplitude for Different Eccentric Moment for Foundation of size 8 m × 5 m × 2 m. H/B ratio
mee = 0.028 N- m fn1 (Hz)
AH1-res (mm)
Ψr1-res (Rad)
fn2 (Hz)
mee = 0.045 N- m AH2-res (mm)
Ψr2-res (mm)
RI PT
Type of rock and soil
fn1 (Hz)
AH1-res (mm)
Ψr1-res (Rad)
fn2 (Hz)
AH2-res (mm)
Ψr2-res (mm)
15.6 11.6 10.4 126 111 107
0.00148 0.0000927 0.0000477 0.000801 0.0000696 0.0000471
0.000125 0.000152 0.0000733 0.000104 0.000103 0.0001
9.97 8.89 9.75 111 106 104
0.000467 0.000161 0.000322 0.00037 0.00018 0.000338
0.000712 0.000142 0.000126 0.000273 0.000131 0.000121
70 66.8 66.8 152 133 152 195 190 195
0.27604 0.16264 0.10983 0.08521 0.12927 0.10756 0.13089 0.11437 0.10756
0.06697 0.09349 0.04903 0.04538 0.06816 0.05124 0.055159 0.058133 0.051419
108 152 122 203 292 248 310 381 339
0.08715 0.0486 0.05702 0.0703 0.05216 0.05896 0.06577 0.05508 0.06058
0.22522 0.06612 0.06926 0.07283 0.0645 0.06595 0.07062 0.06223 0.06272
Soil-rock/weathered rock-rock systems 15.6 11.6 10.4 126 111 107
0.000925 0.0000578 0.0000298 0.000499 0.0000434 0.000294
0.0000781 0.0000953 0.0000603 0.0000648 0.0000647 0.0000628
9.97 8.89 9.75 111 106 104
0.000291 0.000101 0.000201 0.000231 0.000112 0.000211
0.000444 0.000089 0.000079 0.000171 0.000082 0.000076
SC
Weathered rockbasalt (Vs1/Vs2 = 0.45)
0.5 1 1.5 0.5 1 1.5
M AN U
Soil-basalt (Vs1/Vs2 = 0.05)
Rock-rock system
Shale-basalt (Vs1/Vs2 = 0.6) Limestonebasalt (Vs1/Vs2 = 0.8)
0.5 1 1.5 0.5 1 1.5 0.5 1 1.5
70 66.8 66.8 152 133 152 195 190 195
0.000852 0.000502 0.000339 0.000263 0.000399 0.000332 0.000404 0.000353 0.000332
0.0000788 0.00011 0.0000577 0.0000534 0.0000802 0.0000603 0.0000649 0.0000684 0.0000605
108 152 122 203 292 248 310 381 339
0.000269 0.00015 0.000176 0.000217 0.000161 0.000182 0.000203 0.00017 0.000187
TE D
Sandstonebasalt (Vs1/Vs2 = 0.3)
0.000265 0.0000778 0.0000815 0.0000857 0.0000759 0.0000776 0.0000831 0.0000732 0.0000738
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fn1, fn2 = First and second resonant frequencies, AH1-res, AH2-res = First and second resonant amplitudes for horizontal motion, ψr1-res, ψr2-res = First and second resonant amplitudes for rocking motion
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