ABSTRACT

In this paper the influence of the presence of a rigid boundary in the soil mass on natural frequency and resonance amplitude are studied experimentally by conducting model block vibration tests in vertical mode. The tests are carried out in two different pits in the field: one with rigid base and other with a large depth simulating the half space. A concrete block of size 400x 400 x 100 mm is used as the model block and a Lazan type mechanical oscillator is used for inducing vibration in vertical direction. The soil layers of different thickness are prepared in the pits and the test conducted over the surface of each layer. In total 80 tests have been conducted in different thickness of layer and different static and dynamic loading combinations and several important observations are reported. The results obtained from these two tests pits are also compared with the test result of laboratory test in a tank available in the literature. Damping factor and the stiffness are calculated from the test results. The damping ratio for different layer thickness and different pit conditions are presented.

Keywords: Damping ratio, dynamic response, natural frequency, stiffness, vibration.

INTRODUCTION

The design of a machine foundation should be such that it leads to a safe and economical foundation block satisfying the operational requirements of machinery and installations and structural and psychological criteria. Vibrations of machine foundations induce elastic waves in soil (as it generates low strains) which may detrimentally effect surrounding buildings and their effects range from serious disturbances of working conditions for sensitive devices and people to visible structural damage. Therefore, in the design of foundations for such machines, it is important to have a reliable method to predict vibrations of the surrounding soil, structures and equipment. A successful machine foundation design requires a systematic use of principles of soil engineering, soil dynamics and theory of vibration. Machine foundations are constructed as rigid concrete blocks and the response to dynamic loads is completely determined by dynamic properties of the underlying soil. In soil media, natural soil deposition process has often resulted in a formation of a multi-layered system. The nature of dynamic loads and the non-homogeneity of soil make the problem of analysis and design of a foundation somewhat complex. Even though the magnitude of dynamic load is relatively small for machines, it needs special attention while designing since it is applied repetitively over a long periods of time and causes disturbance to the surrounding structures and people.

The criteria for design of machine foundations require that the natural frequency of the foundations should not coincide with the operational frequency of the machines: and the vibration amplitude should not exceed a given value.

Several methods (e.g. Mass-spring-dashpot model, Elastic half space theory, Lumped parameter model, Numerical methods) are available to determine the resonant frequency and peak displacement amplitude. Among the above methods the Lumped parameter model is a popular one because of it’s simplicity. The stiffness and damping parameters for different modes have been presented by (Lysmer and Richart (1966), Whiteman and Richart (1967), Richart et al (1970), Nagendra (1981), Sridharan et al (1990) to name a few. The stiffness and damping parameters are also influenced by the non-linearity of soil. Novak (1971) calculated the non-linear damping and the mass of the system from the experimentally obtained response curves.

Gazetas and Stoke (1991) discussed different types of *experimental* investigations related to vibrating foundations and also discussed the advantage and limitations of each method. They also indicated that case histories and field experiments are best since the propagation of elastic waves is not interrupted by the presence of artificial lateral boundaries as in laboratory tests in which spurious wave reflections on the wall may profoundly affect the measured radiation damping. But in the literature experimental work, case histories and field experiments on foundation vibration are limited.

Baidya and Sridharan (1994), Baidya and Muralikrishna (2001) studied the dynamic response of foundation response on a stratum underlain by a rigid layer using lumped parameter model and also suggested a method to estimate the equivalent stiffness and the equivalent damping for such systems.

Mark R. Svinkin (2002) developed the method for predicting complete vibration records of soil and structures prior to installation of foundations for impact machines. In particular, this method is most useful under non-uniform and complicated soil conditions for determination and verification of safe distance from machine foundations for buildings and facilities sensitive to vibration.

It can be seen from the above review that only a few have studied the response of foundation on natural soil system experimentally. At the same time, most of the literature related to experimental studies are to investigate the effect of shape, size, depth of embedment etc in the laboratory. This laboratory experiments may get influenced due to confining effect. Hence, in the present study, it is aimed to investigate experimentally in the field the dynamic response of the foundation on finite stratum underlain by rigid layer.

EXPERIMENTAL PROGRAMME

In the present study the effect of stratum thickness on the dynamic response of foundation soil system and the influence of presence of rigid boundary is proposed to investigate experimentally. Vertical vibration tests using mechanical oscillator (Lazan Type) on different stratum depths with different static weights, W and different eccentric settings, m_{e}e are proposed to simulate different foundations weight and dynamic load respectively. Detailed programme of the study is presented in Figure 1 and Table 1. Figure 1 shows the footing resting on a stratum of sand in two different pit boundary conditions where as Table 1 present the different stratum thickness considered in the investigation for two different boundary conditions of sand layer. The purpose of using three different boundary conditions is to observe the influence of the presence of rigid layer at a distance.

**Table 1.** Layer thickness and the parameters

considered in the experiments

**Figure 1.** Foundation resting on a sand layer underlain by (I) Rigid base (II) Same sand

TEST SET-UP AND TEST PROCEDURE

Two Test Pits: Pit-I, tests pit with rigid base at a depth, Pit-II, tests pit in the Half-space are constructed in the field to study the effect of presence of rigid boundary at a shallow depth within the soil on foundation response.

Test pit with rigid base at a depth (Pit-I)

Experiments were carried out to study the effect of presence of rigid boundary at a shallow depth within the soil on foundation response. To simulate the condition of soil layering only choice was to conduct the test in a tank or a pit of finite dimension. To conduct model test in the laboratory, an optimization was needed between tank and footing size to minimize the effects caused by restricting lateral boundary. Present study inspired for the field test to minimize such errors. Present investigation was carried out in a pit of size 2m x 2m x 1.9m, excavated at the adjoining area of the S.R. Sengupta Foundation Engineering Laboratory, I.I.T.-Kharagpur which is sufficiently larger (width is 5 times the width of the footing and volume is 7.6 m^{3}) than that required for the static condition. At the bottom of the pit, a 0.3 m concrete slab was casted and cured to represent a rigid base. Side of the pit was made of in-situ soil of density 18.0 kN/m^{2}. After casting and curing of concrete slab at the base, the pit was gradually filled up with sand and test conducted in each sand layer using two different static loads and four eccentric settings.

Tests pit in the 'Half-space' (Pit-II)

A pit of size 2m x 2m square and 1.6 m depths was excavated at the adjoining area of S.R. Sengupta Foundation Engineering Laboratory, I.I.T. Kharagpur. The distance of this pit from Pit-I was approximately 4.5 m the foundation of the laboratory building was at a distance of 2.5 m from the test pit. Suitability of the dimensions of the pit with respect to the size of the footing for possible boundary effects was considered. The side as well as bottom of the pit were made of local soil of density 18.0 kN/m^{2} and is assumed to extend infinite distance. The pit was filled with layer of sand of uniform density 17.0 kN/m^{2} with a different thickness as required in each test series.

Test Material

Sand was chosen as testing materials to form the layer due to the availability in the locality. Further, sand id easy to work with and maintain uniformity while preparing the layered beds.

Preparation of soil layer

The locally available river sand (medium fine sand, f = 36^{o} from direct shear test at density, g=17.0 kN/m^{3} was used to form finite sand layer of different thicknesses. To maintain a uniform condition throughout the test program, the empty pit was filled in steps of 200 mm thick layer and level the surface of sand layer. After leveling each layer was compacted using a compactor by constant compactive effort to achieve a density of approximately 17.0kN/m^{3}. Calculated amount of dry sand for 200 mm depth maintaining uniform density (17.0kN/m^{3}) was poured and compacted to bring it to 200 mm. For the first sand layer in the study, the 400 mm thick sand layer at the bottom was prepared in two steps of 200 mm. Thus five sand layers of different thickness (Table 1) were prepared and tests were conducted over the level surface of each layer.

Experimental Procedure

A model concrete footing of size 400 x 400 x 100 mm and a ‘Lazan type’ mechanical oscillator were used to conduct model block vibration tests in vertical mode. The concrete footing was placed centrally over the prepared soil layer. A rigid mild steel plate was fixed on the concrete footing to facilitate load-fixing arrangement. Oscillator was then placed over the plate and a number of mild steel ingots were placed on the top of the oscillator to provide required static weight. Whole set-up was then connected to act as a single unit. Proper care was taken to maintain the center of gravity of whole system and the footing to lie in the same vertical line. In this investigation, 8.0 and 8.9 kN static weights were used to simulate two different foundation weights and four different eccentric settings (0.0064 N-sec^{2}, 0.008 N-sec^{2} and 0.0096 N-sec^{2}, 0.011 N-sec^{2}) were used to simulate four different dynamic force level.

The oscillator was connected through a flexible shaft to a variable DC motor (3 H.P. frequency range up to 3000 rpm). A B&K piezoelectric-type vibration pickup (type 4370) was placed on top of the footing to measure the displacement amplitude with the B&K vibration meter (type 2511). Figure 2 shows the schematic diagram of the experimental set-up. The oscillator was then run slowly through a motor using speed control unit to avoid sudden application of high magnitude dynamic load. Thus the foundation was subjected to vibration in the vertical direction. Frequency and corresponding displacement amplitude of vibration were recorded by photo tachometer and vibration meter respectively. To obtain a foundation response and locate the resonant peak correctly, the displacement amplitudes were noted at a frequency interval approximately of 25 to 50 r.p.m. A sufficient time between two successive measurements has been given to reach equilibrium, which facilitates accurate measurement of frequency and the corresponding displacement amplitude. Finally, frequencies versus displacement amplitude curves were plotted for different layered systems.

**Figure 2.** Experimental Set-up with rigid-base.

RESULTS AND DISCUSSIONS

Frequency versus displacement amplitude response curves were obtained on different stratum depths for each static weight and each value of eccentric setting. Figure 3(a) presents the typical frequency versus displacement amplitude response curve obtained from the sand layer of depth 800 mm and for static weights of 8.0 kN. Figure 3(b) presents response curves obtained from the same sand layer but for static weight of 8.9 kN. It can be observed from the Figure 3(a) and Figure 3(b) that, with the increase in static weight, the natural frequency and the peak displacement amplitude decrease and with the increase in eccentric setting the natural frequency decreases and the resonant amplitude increases. These qualitative observations follow the already existing findings. But the main objective in the present study is to find out the effect of presence of bedrock at a depth on the dynamic response of the foundation. Nature of response curves, obtained from different sand layers is similar except the magnitude of displacement amplitude. Hence response curves for all depths of sand layers are not presented.

**Figure 3.** Frequency-Amplitude Response curves for 800 mm thick sand layer

(a) for static weight 8.0 kN and

(b) for static weight 8.9 kN

Figure 4 shows the variation of dynamic responses of the foundation with the variation of thickness of sand layer for static weight of 8.0 kN and eccentric setting of 0.0096 N-sec^{2} for the pit having rigid base at a depth and side is made of local soil. Figure 5 shows the variation of dynamic responses of the foundation with the variation of thickness of sand layer for static weight of 8.0 kN and eccentric setting of 0.0096 N-sec^{2} for the pit in half-space. It is observed from the Figures 4 that with the increase in thickness of soil layer above rigid base, the resonance frequency decreases and attain almost a constant value at a depth equal to 1200 mm. The nature of variation of amplitude with the variation of layer thickness is not conclusive though the resonant amplitude is minimum for the layer thickness 400 mm for the rigid base. With the increase in thickness of soil layer above the rigid base, stiffness of the system reduces which result in decrease in natural frequency. It can be seen from Figure 5 that with the increase in the thickness of sand layer at the top there is no significant change in the resonant frequency as well as the amplitude.

**Figure 4.** Frequency vs. Amplitude plots of different
stratum thickness for eccentricities 0.0096 N-sec^{2} and Static weights 8.0 kN over rigid base without side wall.

**Figure 5.** Frequency vs. Amplitude plots of different sand layers thickness for eccentricities 0.0096 N-sec^{2} and Static weights 8.0 kN in the half space.

Analysis of Test Results

The elasticity of the soil and the energy carried into the half space by waves traveling away from the vibrating footing (geometric damping) are accounted for and the response of such a system may be predicted using a mass-spring-dashpot model. For the analysis of dynamic response of footing resting on a finite layer underlain by a rigid layer, mass-spring-dashpot model with suitable modification is used.

**Figure 6.** Depth of Sand layer versus damping ratio for three different boundary conditions for static weight 8.0 kN and (a) eccentric setting 0.0064 N-sec^{2} (b) eccentric setting 0.008 N-sec^{2} (*c*) eccentric setting 0.0096 N-sec^{2} (d) eccentric setting 0.011 N-sec^{2}

Two basic parameters namely stiffness and damping ratio are essential to use mass-spring-dashpot model. Stiffness equation for a finite layer underlain by rigid layer given by Baidya and Muralikrishna (2001) are used. The damping is a mathematical abstraction used to represent the fact that the vibration energy does decay. It is difficult if not impossible to measure directly. The damping plays very important role to calculate the resonant amplitude.

Dynamic response of foundation resting on a relatively thick soil layer can be obtained satisfactorily using single-degree of freedom mass-spring dashpot model if appropriate values of spring and dashpot coefficient are used. It is seen from the experimental results that with the increase in thickness of layer, natural frequency reduces and the change is insignificant when thickness of the layer more than three times the width of the footing (attained almost half space value). Hence, footing resting on 1200 mm thick soil layer can be represented as a footing resting on half space and corresponding governing equation of motion using single degree of freedom mass-spring-dashpot model is given by

(1) |

where *F*(*t*) is the dynamic force = *m*_{e} e^{2} sin(wt), *K* is the spring stiffness, *m* is the mass of vibrating foundation, *c* is the dashpot coefficient, w is the vibrating frequency. Solution for displacement, *x* from the above equation can be obtained as

(2) |

where w_{n} is the natural frequency, *D* is the damping ratio,

(3) |

At frequency ratio

(4) |

the displacement becomes the maximum and it is given by

(5) |

When the damping in the system is low, the displacement becomes maximum at w = w_{n}. Using the peak displacement obtained from a 1200 mm thick soil layer in Eq. 5, damping ratio of the soil foundation system is obtained and the stiffness is obtained from the natural frequency of the foundation soil system as

(6) |

(7) |

Finally shear modulus of the soil is obtained from the natural frequency as

(8) |

Shear modulus of the soil is assumed to be unchanged for the finite layer made of the same soil. However spring and dashpot coefficients of the finite layer will vary with the variation of the thickness of the layer. Using recorded resonant frequency and amplitude in Eq. 5-6, stiffness and damping ratio are obtained for different test conditions of different thickness of sand layer and presented in Tables 2, 3 & 4 for pit over rigid base, tank over rigid base with side brick wall and pit in half space respectively.

**Table 2.** Resonant Frequency, Resonance Amplitude and Damping

for different stratum thickness. (Rigid base, Pit-I)

**Table 3.** Resonant Frequency, Resonance Amplitude and Damping

for different stratum thickness. (Halfspace, Pit-II)

**Table 4.** Resonant Frequency, Resonance Amplitude and Damping

for different stratum thickness in the tank. (Rigid base with side brick wall)

It can be seen from Table 2 that the stiffness reduces with the increase of thickness of the layer and it attains almost constant value when the thickness of the layer exceeds three times the width (3B) of the footing for the pits (pit-I and pit-II). It is observed from Table 3 that the change of stiffness and the damping is insignificant in case of pit in the half space. Further it can be seen that the damping in the entire suite of investigation is low and it’s variation with the variation of thickness of the layer is insignificant. Table 4 presents the stiffness and damping value for the test conducted in a tank in the laboratory. The change of damping ratio in three different test conditions is compared in Fig 6(a-d) & 7(a-d). The change in damping ratio with respect to half space value is around 10% for the pit in rigid base. But in case of the test in a tank in the laboratory, the damping ratio almost double with respect to the half space value.

**Figure 7.** Depth of Sand layer versus damping ratio for three different boundary conditions for static weight 8.9 kN and (*a*) eccentric setting 0.0064 N-sec^{2} (*b*) eccentric setting 0.008 N-sec^{2} (*c*) eccentric setting 0.0096 N-sec^{2} (*d*) eccentric setting 0.011 N-sec^{2}.

CONCLUSIONS

The effect on the dynamic response of the foundation soil system due to the presence of rigid layer is studied experimentally representing the foundation soil system by mass spring dashpot model and following conclusion are drawn.

Three different test pits are used (Rigid base without side wall, Rigid base with side brick wall and Half space) and same trends are observed in all the pits.

With the increase in stratum thickness, the stiffness of foundation soil system decrease significantly and at a larger depth, it attains almost a constant value, which corresponds to the half space value.

Damping values of foundation stratum underlain by rigid layer are significantly low.

The change in damping values obtained from the tests on different boundary conditions can be typically assumed due to the change in radiation damping.

The position of a rigid layer influences the natural frequency significantly, specially when it is at shallow depths, presence of a rigid layer at greater depths (more than 3B) does not have much influence on natural frequency but may have influence on damping.

Hence an overall conclusion that can be drawn from this study is that the natural frequency of the foundation soil system is influenced significantly due to the presence of rigid layer. Damping of a foundation soil system is a complicated parameter, particularly on non-homogeneous soil system. However, the change in damping due to presence of rigid boundary and pit wall could be quantified from the above study.

NOTATION

*c*_{c} = critical damping

*c* = dashpot coefficient

*D* = damping ratio (%)

*e* = eccentricities

*G* = shear modulus

*K* = stiffness

*m* = mass of footing

*m*_{e} = eccentric mass

*r* = equivalent circular radius

*x* = displacement

w_{n} = natural frequency

w = circular frequency of vibration

REFERENCES

Baidya, D.K. and Muralikrishna (2001), “ Investigation of Resonant Frequency and Amplitude of Vibrating Footing Resting on a Layered Soil System”, Geotechnical Testing Journal, ASTM, GTJODJ, Vol. 24, No. 4, December 2001 pp. 409-417.

Baidya, D.K. and Muralikrishna (2001), “Dynamic response of foundation on finite stratum-An experiment investigation”, Indian Geotechnical Journal 30(4), 2000, pp.327-350.

Baidya, D.K. and Sridharan, A. (1994), “The dynamic response of the foundations Resting on a stratum Underlain by a Rigid Layer”, proc. Indian Geotech. Conf., Warangal, India 1, 155-158.

Gazetas, G. and Stokoe II, K.H. (1991): “Vibration of embedded foundatios: Theory vs. Experiment”, Jl. Geotech. Engg., ASCE, 117(9), 1382-1401.

Lysmer, J. and Richart, F.E., Jr., (1966), “Dynamic response of footings to vertical loading”, Jl. Soil Mech. and Found. Engg. Div. ASCE, Vol. 92, SM 1, 65-91.

Mark R. Svinkin (2002), “ Predicting soil and structure vibration from impact machines”, Jl. Of Geotechnical and Geoenvironmental Engg. Vol. 128, No. 7, 2002, pp. 602-612.

Nagendra, M. V. and Sridharan, A. (1981), “Response of Circular Footing to Vertical Vibration”, Jl. Geotech. Engg. Div. ASCE, 107, 989-995.

Richart, F.E Jr., Hall, J.R. Jr. and Woods, R.D. (1970), “Vibrations of soils an foundations”, Printice-Hall, Inc. Englewood Cliffs, New Jersey.

Sridharan, A., Gandhi, N.S.V.V.S.J. and Suresh, S. (1990), “Stiffness coefficients of layered soil system”, Jl. of Geotech. Engg. ASCE, Vol. 116(4), 604-624.

Whitman, R.V. and Richart, F.E., (1967), “ Design Procedure for Dynamically Loaded Foundations”, Jl. Soil Mech. Found. Engg. Div., ASCE, 93(6), 169-173.

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