CHAPTER ONE INTRODUCTION

Transcription

1 CHAPTER ONE INTRODUCTION 1.1 General Liquefaction is a physical process that takes place during some earthquakes. Liquefaction may lead to ground failure. As a consequence of liquefaction, water-saturated, well sorted, fine grain sands and silts behave as viscous fluids. This behaviour is very different than solids behaviour. Liquefaction takes place when seismic shear waves pass through a saturated granular soil layer. These shear waves distort its granular structure, and cause some of its pore spaces to collapse. The collapse of the granular structure increases pore space water pressure. Furthermore, it also decreases the soil s shear strength. If pore space water pressure increases to the point where the soil s shear strength can no longer support the weight of the overlying soil, buildings, roads, houses, etc., then the soil will flow like a liquid. Consequently, extensive surface damage results. 1.2 Damage Cases due to Liquefaction Great number of damage cases had been reported due to soil liquefaction and many examples may be cited over the world. The literature lists eight types of failure commonly associated with soil liquefaction in earthquake: sand boils, flow failure of slopes, lateral spreads, ground oscillation, loss of bearing capacity, buoyant rise of build structures, ground settlement and failure of retaining walls. In the last years, numerous cases of collapsed buildings, cracked streets, failure of bridges and footings were recorded due to rupture water remains and ground damages (liquefaction). Plates (1-1) to (1-4) show examples of the deformations due to earthquakes. 1

2 Chapter One Introduction Plate (1-1) - Liquefaction in the recent subduction zone earthquake in Japan, 211( 211). Plate (1-2) - Liquefaction on Kilmore Street, New Zealand, 211 ( 211). 2

3 Chapter One Introduction Plate (1-3) - Overturning of building on shallow foundation in Turkey from 1999 Izmit earthquake (Rensselaer Polytechnic Institute, 29). Plate (1-4) - Failure of bridge due to liquefaction in Japan from 1995 Kobe earthquake (Rensselaer Polytechnic Institute, 29). 3

4 Chapter One Introduction 1.3 Purpose of the Study The main objectives of this research are: 1- Studying the generation pore water pressure in saturated sandy soils as a result of dynamic loading, especially for the foundation machinery. 2- Studying the possibility of liquefaction as a result of the dynamic loads. 3- Studying the factors affecting the occurrence of liquefaction including frequency and amplitude of the dynamic load, damping and density of sand. 1.4 Layout of Thesis The skeleton of thesis is divided into six chapters including, chapter one. Chapter two includes a survey of methods of dynamic analysis of soil problems with emphasis on saturated soils. The chapter also describes the phenomenon of soil liquefaction and development of pore water pressure during dynamic loading of soils. Chapter three presents recent developments in soil liquefaction engineering which include a definition, mechanisms and factors affecting of liquefaction. Moreover, ground failure resulting from soil liquefaction was reviewed also. Chapter four gives a brief description of the computer program and its capabilities. The governing equations for dynamic problems and finite element formulation were presented. Two previously solved soil dynamic problems were reanalyzed using the program QUAKE/W. Chapter five is devoted to present the results of analysis which include studying the effect of frequency and amplitude of dynamic load on generation of pore water pressure and propagation of liquefaction zones beneath foundations. Finally, Chapter six presents the conclusions and recommendations based on the results of analysis of chapter five. 4

5 CHAPTER TWO REVIEW OF LITERATURE 2.1 Introduction The term "Liquefaction" was originally used to refer to response of saturated loose sand to strains that result in flow slides. Recently, researchers have used this term to refer to a state of stress, especially in the response of sands under dynamic loading. Liquefaction is defined as the transformation of soil from the solid to the liquid state (Lenart, 28). This chapter presents a literature review of the currently accepted understanding of saturated soil behaviour under dynamic loads with emphasis on liquefaction. 2.2 Liquefaction Phenomenon The concept of liquefaction was first stated by Casagrande in Liquefaction happens in loose silt and sand that is below the water table. It does not happen in clay soil, which tends to stick together, and is uncommon in gravelly soils. It only happens in soils that are below the water table, where the spaces between the grains of sand and silt are filled with water. Dry soil does not liquefy. When an earthquake occurs, the shaking is so rapid and violent that the sand and silt grains try to compact, and they compress the spaces filled with water. The water pushes back and pressure builds up in the water until the silt and sand grains float in the water. When that happens, the soil behaves like a liquid and loses its ability to support the weight of whatever is lying above it. The pressurised water is forced up to the ground surface through the easiest way it can find often through cracks or crevasses in the ground. It takes silt and sand with it to create the sand boils or sand volcanoes seen on the ground. Sometimes sand boils are created under roads or lawn without the asphalt or lawn 5

6 Chapter Two Review of Literature breaking. As liquefaction occurs, water, silt and sand rises to the ground surface while the ground surface lowers or subsides. It doesn t do this evenly, which leaves the ground surface lumpy and bumpy. Buildings can sink or tilt to one side and things buried in the ground such as tanks, pipes and manholes can float upwards. Liquefaction does not happen in peat, because it is made up of plant material, rather than sand and silt. However, there may be sand or silt layers above or below the peat that can liquefy (Institute of Professional Engineers of New Zealand, 211). 2.3 Pore Water Pressure Build-up in Saturated Soils The extensive research had been performed in which laboratory specimens of saturated sand under zero driving shear stresses are subjected to either controlled cyclic stresses or strains. Many variables have been investigated including density of the sand, confining pressure, frequency of loading, shape of load cycle, method of sample preparation and cyclic loading history. Excess pore water pressure may build up under cyclic loading conditions that cause the effective stress to decrease. When soil is under an isotropically consolidated condition, the effective stress may reduce to zero when the excess pore water pressure continuously builds up. Seed and Lee (1966) defined initial liquefaction as a point at which the increase of excess pore water pressure is equal to the initial confining stress. A number of approaches and a large amount of laboratory tests have been conducted on the liquefaction potential and prediction of the excess pore water pressure under earthquake loading conditions. One important and commonly accepted approach is the cyclic stress method developed by Seed et al., (1976). In this method, the earthquake loading, expressed in terms of equivalent cyclic shear stress, is compared with the liquefaction resistance of 6

7 Chapter Two Review of Literature the soil, also expressed in terms of cyclic shear stress. When the earthquake loading exceeds the resistance, liquefaction or maximum excess pore water pressure is expected to occur. The level of excess pore water pressure development can also be predicted based on the cyclic stress approach in which the excess pore water pressure is directly related to the amplitude of cyclic stress and the number of the stress cycles. Figure (2-1) illustrates a typical normalized relationship between the cyclic ratio and the pore pressure ratio. The cyclic ratio is the ratio of the number of cycles applied N divided by the number of cycles required for liquefaction N L ; that is, N/N L. Pore Pressure Ratio, ru Cyclic Ratio, N/N L = r N Fig. (2-1) - Rate of pore water pressure build up in cyclic simple shear test (Seed et al., 1976). This curve has been adopted as a convenient basis for predicting the rate of pore pressure generation for the following development. If the excess pore pressure ratio, r u is expressed as the ratio of the peak excess hydrostatic pore-water pressure generated by the earthquake u g to the initial effective confining pressure, σ oˊ acting on the sand and the cycle ratio, r N, the following expression may be shown to fit the characteristic shape of the curves shown in Figure (2-1). 7

8 Chapter Two Review of Literature r N = [ (1 cos π r u )] α (2-1) where α is a function of the soil properties and test conditions. For the dashed line in Figure (2-1), a value of α =.7 provides the best fit value relating r N and r u : in fact, the dashed line was plotted from Equation (2-1) with α =.7. Where required, the pore pressure ratio r u may be expressed in terms of the cycle ratio, r N, by the relationship: 1/ α r u = arc sin (2 r N - 1 )..(2-2) As a consequence of the applied cyclic stresses, the structure of the cohesionless soil tends to become more compact with a resulting transfer of stress to the pore water and a reduction in stress on the soil grains. As a result, the soil grain structure rebounds to the extent required keeping the volume constant, and this interplay of volume reduction and soil structure rebound determines the magnitude of the increase in pore water pressure in the soil. The basic phenomenon is shown schematically in Figure (2-2), (Seed, 1979). Fig. (2-2) - Schematic illustration of the mechanism of pore pressure generation during cyclic loading (after Seed, 1979). 8

9 Chapter Two Review of Literature The mechanism can be quantified so that the pore pressure increases due to any given sequence of stress applications can be computed from knowledge of the stress - strain characteristics and the rebound characteristics of the sand due to stress reduction. The volume changes that would have occured under drained conditions are reflected in a pore pressure increase in accordance with the swelling characteristics of the sand skeleton (Castro, 1987). During cyclic loading, the pore water pressure rises. To find the mean normal effective pressure p during the rise in pore water pressure, the equivalent number of cycles for the earthquake is determined first from the magnitude using Figure (2-3), Then the pore water pressure ratio r u is found from Figure (2-4) using the equivalent number of cycles and the cyclic stress ratio CSR. Also, CSR is used to determine the factor of safety against liquefaction as follows: F.S =. (2-3) where: CRR: Cyclic Resistance Ratio, and CSR: Cyclic Stress Ratio. Fig. (2-3) - Number of equivalent uniform cycles N eq for earthquakes of different magnitudes (Das and Ramana, 21). Fig. (2-4) - Cyclic stress ratio-pore pressure relationship for different numbers of cycles (Ansal and Erken, 1989). 9

10 Chapter Two Review of Literature 2.4 Pore Water Pressure Models The first model for predicting pore water pressures generated by cyclic loading was proposed in 1975 and 1976 by Martin, Finn and Seed. This model was coupled to a procedure for dynamic analysis by Finn, Lee and Martin in 1976, giving the first method for dynamic effective stress analysis. A quantitative relationship between volume reductions occuring during drained cyclic tests and the progressive increase of pore water pressure during undrained cyclic tests had been developed. Using of this relationship enables the build-up of pore water pressure during cyclic loading to be computed theoretically using basic effective stress parameters of the sand (Martin et al., 1975). Seed et al. (1976) carried out analytical procedures making the assumption that, during the period of earthquake shaking, there would be no significant dissipation or redistribution of pore-water pressures in the soil mass. The analytical procedure applies for evaluating the general characteristics of pore-water pressure build up and subsequent dissipation in sand deposits, both during and after a period of earthquake shaking. In layers of fine sand, excess hydrostatic pressures may persist for 1 hr or more after an earthquake. Evidence of subsurface liquefaction may not appear at the ground surface until several minutes after the shaking has stopped and the critical condition at the ground surface may not develop until 1 min-3 min after the earthquake. For coarse sands and gravels with no impedance of drainage due to the presence of sand seams or layers, pore pressures generated by earthquake shaking may dissipate so rapidly that no detrimental buildup of pore pressure or a condition approaching liquefaction can be developed. Improving the drainage capability of a sand deposit by the installation of a highly pervious continuous drainage system may thus provide an effective means of stabilizing a potentially unstable deposit. 1

11 Chapter Two Review of Literature Analysis of the type described also provide the means for determining whether subsurface liquefaction will have any serious effect on structure supported near the ground surface. 2.5 Dynamic Soil Structure Interaction The analysis of the dynamic interaction of soils and structures changes substantially as the specific physical problem changes. There are at least four dynamic soil-structure interaction problems (Christian and Hall, 1982): 1. Vibrating machinery usually is relatively small compared to the extent of the founding soil, and the important conditions are usually those during sustained operation of the equipment. Therefore, a steady state solution for a structure found at or near the surface of a half-space is usually satisfactory. The machinery operates in limited and known frequencies. 2. Blast monitoring or blast protection involves a transient problem with very few, usually one, pulses of overpressure. Steady state response and the behaviour at large times are not especially important. 3. Earthquakes involve long input signals that have rich frequency contents and many cycles of load reversal. Because the waves propagate over a considerable distance, the modelling of large and complicated geometries is an important issue. The input motion is often not well known and is often specified at an unknown or inconvenient location. 4. Large offshore structures experience some transient response, but the steady state response to wave loading is most significant for fatigue problems. The soil behaviour under dynamic loading depends on many factors, including (Daghigh, 1993): 11

12 Chapter Two Review of Literature 1. The nature of the soil (permeability, relative density, fabric, etc...). 2. The environment of the soil (static stress state and water pressure). 3. The nature of the dynamic loading (strain magnitude, strain rate, and number of cycles of loading). 2.6 Methods of Dynamic Analysis of Soils A number of different methods are available for the analysis of the dynamic response of saturated cohesionless soils to dynamic loading. The methods differ in the simplifying assumptions that are made, in the representation of the stress- strain relations of soils, in the manner in which the development of pore water pressure is taken into account and in the methods used to integrate the equations of motion. They may be classified into two main categories; total stress methods and effective stress methods. In total stress methods, no explicit account is taken of the effect of changes in pore water pressures. In effective stress methods, changes in pore water pressures are considered. The effects of dynamic soil - structure interaction (DSSI) shall be taken into account where (load - deflection) effects are significant, for structures with massive or deep-seated foundations (such as bridge abutments and gravity walls, piles, diaphragms and caissons), for slender tall structures and for structures supported on very soft soil, with average shear wave velocity less than 1 m/s. Two types of DSSI are commonly referred to in the literature (Srbulov, 28): 1. Kinematic interaction is caused by inability of a foundation to follow ground motion due to greater foundation stiffness in comparison with ground stiffness. In effect, stiff foundation filters high frequency ground motion to an averaged translational and rotational foundation motion. Average values are smaller than the 12

13 Chapter Two Review of Literature maximum values and therefore kinematic interaction is beneficial except if averaged motion results in significant rotation and rocking of a foundation. 2. Inertial interaction is caused by the existence of structural and foundation masses. Seismic energy transferred into a structure is dissipated by material damping and radiated back into the ground causing superposition of incoming and outgoing ground waves. As a result, the ground motion around a foundation can be attenuated or amplified, depending on a variety of factors. The most important factor in determining the response is the ratio between the fundamental period of a foundation and the fundamental period of adjacent ground in the free field. The ratio of unity indicates resonance condition between foundation and its adjacent ground, which is to be avoided. 2.7 Vibration Analysis of Machine Foundation Although a machine foundation has six degres of freedom, it is assumed to have a single degree of freedom for a simplified analysis. In this case, the mass of the system lumps together the mass of the machine and the mass of foundation. The total mass acts at the centre of gravity of the system. The mass is under the supporting action of the soil. The elastic action can be lumped together into a single elastic spring with a stiffness k. Likewise, all the resistance to motion is lumped into the damping coefficient c. Thus, the machine foundation reduces to a single mass having one degree of freedom Determination of parameters For vibration analysis of a machine foundation, the parameters m, c and k are required. These parameters can be determined as below. 13

14 Chapter Two Review of Literature (1) Mass (m). When a machine vibrates, some portion of the supporting soil mass also vibrates. The vibrating soil mass is known as the participating mass or in-phase soil mass. Therefore, the total mass of the system is equal to the mass of the foundation block and machine m f and the mass m s of the participating soil. Thus: m = m f + m s.. (2-4) Unfortunately, there is no rational method to determine the magnitude of m s. It is usually related to the mass of the soil in the pressure bulb. The value of m s generally varies between zero and m f. In other word, the total mass (m) varies between m f and 2m f in most cases. (2) Spring Stiffness (k). The spring stiffness depends upon the type of soil, embedment of the foundation block, the contact area and the contact pressure distribution. The following methods are commonly used (Arora, 28): (a) Laboratory test. A triaxial test with vertical vibrations is conducted to determine Young's modulus E. Alternatively, the modulus of rigidity G is determined by conducting the test under torsional vibration and E is obtained indirectly from the relation E = where υ is Poisson s ratio. The stiffness k is determined as: k = (2-5) where: L = length of the specimen. (b) Barkan's method. The stiffness can also be obtained from the value of E using the following relation given by (Barkan, 1962).. (2-6) where: A= base area of the machine. 14

15 Chapter Two Review of Literature (c) Plate load test. A repeated plate load test is conducted and the stiffness of the soil in the test (k p ) is found as the slope of the loaddeformation curve. (d) Resonance test. The resonance frequency (f n ) is obtained using a vibrator of mass m set up on a steel plate supported on the ground. The spring stiffness is obtained from the relation: k = 4 f n. m...(2-7) (3) Damping constant c. The damping ratio D for an underdamped system can be determined in the laboratory. Vibration response is plotted and the logarithmic decrement δ is found from the plot as (Arora, 28): δ = log ( )..(2-8) where Z l and Z 2 are amplitudes of two consecutive cycles of an amplitude response curve. The damping ratio D and the logarithmic decrement δ are related as: δ.(2-9) Determination of natural frequency, The natural frequency of the foundation-soil system can be determined using the theory of vibration. The equation of motion, neglecting damping, is:- m + k. z = F o sin ωt. (2-1) m =mass of machine, foundation and the participation soil, and k = equivalent spring constant of the soil. The natural frequency of the system is given by:-. (2-11) 15

16 Chapter Two Review of Literature where : is the natural frequency in radians per second. f n =... (2-12) where f n is natural frequency in cycles per second. Thus =..(2-13) where : m f = mass of machine foundation, and m s = mass of the participating soil mass. Barkan (1962) gave the following relation for the natural frequency. (2-14) where : C u = coefficient of elastic uniform compression, A = contact area of foundation with soil. The maximum amplitude is given by Z max =. (2-15) where : F o = exciting force, and r = frequency ratio / The coefficient of elastic uniform compression C u depends upon the soil type. It can be obtained from the following relation: C u = (2-16) As it is evident, the coefficient varies inversely proportional to the square root of the base area of the foundation. 2.8 Previous Analyses of Soil Dynamics Problems Hamada et al. (1986) introduced a simple empirical equation for predicting the liquefaction induced lateral ground deformations only in terms of ground slope and thickness of liquefied soil layer. This equation 16

17 Chapter Two Review of Literature was based on the regression analysis of 6 earthquake case histories, mostly from Noshiro-Japan, and it was expressed as: D h =.75 H 1/2 θ 1/3. (2-17) where: D h : is the predicted horizontal ground displacement (m), H : is the thickness of liquefied zone (m), (when more than one sub-layer liquefies, H is measured as the distance from the top-most to the bottommost liquefied sub-layers), and θ is the larger slope of either ground surface or liquefied zone lower boundary (%). Elgamal et al. (1998) developed constitutive model to reproduce salient aspects associated with seismically induced soil liquefaction. Attention was mainly focused on the deviatoric (shear) stress-strain response mechanism. Soil shear behaviour during liquefaction is modelled to display a significant regain in stiffness and strength with the increase in deformation during each cycle of applied load. This behaviour appears to play a major role in dictating the magnitude of shear deformations as observed in laboratory tests and manifested in acceleration records from earthquakes and centrifuge experiments. A new constitutive model is developed to model cyclic shear behaviour during liquefaction. Fattah (1999) implement the cap plasticity model and ALTERNAT model to solve some problems in soil dynamics. The ALTERNAT model describes most of the sand behaviour features such as stress dilatancy, cyclic mobility and anisotropy. This model can be used to predict the liquefaction potential of sand subjected to dynamic loading. Dynamic analysis of the standard problem of a horizontal soil layer for depth 15.2 m subjected to a base motion of the N-S component of El-Centro earthquake was carried out. The layer was modelled by ten four noded isoparametric elements. The predicted pore water pressure distribution with depth of the 17

18 Chapter Two Review of Literature layer is shown in Figure (2-5). Pore pressure time history at 5 m depth is presented in Figure (2-6). The ALTERNAT model results continue to oscillate around a constant value after this condition. This can be understood since the ALTERNAT model takes into account the cyclic mobility which takes place in the layer after the initial liquefaction occurs. Besides, dilation was also accounted for in the ALTERNAT model which produces negative pore pressure and helps stabilization of the soil under the load. Dilation occurs with limited strain potential due to the remaining resistance of the soil to deformation. Fig. (2-5) - Pore water pressure profile through a horizontal sand layer subjected to an earthquake at base (after Fattah, 1999). Zienkiewicz et al. (1982). ALTERNAT model. Fig. (2-6) - Pore pressure time history at a depth of 5 m (after Fattah, 1999). 18

19 Chapter Two Review of Literature Jawdat (2) adopting (u-p) formulation, analyzed some soil dynamics problems using different types of dynamic loads, these are cyclic 'repeated' loading and earthquake loading. Comparison between coupled and uncoupled media revealed the importance of coupling between solid skeleton and generated pore pressure, which represents more accurate idealization of soils. The bounding surface model showed a good incapable of tracing the movement of soils under the applied dynamic loads. Bounding surface model is a robustness model for predicting soil behaviour in dynamic loading. In spite of the lack in soil parameters required, its results of displacements gave a logical impression for the soil behaviour, while its result of excess pore pressure show more sensitivity to model parameters. Savage and Safak (21) investigated the potential of finite-element wave propagation methods for modelling seismic response of multistory buildings. Plane strain condition was assumed in the analysis. General sketch of the configuration for finite-element model of wave propagation in a multistory building and layered substrata is shown in Figure (2-7). Fig. (2-7) - General sketch of the configuration for modelling wave propagation in a multistory building and layered substrate (Savage and Safak, 21). 19

20 Chapter Two Review of Literature Vertically propagating waves emanate from the shear wave source was shown in this figure. The source is situated, so that, the generated wave front will become plane and vertically incident at the interface between the multistoried building and its substrate. The problem is considered plane strain. Excluding the 1-story building, the finite element mesh extends from to 4 meters in the x-direction and from to 195 m in the Y- direction and is laterally restrained. The 1-story building is 2 m wide and 36 m high. Here 1-cycle 2-second duration impulsive sinusoidal forces of sufficient magnitude to generate the horizontal nodal displacement history are applied in the x-direction. This results in an essentially plane-wave at the bedrock-soil interface from the similarity of wave histories Figure (2-8) which shows calculated acceleration time histories for the 1st soil and the foundation. With the exception of the foundation, all time histories in Figure (2-8) are from nodes in the upper soil layer and the center of each story. (a) Upper soil (b) Foundation Fig. (2-8) - Acceleration time histories for (a) the upper soil, (b) the foundation (after Savage and Safak, 21). Amini and Duan (22) described a numerical model which is used to study the soil liquefaction resistance at high confining pressures. A 2

21 Chapter Two Review of Literature two-dimensional numerical model was set up. Base accelerations with different magnitudes and frequencies were applied to the model. The pore water pressure and effective stress at different depths in the model were monitored during shaking. It was found that, soil liquefaction resistance increases with the increasing of confining pressure at large depths. At a large acceleration magnitude, the liquefaction can occur at virtually any depth. It was concluded that at a lower frequency, the liquefaction occurred faster at large depths. Ashford et al. (24) described a pilot test program that was carried out to determine the appropriate charge weight, delay, and pattern required to induce liquefaction for full-scale testing of deep foundations. The results of this investigation confirmed that controlled blasting techniques could successfully be used to induce liquefaction in a well-defined, limited area for field-testing purposes. The tests also confirmed that liquefaction could be induced at least two times at the same site with nearly identical results. Excess pore pressure ratios greater than.8 were typically maintained for at least 4 minutes after blasting. The test results indicated that excess pore pressure ratios produced by blasting could be predicted with reasonable accuracy when single blast charges were used. However, for multiple blast charges, measured excess pressures were significantly higher than would have been predicted for a single blast with the same charge weight. The measured particle velocity attenuated more rapidly with scaled distance than would be expected based on the upper bound relationship developed from previous case histories. Settlement was typically about 2.5% of the liquefied thickness, and about 85% of the settlement occurred within 3 min after the blast. Zhang et al. (24) presented the approach that combines available standard penetration test and cone penetration test based methods to 21

22 Chapter Two Review of Literature evaluate liquefaction potential with laboratory test results for clean sands to estimate the potential maximum cyclic shear strains for saturated sandy soils under seismic loading. A lateral displacement index is then introduced, which is obtained by integrating the maximum cyclic shear strains with depth. Empirical correlations from case history data are proposed between actual lateral displacement, the lateral displacement index, and geometric parameters characterizing ground geometry for gently sloping ground without a free face, level ground with a free face, and gently sloping ground with a free face. The proposed approach can be applied to obtain preliminary estimates of the magnitude of lateral displacements associated with a liquefaction-induced lateral spread. Given the complexity of liquefaction-induced lateral spreads, considerable variations in magnitude and distribution of lateral displacements are expected. Generally, the calculated lateral displacements using the proposed approach for the available case histories showed variations between 5 and 2% of measured values. The accuracy of measured lateral displacements for most case histories is about ±.1 to ± 1.92 m. Sitharam et al. (24) studied methods of determining the dynamic properties as well as potential for liquefaction of soils. Parameters affecting the dynamic properties and liquefaction have been brought out. A simple procedure of obtaining the dynamic properties of layered ground has been highlighted. Results of a series of cyclic triaxial tests on liquefiable sands collected from the sites close to the Sabarmati river belt have been presented. Simple method was used to obtain the equivalent modulus of layered system. Cyclic strain-controlled triaxial tests to evaluate the dynamic properties and liquefaction potential of sands have been carried out. It has been brought out that the material immediately beneath the foundation plays a dominant role in controlling the dynamic response. 22

23 Chapter Two Review of Literature Material at a depth greater than twice the width of the foundation plays an insignificant role. A major reduction in the shear modulus and a corresponding increase in the damping of sand occur in the large shear strain range. As the initial densities of sand increase, the shear modulus shows clearly an increasing trend. However, more or less the same values of shear modulus occur beyond.5% shear strain level irrespective of their initial density. As a result of application of cyclic loads on the soils, pore water pressure builds up steadily and reaches initially applied confining pressure depending on the magnitude of cyclic shear strain as well as the density of the soil. At higher cyclic shear strain amplitudes, the pore water pressure builds up fast and there is triggering of liquefaction at lower cycles. Al-Bawwab (25) developed both probabilistically and deterministically based semi-empirical methods for the assessment of liquefaction-induced lateral ground spreading. By employing maximum likelihood, as well as linear and nonlinear regression assessment techniques, new sets of probabilistic models are developed addressing the effects of: i) site boundary conditions, ii) seismic soil liquefaction triggering, iii) inertial forces acting during the earthquake, and iv) gravitational forces acting before, during, and after the earthquake on the liquefaction-induced lateral ground spreading phenomenon. The resultant models were developed for sloping sites with and without free-face, and level sites with free-face boundary conditions enabling to model all sources of uncertainties in the estimation of the magnitude of seismic liquefactioninduced lateral ground deformations. The new probabilistic models were also introduced as a performance-based analysis tool. It was found that: 23

24 Chapter Two Review of Literature Previously available field case history data have been re-evaluated with current knowledge and understanding of seismicity, liquefaction triggering and liquefaction-induced lateral spreading mechanisms, With this greatly enhanced database, higher standards were set for acceptability of case history data, and data not meeting these standards were deleted. The result is an enlarged database of higher overall quality, More advanced probabilistic tools, likelihood analytical methods, were used to develop and evaluate new correlations. These methods allowed for separate treatment of different sources of uncertainty, and also allowed assessment of more contributing variables/parameters than previous studies, Uncertainties within the measurement/estimation of the main descriptive variables were probabilistically assessed and incorporated into the model (i.e.: less accurate case histories had less weight on the overall performance), The probabilistic nature of the proposed model enables an efficient framework in performance-based design decisions (i.e.: at a selected site after a design earthquake, one can estimate the probability of the lateral ground deformations to be greater than a threshold value). Lee (26) conducted test to simulate liquefaction in saturated sandy soil induced by nearby controlled blasts. Involving the modelling of a three-dimensional half-space soil region with pre-defined, embedded, and strategically located explosive charges to be detonated at specific time intervals. A new geo-material model was applied to evaluate the liquefaction potential of saturated sandy soil subjected to sequential blast environments. Explosive charge detonation and pressure development characteristics were modelled using proven and accepted modelling 24

25 Chapter Two Review of Literature techniques. As explosive charges were detonated in a pre-defined order, development of pore water pressure, volumetric (compressive) strains, shear strains, and particle accelerations compared against blast-test data gathered at the Fraser River Delta region of Vancouver, British Columbia in May of 25 to validate and verify the modelling procedure s ability to simulate and predict blast-induced liquefaction events. The baseline model calibrated to blast series 3 was applied to predict liquefaction development for blast series 1 with non-uniformly placed and nearly one-third of the explosives relative to blast series 3. Minor and expected deviations were observed in the comparison between predicted and measured test data due to faulty placement of certain explosives in blast series 1. Dawood (26) studied two different approaches of absorbing boundaries. The first by using infinite elements and the second by using viscous boundaries method. Baghdad metro line was considered. The results were compared for three cases; the first one using finite elements only, the second using mapped infinite elements and the third one using viscous boundaries. It was concluded that: The transmitting boundary absorbs most of the incident energy. The distinct reflections observed in the "fixed boundaries" case disappear in the "transmitted boundaries" case. This is true for both cases of using viscous boundaries or mapped infinite elements. The viscous boundaries are more effective in absorbing the waves resulting from dynamic loads than mapped infinite elements. This is clear when comparing the results of both types with those of transient infinite elements. The type and location of the dynamic load are two controlling factors in deciding the importance of using infinite boundaries. It was found that, the results are greatly affected when earthquake is applied as 25

26 Chapter Two Review of Literature base motion or a pressure load is applied at the surface ground than the case of applying impulse load on the tunnel same result was recorded by (Fattah et al., 28). Anandarajah (28) developed model within the framework of a microstructural theory known as the sliding rolling theory. The resulting model falls within the definition of multimechanism models. The model was shown to satisfactorily represent the drained and undrained behaviours under monotonic loading. The framework used in the model allows extension to describe the behaviour under cyclic loading. Specifically, the model is further developed for representing the undrained behaviour of granular materials under one- and two-way cyclic loading, some of which cause liquefaction resulting in large strain accumulations and the others lead to limited pore pressure and strain accumulations. The validity of the model was verified using triaxial data on Nevada sand. A multimechanism model, developed based on the results of the sliding rolling theory, was presented. It was shown that with minor revisions, the model was capable of capturing the behaviour of granular materials both under monotonic and cyclic loading. In particular, the model is shown to be capable of capturing the behaviour under two-way undrained cycling, which results in liquefaction and strain accumulations, and under one-way undrained cycling, which results in limited pore pressure and strain accumulations. Jin et al. (28) tested samples of saturated loose sand under bidirectional cyclic loading to characterize liquefaction and cyclic failure, by using an advanced soil static and dynamic universal triaxial and torsional shear apparatus. Tests were performed with two cyclic components involving the horizontal shear stress (torsional shear stress) and the vertical shear stress (stress difference between vertical normal stress and horizontal normal stress) to provide an approximate presentation of wave or seismic 26

27 Chapter Two Review of Literature loading conditions. Samples were consolidated under various initial static horizontal shear stresses and subsequently subjected to a specified level of dynamic loading. Three stress levels of dynamic load were concerned. Results showed that the developed pore water pressure decreases linearly as initial static horizontal shear stress increases and decreases exponentially as consolidation stress ratio increases. Worthen (29) carried out a study on the characterization of the liquefaction potential of fine-grained soils, based on plasticity characteristics using the Chinese criteria. Recent research showed that such criteria are ineffective. In addition, the current liquefaction models do not account for the confining pressure of in-situ soil or the strength of the earthquake. The study used the critical state soil mechanics framework, which emphasizes that the shear strength and deformation behaviour of soil depends on changes in volume and confining stress. Depending on their combination, a soil aggregate may fracture into clastic debris, fail with fault planes, or yield plastically. Lu et al. (21) investigated the responses of saturated sand under horizontal vibration loading induced by a bucket foundation. The saturated sand liquefies gradually since the vibration loading is applied on. The maximum displacement on the surface of sand layer occurs near the loading end and in this zone; the sand is compressed and moves upwards. The liquefaction zone was developed from the upper part near the loading side and stopped gradually under the vibrating loading on one side of the saturated sand; liquefaction occurs first near the loading end and then develops faraway. The deformation becomes up-heave near the loading end and degrades faraway gradually. It was found that: 27

28 Chapter Two Review of Literature The liquefaction and the deformation develop fast with the increase of the loading amplitude and the frequency and the decrease of the modulus. The liquefaction may occur under vibration loading on the side from the foundation side to a finite distance. It needs to be considered in the design of platform. 2.9 General Comments It can be concluded from previous studies on liquefaction show that most of them considered on earthquake induced liquefaction and that little studies talked liquefaction caused by mechanical factors. Some equipments or heavy machines used during construction particularly on saturated sandy soil might cause some vibration and consequently, a loose soil will be ready for liquefaction. 28

29 CHAPTER THREE RECENT DEVELOPMENTS IN SOIL LIQUEFACTION ENGINEERING 3.1 Introduction Soil liquefaction is a major cause of damage during earthquakes. Modern engineering treatment of liquefaction related issues evolved initially in the wake of the two devastating earthquakes of the 1964 and 211 Japan. Over the nearly four decades that have followed, a significant progress has occurred. Initially, this progress was largely confined to improve the ability to assess the likelihood of initiation of liquefaction in clean, sandy soils. Additional potential problems associated with both silty and gravelly soils, and the important additional issues of post-liquefaction strength and stress deformation behaviour also began to attract increased attention (Seed et al., 23). 3.2 Liquefaction Definitions Liquefaction is an engineering phenomenon referring to the total and sharp loss of soil shear strength due to rapid pore water pressure build -up. Seismic soil liquefaction occurs when the structure of a loose and saturated soil breaks down due to some rapidly applied loading. As the soil structure breaks down, the loosely-packed individual soil particles attempt to move into a denser and more stable configuration. During an earthquake event, however, there is not enough time for the water within the pores of the soil to be squeezed out or dissipated, but instead, water is trapped in the soil pores and prevents the soil particles moving to a denser state. Simultaneously, this is accompanied by an increase in pore water pressure which reduces the contact forces between the individual soil particles, and consequently, resulting in softening and weakening of the soil deposit to a 29

30 Chapter Three Recent Developments in Soil Liquefaction Engineering considerable extent. Because of this high pore water pressure, the contact forces become very small or almost zero, and in an extreme case, the excess pore water pressure may increase to a level that may break the particle-to-particle contact. In such cases, the soil will have very little or no resistance to shearing, and will exhibit a behaviour more like a viscous liquid than a solid body. If it is expressed in Mohr Columb s soil shear strength formulation, it is given as (Das and Ramana, 21): = с' + σ' v tan ϕˊ.. (3-1) σ' v = σ v u.. (3-2) where: τ : soil shear strength, с' : undrained soil cohesion, σ v σ' v : total vertical stress, : effective vertical stress, u : pore water pressure, and ϕˊ : effective angle of soil internal friction. During an earthquake shaking, the applied stresses will generate an increase in the excess pore water pressure, and if pore pressure and total stresses equates and hence leading to the effective vertical stress to be almost zero (σ v ' ), the soil will loose its consistency and will "liquefy", resulting in significant deformations (Das and Ramana, 21). High excess pore water pressures leading to soil liquefaction can cause pore water to flow rapidly up to the ground surface. This pore water flow can occur both during and after an earthquake event, and if the flow is strong enough, it can carry sand particles through cracks up to the top surface, where they are deposited in the form of sand boils (Al Bawwab, 25). 3

31 Chapter Three Recent Developments in Soil Liquefaction Engineering 3.3 Liquefaction Mechanisms Liquefaction has been used to define two mainly related but different soil behaviours during earthquakes, namely flow liquefaction and cyclic softening. Since both phenomena can have quite similar consequences, it is difficult to distinguish between them. However, the mechanisms behind them are rather different Flow liquefaction Flow liquefaction is a phenomenon in which the equilibrium is destroyed by static or dynamic loads in a soil deposit with low residual strength, which is defined as the strength of soils under large strain levels. Static loading, for example, can be applied by new buildings on a slope that exert additional forces on the soil beneath the foundations. Earthquakes, blasting, and pile driving are all examples of dynamic loads that could trigger flow liquefaction. Once triggered, the strength of a soil susceptible to flow liquefaction is no longer sufficient to withstand the static stresses that were acting on the soil before the disturbance. Failures caused by flow liquefaction are often characterized by large and rapid movements which can lead to disastrous consequences (Youd and Idriss, 21) Cyclic softening Cyclic softening is another phenomenon, triggered by cyclic loading, occurring in soil deposits with static shear stresses lower than the soil strength. Deformations due to cyclic softening develop incrementally because of static and dynamic stresses that exist during an earthquake. Two main engineering terms can be used to define the cyclic softening phenomenon, as follows (Youd and Idriss, 21): Cyclic mobility Requires undrained cyclic loading during which shear stresses are always greater than zero; i.e. no shear stress reversals develop. 31

32 Chapter Three Recent Developments in Soil Liquefaction Engineering Zero effective stress will not develop. Deformations during cyclic loading will stabilize, unless the soil is very loose and flow liquefaction is triggered. Can occur in almost any sand provided that the cyclic loading is sufficiently large in size and duration, but no shear stress reversal occurs. Clayey soils can experience cyclic mobility, but deformations are usually controlled by rate effects (creep) Cyclic liquefaction Requires undrained cyclic loading during which shear stresses reversals occur or zero shear stress can develop; i.e. occurs when insitu static shear stresses are low compared to cyclic shear stresses. Requires sufficient undrained cyclic loading to allow effective stress to reach essentially zero. At the point of zero effective stress, no shear stress exists. When shear stress is applied, pore water pressure drops as the material tends to dilate, but a very soft initial stress strain response can develop resulting in large deformations. Deformations during cyclic loading can accumulate to large values, but generally stabilize when cyclic loading stops. 3.4 Liquefiable Soil Types The first step in engineering assessment of the potential for initiation of soil liquefaction is the determination of whether or not soils of potentially liquefiable nature are present at a site. This, in turn, raises the important question regarding which types of soils are potentially vulnerable to soil liquefaction. 32

33 Chapter Three Recent Developments in Soil Liquefaction Engineering It has long been recognized that relatively clean sandy soils, with few fines, are potentially vulnerable to the seismically induced liquefaction. There has, however, been significant controversy and confusion regarding the liquefaction potential of silty soils (and silty/clayey soils), and also of coarser, gravelly soils (Seed et al., 23). The most widely criteria used for defining potentially liquefiable soil is Modified Chinese Criteria Wang (1979). According to these criteria, fine (cohesive) soils that plot above the A-line are considered to be of potentially liquefiable type and character if: (1) There are less than 15% clay fines (based on the Chinese definition of clay sizes as less than.5 mm), (2) A liquid limit (LL) 35%, and (3) In-situ water content greater than or equal to 9% of the liquid limit. Andrews and Martin (2) re-evaluated the liquefaction field case histories from the database of Wang (1979), as well as a number of subsequent earthquakes, and have transposed the Modified Chinese Criteria to U.S. conventions (with clay sizes defined as those less than about.2 mm). Their findings are summarized in Table (3-1). Table (3-1) - Liquefaction susceptibility of silty and clayey sands (after Andrews and Martin, 2). Liquid Limit < 32 % Liquid Limit 32 % Clay Content < 1% Clay Content 1% Susceptible Further Studies Required (Considering nonplastic clay sized grains) Further Studies Required (Considering plastic non-clay sized grains) Not Susceptible 33

34 Chapter Three Recent Developments in Soil Liquefaction Engineering Figure (3-1) represents interim recommendations regarding liquefiability of soils with significant fines contents. For soils with sufficient fines content that the fines separate the coarser particles and control overall behaviour (Seed et al., 23): (1) Soils within Zone A are considered potentially susceptible to classic cyclically induced liquefaction, (2) Soils within Zone B may be liquefiable, and (3) Soils in Zone C (not within Zones A or B) are not generally susceptible to classic cyclic liquefaction, but should be checked for potential sensitivity (loss of strength with remoulding or monotonic accumulation of shear deformation). Fig. (3-1) - Recommendations regarding assessment of liquefiable soil types (after Seed et al., 23). 3.5 Factors Affecting Liquefaction Many factors have been studied by researchers and they can be summarized as follows: Soil type. Liquefaction usually occurs in cohesionless soils, especially soils of type SP (poorly graded sand). On the other hand, liquefaction does not occur in fine-grained, cohesive soils. Liquefaction 34

35 Chapter Three Recent Developments in Soil Liquefaction Engineering occurs in loose, saturated, clean to silty sands but has also been observed in gravels and nonplastic silts (Arora, 28) Particle size and gradation. Fine, uniform sands are more prone to liquefaction than coarse, well graded sands. Since the permeability of coarse sands is greater than that of fine sands, the pore water pressure is rapidly dissipated in such sands and liquefaction normally does not occur. Ranges of grain size curves separating liquefiable and nonliquefiable soils are proposed as shown in Figure (3-2). The area within the two inner urves in the figure represents sands and silty sands. Fig. (3-2) - Limits in the gradation curves separating liquefiable and nonliquefiable soils (Sitharam, 28) Relative density is one of the most important factors controlling liquefaction which occurs principally in saturated clean sands and silty sands having a relative density less than 5%. For dense sands, however, their tendency to dilate during cyclic shearing will generate negative pore water pressures and increase their resistance to shear stress. The lower limit of relative density beyond which liquefaction will not occur is about 75%. 35

36 Chapter Three Recent Developments in Soil Liquefaction Engineering Both settlement and pore pressure are considerably reduced during vibration with increase in initial relative density (Nayak, 1985) Surcharge loads. The initial effective stress in the sand deposit can be increased by the application of a surcharge load on it. With an increase in the effective stress, the transfer of stress from the soil particles to the pore water is delayed. Thus, the sand deposit would require higher intensity vibrations for greater duration when the surcharge loads are applied to it (Arora, 28) Vertical effective stress. It is well known that an increase in the effective vertical stress increases the bearing capacity and shear strength of soil, and thereby increases the shear stress required to cause liquefaction and decreases the potential for liquefaction (Geotechnical Engineering Bureau, 27) Groundwater table. The condition most conducive to liquefaction is a near surface groundwater table. Unsaturated soil located above the groundwater table will not liquefy (Day, 22) Location of drainage and dimensions of deposit. Sands are generally more pervious than fine-grained soils. However, if a pervious deposit has large dimensions, the drainage path increases and, under quick loading during an earthquake, the deposit may behave as if it were undrained. Therefore, the chances of liquefaction are increased in such deposit (Prakash, 1981) Drainage conditions. If the excess pore water pressure can quickly dissipate, the soil may not liquefy. Thus, highly permeable gravel drains or gravel layers can reduce the liquefaction potential of adjacent soil (Day, 22). 36

37 Chapter Three Recent Developments in Soil Liquefaction Engineering Degree of saturation. Liquefaction does not occur in dry soils. Only settlement, as a result of densification during shaking, may be of some concern. Very little is known on the liquefaction potential of partially saturated sands. Available laboratory test results (Sherif et al., 1977) show liquefaction resistance for soils increases with decreasing degree of saturation, and that sand samples with low degree of saturation can become liquefied only under severe and long duration of earthquake shaking Thickness of sand layer. The liquefied soil layer must be thick enough, so that, the resulting uplift pressure and amount of water expelled from the liquefied layer can result in ground rupture such as sand boiling and fissuring (Dobry, 1989). If the liquefied sand layer is thin and buried within a soil profile, the presence of a nonliquefiable surface layer may prevent the effects of the at-depth liquefaction from reaching the surface Earthquake intensity and duration. In order, to have liquefaction of soil, there must be ground shaking. The character of the ground motion such as acceleration and duration of shaking determines the shear strains that cause the contraction of the soil particles and the development of excess pore water pressures leading to liquefaction. The most common cause of liquefaction is due to the seismic energy released during an earthquake. The potential for liquefaction increases as the earthquake intensity and duration of shaking increases (Day, 1999) Characteristics of vibration. The main characteristics of vibration are its acceleration, frequency, amplitude and velocity. For liquefaction of soils, the first two characteristics, namely, acceleration and frequency are more dominant. In general, the greater the acceleration, the greater are the chances of liquefaction. Liquefaction usually occurs only after a certain number of vibration cycles are repeated. Frequency of vibration is 37

38 Chapter Three Recent Developments in Soil Liquefaction Engineering important if it is close to the natural frequency of the soil-foundation system and resonance occurs (Arora, 28) Seismic strain history. The prior seismic strain history can significantly affect the resistance of soils to liquefy. Low levels of prior seismic strain history, as a result of a series of previous shakings producing low levels of excess pore pressure, can significantly increase soil resistance to pore pressure buildup during subsequent cyclic loading. This increased resistance may result from uniform densification of the soil or from better interlocking of the particles in the original structure due to eliminate of small local instabilities at the contact points without any general structural rearrangement taking place (Arora, 28). 3.6 Ground Failure Resulting from Soil Liquefaction There are several types of ground failure that commonly result from liquefaction. Other phenomena associated with liquefaction include rise of pore water pressure, sand boils and various types of deformation Flow failures Flow failures occur in sloping areas inclined at 5 percent or greater and are basically landslides on a very large scale. These failures commonly displace large masses of soil laterally tens of meters and in a few instances, large masses of soil have traveled tens of kilometers down long slopes at velocities ranging up to tens of kilo-meters per hour (EERI, 1994) Ground oscillation When the ground is too flat to permit lateral movement, liquefaction at depth may create separations at the surface by decoupling overlying soil blocks, allowing the blocks to jostle back and forth on the liquefied layer (Biswas and Naik, 21). 38

39 Chapter Three Recent Developments in Soil Liquefaction Engineering Loss of bearing capacity When the soil supporting a building liquefies and loses strength, a large deformations can occur, leading to large settlements, loss of soil bearing capacity may also occur when liquefaction that has initially developed in a sand layer a few meters below a footing propagates upward through overlying sand layers and subsequently weakens the soil supporting the building Settlement In many cases, the weight of a structure will not be great enough to cause the large settlements associated with soil bearing capacity failures described above. However, smaller settlements may occur as soil porewater pressures dissipate and the soil consolidate after the earthquake. The eruption of sand boils (fountains of water and sediment emanating from the pressurized, liquefied sand) is a common manifestation of liquefaction that can also lead to localiz differential settlements (New Zealand Geotechnical Society Inc, 21). 3.7 Methods to Evaluate Liquefaction Potential of Soil The commonly employed methods are cyclic stress approach and cyclic strain approach to characterize liquefaction resistance of soils both by laboratory and field tests. The cyclic stress approach to evaluate liquefaction potential characterizes both earthquake loading and the soil liquefaction resistance in terms of cyclic stresses. But, in the cyclic strain approach, earthquake loading and liquefaction resistance are characterized by cyclic strains. Cyclic triaxial test, cyclic simple shear test and cyclic torsional shear test are the common laboratory tests. Further, standard penetration test, cone penetration test, shear wave velocity method dilatometer test are some of the in situ tests to characterize the liquefaction 39

40 Chapter Three Recent Developments in Soil Liquefaction Engineering resistance. Figures (3-3) and (3-4) can be employed to determine the cyclic resistance ratio of the in situ soil. Fig. (3-3) - Cyclic resistance ratio causing liquefaction and (N 1 ) 6 values for magnitude 7.5 earthquake for clean sands and silty sands, (Sitharam, 28). Fig. (3-4) - Cyclic resistance ratio causing liquefaction and corrected CPT tip resistance values for magnitude 7.5 earthquake for clean sand and sandy silt, (Sitharam, 28). 4

41 CHAPTER FOUR EQUATIONS OF DYNAMIC ANALYSIS, FINITE ELEMENT FORMULATION AND VERIFICATIONS 4.1 Introduction A large part of geotechnical engineering is directly concerned with soil-structure interaction. The designer of a mat foundation, a pile group, a tunnel lining, or a tied-back wall must consider the behaviour of both structure and soil and their interaction with each other. In some cases, the soil may be modelled in great detail and with complicated relations while the structure is represented very simply. In dynamic problems, the addition of time as a dimension increases the mass of data and the number of options. There is no universal, ideal, or complete way to deal with problems of soil-structure interaction. The choice must be based on the relative importance of the various parts of the problems and on the time and resources available to deal with it. This chapter focuses on description of the finite element equations of dynamic problems. A description of the computer program QUAKE/W is presented with most of its capabilities and features. Two verification soil dynamic problems are then solved by this program. 4.2 Types of Dynamic Loads: Dynamic loads are expressed as a function of time. The main causes of these loads on foundations, soils and structures are due to one or more of the following (Rasheed, 26): (i) Earthquakes (acceleration-time history), (ii) Blast loading (arbitrary load history), (iii) Impact loading such as pile driving in construction operations, 41

42 Chapter Four Equations of Dynamic Analysis, Finite Element Formulation and Verification (iv) (v) (vi) (vii) Loading due to wave action of water, Heaviside function, F(t) =1. (Step loading), Ramp loading, and Harmonic excitation, F (t) = a o + b o sin ωt or F (t) = a o + b o cos ωt. 4.3 Finite Element Equations Motion equation The governing motion equation for dynamic response of a system in finite element formulation can be expressed as (Chopra, 1995): M. + C. + K. = F (4-1) where: [M] = mass matrix, [C] = damping matrix, [K] = stiffness matrix, {F} = vector of loads, = vector of nodal accelerations, = vector of nodal velocities, and = vector of nodal displacements. The vector of loads could be made up by different forces: F = F b + F s + F n + F g... (4-2) where: F b = body force, F s = force due to surface boundary pressures, F n = concentrated nodal force, and F g = force due to earthquake load Mass matrix The mass matrix can be a consistent mass matrix or a lumped mass matrix. The consistent mass matrix is given as follows (Chopra, 1995): 42

43 Chapter Four Equations of Dynamic Analysis, Finite Element Formulation and Verification M = T dυ. (4-3) The lumped mass matrix is: M = ψ dυ.. (4-4) where: ρ = mass density, N = row vector of interpolating functions, and ψ = a diagonal matrix of mass distribution factors Damping matrix It is common practice to assume the damping matrix to be a linear combination of mass matrix and stiffness matrix (Chopra, 1995): C = α M+β K (4-5) where α and β are scalars and called Rayleigh damping coefficients. They can be related to a damping ratio ξ by: ξ = (4-6) where ω is the particular frequency of vibration for the system Stiffness matrix The stiffness matrix is given by the following relation (Zienkiewicz and Taylor, 25): K = B T. D. B. dυ. (4-7) where: B = strain-displacement matrix, and D = constitutive matrix. The superscript (T) refers to matrix transpose. For a two-dimensional plane strain analysis, all elements are considered to be of unit thickness. 43

44 Chapter Four Equations of Dynamic Analysis, Finite Element Formulation and Verification Strain - displacement matrix Engineering shear strain can be used in defining the strain vector as follows (Zienkiewicz and Taylor, 25): ɛ = (4-8) The field variable of a stress/deformation problem is displacement, which is related to the strain vector through: ɛ = B (4-9) where: [B] = strain matrix, and, = nodal displacement in x- and y-directions, respectively. This analysis is restricted to perform infinitesimal strain analyses. For a two-dimensional plane strain problem, is zero and the strain matrix is defined as (Zienkiewicz and Taylor, 25): [B] = N X N y 1 1 N y 1 N X N X N y 8 8 N X 8 N X 8.. (4-1) Elastic constitutive relationship According to the theory of elasticity, stresses are related to strains as follows (Zienkiewicz and Taylor, 25): σ = D. ɛ (4-11) For plane strain problems, D is given as follows : 44

45 Chapter Four Equations of Dynamic Analysis, Finite Element Formulation and Verification D =.(4-12) where: E = Young s modulus, and υ= Poisson's ratio Body forces Body forces applied in both the vertical and the horizontal directions can be modelled in finite element analyses. The body force in the vertical direction, b v, is due to gravity acting on an element. For a given material, the unit body force intensity in the vertical direction is given by its unit weight γ s which is in turn related to its mass density, ρ: γ s = ρ. g.(4-13) where g is the gravitational constant. When the unit weight γ s is non-zero, the following integral is evaluated by numerical integration and applies a vertically downward (negative) force, F b at each node of the element: F b = γ T s. dυ (4-14) Similarly, when the unit body force intensity in the horizontal direction, b h, is non zero, nodal forces in the horizontal direction are computed as: F b = b h T dυ.. (4-15) In a dynamic analysis, the mass density is calculated from the vertical unit body force. 45