Interaction of Soil Static and Dynamic Stiffness and Buried Pipeline Under Harmonic Vibration

Problem statement: When earthquake is occur, many damages were occurr ed in pipelines that San Francisco (1906) and Manson (1908), Kobe ( Japan) and ate are samples of this topic. So many researchers studied on the pipelines and dynamic fo rces. Approach: Determine static and dynamic performance parameters of the pipeline and the surr ounding soil such as static stiffness, dynamic stiffness, damping and additional mass share of soi l wh ch take part with pipe mass in dynamic performance. In the static case relationship between friction f rces and joint deflections in a buried element pipe had be calculated and with using of so me experimental results and results are compared together. For dynamic cases, Dynamic equilibrium eq uation of pipeline element axial vibration in continuous system, with neglecting the effect of so il mass share which participates in producing vibration and with considering of it were abstained and values of displacement and forces were calculated. In continuous, these formulations were process for many cases and were drawn in graphs for comparison. Results: Stiffness for ω/ωn<1 doesn’t change much but for the values more than 1 it increase rising. when ω/ωn<1 the ratio of dynamic stiffness to the static stiff ness is less than unique except in big amount of damping ratio ( ρ>0.5) which the ratio becomes more than 1. Finally for ω/ωn>1, the ratio of dynamic to static stiffness rises ra pidly and by increasing the additional mass, the value of dynamic stiffness in case of ω/ωn>1 would increase highly. Conclusion: The static performance between soil and pipe is nonlinear in a xial direction and when the hysteric dominates grows, the value of force dominates between soil an d pipe and dynamic stiffness would ascend. Also by increasing damping ratio, the dynamic stiffness would increase too however it depends on the static to dynamic stiffness ratio and the damping ratio.


INTRODUCTION
The purpose of this study is to study on the static and dynamic properties of buried pipeline and surrounded soil in axial direction due to harmonic dynamic vibration. The stiffness between soil and pipe in different static and dynamic conditions and cases based on the dynamic distributed formulation, considering soil additional mass share and also refer to numerical experimental results, has been estimated and concluded in the following pages. Poulos and Sim (1979) has presented a method applying elastic theory that axial vibration loading of a pile would increase the porous pressure, decrease the soil modulus and lateral resistance of soil. A quasibifurcation theory of dynamic buckling and a simple flow theory of plasticity are employed to analyze the axisymmetric, elastic-plastic buckling behavior of buried pipelines subject to seismic excitations. Using the seismic records of the 1971 San Fernando earthquake, a series of numerical results have been obtained, which show that, at strain rates prevalent in earthquakes, the dynamic buckling axial stress or strain of a buried pipe is only slightly higher than that of static buckling (Lee, 1984).
A comparative study has been performed to obtain the response characteristics of strains in a buried pipeline section, axial relative displacement and transverse relative displacement. The maximum relative displacement response in the transverse direction is significant even under the design level of earthquakes. The maximum transverse displacement response mainly occurs at the center of the pipeline and shows that the pipeline embedded in soft clay is particularly vulnerable to earthquake excitation. A minor discrepancy is observed between the responses of the pipeline with various angles of lowering and highering bends. In all, the soil type is significant for the response of the pipeline (Do et al., 2009).
The axial performance of pile is approximately close to buried pipeline. The pile under axial static load resists against the loading with side strength in pile column and end pile strength at the end edge of pile. At this research, resistance between pile and surrounding soil is modeled according to Winkler theory with elastic springs. Then, the formulation and experimental methods, to find a method for determining dynamic stiffness (impedance function), damping between the pile and surrounding soil and also the additional mass of soil which contributes with pile in dynamic performance applying harmonic vibration would be presented. The formulation methodology is based on dynamic of structures theory and mechanical vibration theory in time and frequency space applying Fourier transport. An efficient numerical approach based on both boundary and finite element methods is developed in this work. This development is capable of realistic three dimensional analyses of soil-structure interaction problems in the real time domain and is specifically tailored to buried lifelines. Results are gauged against empirical design formulae. It is shown that the seismically induced stress state in a buried pipeline is more pronounced in the case of transverse vibrations than in the case of longitudinal vibrations (Manolis et al., 1995). Study on the Dynamic responses of buried pipelines during a liquefaction (Wang et al., 1990) and subsea pipeline buckling (Neil and Tran, 1996) were done by other researchers.

MATERIALS AND MATHODS
In this research, with using of experimental results which was done by first author, static and dynamic stiffness were calculated in two methods that are explained in later section. Experiment data are for many cases of pipe depth and water absorption. In continuo, with using of formulation, variation of dynamic and static stiffness, mass of pipe, the part of soil which contributes with pipe in dynamic performance, ratio of frequency to natural frequency and damping ratios are compared together.

Study the interaction of soil and pipe in static case:
The relationship between friction forces and joint deflections in a buried element pipe can be shown as Eq. 1: u i and u i+1 are the measured displacements in points i and i+1 of pipe length in laboratory. At this part, the axial stiffness between the soil and pipe has been calculated by Calton and experimental method. In Calton method, the values of stiffness have been estimated by dividing the force values to initial deflections. Sample result of Calton method is shown in Table 1.
Here is a sample of calculation in experimental method:    Calculation of static stiffness for an experiment by experimental method is shown in Table 2 and Comparison of calculated stiffness by two methods in an experiment is shown in Fig. 1. Also forcedisplacement curve for an experiment is shown in Fig. 2.
Effect of the height of the soil over the pipe to the axial stiffness: To study the effect of the height of the soil over the pipe, it has been assumed that all of the conditions are constant and just the height of the soil has changed. It means that, a PVC pipe with diameter of 26 cm and 4 mm thickness has been put in a sand soil with water absorption percentage of 2.97 and one pas compaction. The values of stiffness have been calculated for different heights of 0, 10 and 20 cm. Figure 3 shows the value of stiffness for different heights. Also the process has been done for a different water absorption of 10.2% which is indicated in Fig. 4. Figure 3 and 4 indicates that when the depth of the soil increases, the stiffness of the soil increases in both methods too. The main reason for increase in stiffness is increasing the lateral pressure to the pipeline and then producing stronger friction forces between soil and pipe. Also, by increasing the water absorption percentage of the soil, the value of stiffness would increase too.
Interaction of soil and pipeline in dynamic condition under harmonic vibration: Dynamic equilibrium equation of pipeline element axial vibration in continuous system, neglecting the effect of soil mass share which participates in producing vibration can be formulated as Eq. 2: Equation 2 illustrates the governing equation of a continuous system that applying the boundary and initial conditions and solving the corresponding deferential equation, the displacement function u (x,t) can be obtained up to constant values of k a, axial stiffness between soil and pipe, ρ p special mass of pipe, E p pipe elasticity modulus, A p pipe section area and C a axial damping of soil and pipe.
The dynamic equilibrium equation can be shown in discrete system with multi degree of freedom considering external forces and excluding ground motion as Eq. 3: The axial dynamic equilibrium in the discrete system with two degree of freedom considering additional mass of soil (M add ) can be shown as Eq. 4: Where: M p = The mass of pipe M add = The part of soil which contributes with pipe in dynamic performance u i , i u ɺ , i u ɺɺ = Displacement, velocity and equivalent joint acceleration in joint i C i = the equivalent damping in joint i k ij = The stiffness in joint i to the effect of joint j f i (t) = The equivalent force in joint i Most of the researchers neglect the effect of M add because of its tiny value but it has been considered at this study. Equation 3 can be distributed for more than two degree of freedom. In such a case that Eq. 3 changes to Eq. 5 the factor k d is defined as dynamic stiffness or impedance function: In a single degree of freedom case the frequency dominates impedance function can be shown as Eq. 6: ω is angular frequency and k d (ω) and k d (t) are the foriour transform of each other. Thus they transform the frequency space to the time space and vise versa and are compatible with Eq. 7: The same as the relation between k d (t) and k ad (ω), displacement u(t) in time space and displacement in frequency space u(ω) and also force f(t) in time space with force in frequency space f(ω) can be related to each other by fourier transform as Eq. 8: According to theory of dynamics of structures and fundamental of mechanical vibration and random vibration viewpoint, the dynamic stiffness k d (t) in time dominant is the result of the problem due to unique impulse, that applying Fourier transform the frequency response of the system or the dynamic stiffness in time dominant would be determined. Frequency dominant dynamic parameters are the Fourier transform of time dominant dynamic parameters and hence the frequency space can be transformed to time frequency by Fourier transforms and vise versa.
In static case, the equivalent dynamic stiffness can be determined as static stiffness k d (ω) = k. Then the numerical value of dynamic stiffness can be shown as Eq. 9: Equation 9 can be drawn as a graph for different values of damping ratios (ρ) and different values for the division of hysteric frequency to natural frequency. The minimum of dynamic stiffness is produced when ω = ω n and by increasing damping ratio the dynamic stiffness would increase too. For impacting the pile to the ground or penetration of a stick in the ground it can be assumed that ω = ω n that ground dynamic stiffness would be in the minimum value. When ω<ω n the condition would decline to static and when ω>ω n dynamic stiffness would increase a lot. When M add participates in the equation, because it is a function of ω, Ø, c, ω n , M p , the equation would be a complex equation. In fact M add becomes a function of ω n and ω n becomes a function of M add . According to the different conditions of ω<ω n and ω>ω n small ρ (under critical) and big ρ(super critical) and also with or without M add the dynamic stiffness would be determined and drawn.
To consider the effect of additional mass, it should be considered that, natural frequency of a single degree of freedom system with additional mass can be defined as Eq. 10: If we assume that the additional mass is equal to 0.5 M p and M p , the value of natural frequency is equal to (1/√1.5) ω n , (1/√2) ω n . It can be said that the natural frequency is dependent on the mass of pile and additional mass. When M add interferes a complete couple equation would be produced because M add is function of hysteric frequency, phase changing angle, harmonic force dominant and harmonic displacement dominant and the natural frequency is a function of M add . in the other hand Eq. 11 is available: If the dynamic force represents as a harmonic force, displacement would be harmonic too but with a phase delay: That Φ is output phase delay (displacement) in relation to input (dynamic force). The relation between F 0 and u 0 can be defined as Eq. 14: Where: Then: Hence the value of dynamic stiffness is equal to: The Eq. 15 and 16 can be draw as a graph for different values of ρ and (ω/ω n ).the numerical value of it would be constant always.
The variation of dynamic stiffness versus ω/ω n is the reverse of variation of dynamic displacement versus ω/ω n up to the value of damping; the maximum value has a little distance from ω/ω n point. If M add is considered, the distance would increase, because the situation of ω n would change in that case.
Although Eq. 16 is for a harmonic force and displacement, but is different from the real conditions because it doesn't interfere the value of M add . But it can be used for stating the numerical values of dynamic stiffness. The availability of M add is effective in variation of ω n that by elimination of damping and external force terms, it can not be an ordinary eigen value problem: By eliminating C x and f (t) M add remains dependent to the initial conditions of the problem because ω n is dependant to M add and ρ. The equations show that the numerical value of dynamic stiffness F 0 /u 0 is independent to the amplitude but dependent to the input frequency ω and in the other side ω n that itself considering the addition of M add is against the ordinary structural dynamics and dependent on the input conditions of problem ω, ρ. In the laboratory the values of F 0 , u 0 have been gained and the results are convincing. Experimental results also show that the numerical value of dynamic stiffness is not so sensitive to amplitude but dependent to the input frequency.
The numerical value of dynamic stiffness would never be a negative number because it is summation of quadratic and positive items. As it has been said before M add is function of ω, c, ω n , m. In the other word M add is a function of ω n and vise versa. Therefore the dynamics of structures problem should be solved at parallel.  Table 3 indicates a sample variation of ω/ω n for M add = 0 considering different values for ρ. Figure 5-7 show the variation of dynamic stiffness to static stiffness for the ratio of hysteric frequency to natural frequency and for different values of additional mass and different damping ratios. Figure 8-15 indicate the variation of dynamic stiffness to static stiffness for the ratio of hysteric frequency to natural frequency and for different values of ρ.

RESULTS AND DISCUSSION
As it is shown in figures, dynamic stiffness for ω/ω n<1 doesn't change much but for the values more than 1 it increase rising so that for ω/ω n = 2 it becomes multiple. The dynamic stiffness increases by increasing damping ratio. The figures also show that when ω/ω n<1 the ratio of dynamic stiffness to the static stiffness is less than unique except in big amount of damping ratio (ρ>0.5) which the ratio becomes more than 1. Finally for ω/ω n>1 , the ratio of dynamic to static stiffness rises rapidly and by increasing the additional mass, the value of dynamic stiffness in case of ω/ω n>1 would increase highly.

CONCLUSION
Interaction of additional mass of soil with axial element in dynamic and seismic vibration case have rarely been considered by researchers. Combining the dynamic formulation distribution and dynamic experiments, one can find dynamic properties of a structure buried in soil like buried pipeline. Applying this model we can find the effect of static and dynamic parameters between soil and pipe to the factors like burial depth, pipe thickness, soil compaction, water absorption percentage, amplitude value and hysteric frequency. Some of the results can be concluded as: • The static performance between soil and pipe is nonlinear in axial direction • Increasing damping between soil and pipe, harmonic displacement phase delay and harmonic forces would rise up • when the hysteric dominate grows, the value of force dominate between soil and pipe and dynamic stiffness would ascend • 4-for ω<ω n ratio of dynamic stiffness to static stiffness is less than one. For very great amount of damping ratio, the ratio of dynamic to static stiffness becomes more than one • 5-for ω>ω n the ratio of dynamic to static stiffness rises rapidly • 6-by increasing damping ratio, the dynamic stiffness would increase too • 7-when the additional mass grows the natural frequency decrease • 8-dynamic stiffness increases when additional mass grows