Instability of seabed under wave induced loading

Transcription

1 Instability of seabed under wave induced loading Ali Abbasi, Pooya Allahverdizadeh & Behrouz Gatmiri Deartment of civil engineering- University of Tehran, Tehran, Tehran, Iran ABSTRACT In this study, several models are resented for analysis of sea floor movement roduced by storm waves. They are based on the finite element method, and used to redict dimension of stress and strain in seabed sediment subjected to wave ressure loading. These models based on elastic and elasto-lastic behavior, and they simlify in two-dimensional modeling. Large deformation of soil skeleton occurs below the mud line. The analysis of these models with both linear and nonlinear soils roerties are generated for tyical offshore soils. The results indicate that under relatively huge surface wave, high stress and deformation will be occurred. This deformation can distribute to considerable deths of seabed sediment. These soil s deformations can induce large lateral force on offshore ile and ieline. RÉSUMÉ Plusieurs modèles sont résentés our lanalyse du mouvement au sol de la mer roduite ar les vagues de temête. Ils se sont basés sur la méthode des éléments finis, et sont utilisés our suuter la dimension de contrainte et de déformation dans les sédiments du fonds de marins soumis à une charge donde de ression. Ces modèles se sont basés sur le traitement élastique et élasto-lastique et ils sont simlifiés ar la modélisation à deu dimensions. les grandes déformations du squelette du sol se sont roduites au-dessous de boue. ces modèles analyse à la fois les roriétés des sols linéaires et non linéaires sont générés our les sols tyiques off-shore. Les roriétés emloyées dans ces modèles sont sélectionnées armi des sols tyiques des off-shore. Le traitement de ces sols est analysé à deu manières, linéaire et non-linéaire. Les résultats indiquent sous la charge de grande vagues sont constatées de très grand contrainte et déformations. Ces déformation euvent déveloer au rofondeur considérable. elles euvent aussi causer de grands forces latérales sur ieu off-shore et de ielines. 1 INTRODUCTION Seabed instability owing to gravity, wave, or earthquake forces may result in massive submarine slide. This henomena is one of the imortant factors affecting the safety of the facilities, such as ielines, oil storage tanks, oil roduction latforms in ocean areas. When wave is formed, it roduced a ressure attern etending down to the orous seabed with change in stresses. Customarily, the instability induced in the seabed by the wave has been researched by either total stress or effective stress concets. In both analyses, the wave-induced water ressure on the seabed surface is considered as an eternal force. The hysical meaning of this eternal force, however, is different for these two aroaches. In the total stress aroach, the waveinduced water ressure is treated as surface force acting on the seabed surface, whereas in the effective stress analysis, it is converted to the seeage force which is a body force acting on the soil skeleton. One of the ossible reason for such z difference is attributed to the drainage condition in the seabed. In the case of an imermeable seabed consisting of clay or mud, total stress analysis can be alied (Henkel, 197). In order to investigate effective stress analysis is referable because it is closely related to the deformation and failure of the soil skeleton. There are two solutions for calculating the seabed resonse, the analytical and the numerical solution. In this field, both the solutions are in elastic manner of soil skeleton. Two different mechanisms occur for waveinduce instability, namely, Shear failure and liquefaction. In this aer, both analytical and numerical solution in elastic manner comare with numerical solution based on elasto-lastic behavior of the soil. Instability of the soil which occurred in this modeling is related to shear failure mechanism. THEORETICAL BACKGROUND Limiting the analysis to the flow, ore ressures and effective stresses induced in a orous bed by lane, eriodic waves roagating in water of essentially constant deth d, it suffices to consider the twodimensional roblem illustrated in Fig.1 As the governing equation for flow of a comressible ore fluid in a comressible orous medium the generally acceted form of the consolidation equation (Biot, 1941) or storage equation (Verruit, 1969) is adoted. For twodimensional roblem and treating the orous bed as hydraulically anisotroic with rincial ermeability K and K z in and z-direction, resectively, this equation may be written in the form k k z wn z z w k t k t 1 1 1S r [] k k w w z [1]

2 Gv (1 z ) e z e i t [9] e z e i t [1] PRINCIPAL STRESSES Figure 1. Definition of Wave-seabed interaction roblem. The equation of equilibrium which relate soil dislacements, volume strain and ore ressure, are given by G G u (1 ) G G v (1 ) z z u v z In equation (1)-(5), is the wave-induced ore ressure, γ w is the unit weight of the ore-water, n is soil orosity, β is comressibility of the ore fluid, t is time, ε is the volume strain, K w is true bulk modulus of elasticity of water, P w is the absolute ore-water ressure, S r is degree of saturation, and G is shear modulus. For linear waves roagating over the sea floor a harmonic ressure ositive under the crest and negative under the trough is induced on the mud line; the hydrodynamic ressure can be given as shown by (, t ) e i t [6] where =(γ wh)/(cosh(λd)). Considering a soil deosit infinitely dee limited by a horizontal surface, the waveinduced effective stresses, dislacements and ore ressures for most of the soil can be written as (Yamamoto, 1978, Madsen, 1978) z z i Gu [] [4] [5] = z e -z e i t [7] i z e z e i t [8] The wave-induced shear stress at a oint within the sediment may become large to overcome its shearing resistance, causing it to fail. The actual mode of such instability will deend on the satial distribution of waveinduced shear failure and the shear strength of the sediments. Conventionally, rediction of failure for soils has been based on Mohr-Coulombs failure criterion, which remains the most widely used in geotechnical engineering. Although other criteria of failure have been suggested in the literature (Griffiths, 1986, 199), the Mohr-Coulombs failure criterion is used here because of its simlicity and conservation. Princial stresses only the wave-induced incremental changes in effective stresses and ore ressure within soils from the initial equilibrium have been considered. Thus, the effective normal stresses, and in, y and z directions are given by: ( z ) Kz [11] y y y ( z ) Kz y [1] and z z z ( z ) Kz z [1] where, and in, y and z directions, resectively, while γ w and γ s are the unit weights of water and soil, resectively. In Equations [11]-[1], K is the coefficient of earth ressure at rest and is related to the oisson s ratio μ as K 1 [14] Since the shear stresses on the horizontal and vertical lanes are zero at the initial equilibrium, the effective shear stresses, and, are given as z yz y z [15] yz [16] y [17]

3 For study of the general stresses fields that occur in a comlicated boundary value roblem, it is convenient use rincial stress sace. The rincial stress sace also leads to a convenient geometric reresentation of various failure criteria. The effective rincial stresses, and can be eressed as (Griffiths, 1986) S 1 t Sin - [18] S t Sin [19] 4 MOHR-COULOMB MODEL IN ABAQUS The Mohr-Coulomb failure or strength criterion has been widely used for geotechnical alications. Indeed, a large number of the routine design calculations in the geotechnical area are still erformed using the Mohr- Coulomb criterion. The Mohr-Coulomb criterion assumes that failure is controlled by the maimum shear stress and at this failure shear stress deends on the normal stress. This can be reresented by lotting Mohrs circle for states of stress at failure in terms of the maimum and minimum rincial stresses. The Mohr-Coulomb failure line is the best straight line that touches these Mohrs circles. Thus, the Mohr-Coulomb criterion can be written as S t Sin [] C tan [8] Where 1 s y z [1] where is the shear stress, is the normal stress (negative in comression), C is the cohesion of the material, and is the material angle of friction. The Mohr- Coulomb criterion written above in terms of the maimum and minimum rincial stresses can be written for general states of stress in terms of three stress invariants. These invariants are the equivalent ressure stress as t 6 y y z z y yz z [] 1 trace ( ) [9] 1 Sin in which 1 6J t [] where is the rincile effective stress matri and the Mises equivalent stress as q S : S [] where S is the effective stress deviator matri, defined as y z yz y z z y yz z y J S S S S S S [4] S = I [1] S s [5] and the third invariant of deviatoric stress is 1 9 r : : S S S [] S y s y [6] S z s z [7] Equation [18]-[] ensure that. The Mohr-Coulomb yield surface is then written as F R q tan C [] mc where is the friction angle of the material in the meridional stress lane, C reresents the evolution of the cohesion of the material in the form of isotroic hardening

4 (or softening) and is the Mohr-Coulomb deviatoric stress measure defined as deviatoric ellitic function used by Menétrey and Willam (1995): 1 1 Rmc, Sin Cos tan Cos [4] where is the deviatoric olar angle defined as R mw, e 41e Cos e 1 1e Cos e 1 4 1e Cos 5e 4e R mc, [9] r Cos q in the Mohr-Coulomb Potential flow is assumed, so l l d G d g [5] [6] Where e is a arameter that describes the out-ofroundedness of the deviatoric section in terms of the ratio between the shear stress along the etension meridian ( ) and the shear stress along the comression meridian ( ). The out-of-roundedness arameter, e, is deendent on the friction angle ; it is calculated by matching the flow otential to the yield surface in both triaial tension and comression in the deviatoric lane: Sin e [4] Sin Where is the differential of lastic strain, is the differential of equivalent lastic strain, g can be written as 1 G g : C [7] and G is the flow otential, chosen as a hyerbolic function in the meridional stress lane and a smooth ellitic function in the deviatoric stress lane: mw G C tan R q tan [8] where is the dilation angle measured in the lane at high confining ressure, is the initial cohesion yield stress, and is a arameter, referred to as the eccentricity, that defines the rate at which the function aroaches the asymtote (the flow otential tends to a straight line as the eccentricity tends to zero). This flow otential, which is continuous and smooth in the meridional stress lane, ensures that the flow direction is defined uniquely in this lane. The function asymtotically aroaches a linear flow otential at high confining ressure stress and intersects the hydrostatic ressure ais at 9. The flow otential is also continuous and smooth in the deviatoric stress lane (the -lane); we adot the Flow in the meridional stress lane can be close to associated when the angle of friction,, and the angle of dilation,, are equal and the eccentricity arameter,, is very small; however, flow in this lane is, in general, nonassociated. Flow in the deviatoric stress lane is always nonassociated. Therefore, the use of this Mohr- Coulomb model generally requires the solution of nonsymmetric equations. According the yamamoto formulation, If we want observe only the effect of the wave ressure on soil without gravity load effect, the wave induced stress in lane strain condition is Cos t z e - zsin t Sin t z e - zcos t z e -z z e -z [41] in one length of the wave, the rincile stress is z e -z z e -z [4] in this situation, the maimum effective stress is ma C Cos z e-z [4]

5 From [4] we can calculate the deth in which soil become lastic. 5 D FINITE ELEMENT MODEL According to table 1, four models with two different elastic roerties and two ermeabilities were assumed. The soil in these models is isotroic and homogenous The wave characteristic is showed in table.the value of maimum wave induced ore ressure is similar to some reort from Gulf of Meico. The boundary condition was shown in Figure 1. The amlitude of harmonic ressures on the seabed surface becomes P =7 kn/m. The ressure with time increase than 1s and after this time the ressure is constant. Lateral boundaries have zero ore ressure in all time. Table1. Proerties of soil and dimensions of models. Figure. Boundary condition (Dislacement and ore ressure) Characteristics Model Model 6 EFFECT OF PERMEABILITY AND ELASTICITY Permeability(m/s) 1E- 1E-6 1E- 1E-6 module(n/m ).7E6.7E6.7E7.7E7 μ, Poisson s ratio.... n, Porosity S r, Saturation 1% 1% 1% 1% φ, internal friction angle Delation angle C (N/m ).5E4.5E4.5E4.5E4 Deth(m) Length(m) Soil analysis was used in ABAQUS model. The total time is 1E5 s and one factor multile to ore ressure which is to of the soil boundary condition. This factor change from to 1 as time change from to 1E4 s. according to Equation [4] maimum effective stress is 165 kn/m and the deths of the model that become lastic is 16.6 m and 54.5 m below the mud line. In FE model minimum deth is 1 m and maimum is 55 m. Because we aly the ore ressure slightly at time 847s in m below the mud line, the soil become lastic and after this, the layer of soil become lastic which the thickness of this layer achieve to 4 m. Table. Characteristics of Wave. Fig. shows the layer of soil which became lastic. Characteristics of wave d; Water deth(m) All Models T; Wave eriod(s) 1. H; Wave height(m).7 L; Wave length(m) P (N/m ) 7E5 The seabed was divided into 4 Quadratic element with rough rigid and imermeable base (u=, v=), lateral boundaries is fie in z direction (v=), in Mud line,, u and v is free. Figure. Equivalent lastic strain on model 1. Maimum deth of lastic deformation is 55m below the mud line and minimum deth is 1m below the mud line (t=1e5 s). The maimum dislacements of models in m below the mud line, the deth of the maimum effective

6 σ (effective horizental stress(n/m) σz vertical effective stresses(n/m) stress is shown in Table. In all models soil at time 847 s became lastic..5e+4 Table. Maimum horizontal dislacement at m below mud line.e+4 Characteristics u ma(m) Model E5 Model Total Run time(s) In comaring models with variation of ermeability, the effect of ermeability on elastic and final lastic deformation is negligible. But at same time equal 89 s in all models the ratio of lastic deformation under elastic deformation for K=1E-6 m/s in both module elasticity is 1.16 and this ratio for K=1E- is 1.. Figure 4 to 6 shows stress rofiles at maimum effective stress. About m to 55 m below the mud line the stresses aroimately are constant and equal with 165 kn/m. 1.5E+4 1.E+4 5.E+.E+ Figure 5. Effective vertical stresses in lastic and elastic condition in same time (t=89s)..5e Deth(m) Model Model.5E+4.E+4.E+4 1.5E+4 Model τy shear stresses (N/m ) 1.5E+4 1.E+4 Model Model Model 1.E+4 5.E+ 5.E+.E E Deth(m) Figure 4. Effective horizontal stresses in lastic and elastic condition in same time (t=89s). Deth(m) Figure 6. Effective shear stresses in lastic and elastic condition in same time (t=89s). Figure 7 show dislacement in and z direction (u, v) at maimum dislacement. from m to 55m below mud line the soil become lastic. At deth 55 m, gradient of diagram changed. This oint is the bound between lastic zone and elastic zone.

7 u horizental dislasment(m) 5.E-1 4.5E-1 4.E-1.5E-1.E-1.5E-1.E-1 Model Model 8 REFRENCES ABAQUS, 9. Abaqus Theory Manual 6.9.1, ABAQUS, Inc. Biot, M.A General theory of three-dimensional consolidation, Journal of Alied Physics, 1: Gatmiri, B. 199, A Simlified Finite Element Analysis of Wave-Induced Effective Stresses and Pore Pressures in Permeable Sea Bed, Géotechnique, 4 No. 1: E-1 1.E-1 5.E-.E Figure 7. Horizontal dislacements in lastic and elastic conditions in same time (t=89s). 7 CONCLUSION deth(m) (E=.7e6) (E=.7e6) The result in lastic condition show that in deth m and 55m below the mud line and by attention to module of elasticity, the dislacement until time increase according to total time of analysis. The total time of analysis is a function of total time of storm. Because of convergence of analysis, the loading of model is gently whereas in fact, this behavior acts raidly. The lastic zone in all models with any module of elasticity and any ermeability is in layer with etreme thickness of 8 m. this deth is only in relation with C and φ. This zone actually is the lace which instability occurs on it. The effect of ermeability on final dislacement is negligible, and the ratio of lastic deformation in same time is increase whenever the ermeability increases. In fact, Value of final dislacement deends on modulus of elastic but starting of lastic dislacement is deends on wave characteristic and soil characteristic. According to the result of analysis, dislacement continues until model became non convergent. This manner shows that soil is unstable. In this aer, Models is only under wave load but actually, soil in sea bed is sloing and gravity load act to become it unstable. However weight of soil causes to increase the normal stress and increase the value of maimum shear straight, but the comonent of gravity load with sloe direction decreases the shear straight of soil. The models have not any sloe but the result of analyzing can hel to find the unstable slo at a definite wave. Griffiths, D. V Some Theoretical Observations on Conical Failure Criteria in Princial Stress Sace, International Journal of Solids and Structures, : Griffiths, D. V Failure Criteria Interretation based on Mohr-Coulomb Friction, Journal of Geotechnical Engineering, ASCE. 116, Henkel, D.H The role of Waves in causing submarine landslides, Geotechnique, (1): Jeng, D.S Wave-Induced Seabed Instability In Front of A Breakwater, Ocean Engineering, 4:887. Jeng, D.S., Cheng L.. Wave-Induced Seabed Instability Around a Buried Pieline In a Poro- Seabed, Ocean Engineering, 7:17. Jeng, D.S. 1, Mechanism of The Wave-Instability Seabed Instability In Vicinity of a Breakwater: a Review, Ocean Engineering, 8:57. Madsen, O. S Wave-induced Pore Pressures and Effective Stresses in a Porous Bed, Géotechnique, Vol. 8, No. 4, Menétrey, P.H. and Willam K. J Triaial Failure Criterion for Concrete and its Generalization, ACI Structural Journal, 9: Verruijt, A Storage of Aquifers in Flow Through Porous Media, Chater 8, De Wiest, R. J. M. (ed.). Academic Press, Wright, S.G., Dunham, R Bottom Stability under Wave Induced Loading, Proc. 4th Annual Offshore Technology Conferences, aer 16 1, Houston, Teas, Yamamoto, T Seabed Instability from Waves, 1th Annual Offshore Technology Conferences, aer 6 1, Houston, Teas,