In this paper the theoretical procedure to analyze the dynamic behavior of highly saturated sand bed around a cylindrical block under the cyclic loading of water pressure is developed. The fundamental equations were derived for the axially symmetric coordinates. Then, the finite element equations were developed to solve these fundamental equations numerically. Finally, the numerical method was verified by experiments

Maeno, Shiro

Nago, Hiroshi

Yamamoto, Tetsuo

Okayama University Scientific Achievement Repository

Journal of the Faculty of Environmental Science and Technology, Okayama UniversityVoLl, No.1, pp.l21-133, March 1996Numerical Analysis of Dynamic Behavior of Highly Saturated Sand Bedaround Cylindrical Block under Cyclic Loading of Water PressureShiro MAENO*, Tetsuo YAMAMOTO**, Hiroshi NAGO*(Received January 16 , 1996)In this paper the theoretical procedure to analyze the.dynamic behavior of highlysaturated sand bed around a cylindrical block under the cyclic loading of waterpressure is developed. The fundamental equations were derived for the axiallysymmetric coordinates. Then, the finite element equations were developed to solvethese fundamental equations numerically. Finally, the numerical method was verifiedby experiments.1. INTRODUCTIONMany types of collapse of coastal structures occurs under stormy waves. One of them isthe settlement of armour blocks used for the protection of coastal structures such as breakwater.To understand the mechanism of this settlement phenomenon we have investigated the settlementof a rectangular parallelepiped block on the sand bed under the cyclic loading of water pressureboth experimentally and theoreticallyl),2). The settlement of concrete block and thecharacteristics of the sand movement around the block were made clear by the experiment. Andit is also clarified that the cyclic seepage force which occurs around the block under waterpressure variation plays an important role to the settlement phenomenon. To get pore waterpressure distribution around the block in detail, we derived the fundamental equations to analyzethe dynamic behavior of the sand bed and carried out the vertical two-dimensional finite elementanalysis3),4). The validity of numerical analysis was verified by comparing with the experimentalresults. For the analysis the plain strain state was assumed in consideration of the experimentalconditions.Photo 1 shows one of the experimental results. The colored sand and the standard sandnext to the front side wall of the container get mixed. It can be noticed that the effect of the frontand the back side walls of the container appears. As long as assuming the plane strain state andusing a rectangular parallelepiped block as a sinking object for the experiment, the wall effect can*Department ofEnvironmental and Civil Engineering, **Graduate student of Okayama Univ.121122 1. rac. Em'iron. Sci. and Tech.. Okayama Univ. 1 (l) 1996minCASE1 :11Photo 1 Experiments for rectangular blockPhoto 2 Experiments for cylindrical blocknot be neglected. To reduce this effect, we used a cylindrical block as a sinking object for theexperiment as shown in Photo 2. Very little effect of the side walls of the container is seen in thisphoto. To get more information about the dynamic behavior of the sand bed around a cylindricalblock under the cyclic loading of water pressure, it is necessary to develop a mathematical modelto explain the phenomenon.In this study, from the above point of view, we first derived the fundamental equations toanalyze the dynamic behavior of the sand bed around the block for the axially symmetriccoordinates. Then, the finite element equations were derived to solve the fundamental equationsnumerically. Finally, the validity of this numerical method was verified by the experiment.(2)S. MAENO et al. / Djrnamic Behavior of Highly Saturated Sand Bed around Cylindrical Block2. FUNDAMENTAL EQUATIONS2.1 Fundamental Equations for Cartesian CoordinatesIn the theoretical treatment, following fundamental assumptions are adopted.(1) The sand layer and the water are compressible.(2) The pore water moves in accordance with Darcy's law.(3) The skeleton ofthe sand layer deforms in accordance with Hooke's law.(4) The sand layer is composed ofthree phases; sand, water and air. Then, the porosity Ais the sum ofthe part for the water Aw and the part of the air Aa . That is,A=Aw+Aa (1)Under these assumptions we derived fundamental equations in Cartesian coordinates asfollows3),5). Taking into account the pore water pressure p, equilibrium conditions are expressedby the following equations.aox dty.x: dtn: ap--+--+--=-ax ay az axd'txy aoy d'tzy ap--+--+--=-ax ay az ayd'txz dtyz aoz ap--+--+--=-ax ay az iJzwhere, ox, 0y, Oz are incremental normal stresses (deviations from the initial stress state), and1:xy, 1:yz, 1:n: are incremental shear stresses. These stress components are expressed by thefollowing stress-strain relationship.123Ox = 2GEx + A'e,o y = 2GEY + A'e,Oz = 2GEz + A'e,1:xy = 1:y.x: = Gy xy1:yz = 1:zy = Gy yz1:n: = 1:xz = Gy~ } (3)where, A' is Lame's constant.relation.A/ = vE(1 +v)(1-2v),G is the shear modulus.G= E2(1 + v)They are given by the following(4)(v: Poisson's ratio, E : Young's modulus)Ex, EY' Ez and yxy' yyz' y~ re components of incremental strains. e is the incrementalvolumetric strain according to the following relation.e=Ex+Ey+Ez (5)The strains are related to the displacements by the following expreSSIons In case of smalldeformation.124J. Fac. Environ. Sci. and Tech., Okayama Univ. 1 (1) 1996Ex = dux / aX,Ey = duy / ay,Ez = duz / az,Yxy = dux / ay + duy / axYJa = duy / az + auz / ayYzx = duz / ax + dux / az} (6)where, ux , uy , Uz are displacement components in x, y, z directions respectively.On the other hand, the continuity equation for pore water is3)(~Aw + Aa ) ap + ae = (~)V2PP at at Pwg(7)(8)where, k :permeability coefficient, Pw: density of the pore water, g : acceleration due to gravity,t: time, ~ : compressibility of the water, P : absolute pressure, V2 is vector operation defined by2 a2 a2 a2V =-+-+-ax2 ay2 az2rFig. 1 Stress components incylindrical coordinatesOr, 0e, 0z are stresses in r, e, z directions respec-tively. 'tre, 'tez, 'tzr are shear stresses. The stresscomponents are expressed by the following stress-strain relationship.2.2 Fundamental Equations for Axially Symmetric CoordinatesIn Fig. 1 an cylindrical element is drawntogether with stress components acting on the surfaceof the element. Performing coordinates transforma-tion of equilibrium equations (2) for Cartesiancoordinates, following equilibrium equations forcylindrical coordinates are obtained5),6).aOr 1 d'ter d'tzr or - 0e ap--+---+--+ =-ar r ae az r ard'tre +! aOe + d'tze + 2'tre =! ap (9)ar r ae az r r aed'trz 1 d'tez aoz 'trz ap--+---+--+-=-ar rae az r azOr = 2GEr + A'e, 'tre = 'ter = Gyre}0e = 2GEe + A'e, 'tez = 'tze = Gyez0z = 2GEz + A'e, 'tzr = 'trz = Gy zr(10)(11)S. MAENO et al. / Dynamic Behavior of Highly Saturated Sand Bed around Cylindrical BlockThese equations are equivalent to Eqs. (3) and (5). The strain components are defined as follows.125(Jur£ =-r ar'1 aUe ur£e=--+-,r as r(Juz£ =-z az'1 aUr aUe ueYre=--+---r as ar raUe 1 auzYez=-+--az r asauz aUrYzr=-+-ar az(12)where, Ur , ue, Uz are displacement components in r, S, z directions respectively.In cylindrical coordinates system, the vector operation in Eq. (8) becomes2 1 a a 1 ii a2v =--(r-)+---+-r ar ar r2 as2 az2(13)As shown in Photo 2, the dynamic behavior of the sand bed around the cylindrical blockunder the cyclic loading of water pressure can be treated as an axially symmetric problem. Incase ofaxially symmetric problem, following conditions exist.Ue = 0, (Jur / as = 0, auz / as = 0, ap / as = 0Using these conditions, relationship (9) is reduced to!~(ror)+ ittzr _ 0e = ap }rar az r ar!~(rtrz)+ aoz = apr ar az azEqs. (10), (11) and (12) are reduced toor = 2G£r + A'e, "tre = "ter = 0 }0e = 2GEe + A'e, "tez = "tze = 0Oz = 2GEz + A'e, "tzr = "trz = Gy zre = aUr / ar +Ur / r + (Juz / az(Jur£r = ar' Yre = 0ur£e = -, Yez = 0r(Juz auz (Jur£z = az' Yzr = ar + azAnd the vector operation (13) is reduced to2 1 a a a2v =--(r-)+-r ar ar az2Then the continuity equation expressed as(14)(15)(16)(17)(18)(19)1261. Fac. Environ. Sci. and Tech., Okayama Univ. 1 (l) 1996(20)(21-a)(21-c)(21-b)Eq. (15) and Eq. (20) are the fundamental system of equations to analyze the dynamicbehavior of sand bed around the cylindrical block. There are three unknown quantities in theseequations. Two of them are incremental displacements ur and Uz . Another one is incrementalpore water pressure p. Consequently the governing equations are written as follows..!~(ror)+ i1T:zr _ 0e = apr ar az r ar.!~(n1Z)+ aoz = apr ar az az(13Aw+ Aa ) ap + ae =(~)V2pP at at PwgIt is very difficult to solve above obtained partial differential equations analytically. Here,we adopt the finite element method which has already been verified by authors of its applicabilityfor plane strain state problem. In the following section the governing equations for axiallysymmetric problem are transformed into finite element equations.3. FINITE ELEMENT FORMULAnONThe Galerkin method3),4),7) which is one ofthe weighted residual methods is used to derive afinite element equations of the system of Eqs. (21). Now we put Eqs. (21) into followingequations.1 a i1T:zr 0e apLI(u u p)=--(ro )+-----=0r, z, r ar r az r ar1 a aoz apT_(U u p)=--(n )+--- =0LJZ r, z, :l 1Z a :lr ur ~ uZAa · ap ae k 2~(ur' uz , p) = (13Aw +-)- + - - (-)V P = 0P at at Pwg(22-a)(22-b)(22-c)In the finite element method the analytical region is divided into a set of sub-domains calledelement. The continuous functions ur , Uz and p are approximated by following interpolationformulation over each sub-domain.nUr = ~aj(t)<I>/r,z)j=I(23-a)S. MAENO et al. / Dynamic Behavwr of Highly Saturated Sand Bed around Cylindrical BlocknUz = ~ b/t)cI>/r,z)j=l(23-b)127(23-c)np = ~c/t)cI>/r,z)j=lwhere, n is the number of node in each element. aj, bj and cj are the nodal values ofun Uz and p respectively. cI> j is the shape functions dependent only on coordinates. ApplyingGalerkin method to Eqs. (22),f L1(ur , uz , p) cI>/iV = 0f ~(un uz , p) cI>jdV = 0f ~(un uz , p) cI> jdV = 0where, dV = 2Jrrdrdz .Substituting Eqs. (22) into Eqs. (24).(j = I,2,···,n)(j = I,2,···,n)(j = 1,2,···, n)(24-a)(24-b)(24-c)IIiJ IiJ- - (ror )cI>i 2wdrdz + - (1:zr )cI>i2wdrdzr iJr iJzI os I iJp- --;- cl>i 2wdrdz - iJr cl>i 2wdrdz = 0I I iJ I iJ IiJP- - (nn )cI>i2wdrdz + - (a z )cI>i 2wdrdz - ~i2wdrdz = 0r iJr iJz iJzA I iJp I iJe«(:lAw +~) - cl>i 2wdrdz + ~i2wdrdzP iJt iJtkJ{IiJ(iJP) iJ2p}-- -- r- +- cl>i2wdrdz =0pg r iJr iJr iJz2(25-a)(25-b)(25-c)Applying Green's theorem to the terms involving second order derivatives in rand z direction.(26-a)(26-b)1281. Fac. Environ. Sci. and Tech., Okayama Univ. 1 (1) 1996Aa f ap f ae k {f ap(AA +-) -A.·2wdrdz+ ~·2wdrdz-- n r-A.·2rrdzP W P at 't'l at't'l pg r ar 't'lf ap aep; J ap f ap aep; }- r - --2rrdrdz + nz - ep; 2wdr - - --2rrrdrdz = 0ar ar az az az . (26-c)where, nx and nz are direction cosines in x and z direction respectively. Substituting the stress-strain relationships (16), (17), (18) and interpolation formulations (23) into volumetric integrationterms of above equations, following equations are obtained.(27-a)(27-b)(27-c)~ [{f ~;dV +f! ep;ep .dV} aaj +{f aep j ep;dV} abj~1 ar r J at az atJ=+{~(f aepj aep; dV +f aepj aep; dV)L. + {(I3Aw + Aa)fep .ep;dV}acjpg ar ar az az rJ p J at=~f(nr ap +nz ap)ep;dSpg ar azA finite difference method is applied for the terms with respect to time. In general the finitedifference formulations are as follows.afj (fJ+M - f})-=at I1t (28)where, 8=0 represents an explicit scheme, 8=1/2 represents a centered difference scheme (CrankNicholson scheme) and 8=1 represents a fully implicit scheme in time. Using above relationshipand introducing the matrix notation, Eqs. (27) become as follows.S. MAENO et al. / Dynamic Behavior of Highly Saturated Sand Bed around Cylindrical Block1298Aij8Dij1-H··/)J lJ8Bij8Eij1-[1;./)J lJ8pgCij8pgFijpg ~Aw + Aa / P Q.. + 8kR../)J lJ lJt+M(8 -l)Aij(8 -l)Dij1-H../)J lJwhere,(8 -l)Bij(8 -l)Eij1-[1;./)J lJ(8 -l)pgCij(8 -l)pgFjjpg ~Aw ~a / P Qij + (8 -l)kRij {a j} {Fj(1) }b. + F(2)J Ic. p.(3)J I(29)Ai' = (A' +2G)J a</> j a</>; dV +A'f!</> . a</>; dV +GJ a</>; a</> j dVf) ar ar r J ar az az+(1....' +2G)I _1 </> ·</>;dV +A'f! a</> j </>;dVr 2 J r arBi' = A'J a</>j a</>; dV +GJ a</> j a</>; dV +A'f! a</> j </>;dVf) az ar ar az r azf a</> .c..= ~·dVlJ ar 'flDi' = GJ a</> j a</>; dV +A'f!</> . o</>; dV +A'J a</> j a</>; dVf) az ar r J az ar azEi' = GI a</> j a</>; dV + (A' +2G)J a</> j a</>; dVf) ar ar az azJ0<1> .F;.= ~·dVlJ az 'flJa</>· f 1Hij = ~aJ ;dV + -</>;</> jdVr rJO</>·[1;. = _J "'.dVlJ az 'flQij = f</>j</>;dVR;. =Ja</> j a</>; dV +Ja</> j a</>; dVf) ar ar az azFj(1) =f(nrOr + nz'tzr )</>;dS'Fj(2) = f(nr'trz + nzoz )<P;dS'F;(3) = f (n ap +n ap)",.dS'I r ar z az 'fl(29-a)(29-b)(29-c)(29-d)(29-e)(29-f)(29-g)(29-h)(29-i)(29-j)(29-k)(29-1)(29-m)1301. Fac. Environ. Sci. and Tech., Okayama Univ. 1 (1) 1996Eq.(28) is the element stiffness matrix for each sub-domain element. It is necessary to superposethe element stiffness matrices to create a global system of stiffness matrix. After that, we cansolve a system of linear equations for each time step. In this study, we adopt e= 112considering its stability in the calculation.4. OUTLINE OF NUMERICAL ANALYSIS AND EXPERIMENTFig. 2 Sand bed model in analysis4.1 Numerical ProcedureFor the analysis, a simplified sand bed model asshown in Fig. 2 is treated. Considering the experi-mental conditions, we adopted following boundaryconditions and numerical conditions.Boundary conditions :(a) h = hs(t) at CD(b) iJhl an = 0 at AB, BC, EA(c) nror + nz"tzr = 0, nr"trz + nzoz = 0at CD(d) load due to the pressure h = hs(t) actingthe block at DEwhere, hit) is the oscillating water pressureacting on the sand surface and the block.z-........-block_--.....r_.....-- .....\BcNumerical conditions :Aa =0.005, Aw =0.4, k=0.015(cm/s), 13 = 4.3 xl0-1O (m 2 IN),G=3.5xl07(N/m 2 ), v=0.45In this study we adopted triangular elements to divide the domain. Fig. 3 shows the finiteelement mesh. Its node number is 345 and element number is 623. In the calculation, at first,the initial stress state without the cyclic loading of water pressure is obtained by taking intoaccount the individual weight of the block and the sand. After that, the incremental value due tocyclic loading ofwater pressure is added to the initial one.4.2 Experimental ProcedureFor the experiment the rectangular parallelepiped container shown in Fig. 4 was used. Thewidth of the container is 40 cm. It is filled with highly saturated standard sand (Toyoura standardsand d50 $5 0.25 mm). The water depth from the sand surface is about 110 cm. The oscillatings. MAENO et al. / Dynamic Behavior of Highly Saturated Sand Bed around Cylindrical Block131100unit : cmtOscillating air pressuren---I I200 IE "IPt.l 00 blockoNoonr--------,o ,o 0o ,, ,o ,o ,Fig. 3 Finite element mesh Fig. 4 Experimental apparatusair pressure acts on the water surface. The amplitude of the cyclic water pressure acting on thesand surface is about 40 cm. Its frequency is about 1 Hz. A cylindrical block was placed at thecenter of the sand surface. Its diameter is 20 cm and its specific weight is 3.65. Theincremental pore water pressure were measured at measuring points shown in Fig. 4.5. RESULTS AND DISCUSSIONSFig. 5 (a) and (b) shows the numerical and experimental results of the pore water pressurevariation at each measuring points. Experimental results show that the water pressure on thesand surface propagates into the sand bed accompanied by the damping in amplitude and the phaselag. This tendency is very similar to the results for the rectangular blockl ),2). From thenumerical analysis, the characteristics of the damping and the phase lag of the pore pressurearound the block are in good agreement with experimental results. That is, the validity of themathematical model and the numerical method developed in this paper can be verified.Fig. 6. shows the distribution of the pore water pressure and the seepage force. Fig. 6 (a)is for the state that the water pressure on the sand surface is high (t/T=0.25), and Fig. 6 (b) is forthe low pressure state(tlT=0.75). In this figure, solid lines show the equipotential lines andarrows show the seepage force vector. The pore water pressure propagates into the sand bed132 J. Fac. Environ. Sci. and Tech.. Okayama Univ. 1 (1) 1996around the block with the damping in amplitude and the phase lag cyclically. This pore waterpressure variation around the block causes the cyclic seepage force around the block. This cyclicseepage force at the low water pressure state plays an very important role in the settlement of theblock under the cyclic loading ofwater pressure2).h(cm)50. 0O. 0- 50. 0(a) numerical resulth(cm)50. 0t(sec)O. 0- 50. 0(b) experimental resultt(sec)Fig. 5 Pore water pressure variation with time~~~ i , I i I;~'11 II+ + d + ~ ~ 1•• t••••••••• •• tt •••••••••t •• tt •• t.•• ••t •HHH H H H' •• • • • • •, ,."". ++++++ •• • • • • • •HH++ ++++++ •• t • • + •'TTY"~ 'Tty"~, , , , , , ,, , , , , , , , , , , , , , , ,, , , , ,"~"~Of , , , , ,, , , , , , , , , , , , ,h403530252015105• • • H • •••••• • • + • • • •• • H •• HHH + • • + + • •"...... .t. , j, j, ;, j, • • • • • •• •••• ... .i. j, j, , • • • • • •---- 5• • • • •• • • • •• • • • • •• • • • • • • • • • • • •--h(cm)(a) high water pressure (t/T=O.25) (b) low water pressure (t/T=O.75)Fig. 6 Pore water pressure and seepage forceS. MAENO et al. / Dynamic Behavior of Highly Saturated Sand Bed around Cylindrical Block6. CONCLUSIONSIn this paper, we derived the fundamental equations to analyze the dynamic behavior of thesand bed around the cylindrical block under the cyclic loading of water pressure and evaluated theapplicability of numerical method based on the derived mathematical model through theexperiment. The numerical results by the finite element method showed the good agreement withexperimental results. It can be concluded that the validity of the mathematical model andnumerical method presented in this paper is verified.REFERENCES1) Maeno, S. and Nago, H : Settlement ofa concrete block into a sand bed under water pressurevariation, Proceedings of the International Symposium on Modelling Soil-Water-StructureInteractions, pp. 67-76, 1988.2) Nago, H. and Maeno, S. : Settlement of concrete block into sand bed under cyclic loading ofwater pressure, Proc. of 26th IAHR. Congress, Vol. 3, pp. 317-322, 1995.3) Nago, H and Maeno, S. : Pore water pressure in sand bed under oscillating water pressure,Memoirs of the School ofEngineering, Okayama Dniv., Vol. 19, NO.1 pp. 13-32. 1984.4) Nago, H, Maeno, S., Sasahara, H. and Nishioka, M. : Prevention work of the settlement ofcoastal block, Proceedings of Civil Engineering in the Ocean, Vol. 6, pp. 229-234, 1990, (inJapanese).5) Verruijt, A. : Elastic storage of aquifers, Flow Through Porous Media, Ed. DeWiest, R. 1. M.,Academic Press, New York, pp. 337-344, 1969.6) Timoshenko, S. and Goodier, 1. N. : Theory of elasticity, Second Edition, McGraw-Hill BookCompany, Inc., New York, pp. 304-306, 1951.7) Szabo, B. A. and Lee, G. C. : Derivation of stiflhess matrices for problems in plane elasticity byGalerkin's method, International Journal of Numerical Methods in Eng. Vol. 1, pp. 301-310,1969.133

Numerical Analysis of Dynamic Behavior of Highly Saturated Sand Bed around Cylindrical Block under Cyclic Loading of Water Pressure

Abstract

Similar works

Full text

Available Versions

Okayama University Scientific Achievement Repository

Okayama University Digital Information Repository