13 th World Conference on Erthque Engineering Vncouver, B.C., Cnd August 1-6, 2004 per No. 2663 DYNAMIC EARTH RESSURE SIMULATION BY SINGLE DEGREE OF FREEDOM SYSTEM Arsln GHAHRAMANI 1, Seyyed Ahmd ANVAR 1 SUMMARY The theory of sttic nd dynmic erth pressure by the method of zero extension line developed by the uthors is extended to the plin strin solution of smooth verticl wll retining snd bcfill subjected to horizontl dynmic excittion. The pressure exerted on n element of the wll for the sttic nd dynmic cse is shown to be function of rottion of the element of the wll. It is shown tht this pressure resulting from the combined effect of sttic nd dynmic loding cn be simulted by single degree of freedom system. The stiffness nd mss chrcteristics of this single degree of freedom is evluted nd presented here. It is shown tht with the ngle of internl friction being function of sher strin of the snd, the prmeters of the single degree of freedom system is function of rottion of the wll element. INTRODUCTION The theory of zero extension line ws developed by Roscoe [1] nd his coworers. This theory ws extended by the uthors [2] nd by Ghhrmni nd Clemence [3] for the evlution of the sttic nd dynmic erth pressure on retining wll with snd bcfill. This theory is used to evlute the response history of smooth verticl retining wll subjected to horizontl excittion. The theory will be explined nd the derivtion will be presented. The results re used to simulte the motion by single degree of freedom system. Expressions re developed for the stiffness nd the mss of the single degree of freedom system s function of rottion of the element of the wll. THEORY The full explntion of the theory cn be found in [2] nd [3]. Only summry will be presented here. Consider the retining wll shown in Figure 1. The presenttion is for the pssive cse, but the formul will lso be given for the ctive cse. 1 Shirz University, Deprtment of Civil Engineering
Figure 1. The simple zero extension line field nd its element. The simple zero extension line field for the smooth wll is composed of two zones; Rnin zone nd Coulomb zone. The ngle is π/4 υ/2 where υ is the ngle of diltion of the soil. The ngle of diltion is evluted from the friction ngle φ by the following formul which is proposed in the computer progrm lxis [4]. φ =ν + 35 (1) The ngle dd is the ngle tht trction force t mes with the direction of the zero extension line s shown in Figure 1. As explined in [3] this ngle is relted to ngles of internl friction nd diltion by the following formul tnδd = ( sinν ) / cosν (2) The pressure t depth h cn be evluted by[2] = γh ρh (3) γ + e where is the pssive pressure, γ is the pssive pressure coefficient, γ h is the unit weight multiplied by the depth of the wll element h, e is the dynmic pssive pressure due to ccelertion of the wll element t depth h, nd ρ is the soil density. Expression similr to eqution (3 ) cn be written for the ctive cse where the subscripts p is chnged to. = γh ρh (4) γ + e For the present cse where the boundry is fixed, refer to Figure 3, e, γ nd e re given by the following formuls = &. Expressions for γ,
γ e γ e 1+ = tn( π / 4 + ν / 2 + δd ) tn( π / 4 + ν / 2) = = tn 1 tn( π / 4 + ν / 2) = 2 1 2 = tn ( π / 4 φ / 2) tn( π / 4 ν / 2) = 2 1+ 2 ( π / 4 + φ / 2) It is to be noted tht the zero extension line theory gives exctly the sme formuls for the sttic pssive nd ctive pressure coefficients s those given by the Rnin formul but hs different formuls for the dynmic pssive nd ctive pressure coefficients. SINGLE DEGREE OF FREEDOM SIMULATION It is well nown tht for soil sinφ is function of the sher strin γ. The test result of such correltion reported by Cole [5] for dense snd is shown in Figure 2. He lso showed tht the ngle of diltion remins lmost constnt during shering.. (5) Figure 2. () sinφ vs sher strin γ nd (b) diltion vs sher strin γ[5]. Now if unit element of the wll t depth h is moved into the snd nd if the displcement of the element of the wll is D, then by zero extension line theory the displcement in the Coulomb zone nd in the Rnin zone re given by the following formuls Coulomb zone: U C = / cos( π / 4 + ν / 2) Rnin zone: U = / cos( π / 4 + ν / 2) (6) R Liewise the rottion ngle of the unit element, θ,cn be clculted from the reltive displcement of the element by
θ = (7) z This mens tht t depth h the zero extension line trnsltes into the snd. The sher strin for the slice of zero extension bnd t depth h with rottion θ will then be Sher strin: 2θ γ = (8) cosν This indictes tht the unit bnd of zero extension line undergoes sher relted to the rottion of the element of the wll t depth h nd thus different bnds do not ffect ech other; see Figure 3. Figure 3. () Bnd of Zero Extension Line t depth h undergoing sher strin due to wll element rottion θ nd displcement, nd (b) single degree of freedom system t depth h It should lso be noted tht the trnsltion without rottion of the element of the wll t depth h does not produce sher in the zero extension line bnd nd it is the reltive trnsltion or rottion θ of the element of the wll t depth h which produces sher in the zero extension line bnd. Considering the bove findings, single degree of freedom system cn then be used to simulte the motion of the unit element of the wll t depth h. The stiffness nd the mss m for the single degree of freedom system is evluted by the following formuls from the results of the sttic nd dynmic pressure on the element of the wll t depth h. For the pssive cse m = γh / = γ e ρh (9) nd for the ctive cse
m = γh / γ = e ρh (10) Dynmic property clcultion procedure for single degree of freedom system The wll will be divided into elements of unit height. Knowing the stte of the wll t time t, it is intended to find the stte of the wll t time t + δ t. The clcultion procedure for evlution of stiffness,, nd mss, m, for ech unit element of the wll is then s follows: For ny unit element of the wll t depth h, the wll displcement D nd ccelertion re nown t time t, nd the rottion θ cn be clculted by eqution (7). Then the sher strin cn be clculted from eqution (8). Knowing the sher strin, sin φ nd ν cn be determined by using grphs lie tht of Figure 2() nd (b), respectively. Knowing sin φ nd ν, the pssive nd ctive pressure coefficients for the sttic nd dynmic cse cn be clculted from eqution (5). Finlly nd m cn be clculted from eqution (9) or (10) for the pssive or ctive cse, nd these single degree of freedom prmeters cn be used to clculte displcement t time t + δ t by using the incrementl form of equtions (3) nd (4). This procedure results in the response history of the wll. It is to be noted tht nd m cn be quite nonliner depending on the reltion between sin φ nd sher strin of the soil γ. CONCLUSIONS From the discussion presented it cn be concluded tht using the zero extension line theory nd with reltion between sin φ nd γ being now, the dynmic prmeters of the single degree of freedom system, nd m, cn be clculted s function of wll displcement D nd rottion θ. REFERENCES 1. Roscoe K.H. The influence of strins in Soil Mechnics Geotechnique, Vol.. 20 No. 2, 1970 2. Anvr Seyyed Ahmd nd Ghhrmni Arsln Equilibrium equtions on zero extension lines nd its ppliction to soil engineering, Irnin Journl of Science nd Technology, Vol. 21, No.1, 1997, pp11-34. 3. Ghhrmni A. nd Clemence S.. Zero extension line theory of dynmic pssive pressure Geotechnicl Eng. Division, ASCE, Vol. 106, No. GT6, 1980, pp 631-644. 4. Geotechnicl finite element code for soil nd roc nlyses, LAXIS, Netherlnds,2004. 5. Cole E.R.L. The behvior of soils in simple sher pprtus, thesis presented to the University of Cmbridge, t Cmbridge, Englnd, in prtil fulfillment of the requirements for the degree of Doctor of hilosophy,1967. ACKNOWLEDGEMENT The uthors wish to cnowledge the support of the Den of Engineering Dr. Jvdpur nd the Deprtment of Civil Engineering Chirmn Dr. Abedini nd the support of the vice chncellor in chrge of reserch of Shirz University.