Cloud Publication Intenational Jounal of Advanced Civil Engineeing and Achitectue Reeach 2013, Volume 2, Iue 1, pp. 42-52, Aticle ID Tech-106 Reeach Aticle Open Acce Paive Peue on Retaining Wall uppoting c-φ Backfill uing Hoizontal Slice Method Sima Ghoh and Sumen Deb Civil Engineeing Depatment, National Intitute of Technology, Agatala, Tipua, India Coepondence hould be addeed to Sima Ghoh, ima.civil@nita.ac.in Publication Date: 30 Augut 2013 Aticle Link: http://technical.cloud-jounal.com/index.php/ijacear/aticle/view/tech-106 Copyight 2013 Sima Ghoh and Sumen Deb. Thi i an open acce aticle ditibuted unde the Ceative Common Attibution Licene, which pemit uneticted ue, ditibution, and epoduction in any medium, povided the oiginal wok i popely cited. Abtact An attempt i made to evaluate the paive eath peue on igid etaining wall unde tatic loading condition. The analyi i baed on Hoizontal Slice Method uing c-φ natue of backfill and the analyi geneate non-linea failue uface. The effect of vaiation of all the paamete have been tudied in detail and it i een that the hape of the uptue uface may be hogging o agging in natue. The eult have been given both tabula a well a gaphical epeentation. Keywod Paive Eath Peue, Hoizontal Slice Method, c-φ Backfill, Rigid Retaining Wall, Wall Inclination, Cuvilinea Ruptue Suface 1. Intoduction The pioneeing theoie fo the detemination of lateal eath peue behind etaining wall wee developed by Coulomb (1776) and Rankine (1857). Culmann (1865) intoduced a gaphical method fo the olution of lateal eath peue behind igid etaining wall. Thee analye ae mainly baed on coheion-le backfill and linea failue uface. Bell (1915) equation give the olution of lateal eath peue conideing the effect of coheion. The non-lineaity of the backfill failue uface i taken into account by Tezaghi (1943) conideing a log pial liding uface. A combination of log pial and taight line ha been conideed by Kuma (2001) to how the non-lineaity of failue uface in the detemination of eimic lateal eath peue. Azad et al. (2008) and Ghanbai and Ahmadabadi (2010) have adopted Hoizontal Slice Method (HSM) to calculate the eimic eath peue with the aumption of linea failue uface. In the peent tudy, the Hoizontal Slice Method ha been ued in uch a way that the failue wedge itelf geneate a non-linea uptue uface. The value of paive eath peue coefficient have been optimized conideing the imultaneou effect of unit weight, uchage, coheion and adheion.
IJACEAR An Open Acce Jounal 2. Method of Analyi Let u conide a batteed face igid etaining wall of height H (in the paive zone) uppoting a c-φ natue of backfill a hown in Figue 1. The wall on the paive zone i inclined at an angle, α with the vetical. The uptue uface i non-linea having diffeent value of inclination of failue uface with the vetical at bottom and top. The inclination of failue uface with the vetical at the bottom lice i conideed a θ n, wheea; the inclination of the failue uface at the bottom of the top lice i θ 1 with the vetical. If we conide the numbe of hoizontal lice (of thickne ΔH) a n, then the foce acting on the individual lice unde paive tate of equilibium i hown in Figue 2. Active Zone θ 1 Retaining wall c a SLICE -1 SLICE -2 θ 1+θ c P p R δ W Φ H α θ n Paive Zone LAYER -n Fig. 1 Show the foce acting batteed face etaining wall unde paive tate of equilibium Fee body diagam of etaining wall-backfill ytem unde paive tate of equilibium i elaboated in Figue 2. The foce acting on the wall ha been calculated by conideing the following: H i-1, H i = Hoizontal hea acting on the top and bottom of the i th lice W i = Weight of the failue wedge of i th lice V i-1, V i = Vetical load (UDL) on top and bottom of i th lice Ф = the angle of intenal fiction of oil P i = Paive eath peue on i th lice R i = the eaction of the etained oil on i th lice δ = the angle of wall fiction α = the angle of wall inclination c = Coheion acting between the failue uface c a = Adheion acting between the wall and oil uface Intenational Jounal of Advanced Civil Engineeing and Achitectue Reeach 43
IJACEAR An Open Acce Jounal c a1 c 1 R 1 P 1 δ α θ 1 Ф ΔH W 1 H 1 V 1 SLICE - 1 V i-1 H i-1 H P i c ai δ α θ 1+(i-1)θ c i Ф R i ΔH W i V i H i SLICE - i V n-1 H n-1 c an P n δ α θ W n Ф c n R n ΔH SLICE - n Fig. 2 Show foce component acting on the wedge lice unde paive tate of equilibium 3. Deivation of Vaiou Fomulation duing the Paive Cae of Equilibium Applying the peue equilibium condition, we can olve the equation in the following patten, 0 H P co( ) R i ca in [ co Whee, V 0 i c in{ 1 ( i 1) } co{ 1 ( i 1) } co{ ( i 1) } 2 {( H ) }] i ( n i) tan ( i 1)(tan( 1 ( i 1) ) ( H ) n 1 i tan )) {tan( 1 m )} mi 1 2 Pi in( ) Ri in( 1 ( i 1) ) ( i ) (tan 2 c co( 1 ( i 1) ) ca co tan( 1 ( i 1) )) co( ( i 1) ) co 1 1 (1) (2) Solving thee two equation, we find the value of genealized equation fo Paive eath peue fo i th lice a follow: P p i 2 ( ) {(2i 1)(tan tan( 1 ( i 1) ))( co( 1 ( i 1) )) co co( 1 ( 1) ) 2 N M i ( H ) i in( 1 ( i 1) )} co( 1 ( i 1) ) co in( ( i 1) ) 1 (3) Intenational Jounal of Advanced Civil Engineeing and Achitectue Reeach 44
IJACEAR An Open Acce Jounal Whee, ( n i) tan ( i 1)(tan( 1 ( i 1) ) 2 ( H ) n 1 i H tan )) {tan( 1 m )} N (tan tan( 1 ( i 1) )) mi (4) Whee, tan( 1 m ) 0 fo i = n (5) The paive eath peue coefficient can be implified a, kp n P pi i1 2 2 The value of N and M in Eqn 6 ae given by (6) Whee, N M NcH H M ch H (7) (8) 2c N c (9) H M c 2ca (10) H 4. Dicuion on Reult The optimized value of paive eath peue coefficient (K p ) have been evaluated fo the vaiation of diffeent paamete. The eult ae given tabula fom and a detailed paametic tudy i given in gaphical fom. The vaiou paamete conideed fo the pupoe ae a follow: Φ = 10, 20, 30 and 40 ; δ = 0, Φ/2, Φ; N c = 0.1, 0.2; M c = 0, N c /2, N c and α = +20, 0, -20. Optimization of the paive eath peue co-efficient, k p i done fo the vaiable θ 1 and θ n atifying the optimization citeia. The optimum value denoted a K p i given in Table 1. Table 1: Paive Eath Peue Co-Efficient (K p) N c=0.1 N c=0.2 Φ Δ M c α=-20 α=0 α= +20 α=-20 α=0 α=+20 0 1.906 1.524 1.434 2.506 1.612 1.496 0 N c/2 1.98 1.57 1.467 2.186 1.691 1.544 N c 2.051 1.612 1.496 2.315 1.763 1.584 10 0 2.177 1.66 1.514 2.321 1.741 1.564 Φ/2 N c/2 2.25 1.702 1.54 2.458 1.813 1.605 N c 2.321 1.741 1.564 2.589 1.885 1.639 Φ 0 2.506 1.812 1.596 2.654 1.889 1.638 N c/2 2.581 1.851 1.618 2.795 1.961 1.673 Intenational Jounal of Advanced Civil Engineeing and Achitectue Reeach 45
IJACEAR An Open Acce Jounal 20 30 40 0 Φ/2 Φ 0 Φ/2 Φ 0 Φ/2 Φ N c 2.654 1.889 1.638 2.932 2.029 1.703 0 3.141 2.173 1.814 3.338 2.295 1.925 N c/2 3.241 2.235 1.867 3.528 2.409 2.013 N c 3.338 2.295 1.925 3.71 2.516 2.088 0 4.559 2.749 2.155 4.781 2.866 2.233 N c/2 4.671 2.808 2.195 4.996 2.979 2.304 N c 4.781 2.866 2.233 5.207 3.087 2.369 0 7.325 3.607 2.565 7.599 3.728 2.631 N c/2 7.462 3.668 2.598 7.84 3.847 2.693 N c 7.599 3.728 2.631 8.132 3.962 2.751 0 5.476 3.167 2.377 5.739 3.324 2.46 N c/2 5.608 3.246 2.415 5.996 3.473 2.573 N c 5.739 3.324 2.46 6.247 3.617 2.7 0 12.495 5.083 3.27 12.874 5.253 3.377 N c/2 12.682 5.169 3.325 13.253 5.419 3.479 N c 12.874 5.253 3.377 13.626 5.582 3.575 0 -- 9.868 4.837 -- 10.093 4.939 N c/2 -- 9.98 4.888 -- 10.317 5.038 N c -- 10.093 4.939 -- 10.535 5.136 0 10.62 4.809 3.246 10.985 5.01 3.303 N c/2 10.804 4.91 3.258 11.345 5.204 3.398 N c 10.985 5.01 3.303 11.705 5.392 3.475 0 -- 11.628 5.492 -- 11.9 5.64 N c/2 -- 11.764 5.566 -- 12.172 5.783 N c -- 11.9 5.64 -- 12.44 5.922 0 -- -- 12.58 -- -- 12.77 N c/2 -- -- 12.675 -- -- 12.96 N c -- -- 12.77 -- -- 13.141 4.1. Effect of Inclination of the Wall (α) Figue 3 to 5 epeent the effect of inclination of the wall on the paive eath peue fo diffeent value of δ. Intenational Jounal of Advanced Civil Engineeing and Achitectue Reeach 46
IJACEAR An Open Acce Jounal Fom thee figue, it i een that due to deceae in α, the paive eitance inceae. Fo example, at Φ = 20, N c =0.1, M c = N c and α = -20, the value ae found 3.338, 4.781 and 7.599 fo δ = 0, Φ/2 and Φ epectively, wheea; with the above condition the value ae1.925, 2.233 and 2.631 at Φ=20 and α = +20. When the inclination of wall goe away fom the backfill, then the wall i in a poition to uppot moe load fom the backfill mateial. Thu, the active eath peue inceae and due to thi, the paive eitance i deceaed. 4.2. Effect of Wall Fiction Angle (δ) Figue 6 to 8 epeent the effect of wall fiction angle on the paive eath peue fo diffeent value of α. Fom the plot, it i een that due to the inceae in the value of δ, the paive eitance inceae. The eaon behind thi i the fictional eitance between wall and oil, which inceae with the inceae in the value of δ. Intenational Jounal of Advanced Civil Engineeing and Achitectue Reeach 47
IJACEAR An Open Acce Jounal Fo example, at Φ = 20, N c = 0.1, M c = N c and δ = Φ/2, the value ae 4.781, 2.866 and 2.233 fo α= - 20, 0 and +20 epectively, wheea; the value ae found 7.599, 3.728 and 2.631 fo α= -20, 0 and +20 epectively fo δ = Φ and all above condition emaining unchanged. Thu, it i found that the inceae in δ i inceaing the incement in the paive eath peue coefficient. The inceae in wall fiction angle inceae the fictional eitance of oil-wall ytem, o the paive eitance inceae. 4.3. Effect of Coheion and Adheion Figue 9 how the vaiation of paive eath peue coefficient (K p ) with epect to oil fiction angle (Ф) at M c = 0, N c /2, N c fo N c =0.1, δ=ф/2 and α=20. Intenational Jounal of Advanced Civil Engineeing and Achitectue Reeach 48
IJACEAR An Open Acce Jounal Fo example, at Φ=30, δ=ф/2, α=20 and M c = N c /2, the value of K p ae 3.325, and 3.479 epectively fo N c =0.1 and 0.2. Wheea the value inceae up to 3.377 and 3.757 fo M c =N c with all othe condition emaining unchanged. Again, fom Figue 10, it i alo found that the value of K p inceae with the inceae of N c value. Fo example, at Φ=20, δ=ф/2, α=20 and N c =0.1, the value of K p ae 2.155, 2.195 and 2.233 epectively fo M c =0, N c /2 and N c. wheea; the value inceae up to 2.233, 2.304 and 2.369 fo N c =0.2 with all othe condition emaining unchanged. 4.4. Effect of Height (H) Figue 11 how the vaiation of paive eath peue fo diffeent height of etaining wall (5m, 7.5m and 10m). It i obeved that the paive eitance gadually deceae with the inceae in height of etaining wall. Fo highe value of oil fiction angle (Φ), the paive eitance inceae fo the ame height. Fo Φ=20, α=20, N c =0.1, M c =N c and δ=φ /2, the deceae in K p i 10.2% and 15.3% fo 7.5m and 10m high etaining wall epectively compaed to 5m high etaining wall. Alo, fo Φ=40, α=20, N c =0.1, M c =N c and δ=φ/2, the decement in K p i again 5.8% and 12.1% fo 7.5m and 10m high etaining wall epectively compaed to 5m high etaining wall. With the inceae in height, the value of (2c/γH) and (2c a /γh) deceae, o the paive eitance get deceaed. Intenational Jounal of Advanced Civil Engineeing and Achitectue Reeach 49
IJACEAR An Open Acce Jounal 4.5. Wall Inclination and Nonlineaity of Failue Suface Figue 12 how the nonlineaity of failue uface of backfill fo diffeent value of wall inclination (α = -20, 0, +20 ). It i een that the inclination of failue uface with the vetical (α) inceae with the inceae in the wall inclination (α) angle. Fo example, at Φ=30º, δ=φ/2, N c =0.1, M c =N c and α= -20º, the value of angle, α at bottom i 75º wheea the value of angle, α at top i 84º. Figue 13 how the vaiation of failue uface with epect to height of the etaining wall. Fom the plot it i een that peent tudy epeent non-linea failue uface, wheea, Ghoh and Sengupta epeent linea failue uface. The compaion how that the value of inclination of failue uface with the vetical (θ) i 69º in cae of Ghoh and Sengupta (2012) fo the peviouly mentioned condition. Wheea; in the peent tudy, the angle i 77 at bottom and 84 at top of the failue uface. Thu, it how a cuvilinea hape a it pogee upwad. Alo, on the bai of peent tudy, it i een that due to the non-lineaity, the peent tudy epeent paticipation of moe backfill oil in compaion to Ghoh and Sengupta (2012) analyi. Intenational Jounal of Advanced Civil Engineeing and Achitectue Reeach 50
IJACEAR An Open Acce Jounal 4.6. Compaion of Reult Figue 14 how the vaiation of paive eath peue coefficient with epect to oil fiction angle (Φ) at δ=φ/2, N c =0.1, M c =N c fo α =20. K p inceae unifomly with the inceae in the value of oil fiction angle (Φ). The compaion in tabula fom i peented in Table 2. The compaion how that the value of K p obtained fom the peent tudy i 15% lee than the value obtained fom the analyi of Ghoh and Shama (2012). Table 2: Show the Compaion of Reult fo N c = 0.1, M c = N c Φ δ α K p, Peent tudy K p, Ghoh and Shama (2012) 20 10 0 3.087 3.66 30 15 0 5.582 6.54 40 20 0 12.44 14.66 5. Concluion Conideing the cuvilinea uptue uface and uing the hoizontal lice method, an analyi ha been developed fo the detemination paive foce on the back of a etaining wall uppoting c-φ backfill. Vey inteetingly, it i een that the hape of the uptue uface may be agging o hogging in natue. The above tudy of c-φ oil give a geneal olution fo paive eath peue. The paive eitance inceae with the inceae in the value of Φ, δ, c and c a. Geneally, the paive eitance i in an aveage 15% lowe in compaion to Ghoh and Shama (2012). Thi fact ugget the acceptability of the model which may be extended fo the analyi of etaining wall unde eimic loading condition. Notation Φ = oil fiction angle δ = wall fiction angle α = wall inclination angle with the vetical C = coheion C a = adheion P p = paive eath peue W i = weight of i th lice Intenational Jounal of Advanced Civil Engineeing and Achitectue Reeach 51
IJACEAR An Open Acce Jounal R = oil eaction γ = unit weight of oil N = (H/ΔH)*N c M = (H/ΔH)*M c Refeence Azad A., et al. Active Peue Ditibution Hitoy behind Rigid Retaining Wall. Soil Dynamic and Eathquake Engineeing. 2008. 28 (5) 365-375. Bell A.L., 1915: Lateal Peue and Reitance of Clay and the Suppoting Powe of Clay Foundation, Minute. Poceeding of the Intitution of Civil Enginee, London, 199. Coulomb C.A., 1776: Eai Su Une Application De Regle De Maximi et Minimi a Queque Poblem De Statique Relatif a1 Achitectue, Memoie d Academie Roy Pe. Diveavant. 7. Culmann K. 1866: Die Gaphiche Statik. Maye and Zelle, Zuich. Ghanbai A., et al. Peudo-Dynamic Active Eath Peue Analyi of Inclined Retaining Wall Uing Hoizontal Slice Method. Tanaction A: Civil Engineeing. 2010. 17 (2) 118-130. Ghoh S., et al. Fomulation of Paive Reitance on Non Vetical Retaining Wall Backfilled with c-φ Soil. Civil and Envionmental Reeach. 2012. 2 (1). Ghoh S., et al. Peudo-Dynamic Evaluation of Paive Repone on the Back of A Retaining Wall Suppoting c-φ Backfill. Geo-mechanic and Geo-engineeing: An Intenational Jounal. 2012. 115-121. Kuma J. Seimic Paive Eath Peue Coefficient fo Sand. Can. Geotech. J., Ottawa. 2001. 38; 876-881. Rankine W.J.M., 1857: On the Stability of Looe Eath. Phil. Ta. Royal Society, London. Shama R.P., et al. Peudotatic Seimic Active Repone of Retaining Wall Suppoting c-φ Backfill. EJGE. 2010. 15. Tezaghi K., 1943: Theoetical Soil Mechanic. John Wiley & Son, New Yok, 510. Intenational Jounal of Advanced Civil Engineeing and Achitectue Reeach 52