ANALYTICAL SOLUTION FOR CALCULATION OF BEARING CAPACITY OF SHALLOW FOUNDATIONS ON GEOGRID-REINFORCED SAND SLOPE *

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1 IJST, Transactions of Civil Engineering, Vol. 39, No. C, pp 67-8 Printed in The Islamic Repblic of Iran, 5 Shiraz University NYTIC SOUTION FOR CCUTION OF ERING CPCITY OF SHOW FOUNDTIONS ON GEOGRID-REINFORCED SND SOPE * V. ROSTMI ** ND M. GHZVI Faclty of Engineering, Science and Research ranch, Islamic zad University, Tehran, I. R. of Iran Rostami@iah.ac.ir Dept. of Civil Engineering, K. N. Toosi University of Technology, Tehran, I. R. of Iran bstract The bearing capacity of fondations resting on slopes is commonly calclated sing empirical eqations. In recent years, it has been demonstrated that geosynthetic reinforced soil can enhance the fondation bearing capacity. In this paper, an analytical method for determination of the ltimate bearing capacity of srface strip footing on a sand slope reinforced with geogrid layers is presented. The angle of the slope with the horizontal direction is varied within o to 4 o. In this stdy, the Colomb-type lateral earth pressre theory has been sed to compte the fondation bearing capacity. It is assmed that geogrid layers act sch that the active earth pressres are redced. The reslts obtained from the proposed method are compared with experimental and nmerical reslts. Parametric stdies have also been performed to show the effects of contribting parameters sch as nmber of geogrid layers, locations of reinforcement layers, soil properties and the slope angle on the bearing capacity of strip fondations resting on the reinforced sand. The reslts indicate that the magnitde of bearing capacity of strip footings on the sand slope can be significantly increased by sing geogrid layers. Keywords Geogrid-reinforced slope, strip footing, bearing capacity, shallow fondation. INTRODUCTION Sometimes fondations are bilt on slope or near the slope crest sch as roads or bildings constrcted in hilly regions and fondations for bridge abtments resting on granlar fill slopes. The bearing capacity of footing rests on slopes depends on the stability of slope and the soil bearing capacity. The constrction of a footing on sloping grond sally reslts in the footing bearing capacity redction as compared to that constrcted on horizontal grond srface, depending on the geometry of the footing and slope geometry as well as the location of the footing with respect to the slope crest. The bearing capacity of footings on loose soil may be improved by sing reinforcement objects sch as anchors, stone colmns, steel bars, and geosynthetics. In general, the se of geosynthetics to increase the bearing capacity and decrease the settlement of shallow fondations has been shown to be efficient []. In recent years, many experimental and analytical soltions have been presented for achievement acceptable footing bearing capacity by locating footings on reinforced soils. In this sitation, the se of reinforcement layers may be sefl to compensate the footing bearing capacity redction. Few stdies on the bearing capacity behavior footings resting on reinforced soil slopes have been reported in the literatre. Khing et. al. [] condcted laboratory-model tests to evalate the bearing capacity of a strip fondation spported on a sand layer reinforced with geogrid layers. They showed that, the bearing-capacity ratio at limited levels of settlement is abot 67-7% of the bearing-capacity ratio calclated on the basis of the ltimate bearing capacity. Omar et al. [3] condcted laboratory model test investigating of the ltimate bearing capacity of strip and sqare fondations spported by geogrid Received by the editors Febrary 3, 3; ccepted May 7, 4. Corresponding athor

2 68 V. Rostami and M. Ghazavi reinforced sand. They evalated the critical depth of reinforcement and the dimensions of geogrid layers to achieve the maximm bearing-capacity. Shin et al. [4] condcted small-scale laboratory model tests for fondations rest on geogrid-reinforced clayey soils. They showed that, in general, the ltimate and allowable bearing capacities of fondations can be improved by incorporating geogrid reinforcement. Yoo [5] condcted laboratory model tests and finite element model to determine the bearing capacity of a strip footing on a geogrid-reinforced earth slope. They evalated the effects of geogrid length, nmber of geogrid layers, vertical spacing and depth to topmost layer of geogrid on the footing bearing capacity. Their reslts indicate that the bearing capacity of strip footings on sloping grond can be increased by sing geogrid layers. This increase depends mostly on the nmber of geogrid layers and their locations. Vafaeian and bbaszadeh [6] stdied the behavior of steep reinforced slopes nder the effect of external loading on the srface. They sed small-scale model tests and showed that with increasing the spacing between reinforcement layers, the failre external load decreased. El Sawwaf [7] evalated the behavior of strip footing on geogrid reinforced sand over a soft clay slope. For this prpose, a model footing with 75 mm width and geogrids was sed. Test reslts showed that the inclsion of geogrid layers in the replaced sand improves the footing load-settlement performance. They fond that with increasing the nmber and lengths of geogrid layers significant improvement in the footing bearing capacity was achieved. lamshahi and Hataf [8] performed a series of nmerical and model tests on geogrid reinforcement and grid-anchor layers to evalate the bearing capacity of a strip footing resting on reinforced sand slopes. The reslts showed that load-settlement behavior and bearing capacity of the rigid footing can be considerably improved by sing a reinforcing layer at the appropriate location in the slope. Sommers and Viswanadham [9] performed laboratory tests to evalate the behavior of strip footing on geotextile-reinforced slopes. They sed small-scale physical tests in large beam centrifge to simlate field prototype conditions and examined the stability of geotextile-reinforced slopes sbjected to a vertical load applied to a strip footing. Their reslts showed that the vertical spacing between reinforcement layers has a significant impact on the stability of reinforced slope, and distribtions of peak strains in reinforcement layers de to the strip footing placed on the srface of the reinforced slope were fond to extend p to mid-height of the slope and thereafter they were fond to be negligible. Chodhary et al. [] investigated the bearing capacity behavior of strip footing on reinforced fly ash slope. They condcted laboratory model tests covering a wide range of bondary conditions. The reslts showed that the pressre settlement behavior and the ltimate bearing capacity of footing resting on the top of a fly ash slope can be affected by the presence of reinforcing layers. Dring the last three decades, a large nmber of small-scale laboratory model tests and large scale tests condcted to evalate the ltimate and allowable bearing capacities of shallow fondations spported on sand reinforced with geogrid layers [-3]. Despite significant attempts to date devoted to enhancing the footing load-carrying characteristics, it is still essential to develop soltions for compting the ltimate bearing capacity of strip footings on reinforced slopes. To this aim, fndamental mechanisms governing the load carrying capacity of shallow fondations on reinforced soil mst be investigated. This is essential for practicing engineers in the design and constrction of footings. For this prpose, this paper presents a closed form soltion for determination of the bearing capacity of strip fondations constrcted on geosynthetics reinforced soil sing a limit eqilibrim approach based on the Colomb type lateral earth pressres exerted on a rigid retaining wall. Parametric stdies are also carried ot to determine the effects of soil reinforcement, soil parameters, and slope geometries on the srface footing bearing capacity.. NYSIS OF ERING CPCITY The footing bearing capacity factors may be determined as sggested by Terzaghi [4] in terms of soil parameters sch as internal friction angle of soil and soil cohesion. Eqation () shows Terzaghi bearing capacity formla for strip footing. IJST, Transactions of Civil Engineering, Volme 39, Nmber C Febrary 5

3 nalytical soltion for calclation of bearing capacity of 69 q =cn c + qn q +.5 γ N γ () where N c, N q, and N γ are the bearing capacity factors, c is soil cohesion, q is srcharge, is soil nit weight, and is the fondation width. Richards et al. [5] made a simplification to the Prandtl failre srface as shown in Fig.. This assmption leads to estimating the fan-shaped transition zone and thereby averaging its effect concentrating on the shear transfer at line C, which can then be thoght of as an imaginary retaining wall with the active lateral thrst P from region C, pshing against the passive resistance P P from region CD (Fig. ).Using the force eqilibrim in the horizontal direction on imaginary wall C (Fig. ), the bearing capacity factors may be determined as [5]: q lt η P p δ δ p q = γd η p Prandll D D C H/tanη p Fig.. Simplification to the Prandtl failre srface by Richards et al. [3] N q= K P /K () N K P tan ( ) (3) K where η is angle between line with the horizontal direction (Fig. ), and K and K P are the active and passive lateral earth pressre coefficients, respectively and are given by: K cos and K P cos cos sin( )sin cos cos sin( )sin cos (4) (5) The angles of active and passive wedges with the horizontal direction respectively are: tan / tan (tan cot )( tan cot ) tan tan (tan cot ) (6) P / tan (tan cot )( tan cot ) tan tan tan (tan cot ) (7) In this paper, the above simple mechanism is sed to compte the bearing capacity of strip fondations on geogrid reinforced soil slope. 3. ERING CPCITY OF REINFORCED SOI SOPE In reinforced soil, active and passive wedges in terms of Colomb mechanism and eqilibrim eqations in the horizontal and vertical directions can be written to calclate the bearing capacity. Figre shows a Febrary 5 IJST, Transactions of Civil Engineering, Volme 39, Nmber C

4 7 V. Rostami and M. Ghazavi given reinforcement layer located at from the fondation bottom. This layer tolerates tensile force and ths the active force is redced. For simplicity, it is assmed that the fondation is located at the grond srface with no embedded depth (Fig. ). The pll ot force in geogrid (Fig. 4) is compted from: F tan (8) R v where is the length of geogrid that provides frictional force between soil and geogrid along it, σ v is vertical stress at depth z, φ is soil-geogrid interface friction angle, and F R is the resltant of longitdinal shear stresses acting on both sides of reinforcement per nit width (Fig. 3).The tensile force on the reinforcement layer is determined from: T= F R.w (9) where w is width of reinforcement materials. In Fig. 4, assming =m., =n., and w=, the total pll ot force on geogrid layer is calclated sing: From Fig. 3, H is given by: T ( q ) tan () b q=γd f q lt ctive wedge H Geogrid layer Passive wedge η η p Vertical magic retaining wall β Fig.. Proposed failre mechanism for reinforced soil-footing system H tan () F R Fig. 3. Pll ot force acting on each geosynthetic layer q T P P P Fig. 4. ctive wedge and reinforcement layer IJST, Transactions of Civil Engineering, Volme 39, Nmber C Febrary 5

5 nalytical soltion for calclation of bearing capacity of 7 Having all parameters, the eqilibrim eqation can be written for all forces acting on the active and passive wedges (Fig. 5). The above procedre can be applied to a fondation on soil reinforced with more than one geogrid layer. Figre 5 shows N geogrid layers each of which has length =m.. The top layer of reinforcement is located at depth from the footing bottom. The second layer is located at from the first layer. In this manner, other geogrid layers are located at 3, 4,. N. The lowest geogrid layer is located at H below the fondation where: N =H () ssming: = = 3 = = N the pll ot force for each reinforcement layer is: The pll ot forces for each reinforcement layer is (Fig. 5): T ( q ) tan (3a) T ( ( ) q) tan (3b) T3 ( ( 3) q) tan (3c) T ( (... ) q) tan (3d) N 3 N T ( N q) tan (4) N q tt T T W R T 3 φ η T N C p p F δ Fig. 5. Contribting forces acting on active wedge Taking eqilibrim eqation in X direction, R is compted from: F R cos(9 ) T P cos (5) X i P i The ltimate bearing capacity can be calclated sing: N q lt ssming b=, the vales of K p and η are calclated from: N W PP sin ( T i PP cos ) cot( ) i (6) Febrary 5 IJST, Transactions of Civil Engineering, Volme 39, Nmber C

6 7 V. Rostami and M. Ghazavi K P cos( ) cos sin( )sin( ) cos( ) cos( ) (7) / tan( )(tan( ) cot )( tan cot ) tan( ) tan tan (tan( ) cot ) (8) Sbstitting all pll ot forces from Eq. (4) and assming D f =S and b= gives the ltimate bearing capacity as: N qlt KP tan (sin cos cot( )) tan 4( S mn Ni )tan cot( ) (9) i The bearing capacity factor for layered reinforcement condition is given by: N N K P tan (sin cos cot( )) tan 4( S mn N i ) tan cot( ) () i 4. VERIFICTION In this section, the reslts obtained from the present analytical soltion sing limit eqilibrim method are compared with the reslts obtained from a nmber of reslts reported by others. a) Comparison with experimental reslts Yoo [5] performed laboratory model tests to stdy the bearing capacity behavior of a strip footing on a geogrid-reinforced slope. steel test box with inside dimensions of.8.5 m in plan and. m in height was sed. The box was sfficiently rigid to maintain plane strain conditions in the reinforced slope models. In all tests, an artificial slope of H:.67V was sed and the model footing was loaded sing a hydralic jack at a speed of. mm/min. He sed sand by a raining techniqe with a specially designed hopper system. The effective size (D ), niformity coefficient (C U ), and coefficient of crvatre (C c ) for the sand were.36,.6, and. mm, respectively, the estimated internal friction angle at the relative density of 7% was approximately 4 o. iaxial geogrids were sed that have tensile strength 55 kn/m at a maximm strain of.5%. The nominal thickness and the apertre size are. mm and mm, respectively. The experimental reslts fond by Yoo [5] are compared with the present analytical stdy in Fig. 6. The increase in the ltimate bearing capacity de to the inclsion of reinforcement has been expressed in the form of a non-dimensional term named the bearing capacity ratio, CR=q R /q, where q R is ltimate bearing capacity of footing on reinforced soil and q is ltimate bearing capacity of the same footing on nreinforced soil. It is noted that the footing ltimate bearing capacity was determined as the pressre corresponding to a footing settlement of %. s seen, there is very good agreement between experimental and analytical reslts. Figre 6 shows that when the one geogrid layer is sed the difference between analytical and experimental reslts is abot 3%. For and geogrid layers, these differences accont for abot 5.4% and.6%, respectively In general, there is a satisfactory agreement between these data, especially for more geogrid layers. IJST, Transactions of Civil Engineering, Volme 39, Nmber C Febrary 5

7 nalytical soltion for calclation of bearing capacity of 73 CR / =.3, / =4.5, h/ =.3, b/ = 3 N (Nmber of reinforcements) Fig. 6. Comparison of present analytical reslts with experimental tests by Yoo [5] b) Comparison with nmerical reslts Yoo () Present nalytical Stdy In this section, the reslts obtained from the present analytical approach are compared with nmerical reslts reported by Thanapalasingam and Gnanendran [6] who performed 3D nmerical analyses based on FC3D on footing characteristics near a slope. They sed strain softening failre criterion with nonlinear stiffness behavior sing Dncan s method. Geogrid SEs element given in FC3D program was also sed to model the strctral behavior of geogrid layer on the interaction of soil-geogrid. They sed only reinforcement layers at.5,.5,.75,.,.5,.5 from the footing bottom. Thanapalasingam and C.T. Gnanendran (8).5 Present nalytical nalysis CR (a) /=h/=.5, /=5.5 Thanapalasingam and Gnanendran (8) Present nalyt ical St dy N (Nmber of reinf orcement ) CR.5.5 N (Nmber of reinf orcement ) (b) /=h/=.75, /=5.5 Fig. 7. Comparison between present analytical reslts with nmerical reslts reported by Thanapalasingam and Gnanendran [4] The sandy soil had a Poison s ratio of.3 and an internal friction angle of 4 o. Other properties were copling spring stiffness per nit area=x N/m 3, elastic modls area=x N/m 3, and blk modls exponent=.5, the slope angle was H:V. The vales of CR predicted by the present analytical soltion with those predicted from nmerical analyses are compared. Febrary 5 IJST, Transactions of Civil Engineering, Volme 39, Nmber C

8 74 V. Rostami and M. Ghazavi In Fig. 7, as observed, when N= geogrid layer is sed, the difference between the two methods is abot 9%. For N=, this difference acconts for abot 4.3%. There is generally good agreement between analytical and nmerical data. c) Comparison with experimental reslts lamshahi and Hataf [8] performed a series of model tests on geogrid and grid-anchor layers to evalate the bearing capacity of a strip footing resting on reinforced sand slopes. For this prpose, they performed a series of laboratory model tests in a steel test box having inside dimensions of.3.5 m and.6 m height. The slope was H:.67V, which makes an angle of 33.8 in the horizontal direction. The model steel strip footing was 499 mm long, mm wide, and mm thick. The footing was located on the crest of the slope at.5 from the slope edge. For soil reinforcement, geogrid and grid-anchor made of high-density polyethylene was sed. The geogrid sheets had mesh apertre 7 7 mm and a maximm tensile strength of 5.8 kn/m. Gridanchor also had a mesh apertre 8 6 mm and a maximm tensile strength of 5.8 kn/m. The sand sed was medim to coarse, washed, and dried, and had ronded to sbronded particles. The maximm and the minimm dry nit weights of the sand were fond to be 9 and 3.4 kn/m 3. The niformity coefficient (C ) and coefficient of crvatre (C c ) for the sand were 7.78 and.4, respectively. The sand internal friction angle at 7% relative density was approximately 38 o. They defined the CR vale as the footing pressre on the reinforced slope (q reinforced ) to that of the same footing on the same slope with no reinforcement (q). They considered these pressres corresponding to a settlement eqal to 5% as the footing bearing capacity. s also stated, the bearing capacity for the model footing was determined from the load-settlement variation sch that after these pressres the footing was assmed to collapse. Figre 8 shows reslts obtained from the present analytical method and experimental reslt performed by lamshahi and Hataf [8]. s seen, when one reinforcement layer is sed, the difference between the footing bearing capacity predicted by the present method and that reported from experimental reslts is.56%. For layers, this difference acconts for abot.8% CR.5 lamshahi and Hat af (9) (Experiment al Met hod) Present nalyt ical St dy b/ =.5. / =.75, / =3.5.5 N (Nmber of reinf orcement ) Fig. 8. Comparison between present analytical reslts with experimental reslts reported by lamshahi and Hataf [8] a) Effect of footing distance from slope crest 5. PRMETRIC STUDIES ased on the present analytical method, the effect of the distance of the footing from the slope crest is investigated as shown in Fig. 9. It is obvios that when this distance increases, the passive wedge is enlarged, reslting in increasing the footing bearing capacity. To evalate the bearing capacity of a footing close to the slope crest, a simplification may be made as shown in Fig. 9. For this prpose, passive CM wedge may be replaced with M. For this wedge, it is assmed that C extends from the footing edge IJST, Transactions of Civil Engineering, Volme 39, Nmber C Febrary 5

9 nalytical soltion for calclation of bearing capacity of 75 to the slope toe and makes angle β. In fact, in this simplification, the effect of wedge CM is similar to wedge M, noting that both wedges have the same area and ths the same weight. ngle β can be compted from: (tan as tan p).sin.sin p tan a.sin( p ) (tan as tan p).sin.cosp tan [ ] () where η a and η p are the angles of active and passive wedges with the horizontal direction respectively and s=b/ is the distance ratio. To compte q lt for the footing shown in Fig. 9, Eq. () may be sed with β instead of β. Figre shows the variation of β verss s=b/. s seen, by increasing s, the vale of β decreases, the s vale decreases and ths the footing bearing capacity decreases. s seen in Fig., for a slope angle of β=3, when φ=3, by increasing the distance ratio p to 3, β decreases to almost zero and this means that the effect of slope angle has very negligible effect on the footing bearing capacity. For φ=4, increasing the distance ratio p to 5, β decreases to almost zero. This also means that the edge distance affects the footing bearing capacity when it is smaller than five times the footing width. The above finding was spported by others. b q lt ctive wedge Eqivalent passive wedge H Geogrid layer Passive wedge C Vertical magic retaining η a Fig. 9. Passive wedge simplification For example, Meyerhof [7] reported that the footing bearing capacity factors for a strip fondation in prely cohesive and cohesionless materials decrease with greater inclination of the slope and increase rapidly with greater fondation distance from the edge of the slope. eyond a distance of abot to 6 times the fondation width (depending on φ and footing embedment), the bearing capacity is independent of the inclination of the slope and becomes the same as that of a fondation on the horizontal grond srface. M η p β D β C 3 5 β 5 β= s Φ=3 Φ=3 5 Φ=4 Fig.. Variation of of β verss s Febrary 5 IJST, Transactions of Civil Engineering, Volme 39, Nmber C

10 76 V. Rostami and M. Ghazavi ee and Manjnath [8] also performed a series of tests to evalate of effect of edge distance of the footing from the slope crest. For this prpose, five different distances for the footing on both reinforced and nreinforced soil with H:V slope were sed. These distances were b=,, 3, 4 and 5. =.5 was sed for reinforcement embedment depth. Their reslts indicate that for both reinforced and nreinforced slopes, the ltimate bearing capacity increased with increasing the edge distance. They showed that at edge distance of 5, the ltimate bearing capacity of the footing on the slope approached that of a footing on a flat srface for both reinforced and nreinforced soil. They also fond that the effect of the slope was minimized when the footing was placed at an edge distance beyond 5. In addition, it was shown that for other smaller edge distances, the footing bearing capacity on the reinforced slope was considerably greater than that of the footing on nreinforced slope. Chodhary et al. [] condcted tests on strip footing resting on flyash slope at varying edge distances from the slope crest. These distances were b=,, and 3 for both nreinforced and reinforced soil. They showed that the footing ltimate bearing capacity for a given slope remained almost constant and identical to that of the footing on the horizontal srface when the edge distance increased beyond 3. EI Sawwaf [7] condcted tests and reported that if the footing was placed at a distance of or frther from the slope crest, the bearing capacity improvement (CR) becomes insignificant. lso, it showed that the CR vale decreased with increasing b from.5 to 3 [5]. ased on the crrent research, the maximm vale of b to have the complete passive wedge is eqal to (tanη a /tanη p). Ths, the footing distance from the slope crest depends on soil properties, slope geometry, and footing width. When the friction angle of cohessionless material, φ, varies in the practical range of 3-4, the edge distance to form complete passive edge from the slope crest varies in 3. to 5.4 range. This range is in good agreement with above findings reported by others. Table shows b/ ratio range reported, obtained from experimental and analytical stdies performed on reinforced and nreinforced soil slope. s seen, there is very good agreement between the present analytical prediction and data reported by others. Table. Vales of b/ from some references Reference Method of nalysis Footing Type Reinforcement Condition Meyrhof [7] nalytical Strip Unreinforced ee and Manjnath [8] Chodhary et al. [] Experimental Experimental Strip Strip Reinforced and Unreinforced Reinforced and Unreinforced Soil Type Sand with φ=3-4 b/ 5 Sand with φ=38 5 Flyash with c= kpa and φ=4 3 EI Sawwaf [7] Experimental Strip Reinforced Sand with φ=43 Wing Ip [9] Experimental Strip Unreinforced Sand with φ=3 6 Shin and Das [4] Experimental Strip Reinforced Clay 3 Castelli and Motta [3] nalytical Circlar Unreinforced Present Stdy nalytical Strip b) Effect of reinforcement length on bearing capacity Reinforced and Unreinforced Undrained clay and sand with φ=-4 IJST, Transactions of Civil Engineering, Volme 39, Nmber C Febrary 5-6 Sand with The length of reinforcement layer has a significant effect on the footing bearing capacity. Figre shows the reslts obtained from the developed analytical method. s seen, the reinforcement layer can vary between the slope face to the other side of the active wedge (see in Fig. ). Therefore, if the

11 nalytical soltion for calclation of bearing capacity of 77 reinforcement length becomes greater than this distance, no improvement in the footing bearing capacity wold be expected de to the fact that 4 wold not be effective (Fig. ). Figre shows that the total effective length is determined from: total = ++b+ 3 () H= tan(η a ) (5 repeated) =H/tan(η p ) (3) tan tan p (4) Therefore: ( H ) (5) H From the slope geometry, the amont of 3 is calclated as: q l b 3 4= Ineffective length Geogrid layer total H ctive wedge Passive wedge η p C Magic retaining wall β Fig.. Effective length of reinforcement layer [tan n] tan p (6) 3 = /tanβ (7) ssming =n, and b=s: n tota l [ [tan n] ( s) ] tan (8) tan p Eqation (8) indicates that the total effective length of the reinforcement layer depends on soil properties sch as φ, fondation distance from slope crest, reinforcement location, and slope geometry. Figre shows the effective reinforcement length verss φ for s=, and β=3. s seen, for a given vale of φ, for example 35, the effective length decreases with increasing. If increases from.3 to.7, the reinforcement length decreases from 7 to 6.3. Figre3 shows effective reinforcement length verss φ for s= and β=5. s observed, for φ=35, the effective length decreases from 8.5 to 7.8 with increasing =.3 p to =.7. Yoo [5] stated that for optimm improvement in the ltimate bearing capacity of strip footing on a reinforced sandy slope, the length of reinforcement shold be in the range of Febrary 5 IJST, Transactions of Civil Engineering, Volme 39, Nmber C

12 78 V. Rostami and M. Ghazavi total / =.3 =.7 5 (s=, β=3) φ Fig.. Effective reinforcement length for s=, β=3 9 8 total / 7 =.3 =.7 6 s=, β= Fig. 3. Effective reinforcement Φ length, s=, β=5 ee and Manjnath [8] performed a series of nmerical and small scale model tests to evalate the bearing capacity of a strip footing on geogrid reinforced sand slope. They showed that the optimm reinforcement length was approximately eqal to the edge distance pls 8, which is measred from the face of the reinforced slope. lso, they performed nmerical analysis and tests and obtained 8.9. ased on the developed analytical stdy, for soil with φ=36, slope angle of β=6.56, and =.5 for s=, the total effective reinforcement layer becomes The present analytical method gives effective lengths varying from for friction angles varying 3-4 and for =.7, β=3, and s=. These are in agreement with others as discssed. c) Effect of soil relative density on reinforced bearing capacity In this section, the CR factor is sed to identify the benefits of footing-soil reinforcement. Figre 4 indicates that the CR vale decreases with increasing the soil internal friction angle p to abot 35, beyond which no significant changes occr in the CR vales..5.5 CR.5 (b/=, /=.4, β=5 ) /=4 /= Φ Fig. 4. Effect of reinforcement length on bearing capacity IJST, Transactions of Civil Engineering, Volme 39, Nmber C Febrary 5

13 nalytical soltion for calclation of bearing capacity of 79 d) Effect of nmber of reinforcement layers In this section, some analyses are condcted to stdy the effect of N (nmber of reinforcement layer) on the bearing capacity of footings on the slope as presented in Fig. 5. Here, the reinforcement length and the distance of the footing from the slope edge were kept constant. s observed, with increasing N, the CR increases, especially for lower soil friction angle Φ=3 8 Φ=4 (/ =4, / =.5,β=3, b/ =) CR N ( Nmber of reinf orcement ) Fig. 5. Effect of variation of the nmber of reinforcement layer The CR vale increases with increasing the nmber of geogrid layers located within the active and passive zones. Figre 3 shows that there is a limited zone depth for inclsion of reinforcement layers in order to have the footing bearing capacity improvement. This depth depends on soil properties, footing geometry, and slope geometry. s seen, the depth that reinforcement layers can be located varies within thickness H, i.e., between to C points. This is in accordance with the findings of others. lamshahi and Hataf [8] reported the increase in CR only for two reinforcement layers beyond which the CR vale remained constant with increasing the nmber of reinforcement layers. Similar trends were reported by ee and Manjnath [8]. El Sawwaf [7] performed two series of stdies in order to investigate the effect of the nmber of geogrid layers on the footing-slope performance. The test series were carried ot, a depth of replaced sand of.5 along with geogrid length, location, and spacing, was kept constant bt the nmber of geogrid layers was varied. The reslts demonstrate that reinforcing a sand layer more than three layers of geogrid cold bring abot an improvement in bearing capacity greater than that obtained. El Sawwaf [7] performed tests on footing constrcted on slopes which consisted of two layers. The pper layer was sand and the lower layer was clay. The sand thickness was.5. He reported that if / and b/ were constant, the footing bearing capacity did not have considerable increase when more than three geogrid layers were sed. In the literatre, there is no evidence that more than three geogrid layers have been sed. This is becase there exists a limited zone beneath the footing on the slope soil for reinforcement to achieve benefits on the CR vale. Figre 6 shows that for a certain slope geometry, limited reinforcement layers can be placed to have the footing bearing capacity improvement. Yoo [5], ee and Manjnath [8] reported that there are a critical nmber of geogrid layers beyond which the CR becomes constant. In this stdy, as depicted in Fig., the reinforcement layers can only be located within C zone to have CR increase de to the soil reinforcement. This is becase C zone is eqal to H= tan η. For a slope angle of 3 o, if φ varies within a practical range of 3-45 for footing-soil reinforcement, H varies in.9-.5 range. The above limitation was spported by lamshahi and Hataf [8] in their tests. h=.75 was sed for spacing of two reinforcement layers. They sed only geogrid layers within thickness C in Fig.. Their reslts show that the third reinforcement layer did not have any effect on the footing bearing capacity. ased on analytical stdy present here, third reinforcement layer was located ot of thickness C, so it cannot affect the bearing capacity. Febrary 5 IJST, Transactions of Civil Engineering, Volme 39, Nmber C

14 8 V. Rostami and M. Ghazavi e) Effect of slope angle on CR Most research on the CR of footings resting on reinforced soil slop sed small scale tests with constant slope angles and the slope angle variation on the CR variation has not been considered [ ]. This paper considers this effect by sing,, and 3 o for the slope angles and the footing bearing capacity ratio verss these vales are shown in Fig. 6. s seen, the CR vale increases with increasing the slope angle. It is interesting to note that the footing bearing capacity decreases obviosly with increasing the slope angle. On the other hand, the se of reinforcement leads to increasing the bearing capacity of footings on slopes. In general, with increasing the slope angle, the latter grows more rapidly than the former, especially for steeper slopes. s a reslt, with increasing the slope angle, the footing bearing capacity ratio increases..5.5 CR.5 Φ=35, /=.4, b/= / β (Degree) Fig. 6. Effect of slope angle on CR f) Determination of bearing capacity coefficient on reinforced sand slope Eqation () has been derived from the crrent analytical soltion. This eqation incorporates the effects of soil parameters, reinforcement material parameters, slope geometry, reinforcement layers, and reinforcement locations, and footing geometry. Figres 7 and 8 show the bearing capacity coefficients for reinforced soil slope for reinforcement lengths of 4 and 5, respectively. s seen, the N vale increases with decreasing the slope angle. 8 6 Nγ 4 β= β= 5 β=3 /=4, /=.5, b/= Φ (Degree) Fig. 7. Variation of N verss for footing on reinforced sand slope (/=4) 8 6 Nγ 4 β= β= 5 β= 3 /=5, /=.5, b/= Φ (Degree) Fig. 8. Variation of N verss for footing on reinforced sand slope (/=5) IJST, Transactions of Civil Engineering, Volme 39, Nmber C Febrary 5

15 nalytical soltion for calclation of bearing capacity of 8 6. CONCUSION In this paper, an analytical method for compting the ltimate bearing capacity of srface strip fondation on sandy slopes reinforced with geogrid layer are presented in which the Colomb-type lateral earth pressre theory has been sed. ased on the reslts predicted by this method, the following conclsions may be cited:. The magnitde of bearing capacity of strip footings on sandy slopes can be significantly increased by sing geogrid layers.. The bearing capacity of footing decreases with increasing the slope angle for both nreinforced and reinforced sandy slopes. lso, the bearing capacity ratio increases with increasing the slope angle. This proves the reinforcement effect on the bearing capacity of footings on slopes. 3. The location of geogrid layers at a depth greater than.5 times the footing width does not have significant improvement on the bearing capacity. This stdy shows that the better depth for reinforcement location varies with soil properties, footing geometry and slope angle. 4. The slope geometry, reinforcement properties sch as length, layer nmber, layer location, and soil properties has been fond to have significant effects on the footing bearing capacity. REFERENCES. Ghazavi, M. & avasan,.. (8). Interference effect of shallow fondations constrcted on sand reinforced with geosynthetics. Geotextiles and Geomembranes, Vol. 6, No. 5, pp Khing, K. H., Das,. M., Pri, V. K., Cook, E. E. & Yen, S. C. (993). The bearing capacity of a strip fondation on geogrid reinforced sand. Geotextiles and Geomembranes, Vol., pp Omar, M., Das,., Pri, V. & Yen, S. (993). Ultimate bearing capacity of shallow fondations on sand with geogrid reinforcement. Canadian Geotechnical Jornal, Vol. 3, pp Shin, E. C., Das,. M., Pri, V. K., Yen, S. C. & Cook, E. E. (993). earing capacity of strip fondation on geogrid-reinforced clay. Geotechnical Testing Jornal, STM, Vol. 6, No. 4, pp Yoo, C. (). aboratory investigation of bearing capacity behavior of strip footing on geogrid-reinforced sand slope. Geotextiles and Geomembranes, Vol. 9, pp Vafaeian, M. & bbaszadeh, R. (6). aboratory small scale tests to stdy the behavior of reinforced soil wall. In: Kwano, J., Kseki, J. (Eds.), Proceedings of Eighth International Conference on Geosynthetics, Vol. 4. Millpress Science, Rotterdam, pp EI Sawwaf, M. (7). ehavior of strip footing on geogrid reinforced sand over a soft clay slope. Geotextiles and Geomembranes, Vol. 5, pp lamshahi, S. & Hataf, N. (9). earing capacity of strip footings on sand slopes reinforced with geogrid andgrid anchors. Geotextiles and Geomembranes, Vol. 7, pp Sommers,. N. & Viswanadham,. V. S. (9). Centrifge model tests on behavior of strip footing on geotextile reinforced slopes. Geotextiles and Geomembranes, Vol. 7, pp Chodhary,. K., Jha, J. N. & Gill, K. S. (). aboratory investigation of bearing capacity behavior of strip footing on reinforced flyash slope, Geotextiles and Geomembranes, Vol. 8, pp Michalowski, R.. (4). imit loads on reinforced fondation soils. Jornal of Geotechnical and Geoenviromental Engineering, SCE, Vol. 3, No. 4, pp Hang, C. C. & Menq, F. Y. (997). Deep-footing and wide slab effect in reinforced sandy grond. Jornal of Geotechnical and Geoenvironmental Engineering, SCE, Vol. 3, No., pp Sharma, R., Chen, Q. & b-farsakh, M. (8). nalytical modeling of geogrid reinforced soil fondation. Geotextiles and Geomembranes, Vol. 7, pp Febrary 5 IJST, Transactions of Civil Engineering, Volme 39, Nmber C

16 8 V. Rostami and M. Ghazavi 4. Gido, V.., Kneppel J. D. & Sweeny, M.. (987). Plate loading tests on geogrid reinforced earth slabs. Proceedings Geosynthetics 87, New Orleans, pp Hang, C., Tatsoka, F. & Sato, Y. (994). Failre mechanisms of reinforced sand slopes loaded with a footing. Soils and Fondations, Vol. 4, No., pp dams, Mike, T. & Collin, James, G. (997). arge model spread footing load testson geosynthetic reinforced soil fondations. Jornal of Geotechnical Engineering, SCE, Vol. 3, No,, pp Patra, C. R., Das,. M., hoi, M. & Shin, E. C. (6). Eccentrically loaded strip fondation on geogridreinforced sand. Geotextiles and Geomembranes, Vol. 4, No. 4, pp asdhar, P. K., Saha, S. & Deb,. (7). Circlar footings resting on geotextile-reinforced sand bed. Geotextiles and Geomembranes, Vol. 5, No. 6, pp Sakti, J. & Das,. M. (987). Model tests for strip fondation on clay reinforced with geotextile layers. Transportation Research Record No. 53. National cademy of Sciences, Washington, DC, pp Chen, H. T., Hng, W. Y., Chang, C. C., Chen, Y. J. & ee, C. J. (7). Centrifge modeling test of a geotextile-reinforced wall with a very wet clayey backfill. Geotextiles and Geomembranes, Vol. 5, No. 6, pp b-farsakh, M., Chen, Q., Sharma, R. & Zhang, X. (). arge-scale model footing tests on geogrid reinforced fondation and marginal embankment soils. Geotechnical Testing Jornal, STM, (in press).. Kmar,. & Saran, S. (3). earing capacity of rectanglar footing on reinforced soil. Geotechnical and Geological Engineering, Vol., pp Mosallanezhad, M., Hataf, N. & Ghahramani,. (). Three dimensional bearing capacity analysis of granlar soils reinforced with innovative grid-anchor system. Iranian Jornal of Science & Technology, Transaction : Engineering, Vol. 34, No. 4, pp Terzaghi, K. (943). Theoretical soil mechanics. John Wiley and Sons, New York. 5. Richards, R, Elms, D. G. & dh, M. (993). Seismic bearing capacity and settlement of fondations. Jornal of Geotechnical Engineering, Vol. 9, No. 4, pp Thanapalasingam, J. & Gnanendran, C. T. (8). Predicting the performance of fondations near reinforced sloped fills. th International Conference of International ssociation for Compter Methods and dvances in Geotechnics, Goa, India, pp Meyerhof, G. G. (957). The ltimate bearing capacity of fondations on slopes. Proceedings of the 4th International Conference on Soil Mechanics and Fondation Engineering, Vol., pp ee, K. M. & Manjnath, V. R. (). Experimental and nmerical stdies of geosynthetic-reinforced sand slopes loaded with a footing. Canadian Geotechnical Jornal, Vol. 37, pp Wing, Ip. (5). earing capacity for fondation near slope. Mse. tesis, Concordia niversity, Montreal. Qebec, Canada 3. Castelli, F. & Motta E. (). earing capacity of strip footings near slopes. Geotechnical and Geological Engineering, Vol. 8, No., pp IJST, Transactions of Civil Engineering, Volme 39, Nmber C Febrary 5