BEHAVIOUR OF LATERALLY LOADED RIGID PILES IN COHESIVE SOILS BASED ON KINEMATIC APPROACH

Transcription

1 LOWLAND TECHNOLOGY INTERNATIONAL Vol., No., 7-4, Jne 8 International Association of Lowland Technology (IALT), ISSN BEHAVIOUR OF LATERALLY LOADED RIGID PILES IN COHESIVE SOILS BASED ON KINEMATIC APPROACH V. Padmavathi, E. Saibaba Reddy and M. R. Madhav ABSTRACT: Piles are sefl in lowland areas in order to improve the properties of the soil. Several methods are available to predict the ltimate lateral resistance of a rigid pile in clays. The existing soltions for ltimate lateral resistance of rigid piles in clays are either semi-empirical in natre or based on approximate analysis with several simplifications. In most of these methods, the behavior of soil is assmed as plastic throghot the analysis inclding at the point of rotation. Even thogh the ltimate lateral loads predicted by these methods are somewhat comparable with the measred vales, the lateral pressre distribtions are not consistent. A new approach based on kinematics and non-linear sbgrade (hyperbolic) response has been developed to stdy the load-displacement response of a single rigid pile with free-head in cohesive soils. The predicted ltimate lateral capacities of the piles as well as the lateral soil pressre distribtions along the pile length compare well with the reslts of available theories and experimental test reslts. Keywords: Lateral load, rigid pile, non-linear sbgrade response, cohesive soil, kinematics, ltimate load INTRODUCTION Design of fondations sbjected to large lateral forces cased by wind, wave action, earthqake and lateral earth pressres is a challenging task especially if fonded in soft grond sally prevalent in lowland regions of the world. Strctres sch as transmission line towers, bridges, tall bildings, which are sally fonded on piles, mst be designed to spport both axial and lateral loads and moments. The lateral resistance of piles is governed by several factors, the most important being the ratio of the strctral stiffness of the pile to the soil stiffness. The relative stiffness of the fondation element with respect to the soil controls the mode of failre and the manner in which the pile behaves nder an applied lateral load. Extensive theoretical and experimental stdies have been carried ot by several investigators on laterally loaded piles to determine their ltimate lateral loads and displacements, nder working loads. Matlock and Reese (96) define the relative rigidity of a laterally loaded pile in terms of the ratio of the flexral stiffness of the pile, EI, and the coefficient of lateral sbgrade reaction, k s. Their method is applicable only if the deflection of piles is within the range of linear deformation of the soil. Vesic (96), Davisson and Gill (963), Broms (964), Banerjee and Davies (978) and Polos and Davis (98) define and tilie a stiffness ratio, K r, as K 4 r = E p I p Es L () where E s - is the deformation modls of the soil, E p - the modls of elasticity of the pile material, I p the moment of inertia of pile cross-section and L - the length of the pile. The pile sally behaves as a rigid one for K r vales greater than -. Kasch et al. (977) have proposed that embedded length to diameter ratio, L/d, of the pile also be sed to assess the flexral behavior of the pile and conclded that in order to ensre rigid behavior, the L/d ratio shold not exceed abot 6. However, nder some conditions, a fondation can have L/d ratio as high as and still behave as a rigid one, bt, for flexible behavior L/d shold be in excess of. Based on the stdies of Meyerhof et al. (98, 985), the pile behavior is rigid even thogh L/d is as high as 6. From the above stdies, it is felt that assessing the rigid behavior of the pile throgh relative stiffness factor, K r, wold be a better option. Failre of rigid or short shafts takes place when the lateral earth pressre reslting from lateral loading attains the limiting lateral resistance of the spporting soil along the fll length of the member. The rigid pile is assmed to be infinitely stiff and the only motion allowed is pre rotation of the shaft as a rigid body abot Dept. of Civil Engineering, JNTU College of Engg., Hyderabad, Andhra Pradesh, INDIA Dept. of Civil Engineering, JNT University, Kkatpally, Hyderabad, Andhra Pradesh, INDIA Note: Discssion on this paper is open ntil December 3, 8

2 Padmavathi, et al. some point on the axis of the shaft. For rigid body motion, the rotation of the shaft and the displacement at the grond line define the deformed position of the pile. Broms (964) assmed that the ltimate soil resistance is developed all along the length for a rigid pile, whereas for a flexible pile, the ltimate load-carrying capacity is arrived at also from the consideration of failre of the pile itself becase of the development of a plastic hinge in the pile. Barber (953) evalated lateral deflections of piles at working loads sing the concept of sbgrade reaction considering the bondary conditions both at the grond srface and at the tip of each individal pile. The behavior at working loads was analysed by assming the laterally loaded pile to behave as an elastic member and the spporting soil as a linearly deforming material. Extensive theoretical and experimental stdies on laterally loaded piles in clays have been carried ot by Broms (964), Drery and Fergson (969), Bhshan et al. (979), Briad et al. (983), Randolph and Holsby (984), Meyerhof et al. (985), Lai and Booker (989), Narasimha Rao and Mallikarjna Rao (995), Prasad (997), McDonald (999), etc. All the theoretical stdies for the estimation of the ltimate capacity of a single pile were based on mobiliation of ltimate lateral soil resistance over the fll length of the pile for plastic conditions. In the present stdy, the kinematics of pile movement is copled with non-linear response of the insit soil to predict the response of the pile nder lateral load. BEHAVIOUR OF PILE AND SOIL The considered failre mechanisms and the reslting distribtions of mobilied soil reactions at failre along a laterally loaded free-head pile driven into a cohesive soil for the estimation of its ltimate lateral capacity is shown in Fig.. Soil located in front of the loaded pile close to the grond srface may heave pwards, i.e. the direction of least resistance, while the soil located at Lateral Load c d Normalised Depth, /L.5 6 Hays Matlock Hansen Ultimate Lateral Soil Pressre, p (kpa) some depth below the grond srface moves laterally arond the pile. The pile may get separated from the soil on its rear down to a certain depth below the grond srface. The ltimate soil resistance for piles in prely cohesive soils increases with depth from c at the grond srface to 8 to times c at a depth of abot three pile diameters below the srface (Polos 98). ULTIMATE SOIL REACTION Reese McDonald Present Fig. Ultimate Soil Reaction for Piles in Cohesive Soil by Different Approaches for Typical Vales c = kpa, γ = kn/m 3, d=.9 m and L=6. m Hansen (96), Matlock and Reese (96), Reese (958), Broms (964), Hays et al. (974), McDonald (999) have proposed different approaches to compte the ltimate soil reaction, p, with depth. The proposed relationships are:. Hansen: p = K c.c d () Soil Move- ment Approximately. Matlock: p = [3+(γ /c )+(.5/d)]c d (3) 3. Reese: p = [3+(γ /c )+(.83/d)]c d (4) 4. Hays: p = η c d + α (5) 5. McDonald: p = c d + (.33/d) c d (6) 8c d to c d Fig. Pile soil behavior and postlated distribtion of ltimate soil resistance in which p is the ltimate soil reaction, γ nit weight of the overbrden material, - depth below the grond line, c - ndrained shear strength of the soil, d - pile diameter, K c - earth pressre coefficient given by

3 Behavior of laterally loaded rigid piles in cohesive soils based on kinematic approach Hansen, η - soil strength redction factor and α - slope of a reaction crve. All these theories recommend a limiting vale for the soil reaction at a critical depth below the grond srface. The limiting vale for the soil resistance according to Matlock, Reese, Hays et al. and McDonald theories is 9c d while Hansen defines the limit as 8.4c d. The ltimate soil reaction by different methods is shown in Fig. for typical vales of c = kpa, γ = kn/m 3, d =.9 m and L = 6. m. LATERAL PRESSURE DISTRIBUTION ACROSS THE DIAMETER In most of the pblished theories, it is assmed that the soil pressre acts niformly along the projected width of the pile, which, for a circlar pile, is its diameter as shown in Fig. 3(a). However, the lateral pressre varies along the pile srface (Bierschwale 98) having a maximm in the direction of applied lateral load and decreasing significantly beyond an angle of 3 as shown in Fig. 3(b). Similar type of behavior was reported by Reese and Cox (969), Randolph and Holsby (984), and Lai and Booker (989) from the analysis of piles sbjected to lateral loads. Prasad and Chari (999) proposed that the lateral pressre is maximm at the centre, that is, in the direction of the load, bt decreases to almost ero at the two edges. An average pressre of.8 times the maximm pressre is recommended to accont for the non-niform pressre distribtion along the pile periphery. Briad et al. (983) and Zhang et al. (5) also proposed shape factors of.8 for circlar and. for sqare and rectanglar shapes to accont for the nonniform distribtion of soil pressre in front of the pile. LATERAL PRESSURE DISTRIBUTION ALONG THE LENGTH OF THE PILE For a free head rigid pile in a cohesive soil Broms (964) sggested a simplified distribtion of soil resistance as being ero from the grond srface to a depth of.5 times the pile diameter and a constant vale of 9c below this depth as shown in Fig. 4(a). This is sometimes mistakenly interpreted to accont for possible shrinkage away from the pile in the near srface one. For large diameter bored pile fondations, the assmption of ero soil resistance to a depth of.5d is conservative. This is especially tre for short piles having slenderness ratio, L/d, less than 5 (McDonald 999). A more rigoros distribtion (Polos 97), in which the soil resistance is eqal to the nconfined compressive strength (c ) of the soil at the srface increasing to 4.5 times of this vale at a depth eqal to three times the pile diameter, reqires the soltion of a cbic eqation. McDonald (999) presented soltions for short rigid piles in clay sing the soil pressre distribtion (Fig. 4b) of c at GL and 9c as the limiting pressre at a depth of from GL (Polos 97). The reslts from McDonald (999) indicate that Broms estimates of the ltimate pile resistance vales are conservative for smaller L/d vales. In this paper, the soil pressre distribtion along the pile is non-linear as shown in the Fig. 4(c). Meyerhof et al. (984 and 985) and Sastry and Meyerhof (986) present reslts from experimental stdies for the lateral load capacity of short rigid piles in sand and clay. Soil resistance distribtions along the pile length were measred on laterally loaded model piles instrmented with pressre transdcers. STATEMENT OF THE PROBLEM A pile is sbjected sally to a combination of axial and lateral loads and moments. In the present paper, free head rigid pile installed in a cohesive soil sbjected to e H GL.5d c d Deflection, ρ Load (H) L 9 c d 9 c d 9 c d 9 c d d (a) (b) (c) (a) (b) Fig. 3 Pressre distribtion arond the pile (a) before and (b) after application of lateral load Fig. 4 Pressre distribtions along the pile nder ltimate lateral load for (a) Broms (964) and (b) McDonald (999), Polos (97) and at large displacements (c) Present approach

4 Padmavathi, et al. e Η ρ θ GL L H = q d d q d d (8) or H = b t qt d d + q t d d q d d (9) L b L O O or H = d k s ρ k s ρ + q 3 max d d + k s ρ k s ρ + q max d d Stress d (a) (b) L k s ρ k s ρ + q max d d () where q t and q b are the lateral stresses above and below the point of rotation respectively with q t and q t being the lateral stress for depths from GL to and from to the point of rotation respectively. The moment eqilibrim eqation is k s lateral load only is considered. The pile of length, L, and diameter, d, is acted pon (Fig. 5a) by a lateral force, H, at an eccentricity, e, creating a moment, M (=e.h). The pile is nrestrained and rotates throgh an angle, θ, abot a point O at depth,, from the grond srface. The response of the soil is represented by Winkler sing a set of springs (Fig. 5b) with modls of sbgrade reaction, k s, and ltimate lateral soil pressre, q max. The displacement ths varies linearly with depth. The lateral stress, q, is related to the lateral displacement by the hyperbolic relation (Fig. 5c) k s ρ q = k s ρ + q max Displacement, ρ (c) q max Fig. 5 (a) Definition sketch (b) Winklers model Non-linear response of the soil (7) where ρ is the displacement of the pile at depth,, from grond srface. k s is assmed constant with depth and q max varies from c at GL and to a limiting pressre of c (Polos 98) at a depth 3 times the diameter. The force eqilibrim eqation is (c) M = H.e = + L q d d b q d d t q t d d k sd ( )tanθ M = H.e = d ks ( )tanθ + ( + / ) c k d θ L s ( )tan k sd ( )tanθ d + d ks ( )tanθ ks ( )tanθ + + c c () () where ρ = ( ) tanθ and ρ = ( ) tanθ are the displacements at depth,, above and below the point of rotation, O, respectively. Normaliing eqations & and simplifying H ( θ H )tan = = d k s dl μ( )tan θ + + / ( )tan θ ( )tan θ + d d μ( )tan θ μ( ) tanθ + + (3)

5 Behavior of laterally loaded rigid piles in cohesive soils based on kinematic approach M ( )tanθ M = = H.e/L = d 3 ksdl μ( )tanθ + + / ( )tanθ ( )tanθ d + d μ( )tanθ μ( )tanθ + + (4) infinitely large vale of ndrained shear strength, c, or linear response of the grond. Increasing vales of μ correspond to decreasing vales of the ndrained shear strength of the soil and hence the response crves become non-linear, the normalied ltimate lateral load of the pile decreases and is attained at very large displacements. Very low ltimate capacities of the pile where = /L - the normalied depth, = / L - normalied depth of point of rotation d = d/l normalied diameter and H = H ksdl - normalied load. The relative stiffness factor for the soil, μ, is defined as k s L μ = (5) c Normalised Load, H.E-.E-.E-3.E-4 μ = 5 Ideally, the depth of rotation,, and the angle of rotation, θ, are to be estimated for a given lateral force, H, and moment, M. It wold be an iterative process and very tedios. Alternately, for given vales of μ, θ, L/d and e/d, vales and H can be obtained directly by solving Eqs. 3 and 4. Knowing and θ, the normalied displacement at grond level, ρ = tanθ, is calclated corresponding to the normalied applied load, H (Eq. 3)..E Normalised Displacement, ρ Fig. 7 Normalied load, H, vs displacement, ρ for e/d= and L/d = 4.4 μ = RESULTS AND DISCUSSION The variation of H with ρ, based on the proposed model, is presented in Fig. 6, for μ vales ranging from (infinitely strong soil or linear response) to (very soft soil) and for no moment at grond level, i.e. e/d = and L/d = 4. The normalied lateral load, H, increases as expected with normalied displacement, ρ. The load verss displacement crve is linear for μ =, i.e. Normalised Lateral Load, H..6 μ = Normalised Displacement, ρ Fig. 6 Normalied load, H, vs displacement, ρ, at GL for e/d = and L/d = 4 H.7 are attained at relatively smaller lateral displacements only for very large vales of μ, i.e. extremely small ltimate soil resistances. Ultimate lateral resistance of the pile cannot be obtained precisely from many of the crves in Fig. 6. The normalied ltimate lateral load can easily be estimated from the plot for μ vales in the range to. On the other hand, it is difficlt to estimate the ltimate load on the pile for μ vales between and, since the loads are predicted pto normalied displacements of.6. The variation of normalied Rotation, θ Fig. 8 Normalied load, H vs rotation, θ for e/d = and L/d = 4

6 Padmavathi, et al. /L H.E-5 7.E-3.4E μ = Normalised Pressre, p Normalised Depth, /L.5 μ= Fig. 9 Movement of normalied depth to point of rotation, /L with respect to H and μ for e= and L/d = 4 /L /L θ = μ μ = Rotation in Radians, θ Fig. Normalied depth to point of rotation, /L with μ and rotation, θ, for e/d = and L/d = 4 5. Fig. Normalied depth to point of rotation with μ and rotation, θ for e/d = and L/d = 4 Fig. Normalied pressre distribtion with depth, /L, for θ =. for e/d = and L/d = 4 - effect of μ displacement, ρ with H on log scale (Fig. 7) presents the data in a somewhat better manner. The crves presented in Fig. 7 are extrapolated to predict the ltimate lateral loads for vales of μ <. The variation of H with rotation, θ, is depicted in Fig. 8 for e/d = and L/d = 4. The variations of the normalied load with θ for different vales of μ are very similar to those depicted in Fig. 6 with respect to displacements since displacement at the top is linearly proportional to rotation. For small μ vales (μ <), the variation H with θ is almost linear. The ltimate vale of H is attained at smaller vales of rotation for higher vales of μ. Similar trend is observed for other vales of e/d and L/d. Normalised Pressre, p Normalised Depth, /L θ = Fig. 3 Normalied pressre distribtion with depth, /L, for μ =, e/d =, L/d = 4 effect of θ.

7 Behavior of laterally loaded rigid piles in cohesive soils based on kinematic approach q θ = θ = /L /L Fig. 4 Incremental of normalised pressre distribtions µ =, e/d=, and L/d = 4 for θ =.-. p Fig. 5 Incremental of normalied pressre distribtions for μ =, e/d = and L/d = 4 for θ =. -. The variation of the depth to the point of rotation with normalied force, H, for e/d = and L/d = 4 is linear and most interesting (Fig. 9). For smaller vales of μ (μ <5) the depth to the point of rotation is nearly constant (=.667) with increase in the normalied load, H.However, the depth to the point of rotation increases with H, for μ >5, the rate increasing with μ. For μ vale of, the normalied depth to the point of rotation increases from.67 to abot.73 as the ltimate capacity of the pile is attained..5 Figre shows the variation of normalied depth to the point of rotation, /L, with rotation, θ and μ for e/d = and L/d = 4. As mentioned earlier, the depth to the point of rotation is almost constant at a vale of.667 for smaller vales of μ. The rate of increase of the normalied depth to the point of rotation with μ increases with increasing vales of θ. Also, the rate of increase of the depth to the point of rotation with θ increases with increasing μ and attains asymptotically the final vale, e.g..75 for μ =, (Fig. ) for e/d = and L/d = 4. The variation of the normalied pressre, p, with normalied depth, /L, at a rotation of. radians and L/d = 4 and e/d = is shown in Fig. for different vales of μ. The pressre distribtion is almost linear for small μ vales (strong or stiff soils). The point of rotation moves downwards and the normalied pressre distribtions become non-linear with increasing μ vales. While the normalised pressre distribtion is linear and decreases with normalied depth for strong or stiff soils, the normalied pressres increase somewhat with depth in the pper one before decreasing with depth at lower depths. The point that shold be noted is that even at a rotation of., the flly plastic state is never attained for any of the crves presented and the soil pressres are very small near the point of rotation. The distribtions of normalied pressre, p, with respect to normalied depth, /L, with increasing rotation of the pile are depicted in Fig. 3 for L/d of 4, μ of and for ero eccentricity of applied load. For small rotations the pressre distribtion with depth is linear as it reflects the linear response of the soil. With increasing rotation of the pile, the normalied pressres increase as also depth to the point of rotation. The pressre distribtions with depth reflect the rate at which the ltimate lateral resistance of the soil is attained away from the point of rotation of the pile. The resistance offered by the soil at the srface cannot increase frther with θ as it has already reached its limit or the ltimate vale of c. The resistance offered by the soil increases with rotation. However, this increase in the normalied soil pressre at any depth, /L, is controlled by two factors, the distance from the point of rotation and the ltimate soil resistance. The increase in displacements with rotation, θ, decrease towards the point of rotation while the ltimate soil resistance increases with depth, at least p to a depth of three times the diameter. As a reslt of these two effects, the depth to the point of maximm soil resistance increases with θ pto a depth of abot.5 for θ =.. The resistance of the soil developed decreases rapidly to become ero at the point of rotation. The soil resistance once again increases with

8 Padmavathi, et al..e-3 Normalised Load, H depth for points below the point of rotation and tend to attain the ltimate resistance vales. The point of rotation moves downwards and the soil pressre in the one of conter thrst increases rapidly towards the base of the pile. Ths the lateral pressre is less at the top, increases with depth pto the point before becoming ero at the point of rotation as it shold be. At depths below the point of rotation, the normalised pressre increases monotonically with depth as the ltimate soil pressres are constant with depth. The pressre distribtions obtained are mch less than those predicted for the complete or total plastic state assmed by Broms (964) and McDonald (999). To elaborate the above phenomenon, the increments in the vales of soil resistances, Δp, mobilied at different depths with each increment in rotation, Δθ, are presented in Figs. 4 and 5 for increments of. and. in rotation for rotations p to. and. respectively for L/d = 4, e/l = and μ =. The increments in soil resistances decrease monotonically with increase in θ. Fig. 6 traces the points along the non-linear load rotation response crve of the pile for the above case. Normalised Pressre, p L/d= Normalised Depth, /L.. θ = E-4.5. Rotation, θ Fig. 6 Normalied load increments with θ =. -. for μ =, e/d = and L/d = 4 H.E-3 6.E-4.E+ e/d 6 8 L/d=4 Fig. 8 Normalied load, H with e/d effect of L/d for θ =. and μ = Normalied pressre, p= q/k s L, verss /L vales are presented (Fig. 7) for different vales of L/d and e/d=, θ =. and μ =. The normalied pressres, p= q/k s L, increase with increasing vales of L/d. The rate of increase of normalied pressre with depth is high at smaller vales of L/d and decreases with increasing vales of L/d. The pressre distribtion with depth tends to become asymptotic to a somewhat linear distribtion with depth. The normalied pressre distribtion varies above the point of rotation, bt there is no remarkable variation below the point of rotation for L/d > 4. Figre 8 represents the variation of H with normalied eccentricity, e/d, for different L/d ratios of the pile for θ =.. The normalied load, H, decreases with increasing eccentricity for all lengths of the piles. The rate of decrease of H with e/d is relatively more for e/d less than and mch less for e/d greater than. The effect of length of the pile on the load carrying capacity is obvios. Longer piles carry larger loads for all eccentricities. The effect of increasing eccentricities, i.e. increasing vales of e/d, redcing the load, H, carried by the pile, is explained by the plot in Fig. 9 where in the normal stress, p, distribtion with depth was plotted for different e/d vales, and for L/d = 4, μ =, rotation, θ =.. The point of rotation moves p with increasing vales of e/d. The areas of the pressre distribtion crves above the point of rotation decrease while the areas beneath the point of rotation increase with increasing vales of e/d. Ths the net effect is a decrease in the net horiontal force, H, the pile can carry at a given rotation. The rate of change in these areas decreases with increasing e/d with no apparent decrease for e/d greater than 6. Fig. 7 Normalied pressre, p vs /L effect of L/d for θ =., μ = and e/d=

9 Behavior of laterally loaded rigid piles in cohesive soils based on kinematic approach ESTIMATION OF ULTIMATE LATERAL CAPACITY OF THE PILE For cases where the ltimate lateral capacities of the pile is not clear in the normalied load verss displacement plots, hyperbolic plot sggested by Kondner (963) is tilied for the estimation of the ltimate lateral capacities. The ratio ρ / H was plotted against ρ, to obtain a straight line. The reciprocal of the slope of the straight line is the normalied ltimate lateral load, H. The ltimate lateral capacity, H, of the pile is then calclated from the expression H = H k s dl (6) From the stdies on hyperbolic response of the soil, the ltimate lateral capacity wold be the asymptotic vale corresponding to infinite displacement. A redction factor mst be applied in order to obtain a H/c d Normalised pressre, p Normalised Depth, /L e/d= Fig. 9 Normalied pressre, p vs /L effect of e/d for θ =., μ = and L/d=4.5 5 Broms H/cd 8 4 realistic failre load. Based on Jess (), failre load is assmed as times the ltimate load estimated from the hyperbolic plot. In the present paper, the ltimate loads ths estimated from the hyperbolic plots are compared with those predicted by Broms (964) and McDonalds (999) theories. COMPARISON WITH EXISTING THEORIES In the present stdy, Eq. 6 is rewritten and transformed to get normalised ltimate lateral load (H /c d ) which is convenient to compare with existing theories. H c H k sd L = = H ( L / ) μ (7) c d d d e/d= L/d Fig. Ultimate lateral load with L/d effect of e/d The normalied ltimate lateral load (H /c d ) is estimated sing Eq. 7 for each μ and L/d vales. The variation of H /c d with respect to μ and L/d are shown in Fig.. Relative stiffness factor, μ increases with decrease in strength of the soil. Hence the, ltimate load normalied with c d vales increase with decrease in strength of the soil. Bt, the absolte vale of ltimate load, H is less for soft soils (Fig.6) becase of less strength of the soil even thogh the H /c d is very high. Figre gives the variation of H /c d, L/d and e/d for μ=. As eccentricity increases the ltimate vales decrease. McDonald μ = Broms (964). L/d Fig. Variations of ltimate lateral loads with L/d and μ and comparisons with Broms and McDonald approaches for e/d = Vales of H c d from Broms (964) approach and the proposed method for piles with different μ vales are presented in Fig.. Broms (964) approach nderpredicts the ltimate lateral load capacity of the piles for all vales of μ and for L/d vales less than 4. For L/d vales greater than 4 the prediction from Broms approach is close to the estimated vales for μ =.

10 Padmavathi, et al. Broms reslts over-prediction the ltimate lateral loads for strong and stiff clays and nder-predict for soft clays. Broms analysis is conservative for L/d less than 3. The consideration of fll mobiliation of soil pressre close to the point of rotation (Fig. 4a) eventhogh the displacement there, is ero in addition to that the ero soil pressre pto.5d from grond srface are responsible for these differences in the prediction of ltimate lateral resistance of the pile. McDonald (999) McDonald s theory is based on the ltimate soil pressre distribtion shown in Fig. 4(b). Vales of H c d from McDonald s (999) approach are presented and compared with predictions based on the present approach in Fig.. The ltimate lateral load vales from McDonald (999) approach lie very close to the predicted vales for very soft (μ=) soil bt for all other soils (μ<), the former are considerably larger than the latter vales. Prasad (997) Prasad (997) presented correlations between the reslts of static cone penetration, pressremeter and standard penetration test reslts and relevant soil properties of cohesive soils to develop methods of analyses for preliminary estimates of ltimate lateral resistance and grond line displacements of rigid and flexible piles. COMPARISON WITH EXPERIMENTAL RESULTS Estimation of Soil and Pile-Soil Interaction Parameters The vales of modls of sbgrade reaction, k s, can be obtained from an actal in sit load-displacement crve. In the absence of sch data, modls of sbgrade reaction, k s, can be estimated from the relation, k s =f.(c /d) where the parameter, f is known to vary in the range of (8-3) for clayey soil (Polos 98). The for-fold variation cold be de to the dependence of f on the L/d ratio of the pile. In the present stdy, the vales of f considered for the test data based on available correlations are shown in Table. The distribtion of soil pressre arond the cross section of the pile is non-niform as stated earlier. Hence, redction factors of.8 and. are considered for circlar and sqare piles respectively, that is: k s L μ = for circlar piles (8).8 c and ksl μ = for sqare or rectanglar piles (9) c H c d and ths vales H are obtained for known vales of L/d, μ and e/d vales. Drery and Fergson (969) carried ot a series of lateral load tests on free-head model brass piles of 6.35 mm diameter in kaolin to obtain load - displacement crves to failre. Load was applied at eccentricities of 9.5, 5.4 and mm. The projected ltimate lateral load vales are obtained sing hyperbolic relationship. Average ndrained shear strength of the soil was 38.8 kpa. The predicted ltimate pile capacities are compared with those from Broms (964) and McDonald s (999) approaches in Table. The predictions from Broms approach are on the higher side while those from McDonald s and the present methods are close to the measred vales. Briad et al. (983) reported the reslts of three field tests condcted on drilled shafts of lengths 6, 4.5 and 4.5 m and diameters.9,.9 and.75 m respectively at a site in Texas A & M University. The soil was stiff clay with average ndrained shear strength of 95.8 kpa. The L/d ratios varied from 6.67 to 5. For all the three tests the load eccentricity is ero. As the tests were not carried ot pto failre the failre loads were estimated sing the hyperbolic relationship and compared in Table 3 with the predictions based on Broms, McDonald s and the present theories. McDonald (999) presented extrapolated ltimate lateral loads of for piles of diameter.45 m, lengths.48,.93,.4 and. m and load eccentricities.5,.5,.5 and.5 m respectively from field tests in a clay having an average ndrained shear strength of 6 kpa. The test details and comparisons are presented in Table 3. Bhshan et al. (979) presented five field tests on straight piers, with pier No. tested at site A while pier Nos. 4, 6, 7 and 8 tested at site B. The average ndrained shear strength for sites A and B were 6 and 8 kpa respectively. Pier No. was 4.58 m long and. m in diameter. It was tested at an eccentricity of.3 m. For Piers No.s 4, 6, 7 and 8, the lengths and diameters were 3.8, 4.73,.75 and 4.73 m and.,.,.6 and.6 m respectively. All these tests were performed with an eccentricity of.3 m. Table 4 compares the measred ltimate capacities with Broms (964), McDonald s (999), Prasad s (997) and the present theories. Bierschwale (98) presented three field tests in clay with average ndrained shear strength of kpa on piles of different lengths and diameters bt with an eccentricity of.79 m. The comparisons are presented in Table 5.

11 Behavior of laterally loaded rigid piles in cohesive soils based on kinematic approach Table Proposed factor f for the calclation of k s = f.c /d vales from correlations c kn/m f Based on all the above comparisons (Tables to 5), ltimate lateral loads predicted by Broms (964) are conservative if L/d vales are less than 3 and nconservative for high L/d vales which are greater than 6. Pile capacities predicted by McDonald (999) are, in general, in excess of the measred vales for all L/d vales. Capacities predicted by Prasad (997) are close to the measred vales. However, the ltimate vales are based on prely empirical relations. The ltimate capacities from the proposed rational method based on actal kinematics and non-linear response of the soil are close to the measred vales. PREDICTION OF LOAD-DISPLACEMENT RESPONSE OF PILES AT WORKING LOADS Broms (964), Polos (97) or McDonald (999) presented only ltimate lateral loads. Briad et al. (983) reported measred load - displacement nder lateral load for the data given in Table 3. The measred load - displacement crves are compared (Fig. ) with the predictions based on the present method. The load displacement crves compare well with the experimental data. The measred load-displacement crves are shown in Fig. 3 corresponding to shaft I, II and III based on Bierschwale et al. (98) which are already specified Lateral Load, kn 8 4 Shaft Im Shaft IIm Shaft IIIm Shaft Ip Shaft IIIp Shaft IIp 5 Displacement at GL, mm Fig. 3 Comparison of measred vs predicted load deflection crves Bierschwale et al. (98) for shaft I, shaft II and shaft III earlier in Table 5. The load displacement crves are somewhat comparable with experimental crves, bt do not match closely, may be becase of correlated vale of k s from Table-. Instead of predicting k s vales from c, if k s vales are obtained from load-displacement crves of testing piles, then the predicted load-displacement crves are well coincide with the measred vales. PREDICTION OF SOIL PRESSURE ALONG THE LENGTH OF THE PILE Bierschwale (98) also presented lateral pressre distribtion of the soil along the shaft for the same Lateral Load, kn 8 4 case m case m case 3m Case p Case p Case 3p Lateral Soil Pressre, kpa McDonald Broms exp..5 Normalised Depth, /L Displacement at GL, mm Fig. Comparison of measred vs predicted load deflection crves Briad et al. (983) for case, case and case 3 Proposed Fig. 4 Lateral soil pressre distribtion nder ltimate lateral load (75kN) for shaft I Bierschwale et al.(98)

12 Padmavathi, et al. Lateral Soil Pressre, kpa Normalised Depth, /L Proposed.5 exp. McDonald Brom s Fig. 5 Lateral soil pressre distribtion nder ltimate lateral load (667kN) for shaft II Bierschwale et al. (98) Lateral Soil Pressre, kpa Normalised Depth, /L.5 exp McDonald Broms CONCLUSIONS The following conclsions are made based on the work presented:. A new approach or method based on modls of sbgrade reaction approach is proposed to predict the load-displacement response of laterally loaded free head rigid pile considering nonlinear hyperbolic soil response and kinematics which incorporates the physics of the pile-soil interaction.. Broms (964) approach for predicting the ltimate lateral loads is shown to nderpredict the vales for L/d<4 and overpredict for L/d >4 for strong or stiff soils corresponding to μ = in the present theory. 3. Polos(97) or McDonald(999) approach overpredict the ltimate lateral loads for L/d <6. The predictions from their approach for L/d>6 compare well with the vales from the present theory for μ = The present theory, predicts the complete loaddisplacement response of piles at working loads that fit closely with the experimental vales. 5. The distribtion of soil resistances along the pile length predicted by the proposed approach compare well with the measred reslts. 6. An additional correlation is proposed (Table ) to estimate the vales of the modls of sbgrade reaction, k s. Proposed Fig. 6 Lateral soil pressre distribtion nder ltimate lateral load (4kN) for shaft III (Bierschwale et al. 98) parameters as specified in Table 5. These are presented in Figs. 4, 5 and 6 and compared with predictions from Broms, Polos and McDonald. Even thogh the ltimate lateral load vales are somewhat comparable with measred ones, the distribtion of lateral pressres predicted by either Broms or McDonald theories do not match with measred reslts. The proposed approach predicts both the ltimate pile capacities and the lateral soil resistance distribtions with depth, that correlate well with the experimental vales. The predicted vales are in better agreement with experimental vales compared to those from all the other theories. REFERENCES Banerjee P.K. and Davies T.G. (978). The behavior of axially and latterly loaded single piles embedded in non-homogeneos soils. Geotechniqe, 8(3): Barber E.S. (953). Discssion to paper by S. M. Glesser. ASTM., 54: Bhshan K., Haley S.C. and Fong P.T. (979). Lateral load tests on drilled piers in stiff clays. Jornal of the Geotechnical Engineering, ASCE, 5(8): Bierschwale M.W., Coyle H.M. and Bartoskewit R.E. (98). Lateral load tests on drilled shafts fonded in clay. Presented in Special Session on Drilled Piers and Caissons, ASCE National Convention, St. Lois, Mo.: Briad J.L, Smith T.P. and Meyer B. (983). Pressremetre gives elementary model for laterally loaded piles. International Symposim on In-sit Testing, Paris, : 7-.

13 Behavior of laterally loaded rigid piles in cohesive soils based on kinematic approach Broms B.B. (964). Lateral resistance of piles in cohesive soils. J. Soil Mech. Fond. Div., ASCE. 9(): Davisson M.T. and Gill H.L. (963). Laterally loaded piles in a layered soil system. Jornal of Soil Mechanics and Fondation Division, ASCE. 89(3): Drery B.M. and Fergson B.A. (969). An Experimental Investigation of the Behavior of Laterally Loaded Piles. B. E. Thesis, Department of Civil Engineering, University of Sydney, Astralia. Hansen J.B. (96). The ltimate resistance of rigid piles against transversal forces. Blletin No., Danish Geotechnical Institte, Copenhagen, Denmark: 5 9. Hays C.O., Davidson J.L., Hagan E.M. and Risitano R.R. (974). Drilled shaft fondation for highway sign strctres. Research Report D647F, Engineering and Indstrial Experiment Station, University of Florida, Gainsville, Fla. Jesús E.G. (). Development of an Extended Hyperbolic Model for Concrete-to-Soil Interfaces. Ph.D. Thesis, Department of Civil Engineering, Virginia Polytechnic Institte and State University, Blacksbrg, Virginia. Kasch V. R., Coyle H. M., Bartoskewit R. E. and Sarver W. G. (977). Design of drilled shaft to spport precast panel retaining walls. Research Report N.., Texas Transportation Institte, Texas A&M University, College Station, Tex. Kondner R. L. (963). Hyperbolic stress-strain response cohesive soils. J. Soil Mech. and Fond. Div., ASCE. 89(): Lai J. Y and Booker J. R. (989). The behavior of laterally-loaded caissons in prely cohesive soils. Research Report No. R66, The University of Sydney, Astralia. Matlock H. and Reese L.C. (96). Generalied soltions for laterally loaded piles. J. Soil Mech. Fond. Div., ASCE, 86(5): McDonald P. (999). Laterally loaded pile capacity revisited. Proc. of 8 th Astralia and New Zealand Conference on Geomechanics, Hobart, : Meyerhof G.G., Mathr S.K. and Valsangkar A.J. (98). Lateral resistance and deflection of rigid wall and piles in layered soils. Can. Geotech. J., 8: Meyerhof G.G. and Sastry V.V.R.N. (985). Bearing capacity of rigid piles nder eccentric and inclined loads. Can. Geotech. J., : Meyerhof G.G. and Yalcin A.S. (984). Pile capacity for eccentric and inclined load in clay. Canadian Geotechnical Jornal, : Polos H.G. (97). Behavior of laterally loaded piles. I: Single piles. J. Soil Mech. and Fond. Div., ASCE, 97(5): Polos H.G. and Davis E.H. (98). Pile Fondation Analysis and Design. Wiley, New York: Narasimha Rao S. and Mallikarjna Rao K. (995). Behavior of rigid piles in clay nder lateral loading. Indian Geotechnical Jornal, 5(3): Prasad Y.V.S.N. (997). Standard penetration tests and pile behavior nder lateral loads in clay. Indian Geotechnical Jornal, 7(): -. Prasad Y.V.S.N. and Chari T.R. (999). Lateral capacity of model rigid piles in cohesionless soils. Soils and Fond., 39(): -9. Randolph M.F. and Holsby G.T. (984). The limiting pressre on a circlarpile loaded laterally in cohesive soil. Geotechniqe, 34(4): Reese L.C. (958). Discssion of soil modls for laterally loaded piles. by McClelland B. and Focht J.A. Transactions, ASCE, 3: Reese L.C. and Cox W.R. (969). Soil behavior from analysis of tests on ninstrmented piles nder lateral loading. ASTM, STP 444: Sastry V.V.R.N. and Meyerhof G.G. (986). Lateral soil pressres and displacements of rigid piles in homogeneos soils nder eccentric and inclined loads. Can. Geotech. J., 3: Vesic A.B. (96). Bending of beams resting on isotropic elastic solid. Jornal of Engineering Mechanics Division, ASCE, 87(): Zhang L., Silva F. and Grismala R. (5). Ultimate lateral resistance to piles in cohesionless soils. J. Geotechnical & Geoenvironmental Engg., ASCE, 3():

14 Padmavathi, et al. Table Comparisons with Drery and Fergson (969) tests Length of pile in Clay, L (mm) Diameter of the pile, d (mm) Load Eccentricity, e (mm) Average Undrained Cohesion (c) kpa. Estimated Ultimate Load (N) Ultimate Load by Broms (N) Ultimate Load by McDonald (N) Ultimate Load by Proposed Method (N) Table 3 Comparisons with Briad et al. (983) and McDonald (999) field tests Length of the pile in Clay, L (m) Diameter of the pile, d (m) Load Eccentricity, e (m) Average Undrained Cohesion (c) kpa. Extrapolated Ultimate Load (kn) Broms (kn) McDonald (kn) Proposed Method (kn) References Briad et al. (983) McDonald (999) Table 4 Comparisons with Bhshan et al. (979) field tests Length of the pile in Clay, L (m) Diameter of the pile, d (m) Load Eccentricity, e( m) Average Undrained Cohesion (c) kpa. Ultimate Load at rotation (kn) Broms (kn) McDonald (kn) Prasad (kn) Proposed Method (kn) References Bhshan et al. (979) Pier, Site A Pier 4, Site B Pier 6, Site B Pier 7, Site B Pier 8, Site B

15 Behavior of laterally loaded rigid piles in cohesive soils based on kinematic approach Table 5 Comparisons with Bierschwale et al. (98) field tests Length of the pile in Clay, L (m) Diameter of the pile, d (m) Load Eccentricity, e (m) Average Undrained Cohesion (c) kpa. Extrapolated Ultimate Load (kn) Broms (kn) McDonald (kn) Proposed Method (kn) Particlars References Shaft I Shaft II Shaft III Bierschwale et al.(98) LIST OF SYMBOLS α = slope of a reaction crve c = ndrained cohesion of the soil d = diameter of the pile e = load eccentricity (the distance between point of application of load and grond level) E p = modls of elasticity of pile material E s = average horiontal soil modls H = lateral load H = normalied lateral load H = ltimate lateral load H = normalised ltimate lateral load η = soil strength redction factor γ = nit weight of the soil I p = moment of inertia of pile cross section K c = theoretical earth pressre coefficient given by Hansen K r = relative stiffness factor k s = modls of sbgrade reaction L = length of the pile from grond level to toe point M = moment at the grond point M = normalied moment at the grond point μ = non dimensional coefficient p = lateral resistance per nit length q t = the lateral stresses above the point of rotation q b = the lateral stresses below the point of rotation q t = the lateral stress from GL to depth q t = the lateral stress from depth to point of rotation ρ = deflection at grond point ρ = normalised displacement at grond point ρ = deflection above the point of rotation at any depth from grond point. ρ = deflection below the point of rotation at depth from grond point. q max = maximm lateral soil pressre θ = angle of rotation in radians = depth of point nder consideration from grond point = depth of point of rotation from the grond point = normalied depth of point of rotation