Mobilisable strength design for flexible embedded retaining walls

Transcription

1 Diakoumi, M. & Powrie, W. (0). Géotechnique 6, No., [ Mobilisable strength esign for flexible embee retaining walls M. DIAKOUMI an W. POWRIE Soil structure interaction may have an important influence on the behaviour of embee retaining walls, affecting both wall bening moments an groun movements. However, it can be iffic an time consuming to capture in esign, especially in a way that gives a physical insight into the key behavioural mechanisms involve. A calculation proceure has been evelope for retaining walls proppe near the crest that takes into account both the non-linearity of the stress strain behaviour of the soil an the flexibility of the wall. Ress for ifferent pore water pressure conitions, soil strengths an soil an wall stiffnesses are presente in the form of look-up charts, an are compare with those erive from factore limit equilibrium analyses. A imensionless parameter is introuce to represent the relative soil wall stiffness, an its importance is emonstrate. A critical flexibility ratio is ientifie at which the bening moments start to reuce below those given by a conventional limit equilibrium calculation. This ratio is linke to the wall eflection, an is use to istinguish a stiff from a flexible system in soils of ifferent strengths an pore water pressure conitions. The approach is iscusse in relation to previous stuies. KEYWORDS: esign; limit state esign/analysis; retaining walls; shear strength; soil/structure interaction; stiffness INTRODUCTION Limit equilibrium calculations with the soil strength reuce by a factor of safety, F s, are commonly use in practice to ensure that the wall is remote from the imate limit state (ULS). Guielines to avoi the serviceability limit state (SLS) are fewer an less clear than for the ULS, since eformations are often assume to be a seconary problem. Where eformations are calculate, it is normally one using relatively simple programs base on elasticity; in reality, soil stiffness epens on stress history, stress state, stress path an strain, an may well be anisotropic. More rigorous soil structure interaction analysis may be carrie out numerically using finite-element or finite-ifference analysis. However, the number of parameters involve, the potential sensitivity of the ress to the values aopte, the cost an the user expertise require ten to restrict the use of these methos to major projects. Past experience an recore behaviour of retaining walls can provie guiance, but are only really applicable in irectly comparable cases. Rowe (95) showe that the flexibility of an embee retaining wall anchore near the crest will reuce the bening moments in comparison with those calculate in a conventional limit equilibrium analysis. Rowe (955) introuce a imensionless group, mr, to characterise the relative soil wall stiffness (where m is a measure of the soil stiffness, r ¼ H 4 /EI, H is the total wall length, an EI is the wall bening stiffness), an presente look-up charts from which the egree of bening moment or prop loa reuction ue to the relative soil wall flexibility coul be estimate. However, Rowe s analysis is now not often use in routine esign, primarily because it (a) was aime at sheet-pile walls in ry sans, which efine Manuscript receive 5 May 0; revise manuscript accepte 7 May 0. Publishe online ahea of print 4 September 0. Discussion on this paper closes on July 0, for further etails see p. ii. Built Environment & Civil Engineering Division, School of Environment & Technology, University of Brighton, UK. School of Civil Engineering & the Environment, University of Southampton, UK. the range of retaine height to embement ratios consiere (b) was base on a limit equilibrium calculation in which the factor of safety was applie entirely to the passive lateral stresses in front of the wall with fully active stresses behin, in contrast to the moern approach in which the factor of safety is applie equally to the soil behin an in front of the wall (c) use a non-funamental soil stiffness parameter that is iffic to measure. Potts & Fourie (985) presente the ress of finite-element analyses carrie out to investigate the effect of the wall bening stiffness, EI, on bening moments an prop loas in a wall proppe at the crest for ifferent values of pre-excavation earth pressure coefficient. The soil Young s moulus increase linearly with epth but i not vary with strain, an pore water pressures were assume to be zero. For flexible walls, the maximum bening moments an prop loas were less than those etermine from a limit equilibrium calculation with the factor of safety F r applie as propose by Burlan et al. (98), an reuce with increasing wall flexibility. Conversely, for stiff walls an an initial earth pressure coefficient greater than, the maximum bening moments an prop loas exceee the limit equilibrium values. Mobilisable strength esign Mobilisable strength esign is a metho of calculation linking the strength mobilise in the soil aroun a geotechnical structure to the eformation using an iealise isplacement fiel an the conition of equilibrium, enabling the movements uner working conitions to be estimate. It was escribe for rigi embee retaining walls by Bolton et al. (989, 990), who iealise the soil behaviour by means of simplifie kinematically amissible strain fiels propose by Bolton & Powrie (988) an Bransby & Milligan (975); an was terme mobilisable strength esign (MSD) byosman & Bolton (004). Diakoumi (007) an Diakoumi & Powrie (009) applie the MSD metho to flexible walls proppe at the crest, by 95

2 96 DIAKOUMI AND POWRIE introucing new kinematically amissible strain fiels to represent the effect of wall bening both on the lateral stress istribution in the soil an on wall movements. The strain fiels for a flexible wall were introuce by subiviing the soil surrouning the wall into a number of pairs of triangles, representing the superposition of a number of the strain fiels propose by Bolton & Powrie (988) for a rigi wall proppe at the crest, relating to a series of epths own the wall, as shown in Fig.. For the analyses presente by Diakoumi & Powrie (009) an in this paper, the soil is ivie into four zones behin the wall an two zones in front. In principle, the soil coul be ivie into more zones to achieve a smoother approximation to the eflecte shape an/or take account of ifferent soil layers. The triangles are free to slie on vertical an horizontal surfaces, which are assume to be frictionless. Within each pair, the triangle ajacent to the wall (OAJ, OBH, etc.) remains rigi while the complementary triangle making up the square of the mechanism (ARJ, BQH, etc.) unergoes a shear strain relate to the rotation of the wall section uner consieration. Outsie the square of the mechanism (OARJ, OBQH, etc.), the soil is assume not to eform, as changes in stress will be small. The istribute wall bening stiffness EI is iealise into rotational springs at iscrete points own the epth of the wall, separate by rigi wall elements. The rotation of each section of the wall is relate to the shear strain an, by means of a moifie hyperbolic stress strain law, to the mobilise soil strength, in the lower triangle of the ajacent soil zone. The use of the aitional kinematically amissible strain fiels permits the introuction of ifferent mobilise shear strengths, corresponing to ifferent shear strains in each soil zone, consistent with the variation in local rotation of a flexible wall. The equations of global equilibrium for the wall, an local equilibrium at each rotational spring, are then use to etermine the wall prop loas an bening moments uner working conitions. Diakoumi & Powrie (009) presente ress of the metho for a single value of soil strength at failure, j9, an hyrostatic pore pressures below a water table at groun level. This paper evelops the MSD metho presente by Diakoumi & Powrie (009) further, to (a) enable the estimation of the bening moments an prop loas associate with a flexible retaining wall proppe at the crest for a variety of grounwater conitions an a range of values of wall stiffness an j9 in a practical an reasonably accurate manner (b) introuce an appropriate imensionless relative soil wall stiffness number base on conventional an measurable parameters (c) relate both of these to the factore ULS limit equilibrium calculation specifie in moern coes of practice, in a way suitable for use in esign () compare the ress with previously publishe stuies. The calculation steps involve in the evelopment of the MSD metho for flexible retaining walls proppe at the crest presente by Diakoumi & Powrie (009) may be summarise as follows. The esign value of the ratio of the retaine to overall wall height, n es, is etermine by a factore limit equilibrium calculation in the usual way, on the basis of the same, uniform mobilise soil strength (effective angle of shearing) j9 es on both sies of the wall. For this value of n es, the wall rotation is assume to take place in four successive stages, as shown in Fig.. The first stage consists of the rotation of the entire wall (OF) by an amount äł 4, an with it the triangles ODF (behin the wall) an HFL (in front). Accoring to the geostructural mechanism presente by Bolton & Powrie (988), the wall rotation, äł 4, is relate to the incremental shear strain in the associate complementary soil triangle DEF by äª 4 ¼ äł 4 () Triangle LFK in front of the wall will be compresse, an the maximum shear strain increment within it, äª 5, is relate to the wall rotation by äª 5 ¼ äł 4ðh þ Þ () where h is the retaine height, an is the embee epth. Compression is taken as positive. The secon stage of wall movement consists of the further rotation of the wall section OG together with the triangles OCG behin an HGM in front, by an amount äł : The corresponing maximum shear strain increments äª an äª 6 within the associate complementary triangles CPG an MGN are given by äª ¼ äł () äª 6 ¼ äł ðh þ =Þ = (4) Similarly, the thir an fourth stages of wall movement O A B C D h h h δθ δθ 4 δθ δθ L M J R H Q N G P K F E Fig.. Amissible strain fiels for flexible retaining wall proppe at crest rotating in four successive stages

3 MOBILISABLE STRENGTH DESIGN FOR FLEXIBLE EMBEDDED RETAINING WALLS 97 consist of the rotation äł of the wall section OH an triangle OBH behin the wall, an äł of the wall section OJ an triangle OAJ, resing in maximum shear strain increments äª within BQH an äª within ARJ of äª ¼ äł (5) äª ¼ äł (6) The total shear strain within each triangle is then taken as the sum of the incremental shear strains impose on the triangle uring each stage of wall rotation. Thus for triangles DEF, CPG, BQH an ARJ behin the wall the total shear strains ª 4, ª, ª an ª are given by ª 4 ¼ äª 4 ¼ äł 4 (7) ª ¼ äª 4 þ äª ¼ ðäł 4 þ äł Þ (8) ª ¼ äª 4 þ äª þ äª ¼ ðäł 4 þ äł þ äł Þ (9) ª ¼ äª 4 þ äª þ äª þ äª ¼ ðäł 4 þ äł þ äł þ äł Þ (0) k ¼ EI h=4 þ h=4 ¼ EI h (4) Diakoumi (007) shows that this introuces an error of less than 6% in the eflecte shape for simply supporte beams an cantilevers subject to uniform an triangular loas. The rotational spring stiffness k i, given (for example) by equation (4), is also equal to M i /Ł i, where M i is the bening moment an Ł i is the relative rotation between the wall sections at the ith iscrete point. Geometry is use to relate the noe rotations, Ł i, to the rotations of the rigi wall sections äł i : For example, from the above an equation (4), the normalise bening moment M /(ª s H ) at point J as shown in Fig. is M ª s H ¼ k Ł ª s H ¼ ðeiþ ð äł Þ hª s H (5) For triangles LFK an MGN in front of the wall, the total shear strains ª 5 an ª 6 are given by ª 5 ¼ äª 5 ¼ äł 4ðh þ Þ () ª 6 ¼ äª 5 þ äª 6 ¼ äł 4ðh þ Þ þ äł ðh þ =Þ () = This is not a superposition of shear strains within a given physical zone of soil, but the summation of the shear strains associate with the rotation of a given section of the wall, which coul occur in ifferent an non-overlapping zones of soil. The summate shear strain ª i is then relate to the mobilise soil strength j9 mobi associate with the ith section of the wall by means of a moifie hyperbolic stress strain law j9 mobi ¼ sin ª i () A þ Bª i Parameters A an B in equation () can be relate to the rate of increase of shear moulus G with epth, G, an the soil strength at failure, j9, an can be etermine from laboratory element tests. The erivation of the moifie hyperbolic stress strain equation an the parameters A an B for ifferent grounwater conitions is presente in the following section. The active an passive pressures ó hi 9 behin an in front of the retaining wall are assume to vary linearly with epth within each section, with active an passive earth pressure coefficients K a i an K p i (which are ifferent for each wall section as the mobilise strengths j9 mobi are ifferent) calculate accoring to the equations given in Eurocoe 7 (EC7; British Stanars Institution, 004) an Powrie (997). From the equations of global equilibrium for the wall an the iealise lateral stress istribution, the normalise bening moments M i /(ª s H ), where ª s is the soil unit weight, are etermine. These will epen on äł i, j9 an G. A key feature of the approach is that the continuous flexural rigiity of the wall, EI, is iealise into rotational springs at iscrete points own the epth of the wall, separate by rigi wall elements. The iscrete points correspon to the vertices of the rigi triangles (OAJ, OBH, etc.) on either sie, as shown in Fig.. The spring stiffness is EI ivie by the average length of the two rigi sections on either sie. For example, the spring rotational stiffness k,at point J on the wall, is given by ¼ 4äŁ n es ª s r where r ¼ H 4 /EI is the wall flexibility as efine by Rowe (95), H is the overall wall height, an EI is its bening stiffness per meter run. Substitution of the expressions for the normalise bening moments erive from the conition of local equilibrium at each rotational spring (e.g. equation (5)) into the equations erive from the global an horizontal equilibrium for the wall gives a system of five equations in five unknowns: the incremental wall section rotations äł, äł, äł, äł 4 associate with the triangles in the active an passive soil zones (Fig. ), an the prop loa, F. Solution of this system requires knowlege of the soil parameters A, B an the imensionless quantity ª s r. MODELLING SOIL BEHAVIOUR Hyperbolic stress strain relationship Diakoumi & Powrie (009) showe that the hyperbolic equation introuce by Konner (96), Konner & Zelasko (96) an Duncan & Chang (970) for representing the non-linear an stress-epenent behaviour of soils may be rewritten to relate the shear stress t to shear strain ª. Assuming zero volumetric strain ª t ¼ (6) a þ bª (see Appenix ). The parameters a an b are relate to the initial shear moulus G 0, Poisson s ratio í an shear stress at failure, t f, by a ¼ (7) G 0 ð þ íþ b ¼ (8) t f The Duncan & Chang (970) formulation was in terms of total stresses for ata from unraine triaxial tests. Diakoumi (007) shows that it is broaly consistent with effective stress strain relationships propose by Jarine et al. (986), Allman & Atkinson (99) an Smith et al. (99), some of which are apparently rather more complex. In this paper, the hyperbolic stress strain function is interprete in terms of effective stresses, although the assumption of zero volume change (í9 ¼ 0. 5) is retaine; this implies raine conitions an soil straining at constant volume, an is consistent with

4 98 DIAKOUMI AND POWRIE the critical state concept. This assumption is not unrealistic for most soils, an is more conservative than upper-boun analyses that assume an angle of ilation equal to the soil strength at failure (j9 ), which can be maintaine only if an while the soil continues to ilate. Mobilise strength The rate of change of mobilise strength, j9 mob, with shear strain is a useful way of expressing both the evelopment of strength an the stiffness of a soil (Bolton & Powrie, 988). It is shown in Appenix that equation (6) may be rewritten in these terms as j9 mob ¼ sin ª (9) A þ Bª Parameters A an B epen on the pore water pressure conitions, as escribe below an later. CONDITIONS OF ZERO PORE WATER PRESSURE For conitions of zero pore water pressure, it is shown in Appenix that the parameters A an B can be calculate as A ¼ ª s =G (0) B ¼ () sin j9 The shear strain can be relate to the mobilise strength by equation (9), with the parameters A an B etermine from appropriate laboratory element tests. To explore the rotations, normalise bening moments an prop loa associate with a retaining wall proppe at the crest for a range of wall flexibilities embee in a variety of soil types, the system of equations erive from the MSD calculations escribe above was solve numerically using Wolfram Mathematica (version 6.0). Values of j9 between 08 an 408, log(ª s r) between. 86 an. 4, an G between 0 5 an 0 kn/m were consiere, assuming zero pore water pressures. A value of wall friction ä s ¼ j9 mob = was use in the MSD calculations, as recommene by EC7 for sheet-pile walls. Parameter A was calculate for í9 ¼ 0.5 an ª s ¼ 0 kn/m : The wall flexibility values (log[ª s r]) correspon to rigi, iaphragm, sheet-pile an soft retaining walls of total height 0 m with ª s ¼ 0 kn/m, as inicate in Table. The values use for the iaphragm an sheet-pile walls are typical of an uncracke reinforce concrete section of m thickness an a Larssen 4B section respectively, an are consistent with those use by Potts & Fourie (985) an Diakoumi & Powrie (009) (Table ). The rigi an soft walls represent extreme cases, beyon likely practical limits which are probably represente by the iaphragm an sheet-pile walls. Similarly, the maximum value of G may be unrealistically high, but is inclue to represent an extreme case. The maximum bening moments, M, an prop loas, F, obtaine from the MSD SLS calculation are normalise by the values M EC7 an F EC7 calculate on the basis of a uniform mobilise strength, j9 es, accoring to the factore ULS calculation given in EC7. The ratios of M/M EC7 an F/F EC7 are plotte in Figs an for ifferent values of wall flexibility an soil stiffness, for an angle of shearing resistance representative of clays (j9 ¼ 08). In Figs an the origin of axes is at (0, ). The x axis represents equality between the maximum bening moments an prop loas calculate accoring to the MSD metho an EC7. The implication is that where the curve lies above the x axis the EC7 approach might unerpreict, an where the curve is below the x axis might overpreict, the maximum bening moment an prop loa, compare with the MSD metho. In particular, as the wall flexibility r or the soil stiffness G increases, the maximum bening moment an prop loa reuce below their limit equilibrium values. This pattern of reuction is similar for ifferent values of j9 (Diakoumi, 007, an Figs 7 an 8). For a given value of j9 an soil stiffness, a low wall flexibility apparently gives bening moments an prop loas greater than those calculate using the EC7 uniform mobilise strength approach. This is because for a rigi wall, accoring to the geostructural mechanism shown in Fig., the shear strain in the soil in front of the wall is greater (by a factor + h/ or /( n)) than the shear strain in the soil Table. Wall flexibilities for ifferent types of retaining wall Rigi Diaphragm Sheet pile Soft EI:kNm /m r:m /kn log(ª s r) M/ MEC ρ log( γ s. ) G* 0 5 kn/m G* 0 4 kn/m G* 0 kn/m G* 0 kn/m Diaphragm Sheet pile Fig.. Comparison between MSD an conventional limit equilibrium (EC7) maximum bening moments for ifferent values of soil stiffness an wall flexibility for j9 ¼ 08 an conitions of zero pore water pressure (soil wall friction angle ä s ¼ j9 )

5 MOBILISABLE STRENGTH DESIGN FOR FLEXIBLE EMBEDDED RETAINING WALLS 99 F/ FEC7 G* 0 5 kn/m ρ log( γ s. ) G* 0 4 kn/m G* 0 kn/m G* 0 kn/m Diaphragm Sheet pile Fig.. Comparison between MSD an conventional limit equilibrium (EC7) prop loas at crest for ifferent values of soil stiffness an wall flexibility for j9 ¼ 08 an conitions of zero pore water pressure (soil wall friction angle ä s ¼ j9 ) behin. Assuming the same rate of mobilisation of soil strength with shear strain behin an in front of the wall (as in the calculations presente here) will res in the mobilisation of a strength greater than j9 es in the soil in front of the wall, an a strength less than j9 es in the soil behin. This leas to higher lateral stresses behin the wall, an hence increase bening moments an prop loas. However, it is unlikely to occur in practice, as the rate of mobilisation of soil strength with shear strain or wall rotation is likely to be greater behin the wall than in front (Powrie et al., 998). Also, the very high wall an soil stiffnesses inclue in Figs an are, as alreay mentione, unlikely to be encountere in practice. For the likely range of real wall flexibilities, a reuction in both the maximum bening moment an the prop loa is shown in Figs an, except for stiffer walls with G, 0 kn/m : F h LINEAR SEEPAGE FROM AN ORIGINAL GROUNDWATER TABLE AT GROUND LEVEL Figure 4 shows the iealise linear seepage pore water pressure istribution often aopte to represent steay-state grounwater flow between the grounwater level on the active sie to excavation level on the passive sie (Symons, 98); the excess (total) hea ifference between the soil surfaces behin an in front of the wall is assume to issipate uniformly along the flow path. In Fig. 4 h,, u an F enote the retaine wall height, wall embement epth, pore water pressure an prop loa respectively. For a hea rop aroun the wall of h an a flow path length l ¼ + h, the pore water pressure at the toe, u toe,is given by u toe ¼ ª w ðhþ Þ () þ h where ª w is the unit weight of water. The seepage pore pressure graient behin the wall, (u/ z) a,is u z a ¼ ª w ð nþ n an that in front of the wall, (u/z) p,is u ¼ ª w z n p where n ¼ h/h. It is shown in Appenix that () (4) u Fig. 4. Pore water pressure istribution for embee retaining wall with steay-state seepage between grounwater tables at groun level behin an in front A ¼ ª s u=z G (5) B ¼ sin j9 (6) with u/z behin an in front of the wall given by equations () an (4) respectively. The system of equations erive from the MSD calculation was solve numerically for the same ranges of values of j9, log(ª s r) an G use in the analysis for zero pore water pressures, an with ä s ¼ j9 mob =: Parameter A was calculate for í9 ¼ 0. 5, ª w ¼ 0 kn/m an ª s ¼ 0 kn/m : The ratios M/M EC7 an F/F EC7 are plotte in Figs 5 an 6 for j9 ¼ 08: For very stiff walls, the MSD maximum bening moments an prop loas slightly excee those calculate using the conventional EC7 limit equilibrium approach; conversely, as the wall flexibility or the soil stiffness increases, they reuce below the limit equilibrium values. The pattern of reuction is similar to that for zero pore water pressure conitions (Figs an ), although the rate of reuction with increasing wall flexibility or soil stiffness is slightly less pronounce. Again, for the range of wall flexibility values commonly aopte in practice, a reuction in both the maximum bening moment an prop loa is apparent. u

6 00 DIAKOUMI AND POWRIE M/ MEC7 G* 0 5 kn/m ρ log( γ s. ) G* 0 4 kn/m G* 0 kn/m G* 0 kn/m Diaphragm Sheet pile Fig. 5. Comparison between MSD an conventional limit equilibrium (EC7) maximum bening moments for ifferent values of soil stiffness an wall flexibility for j9 ¼ 08 an steay-state seepage into excavation from water table at original groun level (soil wall friction angle ä s ¼ j9 ) F/ FEC ρ log( γ s. ) G* 0 5 kn/m G* 0 4 kn/m G* 0 kn/m G* 0 kn/m Diaphragm Sheet pile Fig. 6. Comparison between MSD an conventional limit equilibrium (EC7) prop loas for ifferent values of soil stiffness an wall flexibility for j9 ¼ 08 an steay-state seepage into excavation from water table at original groun level (soil wall friction angle ä s ¼ j9 ) RELATIVE SOIL/WALL STIFFNESS It is remarkable that the curves presente in each of Figs,, 5 an 6 for ifferent values of G have the same shape, an are separate along the log(ª s r) axis by one orer of magnitue. By plotting M/M EC7 or F/F EC7 against log(g r) for either zero pore water pressures or a state of linear seepage, a single curve is obtaine for a given value of j9 (Figs 7 an 8). The presence of pore water pressures ress in a less pronounce rate of reuction in both the maximum bening moment an prop loa with increasing relative soil wall stiffness than when the pore water pressures are zero. This is because increasing the pore water pressure reuces the influence of wall flexibility, in that only the effective stress component of the total lateral stress acting on the wall can be reistribute by wall flexibility effects (Powrie, 997; Bourne-Webb et al., 007). For the same reason, ecreasing the soil strength has a greater effect on the rate of reuction of maximum bening moment an prop loa with increasing relative soil wall stiffness when the pore water pressures are low (zero). As alreay iscusse, for stiff walls with a relatively low value of log(g r), the calculate increase in bening moments an prop loa above the EC7 values is perhaps unfeasibly high for j9 ¼ 08: In this case, it will be important to take into account the likely ifference in 0 zero pwp M/ MEC ρ log( G* ) 0 linear seepage 0 zero pwp 0 linear seepage 0 7 Fig. 7. Comparison between MSD an conventional limit equilibrium (EC7) maximum bening moments for ifferent values of log(g r) an ifferent conitions of pore water pressure when j9 ¼ 08 an j9 ¼ 08 (soil wall friction angle ä s ¼ j9 )

7 MOBILISABLE STRENGTH DESIGN FOR FLEXIBLE EMBEDDED RETAINING WALLS 0 F/ FEC ρ log( G* ) 0 zero pwp 0 linear seepage 0 zero pwp 0 linear seepage Fig. 8. Comparison between MSD an conventional limit equilibrium (EC7) prop loas for ifferent values of log(g r) an ifferent conitions of pore water pressure when j9 ¼ 08 an j9 ¼ 08 (soil wall friction angle ä s ¼ j9 ) strength mobilisation rates with shear strain behin an in front of the wall. Nonetheless, Figs 7 an 8 show that the MSD metho as evelope in this paper offers generic moment an prop loa reuction curves for possible use in esign. It also emonstrates that G r ¼ G H 4 /EI, where G is the rate of increase in shear moulus with epth, H is the overall wall height an EI is its bening stiffness per metre run, is the appropriate imensionless group for characterising relative soil/wall stiffness (Li, 990; Powrie, 997). Although the units are inevitably similar, the group an the concept unerlying it are ifferent from those propose by previous authors, such as Clough et al. (989), who use the unit weight of water rather than the rate of increase in soil stiffness with epth; Rowe (955), whose soil stiffness parameter, m, was ifferent; an Potts & Bon (994), who use E s,av H 4 /EI, where E s,av was the average Young s moulus over the epth of the wall, an EI was ostensibly in kn m rather than kn m /m. δ e δ eb F δ er h CRITICAL FLEXIBILITY NUMBER For a retaining wall proppe at the crest, wall eformation occurs partly as a res of rigi boy rotation about the prop an partly as a res of bening, as shown in Fig. 9. On the basis of tests on moel embee walls anchore near the crest retaining ry san, Rowe (95) foun that the lateral stress istribution in front of the wall epene on the relative importance of the bening component of wall eformation, an hence on the bening stiffness of the wall. If the wall was stiff, so that the eflection at the excavation level was less than that at the toe, the stress istribution in front of the wall was approximately linear, an there was no reuction in the bening moments or anchor loas compare with those calculate in a limit equilibrium analysis with fully active lateral pressures an passive pressures reuce by a factor F p : If the wall was more flexible, so that the eflection at the excavation level was greater than that at the toe, the centroi of the stress istribution in front of the wall was raise, an a reuction in both the bening moments an anchor loas was observe. A stiff wall coul be efine as a wall in which bening eflections are small enough in comparison with isplacements ue to rigi boy rotation not to affect the linearity of the lateral stress istribution. Transition from a stiff to a flexible wall is inicate by a critical value of wall flexibility r ¼ H 4 /EI, enote r c : Accoring to Rowe (95), r c is reache when bening effects are such that the total eflection at the excavation level becomes equal to that at the toe. From Fig. 9, the eformations (normalise by the overall wall height H) at the toe of a retaining wall proppe at the crest, ä t, an at excavation level, ä e, can be ivie into δ t Fig. 9. Components of wall eformation ue to rigi boy rotation an wall bening components ä tr an ä er ue to rigi boy rotation, an ä tb ( ¼ 0) an ä eb ue to wall bening. From the soil isplacement fiels shown in Fig., the normalise components of wall eformation, ä t, ä er an ä eb, are relate to the wall rotations at the crest by ä t ¼ äł 4 (7) ä er ¼ äł 4 n (8) ä eb ¼ ðäł þ äł Þn (9) where n ¼ h/h. ä t is plotte against the imensionless quantity ª s /G in Fig. 0, an ä eb is plotte against the imensionless quantity ª s r in Fig., for j9 ¼ 08, ä s ¼ j9 = an zero pore water pressures. Figures 0 an show that, following Powrie (997) an Li (990), the maximum eformation ue to rigi boy rotation is proportional to the soil stiffness parameter ª s /G, an the maximum eformation ue to wall bening is proportional to the wall flexibility ª s r but inepenent of the soil stiffness. These observations also apply for ifferent values of j9 : Figure shows ä t an ä eb plotte against the relative soil/wall stiffness, log(g r), for j9 ¼ 08, ª s /G ¼ 0 4 an zero pore water pressures. The value of G r at which ä t ¼ ä eb in Fig. is that beyon which the

8 , 0 DIAKOUMI AND POWRIE δ t γ s γ s 0 γ s eformation at excavation level, ä e the criterion aopte by Rowe (955). Further calculations emonstrate that R c is inepenent of ª s /G, but Fig. shows that R c is proportional to j9, an for a given value of j9 increases as the pore water pressures are reuce. The effect of pore water pressure becomes more pronounce as j9 increases γ s / G* Fig. 0. Normalise eformation ue to rigi boy rotation at toe, ä t, as function of ª s /G (soil wall friction angle ä s ¼ j9 ) δ eb G* 0 kn/m 4 G* 0 kn/m G* 0 kn/m γ s Fig.. Normalise component of eformation ue to wall bening at excavation level, ä eb, as function of ª s r (soil wall friction angle ä s ¼ j9 ) maximum bening moments calculate accoring to the MSD approach begin to reuce below those calculate in a conventional limit equilibrium calculation accoring to EC7 in Fig. 7. This value of G r will be terme the critical flexibility number, an will be enote R c : R c is reache when the eformation ue to rigi boy rotation at the toe, ä t, is equal to the component of eformation ue to wall bening at excavation level, ä eb, rather than to the total COMPARISON OF MSD APPROACH WITH PREVIOUS STUDIES Comparison with Rowe s analysis of anchore sheet-pile walls (Rowe, 955), which le to his moment reuction curves, is complicate, an is not attempte here because of (a) ifferences in the comparator limit equilibrium calculations (fully active pressures behin the wall an factore passive pressures in front, compare with the same factor of safety on soil strength on both sies of the wall as specifie in EC7), an (b) the starting or pre-excavation stress state of the soil (the MSD approach as presente in this paper has assume a starting conition of zero mobilise strength at zero strain, i.e. a lateral earth pressure coefficient of unity; in Rowe s tests an analyses, the initial conition woul have been closer to a lateral earth pressure coefficient of perhaps 0. 5). Potts & Fourie (985) presente the ress of finite-element analyses on the effect of wall flexibility on wall movements, bening moments an prop loas. Pore water pressures were set to zero, an the pre-excavation earth pressure coefficient, K 0, was assigne a value of 0. 5,,. 5or. Tables an etail the soil an wall parameters aopte in the Potts & Fourie (985) analyses. For ä s ¼ j9, a factor of safety F r ¼ correspons to a factor of safety applie equally to the soil strength behin an in front of the wall, F s ¼.6, which is similar to the value recommene by EC7. The rate of increase of the soil Young s moulus with epth, E s, Poisson s ratio, í, an G were assume not to vary with soil strain. For consistency with the MSD approach, the units of r shown in Table are m /kn, an correspon to wall bening stiffness (EI) inknm /m. The MSD ress may be compare with the finite-element ress for a pre-excavation lateral earth pressure coefficient 0 0 δ t Zero pwp δ eb 40 Linear seepage t eb δ δ R c : egrees Stiff Flexible log ( G* ) log R c Fig.. Determination of critical flexibility number for j9 ¼ 08, ª s /G 0 4 an zero pore water pressure (soil wall friction angle ä s ¼ j9 ) Fig.. Critical flexibility number R c against j9 for ifferent pore water pressure conitions (soil wall friction angle ä s ¼ j9 ) Table. Soil strengths an geometrical parameters investigate by Potts & Fourie (985) j9: egrees ä s : egrees ª s : kn/m n ¼ h/h H:m F r for ä s ¼ j9 F s for ä s ¼ j

9 MOBILISABLE STRENGTH DESIGN FOR FLEXIBLE EMBEDDED RETAINING WALLS 0 Table. Soil an wall stiffness parameters investigate by Potts & Fourie (985) Value range ª s : kn/m r:m /kn E s : kn/m í G : kn/m log(g r) Minimum Maximum K 0 ¼, as this correspons to the assumption in the MSD analyses presente in this paper of zero mobilise soil strength (j9 mob ¼ 0) at zero strain. Figures 4 an 5 compare the ratios of maximum bening moments an prop loas with the values erive from a conventional limit equilibrium calculation, plotte against log(g r), calculate using the MSD approach for the parameter values given in Tables an with those from the Potts & Fourie (985) finite-element analysis for F r ¼ (corresponing to F s ¼. 6) an K 0 ¼. The MSD metho gives a smaller reuction an a smaller rate of reuction in bening moments with increasing relative soil wall stiffness than the finite-element ress (Fig. 4). The calculate prop loa reuctions (Fig. 5) seem consistent between the two methos at high values of G r, but iverge as G r ecreases. For stiff walls, Potts & Fourie (985) report prop loas exceeing the limit equilibrium values. For j9 ¼ 58, full soil/wall friction (ä s ¼ j9 ) an conitions of zero pore water pressure, the MSD metho an finite-element analysis give critical flexibility numbers, R c, of. an.9 respectively (Fig. 4), a ifference of about 0%. The ifferences between the methos apparent in Fig. 4 may arise from a combination of the following factors. M/ MEC7 F/ FEC7 0 4 MSD 0 Potts & Fourie (985) log ( G* ) Fig. 4. Comparison between MSD an Potts & Fourie (985) bening moment reuction curves: M/M EC7 against log(g r) MSD 0 Potts & Fourie (985) log ( G* ) Fig. 5. Comparison between MSD an Potts & Fourie (985) prop loa reuction curves: F/F EC7 against log(g r) (a) (b) (c) In the Potts & Fourie (985) analyses, the soil stiffness oes not vary with strain whereas in the MSD approach it oes, accoring to the hyperbolic equation. The ecrease in soil stiffness with increasing strain will ten to reuce the extent to which high local bening affects the linearity of the stress istribution (Powrie, 997), an hence the reuction in bening moments. The effect of shear stresses on the back of the m thick wall in the finite-element analysis will reuce bening moments compare with those calculate for an iealise wall of zero thickness (Powrie & Li, 99; Day & Potts, 99). It is generally accepte that limit equilibrium methos ten to unerestimate prop loas compare with full soil structure interaction analyses (Gaba et al., 00), although Rowe s ress (Rowe, 95, 955) i not show that tren. () The iealise strain fiels use in the MSD approach are approximate in nature. SUMMARY AND CONCLUSIONS Kinematically amissible soil strain fiels have been introuce to enable the mobilisable strength esign (MSD) metho to be applie to flexible retaining walls proppe at the crest. A calculation proceure has been evelope for etermining the eflecte shape, bening moments an prop loas for such a wall. The MSD approach gives funamental insights into the relative soil wall stiffness problem, an enables the factors affecting it to be investigate from a physical point of view. It also offers a straightforwar an rational way of relating the mobilise soil strength to wall an soil behaviour, taking into account both the wall flexibility an the variation in soil stiffness with epth an strain. It is shown that rigi boy rotation of the wall is proportional to the imensionless parameter ª s /G, an bening eformation to ª s H 4 /EI, an hence that their relative importance (an the effects of soil wall flexibility) are appropriately characterise by the imensionless relative soil wall stiffness parameter R ¼ G H 4 /EI. The maximum bening moments an prop loas obtaine from the MSD calculation have been compare with those calculate on the basis of a uniform mobilise strength, following the factore ULS limit equilibrium calculation given in EC7. A range of pore water pressure conitions, soil strengths an soil an wall stiffnesses have been consiere. Remarkably, a single curve is obtaine for a given value of soil strength an given pore water pressure conitions, when either the normalise maximum bening moment or the normalise prop loa is plotte against the logarithm of G H 4 /EI. As in previous stuies, the maximum bening moment an prop loa reuce below the limit equilibrium values as the relative wall flexibility increases. The reuction in bening moment an prop loa is less pronounce an less epenent on the soil strength at higher pore water pressures. A critical relative soil wall stiffness has been ientifie at which the bening moments start to fall below those calcu-

10 04 DIAKOUMI AND POWRIE late in a limit equilibrium calculation as a res of wall where å vol an å a are the volumetric an axial strains. Assuming bening effects, as occurring when the bening eflection at zero volume change excavation level is equal to the eflection ue to rigi boy å vol ¼ 0 (7) rotation at the toe. This may be use to istinguish a stiff from a flexible system: its numerical value has been shown Thus to epen on the soil strength an the pore water pressure ª ¼ :5å a (8) regime. The critical flexibility number erive from the From equations (4) an (8), the hyperbolic equation (0) can be MSD approach is broaly in agreement with that inferre rewritten in the form from previous finite-element analyses. ª The generic bening moment an prop loa reuction t ¼ (9) curves evelope will facilitate the rapi estimation of wall a þ bª flexibility effects on these structural stress resants in a way that is not currently possible. In equation (9) the parameters a an b are relate to the initial shear moulus G 0, Poisson s ratio í an shear stresses at failure, t f, by ACKNOWLEDGEMENTS The research escribe in this paper was sponsore by an carrie out in the School of Civil Engineering an the Environment, University of Southampton. a ¼ G 0 ð þ íþ b ¼ t f (40) (4) APPENDIX Duncan & Chang (970) propose a simple relationship for representing the soil behaviour, base on the hyperbolic equation introuce by Konner (96) an Konner & Zelasko (96) ó ó ¼ å a a þ bå (0) where ó an ó are the major an minor principal stresses respectively, å a is the axial strain, an a an b are constants, which can be erive from experimental ata. Accoring to Duncan & Chang (970), constants a an b can be relate to the initial tangent Young s moulus, E s0, an the asymptotic value of stress ifference, (ó ó ), which the stress strain curve approaches at large (infinite) strain respectively a ¼ E s0 () b ¼ ðó ó Þ () From the geometry of the Mohr circle of stress shown in Fig. 6, j9 mob is given by j9 mob ¼ sin t s9 () where t is the raius of the Mohr circle t ¼ ð ó 9 ó 9Þ (4) an s9 is the average effective stress s9 ¼ ð ó 9 þ ó 9Þ (5) In a compression triaxial test, the maximum shear strain is given by ª ¼ ð å a å vol Þ (6) τ mob t σ s σ σ Fig. 6. Mohr circle of stress for triaxial test APPENDIX For the soil behin the wall (active sie), from equations (), (4), (5) an (9) j9 mob ¼ sin ª (4) aó 9 þ ª ðó 9b Þ For the soil in front of the wall (passive sie), from equations (), (4), (5) an (9) j9 mob ¼ sin ª (4) aó 9 þ ª ð þ ó 9bÞ Equations (4) an (4) can be written more simply in the form of equation (9), where for the active sie A ¼ aó 9 (44) B ¼ ó 9b (45) an for the passive sie A ¼ aó 9 (46) B ¼ þ ó 9b (47) Parameters a, b are given by equations (40) an (4). From Fig. 6 ó 9 ó 9 ¼ ó 9 sin j9 þ sin j9 (48) ó 9 ó 9 ¼ ó 9 sin j9 (49) sin j9 Therefore, from equations (4), (48) an (49), b can be expresse as b ¼ ó 9 ó 9 ¼ þ sin j9 (50) ó 9 sin j9 b ¼ ó 9 ó 9 ¼ sin j9 ó 9 sin j9 (5) For conitions of zero pore water pressure, parameters A an B can be calculate as follows. For the soil behin the wall (active sie) substitution of a given by equation (40) an í9 ¼ 0. 5, G ¼ G/z, ó 9 ¼ ó 9 v ¼ ª s z, where z is the epth from the original groun level, into equation (44) gives A ¼ ª s G (5) Substitution of b given by equation (50) into equation (45) gives: B ¼ (5) sin j9

11 MOBILISABLE STRENGTH DESIGN FOR FLEXIBLE EMBEDDED RETAINING WALLS 05 For the soil in front of the wall (passive sie), substitution of a given by equation (40) an í9 ¼ 0.5, G ¼ G/(z h), ó 9 ¼ ó v 9 ¼ ª s (z h) into equation (46) gives equation (5). Substitution of b given by equation (5) into equation (47) gives equation (5). From the above, parameters A an B can be expresse by equations (5) an (5) for both the active an passive sies. For pore water pressure conitions corresponing approximately to a state of linear seepage aroun the wall into the excavation from an original water table at groun level, parameter B is erive from equation (5) as alreay explaine, an parameter A is calculate as follows. For the soil behin the wall (active sie) " ó 9 ¼ ó v 9 ¼ z ª s u # (54) z a From equations (40), (44) an (54) A ¼ z ª s ð u=z Þ a (55) G 0 ð þ íþ For í9 ¼ 0.5 an G ¼ G/z, equation (55) gives A ¼ ª s ð u=z Þ a G (56) For the soil in front of the wall (passive sie) ó 9 ¼ ó v 9 ¼ (z h)[ª s (u=z) p ] (57) From equations (40), (46) an (57) " A ¼ ðz hþ ª s ð u=z Þ # p (58) G 0 ð þ íþ For í9 ¼ 0. 5 an G ¼ G/(z h), equation (58) gives A ¼ ª s ð u=z Þ p G (59) NOTATION A, B parameters use in transforme hyperbolic stress strain relationship b parameter use in hyperbolic stress strain relationship as efine by Duncan & Chang (970) embement epth of retaining wall E Young s moulus of retaining wall E s, E s Young s moulus, an rate of increase of Young s moulus with epth F prop loa F p factor of safety on the passive pressure F r factor of safety as efine by Burlan et al. (98) F s factor of safety accoring to Eurocoe 7 G, G shear moulus, an rate of increase of shear moulus with epth H overall height of a retaining wall h retaine height of retaining wall I secon moment of area K earth pressure coefficient k rotational spring stiffness l length of flow path M bening moment M/(ª s H ) normalise bening moment m soil stiffness parameter as efine by Rowe (955) n retaine height ratio ¼ h/h R flexibility number s9 average effective stress t shear stress u pore water pressure u/z seepage pore pressure graient z Æ epth coorinate parameter use in hyperbolic stress strain relationship as efine by Duncan & Chang (970) ª shear strain äª incremental shear strain ª s unit weight of soil ª w unit weight of water ä eformation normalise by overall wall height ä s angle of soil wall friction å strain Ł relative rotation between wall sections äł relative rotation í, í9 unraine an raine Poisson s ratio r wall flexibility as efine by Rowe (95, 955) ó total stress ó9 effective stress ô shear stress ö9 soil strength Subscripts a active; axial av average b ue to wall bening c critical es esign value EC7 value calculate accoring to Eurocoe 7 e at excavation level f value at failure h horizontal i value at ith point mob mobilise p passive r ue to wall rigi boy rotation t, toe at toe of wall imate v vertical vol volumetric 0 initial value; pre-excavation value major principal minor principal REFERENCES Allman, M. A. & Atkinson, J. H. (99). Mechanical properties of reconstitute Bothkennar soil. Géotechnique 4, No., 89 0, Bolton, M. D. & Powrie, W. (988). Behaviour of iaphragm walls in clay prior to collapse. Géotechnique 8, No., 67 89, Bolton, M. D., Powrie, W. & Symons, I. F. (989). The esign of stiff in-situ walls retaining overconsoliate clay. Part : Short term behaviour. Groun Engng, No. 8, an, No. 9, Bolton, M. D., Powrie, W. & Symons, I. F. (990). The esign of stiff in-situ walls retaining overconsoliate clay. Part : Long term behaviour. Groun Engng, No., 8. Bourne-Webb, P. J., Potts, D. M. & Rowbottom, D. (007). Plastic bening of steel sheet piles. Proc. Inst. Civ. Engrs Geotech. Engng 60, No., Bransby, P. L. & Milligan, G. W. E. (975). Soil eformations near cantilever sheet-pile walls. Géotechnique 5, No., 75 95, British Stanars Institution (004). Eurocoe 7: Geotechnical esign, BS EN 997-:004: Part. Milton Keynes, UK: BSI. Burlan, J. B., Potts, D. M. & Walsh, N. M. (98). The overall stability of free an proppe embee cantilever retaining walls. Groun Engng 4, No. 5, 8 7. Clough, G. W., Smith, E. M. & Sweeney, B. P. (989). Movement control of excavation support systems by iterative esign. In Founation engineering: Current principles an practices (e. F. Kulhawy), Vol., pp New York, NY, USA: American Society of Civil Engineers. Day, R. A. & Potts, D. M. (99). Moelling sheet pile retaining walls. Comput. Geotech. 5, No., 5 4. Diakoumi, M. (007). Relative soil/wall stiffness effects on retaining walls proppe at the crest. PhD thesis, University of Southampton, UK. Diakoumi, M. & Powrie, W. (009). Relative soil/wall stiffness effects on retaining walls proppe at the crest. Proc. n Int. Conf. on New Developments in Soil Mechanics an Geotechnical Engineering, Nicosia,

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