Upper And Lower Bound Solution For Dynamic Active Earth Pressure on Cantilever Walls

Transcription

1 Upper And Lower Bound Solution For Dynamic Active Eart Pressure on Cantilever Walls A. Scotto di Santolo, A. Evangelista Università di Napoli Federico, Italy S. Aversa Università di Napoli Partenope, Italy SUMMARY: In tis paper te lower and upper bound tecniques of limit analysis are applied to determine te seismic eart pressure on cantilever walls witin a pseudo-static approac. In design practice, te active trust on cantilever retaining wall is evaluated on te ideal vertical plane passing troug te eel of te wall. If te wall presents a long eel, failure planes do not interfere wit te vertical stem, so tat te limit Rankine conditions can develop freely in te backfill. In tis case, under vertical static actions, te inclination of trust is constant and is equal to te slope of te ground level. In dynamic condition tese structures are considered equivalent to a gravity type and te well known M-O approac is used considering te ideal plane. Te present paper describes te application of limit analysis to obtain te eart pressure acting on a cantilever retaining wall supporting a coesionless backfill. Te comparison between different procedures is analyzed. Keywords: eart pressure, retaining walls, seismic actions, limit analysis, pseudo-static conditions. 1. INTRODUCTION Te cantilever walls are usually built to support backfill eigts wen gravity walls are uneconomical. In tese walls, te weigt of te soil over te inner foundation slab provides a significant stabilising effect on te retaining structure. It is also responsible of te eart pressure transmitted to te wall. Te dual contrasting function of te soil above te internal part of foundation base makes te interpretation of te structure beaviour difficult, especially in te presence of seismic loads. In general, construction codes refer mainly to gravity wall, providing guidance and rules intended specifically for teir design (Eurocode, 7 and 8; NTC, 8). Unlike tis, several recent contributions on te beaviour of tese structures in te tecnical literature (Greco, 1999; O Sullivan and Creed, 7; Evangelista et al., 1; Kloukinas and Mylonakis, 11). In design practice, te active eart pressure on cantilever retaining walls is evaluated wit respect to vertical plane passing troug te eel of te wall (Fig. 1). In a cantilever wall wit a long eel, b, te limit Rankine conditions can develop freely in te backfill, because failure planes do not interfere wit te vertical stem of te wall. In static conditions te inclination of lateral actions on virtual back, δ, giving rise to te trust s inclination, is assumed to be constant and equal to te slope of te ground level, ε, (limited by te friction angle, ϕ', of te soil). Under seismic actions some codes (Eurocode 8; NTC 8) suggest to assume δ= ε wit ε ϕ and to use te Mononobe e Okabe metod (Mononobe and Matsuo, 199; Okabe, 194) to evaluate te active trust (Fig. a). As sown by Evangelista et al (1), in te presence of te eartquake te inclination of lateral pressure canges. Pysical model tests of gravity retaining walls on saking table (e.g. Aitken, 198; Simonelli et al., 1998) sowed tat, once te seismic acceleration reaces te yield level, te structures exibits permanent displacements and a block failure mecanism can be easily adapted. Tis is associated wit te appearance of a sear band in te backfill. Te inclination of tis sear band is dependent also on te input excitation and canges during te eartquake, according to Green and Micalowski (6). For te cantilever retaining walls a two block failure mecanism can be taken in account in agreement wit recent pysical model results on 1g saking table (Kloukinas et al., 1).

2 ε V (α+β) = (9-ϕ) α β π ϕ ε ξ α = + 4 π ϕ ε ξ β = S a δ D z B b A z a x Figure 1. Cantilever wall: definition of te symbols (see also Fig. 4) Evangelista et al. (1) proposed a metod to evaluate te active eart trust (magnitude, S ae, inclination δ e and point of action z a ) on a cantilever wall retaining coesionless soil wit a regular plane and a long eel by means of te lower-bound solution of te limit analysis, wic is valid for static and pseudo-static conditions. A validation of te procedure as been obtained troug a numerical analysis performed wit FLAC D (Itasca, ) assuming an elastic perfectly plastic constitutive law for te soil. Te metod is based on te traditional assumptions of rigid-plastic beaviour of soil and terefore disregards te magnitude of te displacements necessary to te mobilization of te sear strengt of soil. Te results are compared wit te previous lower bound solution and wit te commonly used Mononobe and Okabe metod.. LIMIT ANALYSIS THEORY Te upper and lower bound teorems of plasticity are widely used to analyze te stability of geotecnical structures. By using te two teorems, te range, in wic true solution falls, can be found. Tis range can be narrowed by finding te closest possible lower and upper bound solutions. Te main ypotesis is a rigid perfectly plastic beaviour wit associated flow rule. As well known, te upper bound teorem states tat, if te work done by external loads in an increment of displacements for a kinematically admissible mecanism equals te energy dissipated by internal stresses, te external loads are not lower tan te true collapse loads. For tis reason tey represent an upper bound on te actual solution. Te lowest possible upper bound solution is sougt wit an optimization sceme by trying various possible kinematically admissible failure surfaces. Te lower bound teorem states tat, if an internal stress field is in equilibrium wit external loads witout violating te yield criterion anywere in te soil mass, te external loads are not iger tan te true collapse loads. Te igest possible lower bound solution can be sougt by trying various possible statically admissible stress fields. In tis paper te upper bound teorem of limit analysis is used to determine te pseudo-static active eart pressures on cantilever retaining wall. Tis solution is compared wit te lower bound solution already determined by Evangelista et al. (1).

3 a) b) V B C V B kvw kvw kw kvw W kw W Sae e kvw kw W kw W Sae e A A Figure. Dynamic trust for cantilever walls: a) typical approac; b) proposed approac 3. CALCULATIONS OF ACTIVE EARTH PRESSURES 3.1. Upper bound Consider te problem of predicting te pseudo-static trust S ae on a cantilever retaining wall. Te backfill is a omogeneous deposit of infinite lateral extent (plane strain conditions) of a dry coesionless soil wit unit weigt, γ, and angle of friction, ϕ. Vertical and orizontal accelerations k g and k v g, wic are uniform trougout te deposit, are introduced subjecting te deposit and te wall to constant inertial body forces k γ and k v γ. For te sake of simplicity, te ground is considered orizontal (ε=) and te vertical component of te seismic acceleration is not taken into account (k v = ). Based on upper bound teorem a translational wall movement and a plane surface mecanism of failure are considered, as sown in Fig. 3. Te active eart pressure corresponds to an outward motion of te wall, caused by te eart pressure and inertial force k W. Te mecanism is composed of two rigid bodies. As a result of te normality rule te sliding increment of displacement along AB (dδ I cosϕ) must be accompanied by a normal increment of displacement of magnitude dδ I sinϕ, so tat gap opens up along te sear plane. Figure 3 sows te wedge I wic is caracterized by an angle of (9-ϕ) between te two slip lines in A and wedge wic is forced to move orizontally (dδ ). Wedge I, accordingly to te associated flow rule, will ave dδ I displacement. Wit te symbol dδ I- it is indicated te relative displacement between wedge I and wedge. On te basis of te geometry te following relationsips exist: W I b = tan( 9 ϕ); b = tan( α * 1 ϕ 1 [ tan( 9 α *) + tan( α ϕ )] = WI 1 + WI = γ * ) (3.1) (3.) [ tanα + tan( 9 α * + ϕ )]; dδ = dδ tan( 9 α ϕ ) d δ = dδ * + (3.3) v v

4 C A b V b 1 D B W I wedge I W I1 ϕ wedge δ I- 9 ϕ δ I ϕ W α* α b CA dδ ΙΙ 9 α 9 α+ϕ * dδ Ι ΙΙ 9+ϕ dδ Ι dδ v dδ Figure 3. Mecanism of failure of a cantilever wall wit orizontal backfill were is te eigt of te wall, γ is te unit weigt of backfill material, and α* is an angle of te sliding plane to te orizontal, as illustrated in Fig. 3. Due to te assumption of a coesionless soil, te energy dissipated by internal stresses is zero (e.g. Cen, 1975). Te rate of external work is composed of two parts, te rate of work done by te soil weigt moving downward and te rate of work done by te pseudo-static active eart pressure S ae moving orizontally: dl = S + k W + W + k W (3.4) e ae I Equating te rate of internal energy dissipation to zero (i.e., te rate of te external work), it is possible to obtain v I S k W + W + k W I v I = (3.5) ae dδ Te trust on te surface AV can be obtained subtracting te pseudostatic inertia forces on te Wedge and on te Wedge I: S ae AV WI v + k WI = dδ k W I (3.6) By substituting te values of weigt and displacements, it results:

5 ( α *) + tan( α * ϕ ') [ 1+ k cot( * ')] tan( * ) α ϕ k α ϕ α * + cot( α * ϕ ') cot S = 1 a γ ' (3.7) ecd tan If expressed in terms of equivalent coefficient of seismic active eart pressure K ae it results: K ae S CD = ae (3.8) γ Te angle α* is variable and te critical one will be cosen in correspondence of te maximum value of te active trust (or K ae ) equating to zero te first derivative of expression 3.7 (or 3.8): ds ae dα * = 4 * tan( ϕ' ) k tan( ϕ' ) ± tan( ϕ' ) k tan( ϕ' ) k αc = arctan k tan( ϕ' ) + k + tan( ϕ' ) + tan( ϕ' ) (3.9) (3.1) Similar results can be obtained removing te ypoteses of orizontal ground surface and k v =, but te equations are not reported in tis paper. 3.. Lower bound Te lower bound solution can be obtained by considering a stress distribution wic is in equilibrium wit te external loads, not exceeding te failure stresses. In te case of inclined backfill of slope ε at te generic dept z a possible statically admissible stress filed wic satisfied all stress boundary conditions is: σ ε = γ cos ε z * k γ sin ε z * (3.11) τ ε = γ sin ε z * + k γ cos ε z * (3.1) were σ ε and τ ε are te normal and sear stress acting on plane parallel to te slope at dept z, were z * = z cosε. Considering te Mor circle passing troug te stresses (σ ε, τ ε ) and tangent to te failure surface, reported in Fig. 4, te following relation could be obtained: K ae cosε cos = cosδ e ( ε + θ ) 1 sinϕ' cos( ξ ε + θ ) ( ) cosθ 1+ sinϕ'cos ξ ε + θ (3.13) sinϕ'sin δ = tan 1 e 1 sinϕ' cos ( ξ ε + θ ) ( ) ξ ε + θ (3.14) ( ) k Were = tan 1 sin ε + θ θ and sinξ =. Te corresponding construction to derive tese 1 kv sinϕ' equations is given by Evangelista and Scotto di Santolo (11) and in a closed form solution by Kloukinas and Mylonakis (11).

6 α β S Figure 4. Active trust on te vertical virtual back AV wit inclined backfill 4. RESULTS Te variation of active eart pressure coefficient Kae, obtained wit te lower bound approac, wit canges in k for and kv = and different values of ε are presented in Fig. 5 for ϕ = 3 and in Fig. 6 for ϕ = 4 according to Eqn and It can be seen tat te magnitude of active eart pressure coefficient increases continuously wit an increase in te magnitude of k. Te variation of inclination of trust δ e are presented in Figure 5b for ϕ = 3 and in Fig. 6b for ϕ = 4 and kv = as function of te orizontal seismic acceleration and slope inclination ε. It can be observed tat te inclination of te trust canges wit seismic acceleration and increases wit k and wit te slope of te backfill approacing to a limit value of ϕ (at most). In te static case, k =, it is equal to te slope inclination according to limit Rankine conditions. As pointed out by Evangelista et al. (1) te assumption of δ = ε recommended in te many seismic code is incorrect. Tis assumption get a less orizontal component of trust and iger vertical ones and tus in te safe side. Te variations of Kae wit canges in k for kv = ±.5k (according to Italian code, NTC 8) and different values of ϕ are presented in Fig. 7 for instance for ε=. It can be seen tat te values of Kae continuously increase wit increase in te magnitude of k and kv. It is worty to observe ere tat te increase in te magnitude of Kae wit increase in kv becomes iger as te magnitude of k increases. Te same results are obtained by using, te upper bound approac, te Eqn. 3.8 after evaluating α* troug Eqn For instance in static condition for ϕ= 3 and k = it is obtained by te Eqn. (3.1) from Eqn. 3.8 * α c = 63 and K ae =.333 according to Rankine. In pseudo static condition wit k =.1 and k v = by Eqn. 3.1 is obtained α = 51.4 and from Eqn. (3.8) * c K ae =.353 according to Evangelista and co- workers (1) and Kloukinas and Mylonakis (11). Tis means tat te lower bound value, being equal to te upper bound one, is te solution of te plastic collapse problem and te evaluated eart pressure is te true value. As pointed out by Evangelista et al. (1) tis solution coincides wit tose of Mononobe & Okabe approac if te same inclination of te trust is utilized. Te advantage of te lower bound procedure is tat allows to evaluate not only te magnitude of te trust but also te inclination and te point of action. It can be also used in te case of non-omogeneous ground, wit strata parallel to te ground surface.

7 a),9 ε=1 ε= 5,6 ε= Kae,3 ϕ= 3 k v =,1,,3,4 k b) 4 3 δe 1 ε=1 ε= 5 ε= ϕ= 3 k v =,1,,3,4 k Figure 5. Variation wit k of Kae (a) and inclination of trust δ e b) for ϕ = 3 and kv =

8 a),9 ε= ε=15 Kae,6,3 ε=1 ε= 5 ε= ϕ= 4 k v =,1,,3,4 k b) 4 3 δe ε=1 1 ε= 5 ϕ= 4 ε= k v =,1,,3,4 k Figure 6. Variation wit k of Kae a) and inclination of trust δ e b) for ϕ = 4 and kv = a)

9 ,9 ε= ϕ= 3,6 ϕ= 4 Kae,3 k v 3 = -,5 k 3 k v =,1,,3,4 k b) 4 3 ε= ϕ= 4 ϕ= 3 δe 1 k v 3 = -,5 k 3 k v =,1,,3,4 k Figure 7. Variation of Kae a) and δe b) wit k for ϕ = 3 and 4, kv = and.5 and ε=. 5. CONCLUSIONS In tis paper it is reported te application of te lower and upper bound tecniques of limit analysis of perfect plasticity to evaluate te active pressure on cantilever walls under pseudo static loads. Te structure supports a dry coesionless backfill wit a planar ground surface and te lengt of te

10 internal base allows te development of Rankine failure surfaces inside te backfill. In design practice, te active eart pressure on cantilever retaining wall is evaluated on te ideal vertical plane passing troug te eel of te wall. If te wall presents a long eel, failure planes do not interfere wit te vertical stem, so tat te limit Rankine conditions can develop freely in te backfill. In static conditions te inclination of trust is constant and depends on te geometry of te ground level as well as by ϕ'. In dynamic condition tese structures are considered equivalent to a gravity type and te well known M-O approac is used considering te ideal plane. Evangelista et al. (1) recently proposed a metod to evaluate te active eart pressure coefficient due to seismic loading and its inclination wit a pseudo-static stress plasticity solution. Tis inclination canges during an eartquake. Te present paper describes te application of te upper bound tecnique of limit analyses to obtain te eart pressure acting on a cantilever retaining wall supporting a ϕ soil backfill. In te assumed ypotesis, for sliding failure mecanism and orizontal backfill, te two solutions coincide and tus, te proposed solution (Eqn. 3.13) is te exact solution. AKCNOWLEDGEMENT Te work as been conducted as part of te Italian ReLUIS project (Rete dei Laboratori Universitari di Ingegneria Sismica) REFERENCES Aitken, G.H. (198). Seismic response of retaining walls. MS Tesis, University of Canterbury, Cristcurc, Nex Zealand. Eurocode 7, Geotecnical Design, Part 1: General Rules, Britis Standards Institution, ENV Eurocode 8, Design provisions for eartquake resistance of structures, Part 5: Foundations, retaining structures and geotecnical aspects, Britis Standards Institution, ENV Evangelista, A., Scotto di Santolo, A. and Simonelli A.L. (1). Evaluation of pseudo-static eart pressure coefficient of cantilever retaining walls. J. Soil Dynamics and Eartquake Engineering, 3: 11, Cen, W.F. (1975). Limit Analysis and Soil Plasticity, Elsevier Science, Amsterdam. Greco, V.R. (1999). Active Eart Trust on Cantilever Walls in General Conditions, Soils and Foundations, Vol. 39: 6, Green, R.A. and Micalowski, L. (6). Sear band formation beaind retaining structures subjected to seismic exitation. Foundations of Civil Environmental Engineering, 7, Huntington, W. (1957). Eart pressures and retaining walls. New York: Jon Wiley. Itasca (). FLAC (Fast Lagrangian Analysis of Continua). Minneapolis: Itasca Consulting Group, Inc. Kloukinas, P., and Mylonakis, G. (11). Rankine solution for seismic eart pressure on L-saped rretaining walls. Proc. 5 t Int. Conf. on Eartquake Geotecnical Engineering, January 1-13, 11 Santiago, Cile. Kloukinas, P., Penna, A., Scotto di Santolo, A., Battacarya, S., Dietz, M., Dioru, L., Evangelista, A. Simonelli, A.L., Taylor, C., Mylonakis, G. (1). Experimental Investigation of Dynamic Beaviour of Cantilever Retaining Walls. Proc. nd Int. Conf. On Performance-Based Design In Eartquake Geotecnical Engineering, May 8-3, 1 - Taormina (Italy). Mononobe N., Matsuo H. (199). On te determination of eart pressure during eartquakes. Proc. World Engineering Congress, Tokyo, Vol. IX, p Murpy, V.A. (196). Te effect of ground caracteristics on te aseismic design of structures. Proc. World Conf. on Eart. Eng., Tokyo, Vol. 1, Newmark N.M. (1965). Effect eartquakes on dams and embankments. Géotecnique, 15:, Norme Tecnice per le Costruzioni (NTC) 8. DM 14 gennaio 8. G.U. n. 9, 4 febbraio 8 n. 3. Okabe S. (194). General teory on eart pressure and seismic stability of retaining wall and dam. J. Jpn Civ. Engng Soc. 1: 5, O Sullivan, C. and Creed, M. (7). Using a virtual back in retaining wall design, Geotecnical Engineering, ICE, 16: GE3, Simonelli, A.L., Scotto di Santolo, A. and Crewe, A. (1998). Saking table tests of scale models of gravity retaining walls). Proc. 6t SECED Conference on Seismic Design Practice into te Next Century. Oxford, 6-8 Marc 1998, Balkema, Rotterdam.