Abstract
This paper describes an experimental investigation of the behaviour of embedded retaining walls under seismic actions. Nine centrifuge tests were carried out on reduced-scale models of pairs of retaining walls in dry sand, either cantilevered or with one level of props near the top. The experimental data indicate that, for maximum accelerations that are smaller than the critical limit equilibrium value, the retaining walls experience significant permanent displacements under increasing structural loads, whereas for larger accelerations the walls rotate under constant internal forces. The critical acceleration at which the walls start to rotate increases with increasing maximum acceleration. No significant displacements are measured if the current earthquake is less severe than earthquakes previously experienced by the wall. The increase of critical acceleration is explained in terms of redistribution of earth pressures and progressive mobilisation of the passive strength in front of the wall. The experimental data for cantilevered retaining walls indicate that the permanent displacements of the wall can be reasonably predicted adopting a Newmark-type calculation with a critical acceleration that is a fraction of the limit equilibrium value.
The seismic design of earth-retaining structures is conventionally carried out using the pseudo-static approach, in which dynamic actions are represented as static D'Alembert forces proportional to an equivalent acceleration, and the performance of the system is quantified conventionally in terms of a static safety factor against an assumed collapse mechanism. However, as noted by many authors (Iai & Ichii, 1998; Iai, 2001; Callisto & Soccodato, 2010), the instantaneous occurrence of limit conditions in the system during an earthquake does not necessarily imply the collapse of the structure, but rather the occurrence of permanent displacements, provided that the behaviour of the system is ductile.
If the structure is designed using equivalent static actions computed using the maximum acceleration expected at the site, no movement of the wall will occur during the earthquake, as the system will never attain limit equilibrium conditions. This is safe, but may lead to unnecessarily conservative and expensive design. On the other hand, if the pseudo-static actions are taken to be proportional to a reduced equivalent acceleration, during the earthquake there will be time intervals in which the safety factor is equal to 1, and the structure will experience permanent displacements. If the permanent displacement at the end of the earthquake is taken as a performance indicator, in the framework of performance-based design the choice of the equivalent acceleration to be used in the pseudo-static calculations could be related to the maximum displacements that the structure can sustain, with respect to different levels of design earthquake motions.
The possibility of adopting a performance-based design for retaining structures, even in the simple form just outlined – that is, by an appropriate reduction of the acceleration to be used in pseudo-static calculations – depends crucially on the ability to predict the displacements experienced by the wall during an earthquake. For gravity retaining walls, the displacements are usually computed through the well-established Newmark (1965) rigid-block analysis (Richards & Elms, 1979; Whitman, 1990; Zeng & Steedman, 2000). Following this method, the relative displacements between the gravity wall and the soil are computed by integrating twice the relative acceleration, a(t) − ac, over the time intervals in which the relative velocities are non-zero, where ac is a threshold or critical acceleration. This critical acceleration, defined with respect to an assumed failure mechanism (generally sliding of the wall on its base), corresponds to the complete mobilisation of the soil strength (Elms & Richards, 1990), and for a rigid-perfectly plastic contact between the foundation of the wall and the soil, depends solely on the geometry of the system and on the strength of the soil.
Experimental dynamic tests carried out on reduced-scale models (Neelakantan et al., 1992; Richards & Elms, 1992) and dynamic numerical analyses (Callisto & Soccodato, 2010) have shown that a Newmark-type calculation may be adopted, at least qualitatively, to also interpret the dynamic behaviour of cantilevered walls or retaining walls with one level of props, where the wall can rotate when a state of limit equilibrium is attained in the adjacent soil. In Newmark's analysis, as the internal forces in the structural members cannot exceed the maximum value attained for a = ac, the increments of permanent displacements computed for a > ac correspond to rigid rotations of the wall. For accelerations a < ac, corresponding to which the soil strength is not yet fully mobilised, any displacements are associated solely with the elastic bending deflection of the wall, induced by the dynamic increment of the horizontal contact stresses in the soil. It is evident that this kind of displacement cannot be taken into account by the rigid-block analysis (Zeng & Steedman, 1993). However, Callisto & Soccodato (2010) have shown that, for credible values of their bending stiffness EI, cantilevered walls can be considered, for all practical purposes, as infinitely rigid. Therefore, according to the rigid-block model, both the permanent displacements of the walls and the maximum internal forces induced by an earthquake depend on the critical acceleration ac, which describes intrinsically the strength of the system.
Three different methods are proposed in the literature to compute the critical acceleration of embedded retaining walls, all of them derived from a limit equilibrium analysis. As far as anchored sheet pile walls are concerned, Towhata & Islam (1987) compute the critical acceleration assuming a translation mechanism of the wall and of the retained soil wedge, whereas Neelakantan et al. (1992) assume a rigid rotation of the wall about the anchor system. Finally, for cantilevered walls, Callisto & Soccodato (2010) compute the critical acceleration with the Blum (1931) method, assuming a rigid rotation of the wall about a point close to the toe.
The rigid-block method provides a powerful tool to compute the permanent displacements of retaining walls during an earthquake. However, it is apparent that, when applied to embedded retaining structures, it does not describe the whole observed behaviour satisfactorily. Centrifuge dynamic tests on reduced-scale models of cantilevered (Zeng, 1990) and anchored (Zeng & Steedman, 1993) sheet pile walls, during which successive earthquakes were applied to the models, have shown that embedded walls may accumulate significant permanent displacements concurrently with an increase of the internal forces in the structural members. As these permanent displacements correspond mainly to rigid rotations of the walls (Madabhushi & Zeng, 2007), it follows that rigid permanent displacements of embedded walls may occur even before the strength of the soil is completely mobilised – that is, in the rigid-block scheme, before the critical acceleration is attained. Also, Callisto & Soccodato (2007) have shown that the threshold acceleration that has to be considered in a Newmark-type integration (to match the permanent displacements resulting from dynamic numerical analyses of cantilevered walls) is smaller than the critical value computed from a limit equilibrium analysis.
This research is an experimental investigation of the physical phenomena that control the dynamic behaviour of embedded retaining walls, aimed at developing suitable simplified procedures to compute permanent displacements of this type of structure under seismic loading. Tests were carried out on reduced-scale models of pairs of retaining walls in dry sand, either cantilevered or propped against each other by one level of support near the top. Construction sequences were not modelled, as the scope of the work was not to simulate exactly the behaviour of a structure in the field, but rather to identify the main features of the response of this type of structure to seismic actions.
The experimental programme was carried out in the 10 m diameter Turner beam centrifuge of the University of Cambridge (Schofield, 1980); it included a total of nine tests on models of pairs of retaining walls, in dry sand reconstituted at different values of relative density, six of which cantilevered and three of which propped against each other, at centrifugal accelerations of 80g and 40g respectively. The models were prepared within two equivalent-shear-beam containers (Zeng & Schofield, 1996; Brennan & Madabhushi, 2002). The main geometrical quantities defining the problem under examination, and the relative densities of the models are reported in Table 1 and Fig. 1. Two different values of the width of the excavation, B, were considered, as the results of numerical analyses indicate that the distance between the walls can affect the behaviour of the system during the dynamic transient (Callisto et al., 2007).
|
| Test | Dri: % | Drf: % | h: mm | d: mm | s: mm | tw: mm | Z: mm | B: mm |
|---|---|---|---|---|---|---|---|---|
| CW1 | 84 | 85 | 50 [4] | 50 [4] | – | 3·14 [0·25] | 200 [16] | 75 [6] |
| CW2 | 53 | 59 | 50 [4] | 50 [4] | – | 3·14 [0·25] | 200 [16] | 75 [6] |
| CW3 | 73 | 74 | 50 [4] | 50 [4] | – | 3·14 [0·25] | 200 [16] | 100 [8] |
| CW4 | 55 | 62 | 50 [4] | 50 [4] | – | 3·14 [0·25] | 200 [16] | 100 [8] |
| CW5 | 49 | 53 | 50 [4] | 50 [4] | – | 3·14 [0·25] | 200 [16] | 75 [6] |
| CW6 | 69 | 75 | 50 [4] | 50 [4] | – | 3·14 [0·25] | 200 [16] | 100 [8] |
| PW1 | 78 | 81 | 140 [5·6] | 60 [2·4] | 9 [0·3] | 6·0 [0·24] | 400 [16] | 150 [6] |
| PW2 | 42 | 52 | 140 [5·6] | 60 [2·4] | 9 [0·3] | 6·0 [0·24] | 400 [16] | 150 [6] |
| PW4 | 44 | 59 | 140 [5·6] | 60 [2·4] | 9 [0·3] | 6·0 [0·24] | 400 [16] | 200 [8] |
* Figures in brackets [ ] are prototype scale: m.
Fig. 1. Problem geometry
Retaining walls were modelled using aluminium alloy plates with a bending stiffness at prototype scale similar to that of a prototype tangent concrete pile wall with a diameter of 400 mm. For propped walls, two square aluminium rods with an axial stiffness of about 1 × 106 kN/m at prototype scale, connected to the walls by cylindrical hinges allowing rotation in the vertical plane, were located at a distance of 195 mm from each other; see Fig. 2.
Fig. 2. Test PW2: photograph of model from the top, showing location of props
A standard fine silica sand was used to form the models, namely Leighton Buzzard, Fraction E Sand 100/170. The specific gravity of the sand is GS = 2·65, its maximum and minimum void ratios are 1·014 and 0·613 respectively, and its critical friction angle is φcv = 32° (Tan, 1990; Jeyatharan, 1991). Further details of the mechanical behaviour of the sand under monotonic, cyclic and dynamic loading conditions can be found in Visone & Santucci de Magistris (2009) and Conti & Viggiani (2011).
Instrumentation was used to measure accelerations of the walls and at different locations in the model and on its boundaries, bending moments and horizontal displacements of the walls, and axial loads in the props. Ground and container accelerations during the dynamic stages of the tests were measured using miniature piezoelectric accelerometers; horizontal and vertical accelerations of the walls were recorded using MEMS accelerometers, capable of measuring the dynamic acceleration as well as the static acceleration due to gravity and centrifuge swing-up. The bending moments and the horizontal displacements of the walls were measured using six strain gauges glued to the middle section of each wall, and linear variable differential transducers (LVDTs). The axial load in the props was measured by two miniature load cells located in the mid section of each square rod.
The dynamic input was provided by a stored angular momentum (SAM) actuator (Madabhushi et al., 1998b). During each test, the model was subjected to a series of trains of approximately sinusoidal waves (Table 2) with different nominal frequencies f and amplitudes amax, and a constant duration of 32 s for cantilevered walls and 16 s for propped walls, at prototype scale. As an example, Fig. 3 shows the acceleration time histories and the Fourier amplitude spectra of the five earthquakes that were applied at the base of model PW2; the nominal frequencies were between 1 Hz and 1·5 Hz, and the maximum acceleration, corresponding to earthquake EQ5, was equal to 0·41g. The applied signal is not perfectly harmonic, both because its amplitude is not constant and because, although the nominal frequency is predominant, significant energy content is associated also with secondary frequencies. Brennan et al. (2005) report that such an extended frequency content corresponds to effective mechanical actions on the model, related to the higher vibration modes of the SAM actuator.
|
| Test | CW1 | CW2 | CW3 | CW4 | CW5 | CW6 | PW1 | PW2 | PW4 | |
|---|---|---|---|---|---|---|---|---|---|---|
| EQ1 | f: Hz | 0·50 | 0·38 | 0·50 | 0·38 | 0·50 | 0·50 | 1·00 | 1·00 | 1·00 |
| amax: g | 0·08 | 0·05 | 0·07 | 0·07 | 0·09 | 0·06 | 0·23 | 0·21 | 0·22 | |
| Ia: m/s | 0·34 | 0·23 | 0·45 | 0·30 | 0·42 | 0·30 | 1·36 | 0·95 | 1·29 | |
| EQ2 | f: Hz | 0·75 | 0·50 | 0·75 | 0·50 | 0·75 | 0·75 | 1·50 | 1·50 | 1·50 |
| amax: g | 0·17 | 0·07 | 0·16 | 0·07 | 0·16 | 0·11 | 0·36 | 0·30 | 0·31 | |
| Ia: m/s | 1·75 | 0·38 | 1·79 | 0·46 | 2·04 | 1·13 | 5·40 | 3·09 | 3·85 | |
| EQ3 | f: Hz | 0·63 | 0·63 | 0·63 | 0·63 | 0·63 | 0·63 | 1·25 | 1·25 | 1·25 |
| amax: g | 0·10 | 0·13 | 0·11 | 0·12 | 0·11 | 0·11 | 0·38 | 0·36 | 0·37 | |
| Ia: m/s | 0·88 | 1·01 | 1·15 | 1·15 | 1·03 | 0·81 | 5·82 | 4·31 | 5·44 | |
| EQ4 | f: Hz | 0·75 | 0·75 | 0·75 | 0·75 | 0·75 | 0·75 | 1·50 | 1·50 | 1·50 |
| amax: g | 0·18 | 0·15 | 0·16 | 0·15 | 0·19 | 0·17 | 0·42 | 0·35 | 0·39 | |
| Ia: m/s | 2·93 | 1·96 | 2·18 | 3·05 | 3·25 | 2·18 | 6·53 | 4·22 | 4·68 | |
| EQ5 | f: Hz | 0·63 | 0·63 | 0·63 | 0·63 | 0·63 | 0·63 | 1·25 | 1·25 | 1·25 |
| amax: g | 0·17 | 0·14 | 0·13 | 0·16 | 0·16 | 0·19 | 0·48 | 0·41 | 0·45 | |
| Ia: m/s | 2·03 | 1·85 | 1·89 | 2·62 | 2·18 | 1·96 | 9·08 | 5·69 | 8·12 |
Fig. 3. Test PW2: acceleration time histories and Fourier spectra of input signals
In this paper, accelerations are positive rightwards, and the horizontal displacements of the walls are positive towards the excavation. Moreover, all results are presented at prototype scale, unless explicitly stated.
Test PW2 was carried out on a loose sand model of propped walls; see Table 1. The sand had an initial relative density Dr = 42% corresponding to a unit weight γd = 14·37 kN/m3 and a void ratio e = 0·84. The relative density measured at the end of the test was Dr = 52%. Fig. 4 shows a cross-section of the model and the layout of the instrumentation, which included 15 miniature piezoelectric accelerometers (A), two LVDTs (LV) on the right wall, at 20 mm and 100 mm from the top of the wall, six strain gauges (SG) on each wall, and two load cells (LC) in the mid-sections of the props, for a total of 31 transducers. The acceleration recorded by accelerometer A1, placed on the lower frame of the ESB box, is considered as the input during the dynamic stages.
Fig. 4. Test PW2: layout of transducers
Figure 5 shows the time histories of the axial load measured in one of the two props (LC1); of the bending moments in the two walls, measured by transducers SG2, SG3 and SG4, located at 2·4 m, 3·5 m and 4·7 m from the top of the walls; and of the free-field accelerations (A4, A5 and A6), during earthquake EQ1. For sake of clarity the plot is limited to the time interval between 5 s and 10 s. The phase shift between the acceleration time histories measured at two different locations can be computed as Δφ=2πfΔt, where Δt is the time for the wave to propagate from one accelerometer to the other, and f is the nominal frequency of the input signal. During earthquake EQ1 the maximum phase shift between accelerometers A5 and A6, located approximately at the bottom and the top of the walls, is only 22°, with a ratio of the height of the wall and the wavelength H/λ = 0·16. As the wavelength depends on the shear stiffness of the soil, λ=Vs/f, the ratio H/λ increases for stronger earthquakes. However, the maximum value of H/λ attained during all the tests in the experimental programme was always less than about 0·2, corresponding to a maximum acceleration phase shift between the bottom and the top of the wall of less than 60°. It follows that, even if the acceleration is amplified, the soil wedge behind the retaining wall is accelerated approximately in phase (Steedman & Zeng, 1990). The bending moments induced at different levels in the retaining walls are substantially in phase with one another, and with the accelerations recorded in the soil at the bottom of the wall (Figs 5(b) and 5(c)). As expected, when the accelerations in the model are maximum (rightwards), the bending moments in the right wall are also maximum, while the bending moments in the left wall are minimum (t = t1). On the other hand, when the model is accelerated leftwards, the bending moments on the left wall increase while the bending moments on the right wall decrease (t = t2). The frequency of the bending moment time histories in the two walls is therefore the same as the nominal frequency of the input signal (1 Hz). As far as the axial load in the props is concerned (Fig. 5(a)), the signal recorded by the load cell is maximum when the acceleration close to the soil surface (A6) is both maximum and minimum. Consistently with the dynamics of the observed phenomenon, axial loads in the props increase when the soil is accelerated both towards the right (maximum inertia forces on the right wall) and towards the left (maximum inertia forces on the left wall). It follows that the frequency at which the internal forces vary in the prop system is twice the nominal frequency of the input signal. Moreover, the observed behaviour implies that the two walls interact with one another during the dynamic event.
Fig. 5. Test PW2, earthquake EQ1: (a) axial load in one prop; bending moment on (b) right and (c) left wall at z = 3·5 m; (d) accelerations in free-field conditions measured between 5 s and 10 s
Figures 6(a) and 6(b) show the bending moment distributions in the two walls at the end of the swing-up stage (static) and after each earthquake (residual); Fig. 6(c) shows the horizontal deflection of the right wall, computed from the LVDT measurements and double integration of the strain gauges recordings; Fig. 6(d) shows the time histories of the axial load in one prop during the five earthquakes applied; and Fig. 6(e) shows the accelerations measured close to the soil surface (A6). As expected, the static bending moments in the walls are symmetric, with a maximum of about 45 kNm/m, while the non-symmetrical distribution observed at the end of the earthquakes (Figs 6(a) and 6(b)) is due to the asymmetry of the accelerations applied to the model. Dynamic bending moments increase with increasing maximum acceleration, such that the maximum dynamic bending moment of EQ3 and EQ5 are smaller than those measured during EQ2 and EQ4 respectively. Residual bending moments progressively increase during the sequence of earthquakes, and are much larger than their static values, with a measured increase of about 60–70% at the end of earthquake EQ1, and about 100–120% at the end of the test. Moreover, while the maximum permanent increments of the bending moments occur after earthquakes EQ1 and EQ2, during subsequent events the maximum increments are about 5% and 10% in the left and right walls respectively. It follows that, during earthquake EQ4, the right wall rotates without significant variations of the internal forces.
Fig. 6. Test PW2: (a), (b) bending moment distributions on the two walls; (c) horizontal displacements of right wall at end of static stage and of each earthquake; time histories of (d) axial load in one prop and (e) accelerations measured close to surface
The horizontal displacements measured on the right wall at the end of the static stage are 6 mm and 13 mm for LV1 and LV2 respectively (Fig. 6(c)). During the dynamic stages the wall rotates progressively towards the excavation, with a final value of measured displacements of 8 mm and 54 mm for LV1 and LV2 respectively. However, whereas the horizontal displacements of the wall increased during earthquakes EQ1, EQ2 and EQ4, earthquakes EQ3 and EQ5 did not produce significant displacements (Δu = 2 mm for LV2), even if their peak accelerations were larger than those of EQ1 and EQ2 respectively. Table 3 reports the maximum accelerations and the Arias intensities of the signals recorded near the top (A6) of the soil layer. The maximum accelerations increase between the first and last earthquake, with EQ3 and EQ5 having substantially the same values of amax as EQ2 and EQ4 respectively, while the Arias intensities measured during earthquakes EQ3 and EQ5 are significantly smaller than those measured during the previous earthquakes EQ2 and EQ4. It appears that the permanent horizontal displacements of the walls depend on the entire acceleration time history, and not just the current earthquake intensity.
|
| EQ | Test PW2 | Test CW1 | ||
|---|---|---|---|---|
| amax: g | Ia: m/s | amax: g | Ia: m/s | |
| 1 | 0·22 | 2·09 | 0·11 | 0·42 |
| 2 | 0·36 | 14·43 | 0·23 | 2·43 |
| 3 | 0·37 | 9·87 | 0·15 | 1·08 |
| 4 | 0·44 | 22·61 | 0·26 | 4·07 |
| 5 | 0·46 | 13·41 | 0·25 | 2·65 |
The axial load in the prop at the end of the static stage (Fig. 6(d)) is about 130 kN. During the five earthquakes, great transient dynamic increments were measured, approximately proportional to the current accelerations applied to the model, but, as already observed in terms of bending moments in the walls, only at the end of earthquakes EQ1 and EQ2 was a significant permanent increase of the axial load observed.
Test CW1 was carried out on a dense sand model of cantilevered walls; see Table 1. The sand had an initial relative density Dr = 84%, corresponding to a unit weight γd = 15·80 kN/m3 and a void ratio e = 0·68. The relative density measured at the end of the test was Dr = 85%. Fig. 7 shows a cross-section of the model and the layout of the instrumentation, which included 16 miniature piezoelectric accelerometers, two LVDTs on the left wall, at 9 mm and 20 mm from the top of the wall, six strain gauges on each wall and one horizontal MEMS accelerometer on the top of each wall. As before, the acceleration recorded by accelerometer A1 is considered as the input during the dynamic stages.
Fig. 7. Test CW1: layout of transducers
Figure 8 shows the time histories of the horizontal displacements of the left wall, together with the free-field accelerations measured close to the soil surface, during the five earthquakes applied. The initial displacements correspond to the end of the swing-up stage (static), and are equal to 29 mm and 25 mm for LV1 and LV2 respectively. During the dynamic stages, the displacements of the wall progressively increase towards the excavation, up to a final value of 78 mm and 63 mm for LV1 and LV2 respectively. As already observed for test PW2, significant displacements occur during earthquakes EQ1, EQ2 and EQ4, whereas during earthquakes EQ3 and EQ5 the walls do not experience significant displacements, even if their peak accelerations were larger than those of EQ1 and EQ2 respectively. However, the maximum accelerations and Arias intensities measured near the top of the soil layer during earthquakes EQ3 and EQ5 (Table 3) are smaller than those of earthquakes EQ2 and EQ4 respectively. This seems to suggest, again, that the permanent displacements of the walls depend on the entire acceleration time history, and not just on the current earthquake intensity.
Fig. 8. Test CW1: (a), (b) horizontal displacements of left wall; (c) accelerations close to the soil surface measured during the five earthquakes
Figure 9(a) shows the horizontal deflection of the right wall, computed from the LVDT measurements and double integration of the strain gauges recordings, and corresponding to a rotation of the wall about a pivot point close to the toe, and Figs 9(b) and 9(c) show the bending moment distributions in the two walls at the end of the swing-up stage (static) and after each earthquake (residual). Static bending moments are not perfectly symmetrical because of experimental problems (e.g. the action exerted by the electrical connections from the strain gauges, rendering bending moments at the top of the walls non-zero), and the non-symmetrical distribution observed at the end of the earthquakes is due mainly to the asymmetry of the input earthquake loading. At the end of the static stage the maximum bending moments are 37 kNm/m and 23 kNm/m, increasing to about 69 kNm/m and 23 kNm/m after the five earthquakes, for the left and right walls respectively. Bending moments show the same trend as displacements. Each time the current earthquake intensity and the current bending moment during shaking are larger than for the previous applied earthquakes, such as for earthquakes EQ1, EQ2 and EQ4, permanent increments of residual bending moments occur. By contrast, even strong earthquakes, such as EQ3 and EQ5, do not produce permanent increments of the internal forces if a stronger earthquake has occurred before. It follows that residual bending moments in the walls, like permanent displacements, depend on the entire acceleration time history.
Fig. 9. Test CW1: (a) horizontal displacements of the left wall; (b), (c) bending moment distributions on the two walls at end of static stage and of each earthquake
Figure 10 shows, for all the centrifuge tests and all the applied earthquakes, the maximum values of the axial force in the props (Fig. 10(a)) and of the bending moments in the two walls (Figs 10(b) and 10(c)), and the horizontal displacements of the walls (Figs 10(d) and 10(e)), as a function of the horizontal acceleration measured close to the soil surface (A6). Structural loads are normalised by the corresponding static values in order to provide a direct comparison between soil models with different relative densities, and hence unit weight and peak friction angle; displacements are expressed as a percentage of the height of the wall. Figs 10(f) and 10(g) also show the critical acceleration, ac, of the propped walls, computed with the method proposed by Neelakantan et al. (1992), and of the cantilevered walls, computed with the Blum (1931) method, for different values of the peak friction angle, φp, and of the relative density, Dr. In both methods the critical acceleration is computed as the value of acceleration corresponding to full mobilisation of the shear strength of the soil and limit equilibrium of moments about the position of the prop, for propped walls, or the pivot point, for cantilevered walls. The peak friction angle was computed as (Bolton, 1986)
Fig. 10. All tests: (a) maximum axial loads in props; (b), (c) maximum bending moments and (d), (e) horizontal displacements of walls, measured during the earthquakes, all normalised by corresponding static values and plotted against maximum acceleration at surface (A6); (f), (g) critical acceleration for different values of peak friction angle and of relative density
where φcv = 32° is the critical-state friction angle of the sand, and p′ = 78 kPa is the mean effective stress at mid-height of the sand layer. In the limit equilibrium analysis, a friction angle δ = 12° was assumed at the contact between the soil and the wall (Madabhushi & Zeng, 2007). The critical acceleration computed for φ = φcv is equal to 0·24g and 0·13g for propped and cantilevered walls respectively, while for the relative densities measured in the centrifuge tests (40% ≤ Dr ≤ 80%) the values of the critical acceleration are between 0·44g and 0·79g and between 0·35g and 0·69g for propped and cantilevered walls respectively. Figs 10(a), 10(b) and 10(c) also show the values of the maximum bending moment and of the axial force computed from the limit equilibrium analysis (dotted lines), as a function of the pseudo-static acceleration a, again normalised by the corresponding static values. As observed above, in all the centrifuge tests the accelerations measured behind the walls are approximately in phase, so that uniform accelerations throughout the soil wedge behind the wall were assumed in the pseudo-static analyses, without taking soil deformability into account (Steedman & Zeng, 1990). Four different values of the friction angle were considered, φ = 32°, 38°, 41°, 50°, representative of the critical-state friction angle and of the peak friction angle at three different values of the relative density of the soil.
As expected, the maximum values of the axial forces in the props and of the bending moments in the walls increase with the amplitude of the accelerations applied to the models (Figs 10(a), 10(b) and 10(c)).
As far as propped walls are concerned, experimental data are in reasonable agreement with the limit equilibrium values for models in both dense and loose sand. Moreover, for accelerations larger than about 0·4g, the maximum values of the structural loads measured in the loose sand models do not experience significant variations. This is in agreement with the limit equilibrium prediction for the same models, as the critical acceleration computed for relative density Dr = 42% is ac = 0·44g; following the pseudo-static approach, for a = ac the passive resistance of the soil in front of the walls is completely mobilised, and the internal forces in the structural members cannot increase further (Callisto & Soccodato, 2010). On the other hand, for cantilevered walls, measured bending moments are always larger than the pseudo-static values, particularly for dense models. This may be because, in this case, the displacements induced in the surrounding soil are generally greater than those experienced by the propped walls, and hence the increase in bending moments during stronger dynamic events is due both to the inertial forces acting on the soil and to the progressive reduction of the mobilised soil friction angle with increasing soil strains, not taken into account in the pseudo-static analysis.
Unlike structural loads, horizontal displacements of the walls do not show an increasing trend with the amplitude of the applied accelerations (Figs 10(d) and 10(e)); moreover, they are not zero, even at accelerations that are smaller than the critical value computed using limit equilibrium solutions. The measured displacements correspond mainly to rigid rotations of the wall, either about a point close to its bottom (CW tests), or about the prop (PW tests). They are accompanied by an increase of the loads in the structural members if the acceleration is below the critical value computed by limit equilibrium, whereas they occur without any changes in the structural loads if the critical value of the acceleration is exceeded, as is the case for some of the earthquakes applied to the models of propped walls.
The fact that the displacements are not proportional to the maximum acceleration is consistent with the observation that strong earthquakes can produce negligible displacements if they are applied after the retaining structure has undergone a stronger event, as shown in detail for tests CW1 and PW2. If the critical acceleration is defined as the value of the acceleration at which the wall starts to rotate, rather than the value of the acceleration at which the soil strength is fully mobilised and the moment safety factor is 1 (limit equilibrium), the observations imply the existence of a sort of ‘hardening' effect, in which the critical acceleration may increase during an earthquake.
Figure 11 shows the acceleration time histories recorded at the soil surface, on the left, and the measured horizontal displacements of the left wall, on the right, for all earthquakes of test CW1. A Newmark calculation was carried out for each earthquake, in which the critical acceleration ac was found by trial and error to match the computed displacement and that measured at the end of each earthquake.
Fig. 11. Test CW1: (a) accelerations measured close to soil surface (A6) and critical acceleration; (b) measured and computed horizontal displacement of wall
Inspection of Fig. 11 reveals that the displacement time histories obtained using a constant value of the critical acceleration are completely different from the observed ones. The only way in which the displacement time histories can be back-calculated using a Newmark analysis is to admit that the critical acceleration varies during the earthquakes. For the first earthquake, the initial value of the critical acceleration is close to zero, whereas for each successive earthquake the initial critical acceleration is the same as the final value computed at the end of the previous earthquake. The data in Fig. 11 also indicate that the critical acceleration required to match the observed displacement time histories increases very rapidly at the beginning of the earthquake (first peak of the imposed acceleration), and then only very slowly during the rest of the earthquake.
The final value of the critical acceleration at the end of the five applied earthquakes is ac = 0·196g, smaller than the limit equilibrium value computed using the peak value of the friction angle (ac = 0·69g). It is evident that, had the Newmark analysis been carried out using the latter, the displacements of the wall would have been zero; in other words, the pseudo-static analysis predicts that the wall is not in a state of limit equilibrium during any of the applied earthquakes, as the available strength of the soil is not fully mobilised. On the other hand, using the critical acceleration corresponding to the friction angle at constant volume (ac = 0·13g) the Newmark displacements computed during earthquake EQ1 would have been zero, while those computed for earthquakes EQ2, EQ4 and EQ5 would have been much larger than measured.
A Newmark-type calculation was also carried out for all earthquakes applied in the tests on cantilevered walls, in which LVDTs measurements were available to obtain the constant values of critical acceleration ac required to match the measured permanent displacement at the end of each earthquake. Fig. 12 shows the computed values of ac as a function of the maximum acceleration amax, both normalised by the limit equilibrium value of the critical acceleration, ac,eq.lim, for those earthquakes in which significant displacement were measured – that is, which were stronger than any previously applied earthquake. The data indicate that the critical acceleration increases linearly with the maximum acceleration applied in the test, and all data plot very closely to the same line, independently of the relative density of the model.
Fig. 12. Newmark analysis of CW test results: required value of critical acceleration as a function of maximum acceleration
The question arises why, for embedded retaining walls, does the critical acceleration increase during an earthquake? Zeng (1990) explained this phenomenon with the densification of the sand under dynamic loading, resulting in a progressive increase of the strength and of the stiffness of the backfill. In the centrifuge tests performed, however, the observed increase of critical acceleration cannot be explained solely in terms of sand densification: as an example, the total variation of Dr measured in test CW1 was only 1%, whereas the variation of relative density required to explain the increase of ac obtained by back-analysis of the observed displacements would be of the order of 20–30%.
It is felt that the reason why the critical acceleration increases during an earthquake is connected to redistribution of earth pressures and progressive mobilisation of the passive strength in front of the wall. In fact, when the active pressure behind the wall increases as a result of seismic loading, equilibrium of moments requires a larger proportion of the passive earth pressure to be mobilised in front of the wall, which can happen only with a progressive rotation of the structure towards the excavation, resulting in finite displacements well before the strength of the soil is fully mobilised. This is completely different from the case of a rigid block or a gravity retaining wall sliding on its base, where the contact is rigid-perfectly plastic, and full mobilisation of the strength occurs with zero relative displacements.
The displacements required to mobilise a larger proportion of the passive strength are inelastic – that is, they are not recovered when the acceleration decreases – and therefore no further rotations occur if the acceleration applied is smaller then the maximum value experienced by the wall. It follows that ac increases rapidly as the acceleration increases from zero to its maximum value (first peak), and then should be constant if the maximum value of the acceleration is not exceeded again. In the dynamic inputs applied in the centrifuge, the maximum acceleration is always reached during the first peak; the small increase of ac observed during successive peaks may well be due to sand densification, as suggested by Zeng (1990). This physical interpretation also explains why the critical acceleration at the beginning of the first earthquake is close to zero (any increase of the active pressure above its static value requires a larger proportion of the passive strength to be mobilised in front of the wall), and why the critical acceleration at the beginning of each successive earthquake is the final value computed for the previous one. Once the passive strength of the soil in front of the wall is fully mobilised, and hence the critical acceleration has reached its maximum, limit equilibrium value, the wall will rotate without any changes of critical acceleration other than those connected to small ‘hardening' due to densification, or even small ‘softening' due to the progressive reduction of soil strength from its peak value towards its critical-state value as displacements increase.
Consistent with previous experimental evidence for gravity (Andersen et al., 1991), cantilevered (Zeng, 1990; Madabhushi & Zeng, 2007) and anchored (Zeng, 1990; Whitman & Ting, 1993; Zeng & Steedman, 1993; Watabe et al., 2006) retaining walls, high residual bending moments were measured after the earthquakes, corresponding to about 80–90% of the maximum values recorded during shaking. According to Whitman (1990) and Zeng (1990), residual moments are associated with the tendency of the sand to densify when vibrated, which in turn implies that the horizontal stresses remain locked behind the wall even after the earthquake, similarly to what happens during compaction of a backfill. Only when the previous maximum horizontal stress is again exceeded – that is, a stronger earthquake is applied to the structure – will permanent deformations be induced that will cause still higher residual earth pressure on the wall (Zeng, 1990). However, as already discussed in terms of displacements, the variations of relative density measured in the centrifuge tests cannot fully justify the observed behaviour of embedded walls. Once again, we believe that the observed behaviour may be justified by stress redistribution and progressive mobilisation of the soil strength on the passive side of the wall produced by the earthquake, as confirmed by preliminary numerical analyses (Conti, 2010).
The experimental results discussed in this paper show that both propped and cantilevered retaining walls experience nearly rigid permanent displacements even for maximum accelerations that are smaller than the critical limit equilibrium value corresponding to full mobilisation of the soil strength. For cantilevered retaining walls the magnitude of the horizontal displacement of the top of the wall can be up to 1–2% of the total height of the wall, whereas for propped retaining walls the horizontal displacement of the toe reaches about 0·5–1% of the total height of the wall, depending on soil relative density. These values are larger than allowable displacements of embedded retaining walls quoted in many recommendations and codes of practice, such as PIANC (2001), the US Navy (Ferritto, 1997) and Italian (NTC, 2008), and cannot be predicted using a Newmark analysis with the limit equilibrium value of the critical acceleration, as in this case zero displacements would be computed.
In this experimental research, the maximum acceleration exceeded the limit equilibrium critical value only for tests on propped walls. In this case the retaining walls experienced even larger permanent displacements, of the order of 1·5% of the total height of the wall, under constant structural loads. A Newmark analysis carried out using the limit equilibrium value of the critical acceleration would yield displacements that are much smaller than observed, as it would overlook the displacements experienced by the wall before the acceleration reaches the limit equilibrium critical value.
In a performance-based design approach, it may be required that the permanent displacement of the structure under an earthquake of given maximum acceleration be computed. A practical solution to the problem of predicting the permanent displacements may be that of using a Newmark analysis with a value of the critical acceleration that is a fraction of the limit equilibrium value. The experimental data for cantilevered retaining walls indicate that the permanent displacements of the wall at the end of each earthquake can be reasonably predicted adopting a Newmark-type calculation with a critical acceleration that is 72% of the limit equilibrium value, independently of soil relative density. However, the data refer only to cantilevered walls in dry sand with approximately the same safety factor under static conditions, and subjected to maximum accelerations less than the limit equilibrium critical value. Further research is required to investigate the appropriate value of critical acceleration to be used in a Newmark-type calculation to predict correctly the permanent displacements of propped walls, of walls with different static safety factors and of walls subjected to maximum accelerations larger than the limit equilibrium critical acceleration. Finally, the experimental work was limited to retaining walls in dry sand, and further testing is required to clarify the role of the presence of the pore water, for either saturated or unsaturated soils.
ACKNOWLEDGEMENT
The work presented in this paper was partly funded by the Italian Department of Civil Protection under the ReLUIS 2005–2008 research project.
|
| A | Fourier spectrum amplitude |
| a | acceleration |
| ac | critical value of acceleration |
| ac,eq.lim | limit equilibrium value of critical acceleration |
| amax | maximum value of acceleration |
| Dr | relative density |
| Drf | final relative density |
| Dri | initial relative density |
| d | depth of embedment |
| EI | bending stiffness of walls |
| e | void ratio |
| f | nominal frequency |
| GS | specific gravity |
| g | gravity acceleration (= 9·81 m/s2) |
| H | total wall height (= h + d) |
| h | excavation depth |
| Ia | Arias intensity |
| M | bending moment |
| Mmax | maximum value of bending moment |
| Mst | static value of maximum bending moment |
| N | axial load |
| Nmax | maximum value of axial load |
| Nst | static value of axial load |
| p′ | mean effective stress |
| s | distance of props from top of the wall |
| t | time |
| Δt | time interval |
| tw | wall thickness |
| u | horizontal displacement |
| Δu | increment of horizontal displacement |
| VS | shear wave velocity |
| Z | thickness of sand layer |
| δ | friction angle at soil/wall contact |
| γd | dry unit weight |
| λ | wavelength |
| φ | friction angle |
| Δφ | phase shift of acceleration time histories |
| φcv | constant-volume friction angle |
| φp | peak friction angle |
Discussion on this paper closes on 1 May 2013, for further details see p. ii.
