Abstract
Gravity walls retaining dry soil are modeled as a system of two bodies: (a) the gravity wall that slides along the wall-foundation soil boundary and (b) the critical soil wedge in the soil behind the wall. The strength of the system is defined by both the frictional and the cohesional components of resistance. The angle of the prism of the critical soil wedge behind the wall is obtained using the limit equilibrium method. The model accounts for changes in the geometry of the backfill soil behind the wall by considering the displacements at the end of each time step under limit equilibrium. The model shows that the standard (single) block model is over-conservative for the extreme case of critical-to-applied-seismic acceleration ratios less than about 0.30, but works well for cases where this ratio ranges between 0.5 and 0.8. The model is applied to predict the seismic displacement of gravity walls (a) tested in the shaking-table and (b) studied numerically by elaborate elasto–plastic analyses.
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Abbreviations
- B:
-
backfill (=retained soil)
- B-cr:
-
the critical soil wedge at the backfill next to the wall
- FEM:
-
Finite element method
- i :
-
increment
- I:
-
interface
- MO:
-
Mononobe-Okabe method
- 0:
-
initial configuration
- RE:
-
the Richards–Elms method
- s :
-
seconds
- sl:
-
sliding-block model
- W:
-
wall
- Notations :
-
- a m :
-
The maximum value of the acceleration in the applied acceleration history
- b Bcr-i :
-
The contact length of the critical soil wedge with the retained soil at increment “i”
- b B-i :
-
The contact length of a soil wedge with inclination not necessarily equal to the critical with the retained soil at increment “i”
- b W :
-
The length of the base of the wall
- c B, c w, c I :
-
The cohesional component of resistance (a) in the retained soil, (b) at the wall- foundation soil interface and (c) at the wall-backfill interface, respectively.
- d :
-
differential
- d i :
-
The length of the soil-retaining wall interface at increment “i”
- g :
-
the acceleration of gravity
- H 0 :
-
The height of the soil behind the wall at the initial configuration (Fig. 1)
- H i :
-
The height of the soil behind the wall at increment “i”
- k c-0 :
-
The critical horizontal acceleration factor for relative motion of the wall-retaining soil system of Fig. 1 (at the initial configuration)
- k c − i :
-
The critical horizontal acceleration factor for relative motion of the wall-retaining soil system of Fig. 1 at increment “i”
- k c-B-i :
-
The horizontal acceleration factor for relative motion of the wall-backfill system at increment “i” when the soil wedge behind the wall that slides has inclination αB- i
- k c-RE :
-
The critical horizontal acceleration factor for relative motion of the wall-backfill system, according the Richards-Elms method (=kc-0)
- k c-sl :
-
The critical horizontal acceleration factor for relative motion of the sliding-block model
- k(t) g :
-
The applied horizontal acceleration history
- k i g :
-
The applied horizontal acceleration at increment “i”
- P a-MO :
-
The lateral force estimated by the MO method
- S W-i, S B-i, S I-i :
-
Dimensionless parameters given by Eq. 7b
- S W-0, S B-0, S l-0 :
-
Dimensionless parameters given by Eq. 7b by replacing the subscript “i” with the subscript “0”
- X i :
-
Dimensionless parameter given by Eq. 7a
- X 0 :
-
Dimensionless parameter given by Eq. 7a by replacing the subscript “i” with the subscript “0”
- t :
-
time
- u W :
-
The absolute value of the distance moved by the wall
- u W-nogech :
-
The wall displacement when changes in geometry are neglected
- u sl :
-
The absolute value of the distance moved by the sliding-block model
- V s :
-
Shear wave velocity
- W Bcr-i :
-
The weight of the critical soil wedge at increment “i”
- W B-i :
-
The weight of the soil wedge with inclination not necessarily equal to the critical at increment “i”
- W w :
-
The weight of the wall
- Z W-i :
-
The factor defined by Eq. 3b
- Z W-0 :
-
The factor defined by Eq. 3b by replacing the subscript “i” with the subscript “0”
- Z Bcr-0 :
-
The factor defined by Eq. 12b
- Greek :
-
- αW :
-
The inclination of the base of the wall
- αBcr-i :
-
The inclination of the base of the critical soil wedge at increment “i”
- αB-i :
-
The inclination of the base of the soil wedge not necessarily equal to the critical at increment “i”
- αBcr-0 :
-
The inclination of the base of the critical soil wedge at the initial configuration
- γ:
-
The unit weight of the soil behind the wall
- δ:
-
Angle defining the inclination of the wall-retained soil interface (Fig. 1)
- \(\Delta \vec{u}_{{\rm W-}i}\) :
-
the distance moved by the wall at increment “i”
- \(\Delta \vec{u}_{{\rm B-cr-}i}\) :
-
The incremental distance moved by the backfill at increment “i”
- θ:
-
The inclination of the soil behind the wall (Fig. 1)
- λ i :
-
The factor defined by Eq. 1
- \(\phi_{\rm B}, \phi_{\rm w}, \phi_{\rm I}\) :
-
The frictional component of resistance (a) in the retained soil, (b) at the wall- foundation soil interface and (c) at the wall-backfill interface, respectively
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Stamatopoulos, C.A., Velgaki, E.G., Modaressi, A. et al. Seismic Displacement of Gravity Walls by a Two-body Model. Bull Earthquake Eng 4, 295–318 (2006). https://doi.org/10.1007/s10518-006-9015-0
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DOI: https://doi.org/10.1007/s10518-006-9015-0