Seismic Analysis of Slopes Seismic Analysis of Slopes George D. George D. Bouckovalas Bouckovalas Achilles Papadimitriou Achilles Papadimitriou National Technical University of Athens School of Civil Engineering Geotechnical Division April 2007 Current Design Practice Current Design Practice
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Seismic Analysis of SlopesSeismic Analysis of Slopes
George D. George D. BouckovalasBouckovalasAchilles PapadimitriouAchilles Papadimitriou
National Technical University of AthensSchool of Civil EngineeringGeotechnical Division
April 2007
Current Design PracticeCurrent Design Practice
H
iθ
W Τ
Ν
Fh
Fv
slidingslope
avWFv =+ = kvWg
cosθFh)sinθFv(Wsinθ] tanφFh)cosθFv[(WcL
FSd+−
−−+=
ahW Fh = = khWg
ah(t)
aV(t)
kh = ah/g
kv = av/g
some times we tend to some times we tend to ……. forget F. forget FVV
8.1 The “Pseudo Static” approach: BASIC CONCEPTS
H
iθ
W Τ
Ν
Fh
Fv
slidingslope
avWFv =+ = kvWg
cosθFh)sinθFv(Wsinθ] tanφFh)cosθFv[(WcL
FSd+−
−−+=
ahW Fh = = khWg
ah(t)
aV(t)
kh = ah/g
kv = av/g
FSd > 1 safe conditions
FSd < 1 slope failure (dynamic) ?
FSd > 1 safe conditions
FSd < 1 slope failure (dynamic) ??
8.2 Dynamic Slope Failure (FSd<1.0): . . . . and so what?
When FSd becomes less than 1.0,
the soil mass above the failure surface will slide downslope
as in the case of a “sliding block on an inclined plane”
HOWEVER,
unlike STATIC FAILURE which lasts for ever,
SEISMIC FAILURE lasts only for a very short period (fraction of
a second), as . . . . . .
“Sliding Block”kinematics
(for the simplified case of sinusoidal motion)
base motion
sliding blockmotion
Seismic failure & downslope sliding
Relative Velocity
Relative Sliding
Computation of Relative Sliding ….Computation of Relative Sliding ….
NEWMARK (1965)NEWMARK (1965)
.
.
.
( )2 CRmax
2max CR
1 aVδ 0.50
a a
−⎛ ⎞⎜ ⎟= ⋅ ⋅⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
2max
2max CR
V 1δ 0.50
a a
⎛ ⎞⎜ ⎟≈ ⋅ ⋅⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
RICHARDS & ELMS (1979)RICHARDS & ELMS (1979)
2max
4max CR
V 1δ 0.087
a a
⎛ ⎞⎜ ⎟≈ ⋅ ⋅⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
E.M.Π. (1990)E.M.Π. (1990)CR
2(1 a )1.15 max
CR
CRmax
V 1δ 0.080 t 1 a
a a
−⎛ ⎞ ⎡ ⎤⎜ ⎟≈ ⋅ ⋅ ⋅ − ⋅⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠
aCR/amax
PE
RM
AN
EN
T D
ISP
LA
CE
ME
NT
(in
)
Newmark - I (1965)
Newmark – II (1965)
Richards & Elms (1979)
Ε.Μ.Π. (1990)
άνω όριογια διάφορα Μ
άνω όριογια διάφορα Μ
Comparison with numerical predictions for actual earthquakes by Franklin & Chang (1977) . . . . Comparison with numerical predictions for actual earthquakes by Franklin & Chang (1977) . . . .
9 cm !!THUSTHUS,if we can tolerate some small down-slope displacements,the pseudo static analysis is NOT performed for the peak seismicacceleration amax, but for the . . . .
HOMEWORK 8.1: Earthquake – induced Permanent displacements of an infinite slope
This HWK concerns an idealized geotechnical natural slope, where5m of weathered (soil-like) rock rests on the top of intact rock. The inclination of the slope relative to the horizontal plane is i=25deg, while the mechanical properties of the weathered rock are γ=18κΝ/m3, c=12.5 kPa and φ=28deg. No ground water is present. Assuming infinite slope conditions:
(a) Compute the static factor of safety FSST.
(b) Compute the seismic factor of safety FSEQ., for a maximum horizontal acceleration aH,max=0.45g accompanied by a maximum vertical acceleration aV,max=0.15g.
(c) In case that FSEQ., comes out less than 1.00, compute the associated downslope displacements, for an estimated predominant excitation period Te=0.50 sec.
HOMEWORK 8.2:The case of Ilarion Dam in Northern Greece