Seismic Analysis of Slopes Current Design Practice

Transcription

1 National Technical University of Athens School of Civil Engineering Geotechnical Division Seismic Analysis of Slopes Current Design Practice George D. Bouckovalas Achilles Papadimitriou April 7

2 8. The Pseudo Static approach: BASIC CONCEPTS i θ F v F h W sliding Τ slope Ν a h (t) a V (t) a h W F h = = k h W g a v W F v =+ = k v W g k h = a h /g k v = a v /g cl + [(W F v )cosθ F h sinθ] tanφ FS d = (W F v )sinθ + cosθ F h some times we tend to.. forget F V i θ F v F h W sliding Τ slope Ν a h (t) a V (t) a h W F h = = k h W g a v W F v =+ = k v W g k h = a h /g k v = a v /g FS d > cl + [(W safe conditions F v )cosθ F h sinθ] tanφ FS d = (W F v )sinθ + F h cosθ FS d < slope failure (dynamic)?

3 8. Dynamic Slope Failure (FS d <.):.... and so what? When FS d becomes less than., the soil mass above the failure surface will slide downslope as in the case of a sliding block on an inclined plane OWEVER, unlike STATIC FAILURE which lasts for ever, SEISMIC FAILURE lasts only for a very short period (fraction of a second), as......

4 Sliding Block kinematics (for the simplified case of sinusoidal motion) base motion sliding block motion Seismic failure & downslope sliding Relative Velocity Relative Sliding

5 Computation of Relative Sliding. NEWMARK (965)... V ( acr ) max δ =.5 a max acr max δ.5 a max acr V RICARDS & ELMS (979) V max δ.87 a 4 max acr E.M.Π. (99) a a V ( a CR ).5 max δ.8 t acr max CR Comparison with numerical predictions for actual earthquakes by Franklin & Chang (977).... PERMANENT DISPLACEMENT (in) άνω άνωόριο για για διάφορα Μ Newmark - I (965) Newmark II (965) Richards & Elms (979) Ε.Μ.Π. (99) a CR /a max

6 Seismic failure & downslope sliding Relative Velocity Relative Sliding δ d 4 V max a max.87 a a max CR = min V max a max.5 a a max CR for EXAMPLE PEAK SEISMIC ACCELERATION a max =.5g PEAK SEISMIC VELOCITY V max =. m/s (T e.8 sec) CRITICAL or YIELD ACCELERATION a CR =.33g (=/3 a max ) Relative Sliding δ d 4 V max a max.87 a a max CR = min V max a max.5 a a max CR 9 cm! TUS,if we can tolerate some small down-slope displacements, the pseudo static analysis is NOT performed for the peak seismic acceleration a max, but for the.... EFFECTIVE seismic acceleration a E = (.5.8) a max

7 8.3 The pseudo static SEISMIC COEFFICIENT why is it much lower than the peak seismic acceleration a max? FIRST.... Using the PEAK seismic acceleration (i.e. TOO conservative. k h a = max g ) is Instead we use the EFFECTIVE seismic acceleration, i.e. a k max h,e = (.5.8 ) g with FS d =.. as this will usually lead to fairly small (< cm) downslope displacements

8 SECONDLY.... observe these numerical results average time histories average acceleration (g) t(sec) g.g average velocity (cm/sec) t (sec) average a max =.6g average (a max,i ) =.6g cm/sec This is because tall earth dams (i.e. > 3m) are flexible and consequently the seismic motion is NOT synchronous all over the the sliding mass: λ Η For common earth dams Η=(3 m) & earthquakes (T e =.3.6s) λ (..) i.e.

9 AS A RESULT OF TESE EFFECTS.... average time histories average acceleration (g) t(sec) g.g average velocity (cm/sec) t (sec) average a max =.6g average (a max,i ) =.6g cm/sec Σχήμα 4. : Υπολογισμός μέσης οριζόντιας επιτάχυνσης και ταχύτητας για την επιφάνεια ολίσθησης AU-8 (Σεισμική Διέγερση ΑΙΓΙΟ 995) K h =. (from the average acceleration time history) =(.5.8) k h =..8

10 8.4 Review & Evaluation of SEISMIC COEFFICIENTS proposed in the literature REVIEW of values, proposed.. on the basis of mere engineering... INTUITION in relation with the FREE FIELD peak ground acceleration (PGA) in relation with the peak seismic acceleration at the DAM CREST EVALUATION in comparison with numerical analyses which take into account: foundation SOIL CONDITIONS dynamic DAM RESPONSE NON-LINEAR YSTERETIC soil response to seismic excitation

11 Numerical Evaluation of : The case of Ilarion Dam in Northern Greece Dam geometry & shear wave velocity distribution. V s (m/sec).e+.e+ 3.E+ 4.E+ 5.E+ 6.E+ 7.E+ 8.E+ 9.E+.E+3 m 7 m/sec 3 m/sec 4 m/sec) 5m/sec m/sec 3 m/sec Numerical Evaluation of : The case of Ilarion Dam in Northern Greece Basic aspects of numerical modeling τ γ

12 Numerical Evaluation of : The case of Ilarion Dam in Northern Greece Typical acceleration time histories.g!.4g.5g Numerical Evaluation of : the case of Ilarion Dam in Northern Greece K h =.56 (from the average acceleration time history) =(.5.8) k h =.8.45 average time histories average acceleration (g) g.56g t(sec) average velocity (cm/sec) ,84 cm/sec t (sec) average a max =.9g average (a max, i )=.9g

13 Numerical Evaluation of : The case of Ilarion Dam in Northern Greece K h =. (from the average acceleration time history) =(.5.8) k h =..8 average time histories average acceleration (g) t(sec) g.g average velocity (cm/sec) t (sec) average a max =.6g average (a max,i ) =.6g cm/sec Ad-hoc values of based on ENGINEERING EXPERIENCE.8 TERZAGI (95). significant earthquakes =. violent earthquakes.5 destructive earthquakes.8.6 "destructive".6 "destructive".4.4. "violent". "violent" "significant" (m) "significant" PGA (g)

14 STANDARD PRACTICE of the 7 s =.. (depending on M) FS d > range of common design values (m) PGA (g) Correlation of with the EGA (effective FREE FIELD acceleration) BRITIS STANDARDS (Charles et al 99) k h,e g EGA/g EGA 3.5 stiff soil or rock / (EGA / g).5 British standards (Charles et al 99).5 soft soil /

15 EUROCODE EC-8 =.5 S S T (EGA/g) k h,e g EGA S = Soil Factor Ground Type V S (m/s) N SPT C U (kpa) M<5.5 S M>5.5 A > B 36-8 > 5 > C D < 8 < 5 < E SALLOW C or D < 5 < EUROCODE EC-8 =.5 S S T (EGA/g) S T = Topography Factor (only for >3m and i>5 o )..4 k h,e EGA S T.

16 EUROCODE EC-8 =.5 S S T (EGA/g) 3.5 stiff soil or rock k h,e g EGA / (EGA / g).5.5 range of EC-8 soft soil / GREEK NATIONAL SEISMIC CODE EAK a max,crest a max,crest Η V s T o =(.5.8) /Vs PGA a max,base T a max,base e 3.5 a max, base =.5 PGA β(το) S a / a max gr.5 A B Γ a max, crest = β(το) a max, base T(sec) T o

17 GREEK NATIONAL SEISMIC CODE EAK a max,crest =.5 β(τ ο ) PGA a max,av EGA a max (/) a max,base =.5 PGA a max,av = a + a ( / ) max,crest max GREEK NATIONAL SEISMIC CODE EAK a max,crest =.5 β(τ ο ) PGA a max,av EGA a max (/) k =.5 [(.5.8 ) a / g ] or he he max,av max,av k = (.5.4 ) a / g a max,base =.5 PGA

18 GREEK NATIONAL SEISMIC CODE EAK 3 / (EGA / g).5.5 stiff soil or rock EAK range k h,e g EGA.5 soft soil / GREEK NATIONAL SEISMIC CODE EAK 3 (EAK) / (analyses) soft soil stiff soil or rock k h,e g EGA (m)

19 GREEK NATIONAL SEISMIC CODE EAK 3 (EAK) / (analyses) soft soil stiff soil or rock k h,e g EGA / GREEK NATIONAL SEISMIC CODE EAK 3 (EAK) / (analyses) soft soil stiff soil or rock k h,e g EGA T o (sec)

20 GREEK NATIONAL SEISMIC CODE EAK 3 (EAK) / (analyses) soft soil stiff soil or rock k h,e g EGA T o / T e Correlation of with the acceleration at the CREST MARCUSON (98) a max,crest and =.33.5 (a max,crest /g) k h =.5.75 (a max,crest /g) k h,e g (k h.5 ). how do you compute a max,crest? k h / (a max,crest /g) Marcusson (98) range soft soil stiff soil or rock /

21 MAKDISI & SEED (978) /3 k h k h = μ (a max,crest /g) a max,crest / g μ MAKDISI & SEED (978) /3 k h k h = μ (a max,crest /g) how do you compute a max,crest?. Makdisi & Seed (978) range a max,crest g k h / (a max,crest /g) soft soil. stiff soil or rock /

22 MAKDISI & SEED (978) /3 k h k h = μ (a max,crest /g) e.g. from EAK 3 a max,crest g (M & S) / (analyses) soft soil stiff soil or rock (m) MAKDISI & SEED (978) /3 k h k h = μ (a max,crest /g) e.g. from EAK 3 a max,crest g (M & S) / (analyses) stiff soil or rock soft soil /

23 MAKDISI & SEED (978) /3 k h k h = μ (a max,crest /g) e.g. from EAK 3 a max,crest g (M & S) / (analyses) soft soil stiff soil or rock T o (sec) MAKDISI & SEED (978) /3 k h k h = μ (a max,crest /g) e.g. from EAK 3 a max,crest g (M & S) / (analyses) soft soil stiff soil or rock T o / T e

24 8.5 C O N C L U D I N G R E M A R K S..... CONCLUDING REMARKS.... The PSEUDO STATIC approach with FS d >. does not prevent slope stability failure. owever, for STABLE SOILS, it ensures that only very small (e.g.< cm) downslope displacements will occur.! for UNSTABLE SOILS (e.g. liquefiable): NEVER USE IT

25 If we can tolerate these displacements, the SEISMIC COEFFICIENT may become much smaller than the corresponding peak seismic acceleration: a max,crest g PGA = (.5.6) PGA/g = (.5.56) a max,crest /g * In general, the higher values are associated with Tall Embankments ( > 3m) & Shallow failure surfaces (/ <.4) When ONLY the PGA is known, a max,crest g PGA you may use: - the BRITIS STANDARDS (conservative for / >.4) - EAK (O.K. for / >.4) The EC-8 is is rather UN-conservative

26 When the maximum CREST ACCELERATION is known (e.g. from seismic analysis of the dam), a max, crest a max,crest g PGA you may use: - MARCUSON (98) [only for very shallow / <.3] - MAKDISI & SEED (978) (overconservative for / =.5.6)

27 OMEWORK 8.: Earthquake induced Permanent displacements of an infinite slope This WK concerns an idealied geotechnical natural slope, where 5m of weathered (soil-like) rock rests on the top of intact rock. The inclination of the slope relative to the horiontal plane is i=5deg, while the mechanical properties of the weathered rock are γ=8κν/m 3, c=.5 kpa and φ=8deg. No ground water is present. Assuming infinite slope conditions: (a) Compute the static factor of safety FS ST. (b) Compute the seismic factor of safety FS EQ., for a maximum horiontal acceleration a, max =.45g accompanied by a maximum vertical acceleration a V, max =.5g. (c) In case that FS EQ., comes out less than., compute the associated downslope displacements, for an estimated predominant excitation period T e =.5 sec. OMEWORK 8.: The case of Ilarion Dam in Northern Greece The dam is constructed on weathered rock formations: V s (m/sec).e+.e+ 3.E+ 4.E+ 5.E+ 6.E+ 7.E+ 8.E+ 9.E+.E+3 m 7 m/sec 3 m/sec 4 m/sec) 5m/sec m/sec 3 m/sec

28 average accelerat t(sec) Compute seismic coefficients k h, for the following potential failure surfaces average velocity ( ,84 cm/sec t (sec) average acceleration (g) g / = t(sec) average (a max, i )=.9g average velocity (cm/sec) cm/sec t (sec) Σχήμα 4. : Υπολογισμός μέσης οριζόντιας επιτάχυνσης και ταχύτητας για την επιφάνεια ολίσθησης average (a max,i ) =.6g AU- (Σεισμική Διέγερση ΑΙΓΙΟ 995) Σχήμα 4. : Υπολογισμός μέσης οριζόντιας επιτάχυνσης και ταχύτητας για την επιφάνεια ολίσθησης AU-8 (Σεισμική Διέγερση ΑΙΓΙΟ 995) / =.9 and the following peak seismic accelerations and predominant periods for the (horiontal) seismic excitation: - Low frequency excitation a max =.37g, T e =. s - igh frequency excitation a max =.g, T e =.65 s