New Analytical l Solutions for Gravitational ti and Seismic Earth Pressures. George Mylonakis Associate Professor
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1 New Analytical l Solutions for Gravitational ti and Seismic Earth Pressures George Mylonakis Associate Professor Department of Civil Engineering, University of Patras, Greece
2 Charles Augustin Coulomb ( )
3 EARTH PRESSURE THEORY: HISTORICAL DEVELOPMENT Bullet, (1691) Papacino, (1781) Vauban, (1704) Gautier, (1717) Couplet, ( ) Belidor, (1729) Coulomb, ( ) Prony, (1802) Mayniel, (1808) Rondelet, (1812) Mϋller Breslau (1906) Mononobe Okabe (1906)
4 WALLS OF CONSTANTINOPLE SELYMBRIA GATE (Towers 35 & 36)
5 WALLS OF CONSTANTINOPLE TYPICAL SECTION
6 CONSTANTINOPLE 1453
7 FORTEZZA OF RETTIMO, CRETE PLAN VIEW
8 FORTEZZA OF RETTIMO, CRETE CROSS SECTION (Ascanio Andreasi 1575)
9 FORTEZZA OF RETTIMO, CRETE CANNON BASTION (Ascanio Andreasi 1575)
10 MARSHAL SEBASTIEN VAUBAN ( )
11 BULLET S THEORY (1691) H 45 o R W P 45 o H K = 1! overestimates thrust 3 times! a
12 COUPLET S THEORY (1726) Dilatancy Theories
13 COULOMB S ANALYSIS
14 COULOMB S OPTIMIZATION SCHEME (Coulomb 1773, 1776)
15 COULOMB S SOLUTION (1773, 1776)
16 MŰLLER BRESLAU SOLUTION (1906)
17 MONONOBE-MATSUO-OKABE (1926) ψ e 1 a h = tan 1 av
18 MONONOBE-MATSUO-OKABE SOLUTION K E = ( + ) 2 cos φ ψe ω ψ ω δ ± ( ψ + ω ) ± 2 cos e cos cos e 1 sin cos ( δ + φ) sin φ ( ψe + β) δ ± ( ψ + ω) cos( β ω) e 2 Citi Critical wedge angle : α φ ψ crit cit tan( φ ψ β) + c tan 1 1 = + c 2 where c1 = tan( φ ψ β) [tan( φ ψ β) + cot( φ ψ ω)] [1 + tan( δ + ψ ± ω)cot( φ ψ ω)] c2 = 1 + [tan( δ + ψ ω) tan( φ ψ β) + cot( φ ψ θ)]
19 DRAWBACKS OF M-O METHOD Estimates are unsafe (active pressures under- predicted, d passive pressures over-predicted) d) Accuracy drops for passive pressures, rough walls, dense soil Stress boundary conditions at soil surface are not satisfied (e.g., failure planes do not always emerge at 45±φ/2 angles in level ground) Formulas are complicated and not symmetric Dynamic effects are not incorporated Distribution of earth pressures along ao the wall not predicted
20 K E = DRAWBACKS OF M-O METHOD ( + ) 2 cos φ ψe ω sin( δ + φ) sin 2 φ ( ψe β) cosψe cos ω cos δ ( ψe ω) 1 + ± + ± cos δ ± ( ψe + ω) cos( β ω) avw ahw ω β W = 45 φ/2 2 δ Pa θ φ R
21 PROPOSED STRESS SOLUTION PROBLEM UNDER CONSIDERATION
22 LIMIT STRESSES ANALYSIS
23 GOAL Determine a stress field that satisfies the following: Equilibrium everywhere in the medium Stress boundary conditions Failure criterion (no violation) Pressure distribution along wall (yes) Wall kinematics (??) Dynamic effects (??)
24 ZONE Α: INFINITE SLOPE
25 ZONE Α: STRESS TENSOR sin sinβ 1 = sinφ
26 ZONE Β: VICINITY OF WALL τ w = σ w tan δ
27 ZONE Β: STRESS TENSOR sin = sinδ 2 sinφ
28 ROTATION OF PRINCIPAL AXES ACTIVE CONDITIONS
29 ROTATION OF PRINCIPAL AXES PASSIVE CONDITIONS
30 ZONE C: STRESS FAN
31 CLOSED - FORM SOLUTION GRAVITATIONAL LOADING 1 P= K qh + K γ H q γ 2 2 cos ( ω β ) cos β 1 sin φ cos ( Δ ) 2 δ K γ = 2 cosδ cos ω 1± sin φcos ( Δ ) 1 ± β cosω Kq = K γ cos ω β ( ) sin Δ = 1 sin β sinφ exp( 2θ tan φ) sin ( ) 2 1 Δ = 2 AB sinδ sinφ 2θ = Δ Δ + δ + β 2ω AB
32 INTEGRATION SCHEME β h tanω tanβ z h tan w w w w p = σ + τ = + δ σ = σ cosδ H h ω L L k q P = dp = 0 0 γ z + ds cosδ cos β h tan ω dh 2 2 w w 1/2 dp = (ó + ô ) ds δ z = s = cos( ω β ) h cosωcos β h tanω s
33 EARTHQUAKE LOADING SIMILARITY TRANSFORMATION
34 CLOSED-FROM SOLUTION EARTHQUAKE LOADING 1 (1 ) 2 E = Eγ v γ + Eq (1 v ) P K a H K a qh 2 ( ω β ) β + ψ φ ( Δ δ ) = exp( 2θE tan φ) ψ δ ω φ β ψe cos ω β cos( β ψe ) 1 sinφ cos 2 δ KE γ 2 * cos e cos cos 1+ sin cos Δ cosω KEq = KE γ cos ( ω β ) sin * sin( β + ψ ) Δ e 1 = sinφ sin Δ = ( ) ( * ) sinδ sinφ 2θ = Δ δ Δ β 2ω ψ E e
35 COMPARISONS ACTIVE CONDITIONS φ = 45 δ = 2φ / 3 1 K =P /( γh 2 A γ A 2 ) β P A δ ω H
36 COMPARISONS (cont d) ACTIVE CONDITIONS φ = 30 ω = K =P /( γ Pγ P H ) 2 P P δ ω H
37 COMPARISONS (cont d) ACTIVE CONDITIONS H δ γ ah P AE γ
38 COMPARISONS (cont d) PASSIVE CONDITIONS H PPE δ γ γ ah
39 COMPARISONS (cont d) ACTIVE CONDITIONS β H P AE δ ω γ ah γ
40 COMPARISONS (cont d) ACTIVE CONDITIONS β H δ γ ah γ P AE
41 GENERALIZED RANKINE CONDITION K γ = 2 ( * cosδ cos ωcosψ 1 sin cos ) e + φ Δ 1 + β + ψe
42 GENERALIZED RANKINE CONDITION critical wall inclination
43 GENERALIZED RANKINE CONDITION critical wall inclination * 1 ωr = ( Δ2 δ ) ( Δ1 β ) ψe 2 e Rankine Condition n φ / ö, Activ ù R ω R / 1,2 1,0 δ H ω 08 0,8 0,6 0,4 0,2 1/4 β P A δ = φ / 2 0 1/2 1/4 β/φ = 1/2 00 0, ö, Passiv ve Rankin ne Conditio on ω ù R R / φ / 2,5 β H 2,0 1,5 1,0 P δ ω 1/4 0 1/2 δ = φ / ,5 1/4 β/φ = 1/2 00 0, Friction Angle, φ ο Friction Angle, φ ο
44 GENERALIZED RANKINE CONDITION critical slope angle ( β ψ ) R e OC S+ Rcos( Δ + β + ψ ) 1 sinφcos( Δ δ 2 ω) 1 e 2
45 GENERALIZED RANKINE CONDITION critical wall roughness tanδ R B C Rsin( Δ2 δ) sinφsin( Δ 1 β + ψe + 2 ω) = = = O' C' S R cos( Δ δ ) 1 sinφcos( Δ β + ψ + 2 ω) 2 1 e
46 CANTILEVER WALL PROBLEM L or inverted Τ - shaped
47 APPLICABILITY OF RANKINE THEORY
48 INTERFACE AT ΑΒ PLANE (Stress Characteristic) K γ = ( ) 1 cos ω cosψ 1+ sinφ cos Δ + β + ψ 2 * char e e φ
49 INTERFACE AT ΑD PLANE (Vertical Interface) ( * ) cos β cos( β + ψ ) 1 sinφ cos Δ β + ψ
50 PRESSURE DISTRIBUTION ON WALL Dubrova solution ( ω β) β φ h ( Δ2 h δ h ) 2 cos ω 1 + sin φ ( h ) cos ( Δ ( h ) + β ) cos cos 1 sin ( ) cos ( ) ( ) σw( h) = γh 1 exp[ 2 θ ( h) tan φ( h)] large outward AB τ ( h) σ ( h)tan δ( h) w w h movement large φ = φ(h) δ ( h) = mφ( h), 0< m< 1 small outward movement small φ
51 φ PROFILES
52 Point of Application of Soil Thrust 0.50 Mode B 0.45 Normaliz zed Point of Action, h a / H Mode C H y d rosta tic Dis tribu tion (1/3) Mode A M o d e D φ = 30 o ω = β = 0 δ = φ / Horizontal Seismic Coefficient, á h
53 COMPARISON WITH EXPERIMENTS 00 (Fang & Ishibashi,1986) θ = 15 x 10-4 rad 0,0 0,1 0,2 0,3 0,4 0,5 Normalized Horizontal Earth Pressures ó w σ(h) w (h) / γ / H γ Η
54 COMPARISON WITH EXPERIMENTS 0,0 (Fang et al.,1994) 1,0 θè = 0.2 rad θè = 0.1 rad θè = 0.05 rad Normalized Horizontal Earth Pressures ó σ w (h)/ / γ γ H Η
55 COMPARISON WITH EXPERIMENTS (Sherif et al.,1982) entricity, h ismic ecce Norm malized se ae /H STRESS LIMIT ANALYSIS SHERIF et al. (1982) F.E.M Horizontal seismic coefficient, a h
56 COMPARISON WITH EXPERIMENTS 2500 (Sherif et al.,1982) m) ÄM AE (N m / STRESS LIMIT ANALYSIS SHERIF et al. (1982) M - O ( 0,6 H ) M - O ( H / 3 ) F.E.M Horizontal seismic coefficient, a h
57 COMPARISON WITH EXPERIMENTS (Bolton & Steedman,1984) STRESS LIMIT ANALYSIS Bolton - Steedman F.E.M. Mae ( N mm / mm ) Horizontal ontal seismic coefficient, ah
58 DYNAMIC EFFECTS u0 Homogeneous layer u.. g
59 DEPTH - VARYING SEISMIC ANGLE ( ω β) β + ψe h φ ( Δ2 δ) 2 * ψe h δ ω ± φ Δ ± ( β + ψ h ) cos cos( ( )) 1 sin cos ph ( ) = exp( 2 θe ( h)tan φ) cos ( )cos cos 1 sin cos 1 e( )
60 FREQUENCY DEPENDENCE h H 0,0 0,2 04 0,4 0,0 Static Static ω ù / / ω ù 1 = ù ω / ù ω 025 0,2 1 = ,4 0,6 gravity only 06 0,6 seismic component 0,8 ω = β = 0 φ = 30 o ; δ = 2φ φ /3 a ho = 0.2 total thrust 0,8 1,0 0,0 0,1 0,2 0,3 0,4 0,5 p E / γ H 1,0 0,00 0,05 0,10 0,15 Äp p E / γ H = (p E p) / γ H E
61 FREQUENCY DEPENDENCE ω / ω 1 ω / ω 1
62 ACCELERATION DEPENDENCE P EA / P A 4,0 35 3,5 3,0 2,5 2,0 1,5 1,0 Static ù / ù 1 = Poin t of appl ication, e / H 0,60 0,55 0,50 0, ,40 0,35 0,5 0,0 0,1 0,2 0,3 0,4 0,5 0,30 0,0 0,1 0,2 0,3 0,4 0,5 horizontal ground acceleration, a h0
63 ULTIMATE LIMIT STATE DESIGN
64 ULTIMATE LIMIT STATE DESIGN
65 LIMIT STATE AGAINST OVERTURNING
66 Equilibrium under Eccentric & Inclined Loading
67 EQUIVALENT CENTRALLY LOADED FOOTING
68 CONCLUSIONS Proposed solution is simpler than Coulomb & M-O Proposed solution is safe i.e. it over-predicts active thrusts and under-predicts passive resistances. For active pressures, accuracy is excellent (maximum observed deviation from numerical data about 10%). The largest deviations occur for high seismic accelerations, high friction angles, steep backfills and large negative wall inclinations. Solution is symmetric i.e. a single formula suffices for both active and passive pressures using pertinent signs for angles φ, δ and ψ e
69 CONCLUSIONS (cont d) For passive case, predictions are satisfactory. However, error is larger - especially at high friction angles. Improvement over M-O predictions is dramatic Stress distribution along the wall can be predicted Dynamic response effects can be incorporated Only the limit states against sliding and bearing capacity failure have an clear physical meaning. The classical safety factor against overturning is arbitrary defined. The ultimate state against bearing capacity is proved to be the most critical mechanism. Kinematics (failure mechanism)?
70 APPLICATION EXAMPLE: ACTIVE CASE PAE kn / m ( ) ( ) 1 sin30 cos sin 30 cos π exp( tan 30) =
71 APPLICATION EXAMPLE: PASSIVE CASE π exp( + 2θE tan 30) =
72 ACKNOWLEDGEMENTS P. KLOYKINAS, Ph.D. Candidate C. ELEZOGLOU, Ph.D. Candidate C. PAPANTONOPOULOS, Professor Grazie!
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