New Analytical l Solutions for Gravitational ti and Seismic Earth Pressures George Mylonakis Associate Professor Department of Civil Engineering, University of Patras, Greece
Charles Augustin Coulomb (1736-1806)
EARTH PRESSURE THEORY: HISTORICAL DEVELOPMENT Bullet, (1691) Papacino, (1781) Vauban, (1704) Gautier, (1717) Couplet, (1726-28) Belidor, (1729) Coulomb, (1773-76) Prony, (1802) Mayniel, (1808) Rondelet, (1812) Mϋller Breslau (1906) Mononobe Okabe (1906)
WALLS OF CONSTANTINOPLE SELYMBRIA GATE (Towers 35 & 36)
WALLS OF CONSTANTINOPLE TYPICAL SECTION
CONSTANTINOPLE 1453
FORTEZZA OF RETTIMO, CRETE PLAN VIEW
FORTEZZA OF RETTIMO, CRETE CROSS SECTION (Ascanio Andreasi 1575)
FORTEZZA OF RETTIMO, CRETE CANNON BASTION (Ascanio Andreasi 1575)
MARSHAL SEBASTIEN VAUBAN (1633-1707)
BULLET S THEORY (1691) H 45 o R W P 45 o H K = 1! overestimates thrust 3 times! a
COUPLET S THEORY (1726) Dilatancy Theories
COULOMB S ANALYSIS
COULOMB S OPTIMIZATION SCHEME (Coulomb 1773, 1776)
COULOMB S SOLUTION (1773, 1776)
MŰLLER BRESLAU SOLUTION (1906)
MONONOBE-MATSUO-OKABE (1926) ψ e 1 a h = tan 1 av
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)]
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
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
PROPOSED STRESS SOLUTION PROBLEM UNDER CONSIDERATION
LIMIT STRESSES ANALYSIS
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 (??)
ZONE Α: INFINITE SLOPE
ZONE Α: STRESS TENSOR sin sinβ 1 = sinφ
ZONE Β: VICINITY OF WALL τ w = σ w tan δ
ZONE Β: STRESS TENSOR sin = sinδ 2 sinφ
ROTATION OF PRINCIPAL AXES ACTIVE CONDITIONS
ROTATION OF PRINCIPAL AXES PASSIVE CONDITIONS
ZONE C: STRESS FAN
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
INTEGRATION SCHEME β h tanω tanβ z h 2 2 2 1 1 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
EARTHQUAKE LOADING SIMILARITY TRANSFORMATION
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 1 + + cosω KEq = KE γ cos ( ω β ) sin * sin( β + ψ ) Δ e 1 = sinφ sin Δ = ( ) ( * ) 2 1 2 sinδ sinφ 2θ = Δ δ Δ β 2ω ψ E e
COMPARISONS ACTIVE CONDITIONS φ = 45 δ = 2φ / 3 1 K =P /( γh 2 A γ A 2 ) β P A δ ω H
COMPARISONS (cont d) ACTIVE CONDITIONS φ = 30 ω = 20 1 2 K =P /( γ Pγ P H ) 2 P P δ ω H
COMPARISONS (cont d) ACTIVE CONDITIONS H δ γ ah P AE γ
COMPARISONS (cont d) PASSIVE CONDITIONS H PPE δ γ γ ah
COMPARISONS (cont d) ACTIVE CONDITIONS β H P AE δ ω γ ah γ
COMPARISONS (cont d) ACTIVE CONDITIONS β H δ γ ah γ P AE
GENERALIZED RANKINE CONDITION K γ = 2 ( * cosδ cos ωcosψ 1 sin cos ) e + φ Δ 1 + β + ψe
GENERALIZED RANKINE CONDITION critical wall inclination
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 0,0 20 25 30 35 40 45 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 / 2 05 0,5 1/4 β/φ = 1/2 00 0,0 20 25 30 35 40 45 Friction Angle, φ ο Friction Angle, φ ο
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
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
CANTILEVER WALL PROBLEM L or inverted Τ - shaped
APPLICABILITY OF RANKINE THEORY
INTERFACE AT ΑΒ PLANE (Stress Characteristic) K γ = ( ) 1 cos ω cosψ 1+ sinφ cos Δ + β + ψ 2 * char e e φ
INTERFACE AT ΑD PLANE (Vertical Interface) ( * ) cos β cos( β + ψ ) 1 sinφ cos Δ β + ψ
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
φ PROFILES
Point of Application of Soil Thrust 0.50 Mode B 0.45 Normaliz zed Point of Action, h a / H 040.40 0.35 0.30 0.25 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 / 2 0.20 0.0 0.1 0.2 0.3 Horizontal Seismic Coefficient, á h
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
COMPARISON WITH EXPERIMENTS 0,0 (Fang et al.,1994) 1,0 θè = 0.2 rad θè = 0.1 rad θè = 0.05 rad 0 1 2 3 4 5 6 7 Normalized Horizontal Earth Pressures ó σ w (h)/ / γ γ H Η
COMPARISON WITH EXPERIMENTS 06 0.6 (Sherif et al.,1982) entricity, h ismic ecce Norm malized se ae /H 0.4 0.2 STRESS LIMIT ANALYSIS SHERIF et al. (1982) F.E.M. 0.0 0.1 0.2 0.3 0.4 0.5 Horizontal seismic coefficient, a h
COMPARISON WITH EXPERIMENTS 2500 (Sherif et al.,1982) m) ÄM AE (N m / 2000 1500 1000 STRESS LIMIT ANALYSIS SHERIF et al. (1982) M - O ( 0,6 H ) M - O ( H / 3 ) F.E.M. 500 0 0.0 0.1 0.2 0.3 0.4 0.5 Horizontal seismic coefficient, a h
COMPARISON WITH EXPERIMENTS (Bolton & Steedman,1984) STRESS LIMIT ANALYSIS Bolton - Steedman F.E.M. Mae ( N mm / mm ) 350 300 250 200 150 100 50 0 0.184 0.241 0.234 Horizontal ontal seismic coefficient, ah
DYNAMIC EFFECTS u0 Homogeneous layer u.. g
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( )
FREQUENCY DEPENDENCE h H 0,0 0,2 04 0,4 0,0 Static Static ω ù / / ω ù 1 =025 1 0.25 ù ω / ù ω 025 0,2 1 = 0.25 0.5 0.5 1 1 0,4 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
FREQUENCY DEPENDENCE ω / ω 1 ω / ω 1
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 = 0.25 0.5 1 Poin t of appl ication, e / H 0,60 0,55 0,50 0,45 040 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
ULTIMATE LIMIT STATE DESIGN
ULTIMATE LIMIT STATE DESIGN
LIMIT STATE AGAINST OVERTURNING
Equilibrium under Eccentric & Inclined Loading
EQUIVALENT CENTRALLY LOADED FOOTING
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
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)?
APPLICATION EXAMPLE: ACTIVE CASE PAE 2 0.82 18 5 185 kn / m ( ) ( ) 1 sin30 cos 43.2 20 1+ sin 30 cos 62.4 + 15 + 11.3 π exp( + 45.5 tan 30) = 0.82 180
APPLICATION EXAMPLE: PASSIVE CASE π exp( + 2θE tan 30) = 6.31 180
ACKNOWLEDGEMENTS P. KLOYKINAS, Ph.D. Candidate C. ELEZOGLOU, Ph.D. Candidate C. PAPANTONOPOULOS, Professor Grazie!