CE 4780 Hurrcane Engneerng II Secton on Floodng Protecton: Earth Retanng Structures and Slope Stablty Dante Fratta Fall 2002 Table of Content Introducton Shear Strength of Sols Seepage Analyss Methods of Slope Stablty Analyss Desgn of Earth Retanng Structures Example Problems Three weeks of classes
Methods of Slope Analyss Two categores of analyss Slope stablty Slope movement Introducton Slope Stablty Methods These methods use lmt equlbrum analyss They requre strength nformaton of the sols They do not provde nformaton about the magntude of movements They yeld a factor of safety They are usually appled n the desgn process 2
Introducton Slope Movement Methods These methods requre stran-stress nformaton about the sol. The soluton s usually found usng fnte element solutons They do not provde factor of safety (drect parameter of stablty) They are usually appled n the preventon of landslde, and rsk emergency analyss. Lmt Equlbrum Analyss They are based n the upper bound theorem Several possble falure surfaces are consdered tll the least favorable s found. For the least favorable surface, the factor of safety s determned. 3
Factor of Safety It s defned as the rato of the shear strength over the appled shear stress (n the local sense) Or the resstance forces over the drvng forces (n the global sense) The least favorable falure surface s determned n the sense of the factor of safety. Methodology Infnte Slopes Isotropc Sols and Unform Slopes b β Shear strength τ = σ tan(φ) Assumed falure plane z E l T l W=γbz b/cos(β) T r E r Infnte slope (and small b): E l =E r ; T l =T r N =γbz cos(β) σ =γz cos 2 (β) S =γβz sn(β) τ =γz sn(β) cos(β) S N β 4
Methodology Infnte Slopes Isotropc Sols and Unform Slopes (cont.) Dry sol at the verge of falure: F horzontal = 0 sn Scos( β) = Nsn( β) S = N cos The maxmum shear resstance s: b b τ l = τ = σ' tan φ cos β cos β ( ) ( ) ( β) = N ( φ) S = τ l N tan tan ( φ) () ( β) ( β) tan Factor of safety tan = N tan ( β) ( ) = N tan( φ) Methodology Infnte Slopes Isotropc Sols and Unform Slopes (cont.) Saturated sol at the verge of falure (no water flow): the effectve stresses and forces are reduced due to the submerged unt weght of the sol. Saturated sol and flow parallel to the slope surface (helpng the sldng mechansms), max. safe slope: γ γ w tan( β) = tan( φ) γ The maxmum shear resstance s: ( φ) () γ γ w tan Factor of safety γ tan w 5
Methodology Fnte Slopes Falure approxmate crcular surfaces (mathematcal convenence). Other surfaces: sprals. Crucal parameters: shear strength and pore pressure dstrbuton Presence of weak layers and heterogenetes are mportant. Water table Sand depost Falure surface Clay layer Methodology Fnte Slopes Problems It s dffcult to determne the weght and center of gravty of the sldng wedge. The problem s statcally ndetermnate. The normal effectve stress along the falure surface s unknown. The moblzed shear strength τ m s unknown (t also vares along the falng surface). The seepage forces are dffcult to determne. 6
Methodology Fnte Slopes Typcal Methods: Basc method: crcular falure surfaces n sotropc saturated sols (undraned shear strength). Smplstc analyss. Fellenus method of slces: crcular falure surfaces n any type of sol. Smplstc analyss. Bshop method of analyss: crcular falure surfaces n any type of sol. Rgorous method, solved usng computer algorthms. Smplfed Bshop method: crcular falure surfaces n any type of sol. Sem-rgorous method. Methodology Fnte Slopes Typcal Methods (cont.): Morganstern-Prce method of slces: arbtrary falure surfaces n all sol types. Rgorous method, solved usng computer algorthms. Spencer method of slces: arbtrary falure surfaces n all sol types. Rgorous method, solved usng computer algorthms. Jambu method of slces: arbtrary falure surfaces n all sol types. Rgorous method, solved usng computer algorthms. Avalable charts smplfy ts use. 7
Methodology Fnte Slopes General Concepts Crcular Falure Surfaces M M resstng drvng τmaxlr Wr + F external 2 Repeats for other falure surfaces r Center of rotaton R r r 2 F external CG W Assumed falure surface, arc length L Falure surface Shear strength τ max = constant Methodology Fnte Slopes General Concepts Method of Sldes Center of rotaton R F external Water table Slde : (free body dagram) b β CG τ Falure surface slde W σ U U + J T + E E + z T z + W l There are up to 3 unknown parameters (N, S, E s, and T s) and 3 equlbrum equatons θ slp surface angle S N 8
Methodology Fnte Slopes Method of Sldes: Bshop s Method Assumptons: Crcular slp surface Colnear E and E + and U and U + N acts on the center of the arc length Summng of vertcal forces: cos( ) + S sn( θ ) W T + T 0 N θ + = N 'cos ( ) = S sn( θ ) + W + T T u l cos( θ ) θ + Methodology Fnte Slopes Method of Sldes: Bshop s Method (cont.) N 'cos( ) = S sn( θ ) + W + T T u l cos( θ ) ub r u = W θ + Pore water pressure rato Bshop s method only consders moment equlbrum: ( θ ) = W ( r ) S sn( θ ) + ( T T ) N 'cos + W r SR = 0 W r S = = W sn R ( θ ) 9
Methodology Fnte Slopes Method of Sldes: Bshop s Method (cont.) τ ( S ) τ f f m = Local factor of safety m N ' tan S S ( φ ) N ' tan( φ ) S = ( θ ) = W ( r ) FS N ' tan FS ( φ ) sn( φ ) ( T T ) N 'cos u + + ( ru ) + ( T T ) tan ( ) ( φ ) sn( φ ) θ + W [ W ( r ) + ( T T )] N ' = + = m u + cos FS Methodology Fnte Slopes Method of Sldes: Bshop s Method (cont.) [ W ( ru ) + ( T T + ) ] tan( φ ) m W sn( θ ) Dsregardng the term (T -T + ) the error s less than % ( ru ) tan( φ ) W sn( θ ) W m Water table bellow slp surface W tan W sn ( φ ) m ( θ ) 0
References and Bblography Abramson, L. W., Lee, T. S., Sharma, S., and Boyce, G. M. (2002). Slope Stablty and Stablzaton Methods. Second Edton. John Wley & Sons. New York. 72 pages. Atknson, J. (993). The Mechancs of Sols and Foundatons. MacGraw-Hll. Budhu, M. (2000). Sol Mechancs and Foundatons. Wley & Sons. Duncan, J. M. (996). Sol Slope Stablty Analyss n Landsldes: Investgaton and Mtgaton, Ed. A. K. Turner and R. L. Schuster. Specal Report 247. Transportaton Research Board. Washngton, DC. pp. 337-37. McCarthy, D. F. (2002). Essentals of Sol Mechancs and Foundatons. Sxth Edton. Prentce-Hall. Upper Saddle Rver, NJ.