Appendix F Notation a b B C c C k C N C s C u C wt C θ D r D 1 D 2 D 10 D 30 Reciprocal of the initial shear stiffness of the interface K si under initial loading; reciprocal of the initial tangent modulus E i of the soil Reciprocal of the asymptotic shear stress τ ult for initial loading of the interface; reciprocal of the asymptotic deviator stress (σ 1 -σ 3 ) of the soil Bulk modulus of the soil Coefficient of curvature Interface stiffness ratio Correction factor for the number of steps in the backside of a rockfounded gravity wall Correction factor for a rock-founded gravity retaining wall with an inclined backfill surface Uniformity coefficient Correction factor for determination of the vertical shear force coefficient during inundation of the backfill Correction factor for inclination of the backside of a rock-founded gravity wall Relative density Thickness of the backfill above the hydrostatic water table Thickness of the submerged backfill above the heel of the wall Particle size diameter corresponding to 10 percent passing in the grain size distribution curve Particle size diameter corresponding to 30 percent passing in the grain size distribution curve F1
D 60 d s dτ i dτ v, dτ o E i E t E ur F v F v,q F v,soil F w F x F x ' G s h H Particle size diameter corresponding to 60 percent passing in the grain size distribution curve Infinitesimal increment in interface displacement Infinitesimal shear stress increment along an inclined stress path Infinitesimal shear stress increments in the orthogonal directions v' and o', respectively Initial tangent (Young's) modulus of the soil Tangent modulus of the soil Young's modulus for unloading-reloading Vertical force or downdrag per unit length of wall Vertical force increment due to surcharge application Vertical earth force due to the self weight of the backfill Hydrostatic force on the wall Total horizontal force per unit length of wall Effective horizontal force per unit length of wall Specific gravity Height of the horizontal earth force above the base of the wall Height measured along a vertical plane passing through the heel of the wall and extending through the backfill H b Total backfill height as measured in Figure 2-5 i i' i'' I Generic name for inclined stress paths; as a superscript in Figures 4-40 through 4-44, it denotes the i th load step in a finite element analysis Generic name for the σ n - s -τ response of the interface to an inclined stress path Generic name for the s -τ response of the interface to an inclined stress path Correction factor for inclination of the stress path F2
[K] K K b k n k s K f -line K h K I K n K o K si K sno K sn ts K sn ys K sn K st K' st I K' st K' s sec K ur K urj K v K v,q Global stiffness matrix Modulus number Bulk modulus number Normal interface stiffness Interface shear stiffness Line joining the points in the p'-q plane that correspond to failure Earth pressure coefficient for effective horizontal forces Dimensionless interface stiffness number for initial loading Normal stiffness of an interface element At-rest pressure coefficient Initial shear stiffness of the interface Normalized interface shear stiffness at zero-stress level Normalized shear stiffness of the interface Transition stiffness number Yield stiffness number Interface tangent stiffness for vertical stress paths (as defined in Clough and Duncan (1971) hyperbolic model for interfaces) Interface tangent stiffness for stress paths of any orientation Interface tangent stiffness at point I Secant interface shear stiffness Unload-reload modulus number for soils Unload-reload stiffness number for interfaces Vertical shear force coefficient Vertical shear force coefficient for sloping backfill and surcharge K v,q,ref Reference value of K v,q obtained for a value of S = 0 F3
K v,soil K v,soil,ref m m k N n n j o' o'' p' p a Vertical shear force coefficient for self-weight of the backfill Reference value of K v,soil obtained for an inclination of the back of the wall θ of 90 degrees Bulk modulus exponent Stiffness degradation parameter Number of steps in the back of a stepped wall Modulus exponent Interface stiffness exponent Generic name for the line of intersection between the initial loading surface and a plane parallel to the σ n -τ plane Generic name for the projection of o' in the s -τ plane (σ 1 '+σ 3 ')/2 Atmospheric pressure = 101.3 kpa q Shear direction parameter; (σ 1 -σ 3 )/2 q s R f R fj S SL O SL ts SL ys v v' Applied surcharge pressure Failure ratio for soils Failure ratio for interfaces Horizontal distance from the vertical plane through the wall heel to the top of the backfill slope Stress level at the origin of unloading-reloading Transition stress level Stress level for current position of yield surface Generic name for vertical stress paths Generic name for the σ n - s -τ response of the interface to a vertical stress path v" Generic name for the s -τ response of the interface to a vertical stress path F4
α γ γ b γ max, γ min γ moist Scaling factor for unloading-reloading Unit weight of the soil Buoyant unit weight of submerged backfill Maximum and minimum density, respectively Moist unit weight of the backfill above the water table γ w Unit weight of water (9.8 kn/m 3 ) δ δ r actual dis meas n s s i-1, s i so sp sr v P { P} { u} γ xy s Peak interface friction angle Residual interface friction angle Actual sliding displacement between soil particles and concrete Deformation of the sand mass due to distortion under the applied shear stresses Displacement measured between the soil box and concrete specimen Displacement normal to the interface Displacement along the interface Consecutive interface displacement readings Interface displacement at the origin of unloading-reloading Interface displacement to peak Interface displacement to residual Displacement normal to the interface; vertical displacement during interface testing H b - H Vector of nodal forces Vector of unknown incremental displacements Shear strain increment Increment of interface displacement F5
ε x, ε y σ x, σ y σ n τ i τ v, τ o τ xy τ φ ε ε v θ ν ν nom σ h σ n ts σ n σ no σ v σ v ' σ 1 σ 1 ' σ 3 Horizontal and vertical strain increments, respectively Horizontal and vertical stress increments, respectively Normal stress increment Shear stress increment between points P and Q Components of the shear stress increment in the orthogonal directions v' and o', respectively Shear stress increment in soils and backfills Shear stress increment in interfaces Reduction in the peak secant friction angle value for a tenfold increase in σ' 3 Axial strain Volumetric strain Angle between the stress path direction and the τ-axis Poisson's ratio Nominal Poisson's ratio Horizontal pressure Normal stress acting on the interface Normal stress corresponding to point TS where the stress path intersects a transition surface Normal stress at the origin Vertical stress Effective vertical stress Major principal total stress Major principal effective stress Minor principal total stress F6
σ 3 ' Minor principal effective stress (σ 1 - σ 3 ) Deviator stress (σ 1 - σ 3 ) f Deviator stress at failure (σ 1 - σ 3 ) ult Asymptotic deviator stress τ τ i-1, τ i τ ts τ f τ o τ ult φ φ' φ cv φ o Interface shear stress Consecutive interface shear stress readings Shear stress at point TS where the stress path intersects a transition surface Interface shear strength Interface shear stress at the origin of unloading-reloading Asymptotic interface shear stress Peak secant internal friction angle of the soil Effective peak secant friction angle of the soil Friction angle at a strain of 15 percent Peak secant friction angle at a confining pressure of 101.3 kpa (1atm) F7