Support forces

A reinforcement (an anchor, a nail, a pile, a geosynthetic sheet or a wall) acts on the sliding mass where it crosses the slip surface. The force depends on the support type (OGR Slip2D has nine) and enters each method as an active or a passive force on the base of the slice crossed. Two related calculations follow: the water thrust in a tension crack and the back analysis of the force a slope needs to reach a target factor of safety.

3.10.1A support on a slip surface#

A support is a straight segment from its head, the end at the slope face where its plate or connection is, to its tail, the anchored end. OGR Slip2D finds the first point where the segment crosses the slip surface, counting from the head, and evaluates there the force the support can deliver, $F(d)$, with $d$ the distance from the head to the crossing. The force acts at that point, on the base of the slice that contains it (Figure 3.10.1). Everything is per metre of slope: the capacity of a discrete element is divided by its out-of-plane spacing $S$.

The force, in the direction its orientation gives it (Section 3.10.3), is resolved on the base of that slice, whose chord is inclined at $\alpha$:

where

$F_x,\ F_y$

components of the support force, $x$ to the right and $y$ upwards (kN/m)

$s$

sense of sliding, ±1, chosen so that $\mathbf{t} = s\,(\cos\alpha,\ \sin\alpha)$ is the unit vector along the base that opposes sliding

$\mathbf{n} = (\sin\alpha,\ -\cos\alpha)$

unit normal pointing from the slice into the ground below its base

$T_S$

tangential component, positive when it resists sliding (kN/m)

$T_N$

normal component, positive when it presses the slice onto its base (kN/m)

A support that never reaches the surface, or that develops no capacity where it crosses it, puts no force on that surface. The factor of safety is then the one of the slope without it, which looks like any other result, so the analysis notes say how many of the supports placed contributed nothing and why.

3.10.2Active and passive supports#

Whether a reinforcement force exists before the mass moves, or develops only as it moves, decides how it enters a factor of safety defined on strength. Following Duncan & Wright (2005), OGR Slip2D offers both readings. An active support, such as a tensioned anchor, is a force already acting on the mass, and it reduces the driving force. A passive support, such as an untensioned nail, a pile or a geosynthetic, is mobilised with the soil, so its tangential force is added to the resistance and divided by the factor of safety like the shear strength. With $R$ and $D$ the resisting and driving terms of a method:

where

$R,\ D$

resisting and driving terms of the method without the support (kN/m, or moments over the radius on a circle)

$T_S,\ T_N$

tangential and normal components of Eq. (3.10.1), summed over the supports and slices crossed

$\phi'$

friction angle at the base of the slice crossed

In both expressions the normal component raises the frictional resistance of the base it presses; only the tangential component changes side. Written for one base, the shear mobilised on a base crossed by a support becomes

with $N$ the base normal force, which already carries $T_N$ through the equilibrium of the slice. The two readings differ only in that line: summed over a mass in force equilibrium, it turns $F = R/(D - T)$ into $F = (R + T)/D$. Each family of methods applies it in its own equations (Section 3.2):

• Spencer, GLE / Morgenstern–Price, the Corps of Engineers methods and Lowe–Karafiath take the normal part of the force as a load on the slice and add the tangential part to the mobilised shear of the base, at $T_S$ if active and $T_S/F$ if passive, as in Eq. (3.10.4).
• Bishop's and Janbu's simplified methods obtain the base normal force from the vertical equilibrium of the slice, so the vertical component of the support force enters that equilibrium and its frictional share is divided by $m_\alpha$ (or $n_\alpha$) like the weight. Janbu's methods, which balance shear along the base, take $T_S\sec\alpha$ on the driving side.
• The ordinary method resolves forces normal to the base, so $T_N$ adds directly to the normal force.
• On a circle, the methods with a moment equation take the support's moment about the centre, divided by the radius, in place of $T_S$; a support tangent to the slip surface takes the arm $R$, like a base shear.

On a planar slip surface the mass is a single free body and the two readings have closed forms,

with $W$ the weight of the wedge and $\beta,\ L$ the inclination and length of the plane. Every method that closes global force equilibrium must give these values; the program's source records that the ordinary method, Janbu's simplified method, the two Corps of Engineers methods and Lowe–Karafiath reproduce them.

3.10.3Orientation of the force#

The direction of $F$ at the crossing is one of six options:

• Tangent to slip surface: along the base of the slice crossed, in the sense that resists sliding.
• Parallel to support: along the support axis from head to tail, the direction in which a bolt in tension pulls the mass. This option depends only on the geometry: a support whose anchor lies downhill of the slip surface pushes the mass, and the program applies it as drawn.
• Bisector: the bisector of the two directions above.
• Horizontal: horizontal, against the direction of failure.
• Perpendicular to pile: perpendicular to the support axis, on the side that resists sliding.
• User-defined angle: at a given angle from the positive $x$ axis.

The resisting senses follow the failure direction set for the model (Section 3.1.5). Four types (the grouted tieback, the grouted tieback with friction, the soil nail and the helical anchor) also declare a shear capacity. When it is not zero, their shear capacity divided by the spacing is added as a second vector, perpendicular to the support axis on the side that opposes sliding, and the resultant replaces the axial force in Eq. (3.10.1).

Head and tail are not interchangeable: the stripping length is measured from the head, the pullout length from the tail, and a parallel force points from head to tail. A support drawn with its head in the stable ground and its tail inside the sliding mass is reported in the analysis notes, because drawn that way round all three are reversed and the factor of safety does not show it.

3.10.4Support types#

OGR Slip2D has nine support types. Table 3.10.1 lists them with their default orientation and application and the force each one delivers at the crossing.

Capacity along a support#

For the types with failure modes, the force available at the crossing is the smallest of the capacities that apply there, so it varies with the position of the crossing along the support. For a grouted tieback, with a bond strength $q_b$ per metre over a bonded length at the tail end:

where

$L_o$

bonded length beyond the crossing, towards the tail: the whole bond when the crossing is in the free length (m)

$L_i$

bonded length between the head and the crossing: zero when the crossing is in the free length (m)

$T,\ P$

tensile capacity of the tendon and capacity of the plate at the head (kN)

The three terms are the pullout of the anchored length, the failure of the tendon and the stripping of the moving mass off the plate together with the bond on the head side. A soil nail is the same with the bond over its whole length, $L_o = L - d$ and $L_i = d$. In the grouted tieback with friction the bond is the integral of an interface strength $\tau$ over the surface of a grout cylinder of diameter $D$,

and in a geosynthetic, which mobilises both faces of the sheet (Berg, Christopher & Samtani, 2009), with $A$ the percentage of the width of slope its strips cover (100 % for a continuous sheet),

with $T$ the tensile capacity per metre of sheet, $C$ the strength of the connection at the face, $L_i$ from the face to the crossing and $L_o$ from the crossing to the embedded end. A sheet whose embedded end is anchored cannot be pulled out, and its first term is dropped.

Interface strength#

Where the bond depends on the stress in the ground, OGR Slip2D samples the support once per analysis in 50 equal segments, from head to tail, and evaluates the interface strength at the middle of each; the integrals above are taken over that profile. The normal stress on the interface is the effective vertical stress at the point: the weight of the soil column above it, plus any ponded water and the vertical pressure of the distributed loads over it, minus the pore pressure. It is the stress of the pullout equation of Berg, Christopher & Samtani (2009). Two envelopes are available:

where

$a,\ \delta$

adhesion (kPa) and friction angle (°) of the interface, the Mohr–Coulomb bond of Jewell (1996)

$a_\infty,\ \phi_0$

limiting strength at large normal stress (kPa) and tangent friction angle at zero normal stress (°), after Esterhuizen, Filz & Duncan (2001)

$\sigma'_v$

effective vertical stress at the point of the support, not less than zero (kPa)

The same pair of numbers describes two different envelopes in the two forms: in the hyperbolic one the adhesion is the limit at large stress, not the strength at zero stress. A geosynthetic can also take $\tau$ as a fraction $C_i$ of the surrounding soil's own strength, evaluated with the material's strength model, or as $\tau = F^{*}\sigma'_v$ with a pullout friction factor $F^{*}$, constant or varying linearly with depth (Berg, Christopher & Samtani, 2009); the scale-effect factor of that guidance is entered multiplied into $F^{*}$.

Helical anchors#

A helical anchor carries steel plates along its shaft, and its capacity steps as the crossing passes each plate. The bearing capacity of one plate is that of a circular footing without the self-weight term (Terzaghi, 1943), as used for helices by Perko (2009):

with $A_p$ the plate area less the area of the shaft, $N_q$ and $N_c$ the factors of Prandtl (1921) and Reissner (1924) ($N_c \to 2 + \pi$ as $\phi \to 0$), and $c,\ \phi$ the equivalent Mohr–Coulomb pair of the soil at the plate, linearised at the effective vertical stress. Pullout is the smallest of three mechanisms, evaluated for the plates beyond the crossing: shallow failure, in which a cylinder of soil is sheared from the crossing to the farthest plate; cylindrical shear between the plates plus the bearing of the nearest one (Mitsch & Clemence, 1985; Mooney, Adamczak & Clemence, 1985); and the sum of the individual bearing capacities. Stripping is the same three mechanisms for the plates between the head and the crossing, plus the capacity of the head assembly. The shear on the cylinders is the soil's own strength at the effective vertical stress, with no separate earth-pressure or adhesion coefficient.

Piles#

A pile resists across its axis. In shear mode it applies a constant force, the shear strength of its section divided by the spacing, wherever the slip surface crosses it. In Ito & Matsui mode the force comes from the soil squeezing between the piles of a row, which loads each pile with a lateral force per unit thickness of layer (Ito & Matsui, 1975):

with

where

$D_1$

centre-to-centre spacing of the piles in the row (m)

$D_2 = D_1 - d_p$

opening between neighbouring piles, with $d_p$ the pile diameter (m)

$c,\ \phi$

equivalent Mohr–Coulomb parameters of the soil at that depth, from the linearisation the methods use

$\sigma'_v$

effective vertical stress, which takes the place of the $\gamma z$ of the original paper (kPa)

For $\phi$ below $10^{-8}$ rad the program evaluates the cohesive solution of the same paper, $p = c\left[D_1\left(3\ln\frac{D_1}{D_2} + \frac{D_1 - D_2}{D_2}\tan\frac{\pi}{8}\right) - 2(D_1 - D_2)\right] + \sigma'_v (D_1 - D_2)$, which is the limit of Eq. (3.10.13) and avoids its cancellation in floating point. The force on the sliding mass, per metre of slope, is the integral of $p$ from the top of the pile down to the crossing, divided by the spacing, as Cai & Ugai (2000) write it:

with $z_s$ the depth of the crossing measured along the pile from its top; the integral, not the samples, is clipped at zero. Its resultant acts at the crossing or at the centroid of the $p$ diagram above it. The theory is written for rigid vertical piles in a dry layer, and the analysis notes say when a model steps outside it: piles that touch ($D_2 \le 0$, where the row applies no force), piles more than 2° off vertical (integrated along the shaft), pore pressure along the pile, and negative pressures near the surface (integrated as published).

Retaining wall (equivalent fluid pressure)#

The wall type does not derive an earth pressure: it integrates a pressure profile $p(s)$ given over its length, from its upper end down to the crossing,

with $\gamma_e$ the equivalent fluid pressure (kN/m³) and $d$ measured from the crest of the wall, whichever end was drawn first (Terzaghi & Peck, 1967). Four profiles are offered: uniform; triangular, zero at the crest; trapezoidal, zero at both ends, rising to $\gamma_e L$ over $(1 - f)/2$ of the length at each end and constant over the fraction $f$ between (60 % by default); and a custom table of pressure against relative distance from the crest, linearly interpolated. A wall is continuous, so its force is already per metre and it has no spacing; the force is horizontal and active by default and can act at the crossing or at the centroid of the diagram above it. A wall drawn with both ends at the same elevation has no crest to measure from and is left out of the analysis, with a note. The couple left by moving the resultant to the centroid only enters the four methods with a moment equation: ordinary, Bishop, Spencer and GLE.

3.10.5Tension cracks#

A tension crack is drawn as a boundary, and the zone above it is soil that cannot carry tension, so no shear strength may be counted along a slip surface inside it. When the crest end of a slip surface lies in that zone, OGR Slip2D cuts the surface at its first crossing with the crack boundary, counting from the crest, and closes the sliding mass with a vertical wall from that point up to the ground surface; the soil between the wall and the slip surface keeps its whole weight (Duncan & Wright, 2005). Only the crest end is cut, because the crack forms at the head of the slide, where the interslice forces go into tension, while the toe is in compression. A surface whose crest lies in the zone but never reaches the boundary lies entirely in soil that cannot resist, and it is discarded.

Water in the crack pushes on the wall with the hydrostatic resultant

where

$h_w$

height of water in the crack, from the crack boundary at the wall up to the water level (m)

$\gamma_w$

unit weight of water, the pore-fluid unit weight of the project (9.81 kN/m³ by default)

The thrust (Terzaghi, 1943) is applied to the outermost slice, next to the wall, and pushes the mass away from the intact ground. The water level is set by one of seven options: dry; filled, which is the default; filled to a percentage of the crack height; filled up to a given elevation; filled to a level a given depth below the top of the crack; at the water table; or at a piezometric line. The level is always clipped to the height of the crack, and a dry crack still truncates the surface.

3.10.6Back analysis of the required support force#

In the back analysis the factor of safety is fixed at a target $F_t$ and the unknown is the horizontal force $T$ that would have to be added, at a chosen elevation, to reach it. With the factor fixed, the resisting and driving sums of the method can be evaluated directly, and Eq. (3.10.2) and Eq. (3.10.3) give the force in closed form, with no iteration:

where

$R,\ D$

resisting sum evaluated at $F_t$, and driving sum, of the method

$a$

arm factor: $(y_c - y_T)/R$ in Bishop's simplified method on a circle, the moment arm of a horizontal force at elevation $y_T$ about the centre over the radius; 1 in Janbu's methods, whose force balance does not depend on where a horizontal force acts

Since $T_{pas} = F_t\,T_{act}$, the passive force is the larger of the two for any target above one, and the two coincide at $F_t = 1$, an identity the program's tests check. Negative values, for a surface that already stands above the target, are reported as zero. Every valid surface of the search is back-analysed and the one that needs the largest force is reported, with both values. Janbu's corrected method solves its uncorrected factor and multiplies it by $f_0$, so its sums and force are taken at $F_t/f_0$.

3.10.7References#

• Berg, R.R., Christopher, B.R. & Samtani, N.C. (2009). Design and Construction of Mechanically Stabilized Earth Walls and Reinforced Soil Slopes, Vol. I. Report FHWA-NHI-10-024. Federal Highway Administration, Washington, DC.
• Cai, F. & Ugai, K. (2000). Numerical analysis of the stability of a slope reinforced with piles. Soils and Foundations, 40(1), 73–84.
• Duncan, J.M. & Wright, S.G. (2005). Soil Strength and Slope Stability. John Wiley & Sons, Hoboken, NJ.
• Esterhuizen, J.J.B., Filz, G.M. & Duncan, J.M. (2001). Constitutive behavior of geosynthetic interfaces. Journal of Geotechnical and Geoenvironmental Engineering, 127(10), 834–840.
• Ito, T. & Matsui, T. (1975). Methods to estimate lateral force acting on stabilizing piles. Soils and Foundations, 15(4), 43–59.
• Jewell, R.A. (1996). Soil Reinforcement with Geotextiles. CIRIA Special Publication 123. CIRIA and Thomas Telford, London.
• Mitsch, M.P. & Clemence, S.P. (1985). The uplift capacity of helix anchors in sand. In Clemence, S.P. (ed.), Uplift Behavior of Anchor Foundations in Soil. ASCE, New York, 26–47.
• Mooney, J.S., Adamczak, S. & Clemence, S.P. (1985). Uplift capacity of helical anchors in clay and silt. In Clemence, S.P. (ed.), Uplift Behavior of Anchor Foundations in Soil. ASCE, New York, 48–72.
• Perko, H.A. (2009). Helical Piles: A Practical Guide to Design and Installation. John Wiley & Sons, Hoboken, NJ.
• Prandtl, L. (1921). Über die Eindringungsfestigkeit (Härte) plastischer Baustoffe und die Festigkeit von Schneiden. Zeitschrift für Angewandte Mathematik und Mechanik, 1(1), 15–20.
• Reissner, H. (1924). Zum Erddruckproblem. Proceedings of the First International Congress for Applied Mechanics, Delft, 295–311.
• Terzaghi, K. (1943). Theoretical Soil Mechanics. John Wiley & Sons, New York.
• Terzaghi, K. & Peck, R.B. (1967). Soil Mechanics in Engineering Practice, 2nd ed. John Wiley & Sons, New York.
• Ukritchon, B. & Keawsawasvong, S. (2017). Error in Ito and Matsui's limit-equilibrium solution of lateral force on a row of stabilizing piles. Journal of Geotechnical and Geoenvironmental Engineering, 143(9), 02817004.