MSE walls

A mechanically stabilized earth (MSE) wall is a block of granular fill with layers of reinforcement, steel strips, steel grids, a geotextile or a geogrid, held at the face by panels, blocks or the reinforcement itself. Rischio checks each layer inside the block, against rupture and pullout, and the block outside, against sliding, the eccentricity of its resultant and bearing, by AASHTO LRFD 10th’s Article 11.10 and FHWA GEC 11. Units, symbols and assumptions are set out in Method and assumptions.

The model

• The wall is $H$ tall, from its base to its top, with its base $D$ below the ground in front.
• The reinforced fill has its own $\gamma_r$ and $\phi'_r$, granular, never above 40°. The retained backfill behind the block is the wall’s backfill strata, drained, level or sloping. The foundation is the wall’s foundation strata. Water inside the block is refused; below its base, the bearing takes it.
• The layers are $S_V$ apart, the top one at $S_V/2$ below the top unless another depth is given, and all of them $L$ long. Each carries the share of the wall between the midpoints to its neighbours, its tributary height.
• The facing is rigid (panels, blocks) or flexible (welded wire, a wrapped face). The reinforcement’s connection to it is optional.
• A surcharge $q$ is uniform over the whole backfill, the block included, dead or live; traffic is a live surcharge.

Each layer’s load

At a layer $z$ below the top, the load the reinforcement carries per metre of wall is

$$T_{max} = S_v\,K\,\sigma_v$$

and AASHTO gives three methods for $K$ and $\sigma_v$ (11.10.6.2.1). Left By the data, the wall takes AASHTO’s method for its reinforcement, in free mode as under AASHTO, and the field says which one and why:

Reinforcement	By the data
Steel strips or steel grids	The Coherent Gravity Method (11.10.6.2.1d)
A geotextile or a geogrid with its secant stiffness $J$	The Stiffness Method (11.10.6.2.1e)
A geotextile or a geogrid without $J$	The Simplified Method (Appendix B11), with a note in the trace

The Simplified Method can also be chosen for any reinforcement, for a designer who designs every wall by one method. A method chosen for a reinforcement it isn’t for is refused: the Coherent Gravity Method is for steel, the Stiffness Method for geosynthetics.

The Coherent Gravity Method. $K$ runs from $k_0 = 1 - \sin\phi'_r$ where the failure surface meets the ground (the top of a level backfill; above it under a slope) to $k_a$ 20 ft (6.1 m) below, and $k_a$ deeper. $\sigma_v$ is Meyerhof’s under the mass above the layer as a rigid body, pushed on its back by the retained backfill’s thrust $F_T$:

$$\sigma_v = \frac{V_1 + V_2 + F_T\sin\beta + qL}{L - 2e}$$

with $V_1$ the reinforced fill above the layer, $V_2$ the soil over it and $e$, never less than 0, the mass’s eccentricity.

The Stiffness Method. For a geosynthetic, after Allen and Bathurst, with $K = k_a\,\Phi$ of a vertical face:

$$T_{max} = S_v\,k_a\,\Phi\left[\gamma_r H D_{tmax} + \frac{H_{ref}}{H}\left(\gamma_f S + q\right)\right], \qquad \Phi = \Phi_g \Phi_{fs}, \qquad \Phi_g = 0.16\left(\frac{S_{global}}{p_a}\right)^{0.26}$$

$S_{global} = nJR_c/H$ is the wall’s global stiffness, with $n$ layers; $H_{ref} = 20$ ft (6.1 m); $\Phi_{fs} = 1.0$, as AASHTO allows for any face. $D_{tmax}$ grows from 0.12 at the top to 1 at $z_b = 0.40\,H^{1.2}$ (in metres) and stays at 1 below. $S$ is the average height of the soil over the block. A live surcharge adds to $\sigma$ whole, as GEC 11 writes it.

The Simplified Method. $K = (k_r/k_a)\,k_a$ with $k_r/k_a$ by the reinforcement (Fig. B11.2-1): 1.7 at the top to 1.2 at 20 ft for strips, 2.5 to 1.2 for grids (their pullout takes the strips’), 1.0 for geosynthetics; $\sigma_v = \gamma_r z + \gamma_f S + q$.

$k_a = \tan^2(45 - \phi'_r/2)$, of the reinforced fill. In every method the live surcharge adds to the tension but not to the pullout (11.10.6.3.2), and a sloping backfill over the block is a soil surcharge.

The failure surface

Each layer is $L_a$ long inside the failure surface and $L_e = L - L_a$ beyond it (11.10.6.3.1). For steel the surface is bilinear: $0.3H_1$ behind the face over the upper half of $H_1$, then a line to the toe, $0.6(H - z)$ below, with $H_1$ the wall’s height up to where the surface meets the ground ($H$ on a level backfill). For a geosynthetic it is Rankine’s plane from the toe, $L_a = (H - z)\tan(45 - \phi'_r/2)$.

Rupture

A steel strip holds $A_c F_y/S_H$ per metre of wall, $A_c = bE_c$ with $E_c$ its thickness after corrosion; a steel grid, its longitudinal bars, $A_c = n\pi D^{*2}/4$ with the corrosion taken off their radius. A geosynthetic holds its long-term strength over its coverage:

$$T_{al} = \frac{T_{ult}}{RF_{ID}\,RF_{CR}\,RF_D}, \qquad FS_{(B)} = \frac{T_{al}R_c}{T_{max}}$$

and steel $FS_{(B)} = A_c F_y/(S_H T_{max})$. Galvanized steel corrodes by AASHTO’s uniform loss model (11.10.6.4.2a): the zinc, 86 µm, goes at about 15 µm a year for two years and 4 µm after, then the steel at about 12 µm a year on each side, about 1.4 mm off a strip over 75 years. A rate given on the strip replaces it.

With the Stiffness Method, each layer’s strain $\varepsilon = T_{max}/(JR_c)$ is also checked against the soil failure limit, 2.5 % (11.10.4.3).

Pullout

Beyond the failure surface, both faces of the layer grip the fill (Eq. 11.10.6.3.2-1):

$$P_r = 2F^*\alpha\,\sigma_v L_e R_c, \qquad FS_{(P)} = \frac{P_r}{T_{max}}$$

with $\sigma_v = \gamma_r z$, plus the soil over the resistant zone under a slope, the surcharges out of it, and $T_{max}$ without the live surcharge. $\alpha$ is 1.0 for steel, 0.8 for geogrids and 0.6 for geotextiles. $F^*$, down to 20 ft (Fig. 11.10.6.3.2-2):

• ribbed strips: $1.2 + \log C_u$, at most 2.0, at the top, to $\tan\phi'_r$ ($C_u = 4$ unless given); a value given replaces it at every depth;
• steel grids: $20\,t/S_t$ at the top to $10\,t/S_t$, with $t$ the transverse bars’ diameter and $S_t$ their spacing;
• geosynthetics: $C_i\tan\phi'_r$, $C_i = 2/3$ unless tested.

The check on the wall is its weakest layer, for rupture and for pullout; each layer shows its own on the drawing.

The connection

Where the wall gives its connection to the facing, steel or geosynthetic, each layer’s load there, $T_o = T_{max}$ (11.10.6.2.2), is checked against its strength $T_{ac}R_c$ (11.10.6.4.4): a steel connector’s after the face’s corrosion, a geosynthetic’s $T_{ac} = T_{ult}CR_{cr}/RF_D$ from its connection tests.

The design curves

Down the wall, every 5 cm, Rischio also gives what a designer sizes the wall with, by the wall’s method and with the model’s limits, edited ones included:

• the length each depth needs for the pullout’s factor of safety, $L_a$ plus the $L_e$ at which $P_r = FS_{(P)}\,T_{max}$;
• for a geosynthetic, the largest spacing each depth allows for the rupture’s, $T_{al}R_c/(K\sigma_v FS_{(B)})$, and a geotextile’s lap length at the face, at least 1 m;
• for strips, the thickness the base needs, plus the corrosion.

Outside: the block

The block is a rigid body $L$ wide, its weight $V_1 = \gamma_r HL$. The retained backfill pushes on its back with Rankine’s thrust, its cohesion left out: $F_1$ from its strata and $F_2 = K_aqH$ from the surcharge. Under a slope the back is $h = H + L\tan\beta$ tall, the thrust at $\delta = \beta$ with its vertical part holding the block, and the soil over the block adds its weight $V_3$; behind a broken back, Rankine’s trial wedge gives it.

• Sliding: $FS = R/(F_1 + F_2)$, $R = V\tan\phi$ on the weaker of the foundation and the fill, at most $\tfrac{2}{3}\tan\phi'_r$ under a geotextile; on a clay, the lesser of that and $c_uL$. The surcharge on the block doesn’t resist.
• Eccentricity: $e = L/2 - (M_R - M_O)/V$.
• Bearing: the uniform pressure $\sigma_v = (V + V_2)/(L - 2e)$, the surcharge over the block included, against $q_n$ by AASHTO’s Eq. 10.6.3.1.2a-1 for a strip $L - 2e$ wide (11.10.5.4), the embedment left out and the load taken vertical, as GEC 11 applies it. Local or punching shear and the strata below enter it as they do a footing’s.

In free mode the checks are factors of safety, unfactored, with FHWA-NHI-00-043’s limits beside them as references (§2.7.c): rupture at $0.55F_y$ for strips ($0.48F_y$ for grids on a rigid face) or $T_{al}/1.5$, pullout 1.5, sliding 1.5, $e \le L/6$ and bearing 2.5. Under AASHTO the thrust is Coulomb’s with $\delta = 0.67\phi'$, every check takes the Strength I combination that governs it, and each has the code’s resistance factor (AASHTO LRFD › MSE walls).

Under an earthquake

With an earthquake among its loads, the wall is checked under it too, by AASHTO’s 11.10.7 in both modes, with $k_h$ set as for any wall. Outside, the retained backfill’s Mononobe and Okabe thrust on the back of the block and the inertia of a mass $0.5H$ wide, in AASHTO’s two combinations, for sliding, eccentricity and bearing. Inside, the active zone’s inertia $P_i = k_hW_a$ shared among the layers, by their resistant lengths for steel and evenly for geosynthetics: rupture on $T_{max} + T_{md}$, and pullout with $0.8F^*$ and without the live surcharge. A connection given is checked under it too.

Barriers and abutments

A traffic barrier at the top of the wall adds its impact to each layer’s rupture and pullout at Extreme Event II (11.10.10.2). A bridge seat on the block, an MSE abutment, is checked as a footing on the reinforced fill and adds its loads to every layer and to the block (11.10.11, 11.10.10.1).

Lateral displacement

Beside the checks, Rischio estimates the wall’s lateral displacement during construction from AASHTO’s Fig. C11.10.4.2-1, by $L/H$, increased for the surcharge: a first-order estimate for simple walls on firm foundations, with no limit to check it against.

Warnings

• The reinforcement shorter than FHWA’s minimum, the greater of $0.7H$ and 2.4 m.
• A layer reaching less than 1 m past the failure surface.
• An embedment shallower than $H/20$ and 0.6 m.
• An $L/H$ outside the displacement chart’s curve.

References

• AASHTO (2024). LRFD Bridge Design Specifications, 10th ed. Article 11.10 and Appendix B11.
• Elias, V., Christopher, B. R., & Berg, R. R. (2001). Mechanically Stabilized Earth Walls and Reinforced Soil Slopes: Design and Construction Guidelines. FHWA-NHI-00-043. Federal Highway Administration.
• Berg, R. R., Christopher, B. R., & Samtani, N. C. (2009). Design and Construction of Mechanically Stabilized Earth Walls and Reinforced Soil Slopes, Vols. I and II. FHWA GEC 11, FHWA-NHI-10-024 and -025.
• Das, B. M., & Sivakugan, N. (2019). Principles of Foundation Engineering, SI edition, 9th ed., sections 17.11 to 17.18. Cengage Learning. The design curves’ lap length and strip thickness.