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# Abutment Earth Pressure \[STG]

The Abutment Earth Pressure workflow item applies a horizontal soil load against the back wall (and, when the abutment has a footing, the footing face) of a selected abutment. Three load types are available: an active (Rankine) earth pressure that grows linearly with depth, a uniform live-load surcharge, and a silo-effect pressure for a backfill confined in a narrow gap between two walls. The load is generated as surface loads on the abutment FE group and becomes active at the specified construction stage.

Supported abutments are the Cantilever, Integral and Semi-Integral abutments. Other abutment types are flagged by validation, because they do not provide the back-wall geometry this load needs.

## General

**Abutment:** Pick the abutment on which earth pressure is applied. The back-wall height and width, the footing, the alignment and the mesh size are read from the selected abutment. The pick must be a Cantilever, Integral or Semi-Integral abutment.

**Stage:** Construction Stage at which the generated surface load becomes active. A stage must be selected.

**Load Type \[Earth Pressure/Live Load Surcharge/Silo Effect (Narrow Backfill)]:** Choose the load to generate:

* **Earth Pressure (default):** A triangular pressure that is zero at the top of the back wall and grows linearly with depth, continuing down the footing face when a footing exists.
* **Live Load Surcharge:** A uniform horizontal pressure over the back wall and footing, representing the effect of traffic on the backfill.
* **Silo Effect (Narrow Backfill):** For a backfill confined between the back wall and a second wall close behind it (for example a 1.5 ft gap). The fill arches onto both walls, so the pressure grows quickly near the top and then levels off at a limiting value instead of growing linearly. Use this when the retained fill is narrow enough that a Rankine triangle would overestimate the pressure at depth.

The formulas for each type are given under [Load Formulation](#load-formulation) below.

**Active Lateral Earth Pressure Coefficient (Ka):** Coefficient that converts vertical soil stress into horizontal pressure for the Earth Pressure and Live Load Surcharge types. The default value is 0.523599. For a level backfill with no wall friction it can be estimated as `Ka = (1 - sin φ) / (1 + sin φ)`; refer to AASHTO LRFD 3.11.5.3 for guidance. Hidden when Load Type is **Silo Effect**, which uses its own coefficient (Ksa) below.

**Unit Weight Source \[Input/From Backfill Soil]:** Where the backfill unit weight comes from. It applies to all three load types.

* **Input (default):** The value typed in **Backfill Soil Unit Weight** is used.
* **From Backfill Soil:** The unit weight is read from a picked Backfill Soil object. For a uniform profile this is the profile's Effective Unit Weight. For a layered profile it is the thickness-weighted average over the profile height (each layer taken at the mean of its two bounding rows), which is the single unit weight that reproduces the layered fill's vertical stress at the base of the profile. The value is used as entered in the soil table; no buoyancy correction is applied here. Rows in the soil table that are not in Soil Properties mode still contribute their stored unit weight (default about 119 pcf), so check the column before relying on this option.

**Backfill Soil:** Pick the Backfill Soil object to read the unit weight from. Visible only when Unit Weight Source is **From Backfill Soil**, and required in that case. Validation reports an issue if the picked soil has no positive unit weight in its profile.

**Backfill Soil Unit Weight:** Unit weight of the retained backfill (typical US units pcf, SI kN/m³). Visible only when Unit Weight Source is **Input**. No load is generated unless the unit weight in use is greater than zero.

**Heel Side Width:** Horizontal heel width used by the Live Load Surcharge type to convert Ka and the unit weight into a uniform back-wall pressure. Visible only when Load Type is **Live Load Surcharge**.

**Load Area Edge Tolerance:** Small margin by which the loaded area is extended beyond each transverse end of the back wall (default 1 in). It keeps the mesh nodes on the wall's end edges strictly inside the load area so they receive their share of the pressure; without it the outermost node columns could receive no load. Because the full load area is distributed, the total lateral force grows by the ratio `(wall width + 2 × tolerance) / wall width` — under 0.3% for 1 in on a 64 ft wall. Set it to 0 to load the exact wall footprint. It cannot be negative.

**Fill Width Between Walls:** Clear horizontal distance `x` between the back wall and the wall behind it, which confines the backfill (default 18 in = 1.5 ft). Visible only when Load Type is **Silo Effect**; must be greater than zero. A narrower gap levels the pressure off sooner and at a lower value.

**Wall Friction Angle:** Friction angle `δ` between the fill and the walls (default 25°). Visible only when Load Type is **Silo Effect**; must be between 0° and 90°, exclusive. Wall friction is what transfers the fill's weight into the walls, so a larger angle gives a lower limiting pressure.

**Lateral Earth Pressure Coefficient (Ksa):** Lateral earth pressure coefficient `K` used by the silo formula (default 0.2913, for φ = 36° and δ = 25°). Visible only when Load Type is **Silo Effect**; must be greater than zero. Take it from the Frydman and Keissar `Ksa` table for the fill's friction angle and the wall friction angle. It is not the at-rest coefficient `K0` (for φ = 36°, `K0` would be 0.412).

**Silo Pressure Band Height:** Target height of one pressure band over the back wall (default 120 in = 10 ft). Visible only when Load Type is **Silo Effect**; must be greater than zero. The silo curve is applied as a stack of straight-line bands (see [Load Formulation](#load-formulation)); the back wall is split into equal bands no taller than this height. A band is never made thinner than the abutment's Mesh Size, because a band thinner than the mesh may contain no nodes and would carry no load. When the two conflict the mesh size wins, so the actual band height can exceed this input by up to one mesh size. Validation warns when this input is smaller than the abutment's Mesh Size.

### Load Formulation

Notation: `γ` = backfill unit weight (from Unit Weight Source), `z` = depth below the top of the back wall, `H_w` = back-wall height, `t_f` = footing thickness (0 when the abutment has no footing), `H = H_w + t_f` = height from the top of the back wall to the bottom of the footing.

The back wall runs from the abutment's deck top elevation down to the bottom of the seat; the footing zone continues from there down to the bottom of the footing. The pressure acts horizontally on the backfill side of the wall, following the abutment's orientation.

#### Earth Pressure

$$
p(z) = K\_a , \gamma , z
$$

* Back wall: a linear ramp from 0 at the top to `Ka·γ·H_w` at the bottom of the back wall.
* Footing (when present): a linear ramp from `Ka·γ·H_w` at the top of the footing to `Ka·γ·H` at its bottom.

#### Live Load Surcharge

$$
p = K\_a , \gamma , h\_{eq}
$$

where `h_eq` is the **Heel Side Width**. The same uniform pressure is applied over the back wall and the footing face. Refer to AASHTO LRFD 3.11.6.4 for equivalent surcharge heights.

#### Silo Effect (Narrow Backfill)

From FHWA NHI-07-071, *Earth Retaining Structures*, Fig. 3-26, after Spangler and Handy (1984):

$$
p(z) = \frac{\gamma , x}{2 \tan\delta} \left( 1 - e^{-2K \frac{z}{x} \tan\delta} \right)
$$

where `x` is the **Fill Width Between Walls**, `δ` the **Wall Friction Angle** and `K` the **Lateral Earth Pressure Coefficient (Ksa)**. Written as `p(z) = p₀ (1 − e^(−λz))`:

* `p₀ = γx / (2 tan δ)` is the limiting pressure the curve levels off at, reached at depth;
* `λ = 2K tan δ / x` is the rate at which it gets there. Near the top `p(z) ≈ K γ z`, the familiar linear pressure.

The resultant over the full height and its lever arm above the base are the closed-form integrals of `p(z)`:

$$
P = p\_0 \left( H - \frac{1 - e^{-\lambda H}}{\lambda} \right)
\qquad
\bar{y} = \frac{p\_0}{P} \left( \frac{H^2}{2} - \frac{H}{\lambda} + \frac{1 - e^{-\lambda H}}{\lambda^2} \right)
$$

**How the curve is applied.** Surface loads in the FE model can be uniform or linear only, so the curve is applied piecewise:

* **Back wall:** split into `n` equal bands. Each band is a linear load from the curve's pressure at its top to the curve's pressure at its bottom. The band count is `n = max(1, min(ceil(H_w / band height), floor(H_w / mesh size)))` — the first term keeps each band no taller than the **Silo Pressure Band Height**, the second keeps it no thinner than the abutment's Mesh Size.
* **Footing (when present):** a single linear load from `p(H_w)` at the top of the footing to `p(H)` at its bottom.

**Accuracy.** Because the curve is concave, each band's straight line runs slightly under it, so the load applied to the back wall is a little below the exact resultant. This chord shortfall is about 1.3% with 6 bands and drops to about 0.1% with 24. A second, mesh-dependent effect acts the other way: nodes near band edges cover slightly less than the full band, which leaves up to about 1.3% of the load unapplied at fine banding. The net effect is typically a 1–1.5% underestimate on the wall; check it against the FE nodal load sums for a critical design. The footing zone errs slightly on the conservative side.


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