AASHTO LRFD Retaining Wall Design Calculations & Methods

1. Scope and Analysis Model

AASHTO LRFD: Section 11 — Abutments, Piers, and Walls; Article 11.6 — Abutments and Conventional Retaining Walls.

The implementation evaluates a conventional retaining-wall section per unit length of wall. The current workflow includes:

• Active earth pressure using equivalent fluid pressure, Rankine, or Coulomb options.
• Passive pressure, surcharge, fence wind, wall and footing weights, and seismic lateral effects.
• Service and factored vertical force, horizontal force, stabilizing moment, and overturning moment.
• Resultant eccentricity, soil or rock bearing pressure, and sliding resistance.
• Foundation toe and heel flexure and shear, plus wall-stem flexure and shear.

Overall or global slope stability is outside this one-dimensional section model and must be checked separately when required.

2. Input Variables and Units

3. Effective Height and Active Earth Pressure

AASHTO LRFD: Article 3.11 — Earth Pressure: EH, ES, LS, and DR; Articles 3.11.5.1 through 3.11.5.5 for lateral earth pressure, active/passive coefficients, and equivalent-fluid pressure.

3.1 Effective height

The retained height used for the main earth-pressure calculation is:

3.2 Equivalent fluid pressure

AASHTO LRFD: Article 3.11.5.5 — Equivalent-Fluid Method of Estimating Rankine Lateral Earth Pressures; Eq. 3.11.5.5-1 and related provisions.

For the equivalent-fluid option, the entered pressure intensity is used directly:

The triangular resultant acts at $H_{eff}/3$ above the base.

3.3 Rankine active pressure

AASHTO LRFD: Articles 3.11.5.1 and 3.11.5.3; Rankine application is addressed in the commentary and equivalent-fluid provisions where the assumed failure geometry is appropriate.

For a sloping backfill, the implementation uses:

The code warns when $\phi \le 0$, $|i| \ge \phi$, the square-root argument is invalid, or a denominator approaches zero.

3.4 Coulomb active pressure

AASHTO LRFD: Article 3.11.5.3 — Active Lateral Earth Pressure Coefficient, including Eq. 3.11.5.3-1 and Eq. 3.11.5.3-2; Commentary C3.11.5.3 discusses Coulomb versus Rankine application.

The code stores wall batter as $\omega$, measured from vertical, and uses the backfill slope $i$. The implemented coefficient is:

Then $P_a=K_A\gamma$ and $H_a=P_aH_{eff}^2/2$. Invalid angle combinations return warnings and prevent a misleading nonzero pressure result.

4. Passive Pressure, Surcharge, and Wind

AASHTO LRFD: Article 3.11.5.4 — Passive Lateral Earth Pressure; Article 3.11.6 — Surcharge Loads (ES and LS); Section 3 for applicable wind load provisions.

4.1 Passive resistance

Passive height includes front soil, footing thickness, and key height:

With no ignored top height:

If a passive top portion is ignored, the code applies the entered $P_p$ to the remaining trapezoidal portion and caps the ignored height at the total passive height.

4.2 Surcharge

The surcharge resultant is rectangular and acts at $H_{eff}/2$.

4.3 Fence wind

The wind moment uses the entered fence height and wall geometry as implemented in the load-moment block.

5. Weights and Moments About the Toe

AASHTO LRFD: Article 3.4.1 — Load Factors and Load Combinations; Article 3.5 — Permanent Loads; Article 3.11 — Earth Pressure and Surcharge Loads.

Each vertical load is multiplied by its horizontal lever arm to form $M_1$ through $M_4$, $M_{sur}$, $M_{VDL}$, and $M_{VLL}$. The principal lateral moments are:

6. Seismic Earth Pressure

AASHTO LRFD: Article 3.11.4 — Effect of Earthquake; Appendix A11 — seismic earth-pressure procedures including Mononobe–Okabe for applicable gravity and semigravity wall conditions; Section 11 seismic wall provisions.

6.1 General Description

The Mononobe–Okabe (M–O) method is a pseudo-static extension of Coulomb earth-pressure theory. It represents the retained soil as a seismic wedge bounded by the wall, the backfill surface, and a planar failure surface. Horizontal and vertical inertial forces are applied to that wedge through seismic coefficients, and the active resultant is obtained when the wall movement and soil deformation are sufficient to mobilize active conditions.

The method is intended for a relatively uniform retained soil mass, a planar potential failure surface, and a wall system for which active pressure can develop. It does not replace project-specific assessment of drainage, global stability, dynamic soil response, or liquefaction.

6.2 Seismic Inertia Coefficients

The conventional pseudo-static coefficients are defined as:

where $a_h$ and $a_v$ are the horizontal and vertical design accelerations and $g$ is gravitational acceleration. With the common sign convention that upward vertical acceleration is positive, the seismic inertia angle is:

A positive $k_v$ therefore reduces the effective downward gravity term in this convention; a negative $k_v$ increases it. The selected sign convention must remain consistent with the adopted design criteria.

PaiaRetain implementation: PaiaRetain currently considers horizontal seismic acceleration in the Mononobe–Okabe analysis and assumes the vertical seismic acceleration coefficient, $k_v$, is zero. Accordingly, the seismic inertia angle is calculated as $\theta=\tan^{-1}(k_h)$. No independent vertical seismic coefficient is currently supplied to this calculation.

6.3 Seismic Active Earth Pressure

For the conventional M–O formulation, the total seismic active earth-pressure resultant and its incremental component may be written as:

$P_{AE}$ is the total seismic active earth pressure, while $\Delta P_{AE}$ is the additional seismic component above the static active pressure $P_A$. In a load combination, static earth pressure must not be counted twice: the combination must use either the total seismic resultant where appropriate or the static resultant plus only the incremental seismic component.

PaiaRetain does not separately calculate a user-facing total $P_{AE}$ using a vertical acceleration term. In its M–O branch, the implemented seismic increment is represented by $K_E=K_{AE}-K_A$ and:

The static active resultant remains in the ordinary active-pressure terms. The seismic load added by this branch is the incremental $H_{eq}$ above; it is not a second copy of the static active pressure. If the M–O inputs are invalid, or if $K_{AE}\leq K_A$, the implementation sets the seismic increment to zero and records a warning.

6.4 Seismic Active Earth Pressure Coefficient

Generalized equation used by PaiaRetain: with $\beta$ defined as wall batter measured from vertical and $i$ defined as backfill slope, the implementation evaluates:

Here $\phi$ is the soil internal friction angle, $\delta$ is the wall–soil interface friction angle, $\beta$ is wall batter measured from vertical, $i$ is backfill slope measured from horizontal, $\theta$ is the seismic inertia angle, and $K_{AE}$ is the seismic active earth-pressure coefficient. In the current input convention, positive $\beta$ and positive $i$ are used in the signed trigonometric terms shown in the equation.

For a vertical wall with horizontal backfill, set $\beta=0$ and $i=0$. With the implementation's $k_v=0$ assumption, the generalized equation reduces to the following illustrative special case:

Thus the simplified equation is mathematically consistent with the implemented generalized equation under those stated conditions. It should not be extended to battered walls or sloping backfill without the corresponding geometry terms and a confirmed sign convention. The implementation checks the square-root argument and denominator; invalid configurations produce warnings and a zero seismic increment.

6.5 Distribution and Point of Application

The static and incremental seismic resultants should be distinguished when forming moments. A general moment expression is:

$h_A$ is the moment arm of the static active resultant and $h_{\Delta AE}$ is the moment arm of the incremental seismic component, both measured from the selected reference point. These arms directly affect overturning moment, foundation eccentricity, and bearing pressure.

In PaiaRetain, the configured input eq_resultant_loc is a fraction of $H_{eff}$, so the seismic pressure moment arm is $h_{\Delta AE}=eq\_resultant\_loc\,H_{eff}$. The corresponding moment contribution is $H_{eq}h_{\Delta AE}$. The same lateral-moment calculation also includes horizontal inertial forces on the wall and soil weights. These contributions are carried into overturning and foundation checks; the configured location should be reviewed with the project inputs and report output.

6.6 Limitations and Engineering Considerations

• The conventional M–O method assumes sufficiently homogeneous soil properties and a planar failure surface.
• The acceleration model is pseudo-static and does not fully represent dynamic soil–wall interaction, frequency effects, or transient response.
• Active pressure requires enough wall displacement and soil deformation to mobilize the active condition; a restrained wall may require a different pressure model.
• Dynamic soil response, pore-pressure generation, and liquefaction require separate project evaluation.
• Results are sensitive to the combination of $\phi$, $\delta$, wall batter, backfill geometry, and seismic coefficients. Invalid angle combinations or square-root arguments are outside the method's applicable range.

This section is limited to the conventional M–O active earth-pressure method. Project-specific geotechnical and seismic criteria govern the final selection of parameters and pressure procedure.

6.7 Application to PaiaRetain

When the Mononobe–Okabe option is selected, PaiaRetain calculates static $K_A$, seismic $K_{AE}$, and the nonnegative increment $K_E=K_{AE}-K_A$. The resulting seismic lateral load is incorporated into the applicable retaining-wall load combinations and design checks.

The seismic load can affect foundation eccentricity and overturning effects, bearing pressure, sliding resistance, and the footing and wall-stem structural demands. The controlling load combination, resistance factors, seismic inputs, and design checks remain subject to engineering review and the adopted project criteria.

7. Service Totals and LRFD Load Combinations

AASHTO LRFD: Article 3.4.1 and Tables 3.4.1-1 and 3.4.1-2 — Load Factors and Load Combinations. Permanent-load factors for DC, EV, and EH and transient-load factors for LS/CT shall be selected for the controlling limit state.

Service totals are:

For each combination:

8. Eccentricity and Overturning Check

AASHTO LRFD: Article 11.6.3 — Bearing Resistance and Stability at the Strength Limit State, particularly Article 11.6.3.3 for eccentricity; Article 11.6.5.1 for Extreme Event eccentricity requirements.

For each factored combination, the resultant location measured from the toe is:

The current implementation compares $e$ with $B/4$ for Strength combinations and $B/3$ for Extreme combinations. The report labels the result OK or NG.

9. Bearing Resistance

AASHTO LRFD: Articles 11.6.3.1 through 11.6.3.3 and Section 10 foundation provisions, including Articles 10.6.3.1 and 10.6.3.2 for bearing resistance and effective footing dimensions. Applicable resistance factors are taken from Section 11/Section 10 as required by foundation type and limit state.

9.1 Soil bearing

For the soil-bearing option, the effective compression length and maximum pressure are:

The result is adequate when $q_{max}\le q_{allow}$.

9.2 Rock bearing

For rock bearing, the implementation uses a linear pressure distribution when $|e|