Geotechnical Retaining Structures Formula Sheet

This formula sheet collects first-pass relationships for retaining walls, excavation support, temporary works, drainage checks, and monitoring review. These equations are screening tools. Site-specific design requires a ground model, groundwater data, construction sequence, code basis, material parameters, support details, and engineering review.

Unit Weight and Vertical Stress

Unit weight:

Vertical total stress at depth z:

Layered vertical stress:

Factored design load:

where F_k is characteristic load and \gamma_F is load factor.

Hydrostatic Pressure

Hydrostatic pressure:

Pressure head:

Resultant water force on a vertical wall of height H:

Location of the resultant from the base:

Use vertical water depth, not slope length. Add water pressure when drainage is not assured or when blocked drainage is a credible condition.

Earth Pressure Coefficients

Rankine active coefficient for level backfill:

Rankine passive coefficient:

At-rest coefficient approximation for normally consolidated soil:

Choose the pressure model from wall movement, soil type, drainage, construction method, and design standard.

Lateral Earth Pressure

Active lateral pressure at depth z:

At-rest lateral pressure:

Resultant active earth force for dry level backfill:

Resultant location from base for triangular pressure:

For restrained basement walls, at-rest pressure may be more appropriate than active pressure.

Surcharge Loading

Uniform surcharge contribution to lateral pressure:

Resultant lateral force from uniform surcharge:

Resultant location from base for rectangular pressure:

Surcharge may come from traffic, construction equipment, stockpiles, adjacent footings, occupancy loads, and compaction.

Sliding Check

Driving horizontal action:

Base friction resistance:

where N is effective vertical normal force and \delta is base friction angle.

Sliding factor of safety:

Use passive resistance cautiously when future excavation, erosion, frost, or utility work can remove it.

Overturning Check

Overturning moment:

Resisting moment:

Overturning factor of safety:

Resultant location from the toe, when moments are summed about the toe:

Centroidal eccentricity for base width B:

Keep sign conventions clear. With the convention above, positive e means the resultant shifts toward the toe. The resultant should remain within the acceptable bearing zone defined by the design basis.

Bearing Pressure

Average base pressure:

For a strip footing with centroidal eccentricity e:

No-tension middle-third condition:

Bearing checks should include settlement, eccentricity, global stability, and construction-stage conditions.

Wall Bending and Shear

Maximum bending stress, elastic form:

Section modulus form:

Average shear stress screening:

Serviceability deflection is often as important as strength because nearby utilities, pavements, and structures can be movement-sensitive.

Movement and Serviceability Screening

Wall movement ratio:

Differential movement between two monitoring points:

Angular distortion:

Deflection margin:

Movement checks should state datum, survey method, trigger level, adjacent structure tolerance, utility tolerance, and construction stage.

Bracing and Strut Buckling

Euler buckling load:

Axial stress:

Utilization:

Bracing checks should include waler bending, connection capacity, eccentricity, installation tolerance, preload, and removal sequence.

Support Load Distribution

Tributary support load from apparent pressure:

Inclined anchor or brace axial force:

Lock-off force ratio:

Support load margin:

Support loads should be checked against excavation stage, preload, connection eccentricity, temperature, relaxation, wall movement, and load-cell calibration.

Jacking Force

Ideal hydraulic jacking force:

Corrected force using calibration factor C_j:

Average jacking stress over bearing area:

Jacking plans should define load increments, displacement limits, hold points, gauge calibration, load-cell verification, and contingency actions.

Construction-Stage Load Change

Load change between stages:

Movement change between stages:

Stage utilization:

Construction-stage checks should track excavation depth, groundwater level, support installation, surcharge position, wall movement, and observed load changes.

Drainage and Permeability

Darcy flow:

Hydraulic gradient:

where k is permeability, A is flow area, and \Delta h is head loss over flow length L.

Infiltration volume estimate:

where C_r is runoff or infiltration coefficient, P is precipitation depth, and A_c is contributing area.

Drainage checks should include filter compatibility, clogging, outlet protection, maintenance access, and blocked-drain scenarios.

Monitoring Trends

Displacement increment:

Average movement rate:

Acceleration of movement:

Load-cell change:

Trigger action levels should state the measurement, threshold, inspection, communication path, stop-work authority, and required engineering response.

Risk and Validation

Risk priority number:

where S is severity, O is occurrence, and D is detection ranking.

Simple risk expression:

Model-to-measurement relative error:

Validation should compare design assumptions with as-built geometry, groundwater readings, support loads, wall movement, drainage condition, and observed ground behaviour.

Minimum Review Checklist

Before accepting a retaining or excavation support calculation, confirm:

1. Ground model, groundwater level, and soil parameters are stated.
2. Wall movement condition matches the selected earth pressure model.
3. Surcharge and construction stages are included.
4. Water pressure and blocked-drain cases are considered.
5. Sliding, overturning, bearing, structural strength, and deflection are checked.
6. Bracing, anchors, jacking, and removal stages are defined where applicable.
7. Monitoring thresholds are tied to specific actions.

The formulas are only as reliable as the construction sequence and ground model behind them.