# Passive Earth Pressure (Rankine)

Calculate the Rankine passive earth pressure on a smooth vertical wall backfilled with cohesionless soil. K_p = tan²(45°+φ/2). Optional uniform surcharge q. Returns the base pressure p₀, the total force P, and the resultant height. Note: theoretical K_p is an upper bound; designers often apply a reduction factor for wall friction.

> Canonical page: https://elysiatools.com/en/tools/passive-earth-pressure

- **Category:** Math & Numbers

- **Keywords:** passive earth pressure, Rankine, K_p = tan²(45+φ/2), retaining wall, lateral earth pressure, cohesionless soil, geotechnical, civil engineering, passive resistance

## Overview

The Passive Earth Pressure (Rankine) calculator computes the lateral passive resistance exerted by cohesionless soil backfill against a smooth vertical retaining wall. By inputting the soil friction angle, wall height, soil unit weight, and an optional uniform surcharge, geotechnical and civil engineers can instantly determine the passive earth pressure coefficient, base pressure, total force, and the height of the resultant force.

## Inputs

- **Friction Angle φ (°)** (number): e.g. 30
- **Wall Height H (m)** (number): e.g. 2
- **Soil Unit Weight γ (kN/m³)** (number): e.g. 18
- **Surcharge q (kPa, optional)** (number): Optional uniform surcharge on the backfill surface.
- **Decimal Places** (number)

## When to use

- When designing or verifying the stability of retaining walls, sheet piles, or abutments against lateral passive soil resistance.
- When evaluating the maximum lateral resistance of cohesionless backfill under Rankine's theory.
- When assessing the impact of uniform surcharge loads on the passive earth pressure distribution.

## How it works

- Enter the soil friction angle to calculate the Rankine passive earth pressure coefficient using the formula K_p = tan²(45° + φ/2).
- Input the wall height, soil unit weight, and any optional uniform surcharge acting on the backfill surface.
- The calculator computes the base pressure and integrates the pressure distribution to find the total passive force and its resultant height above the base.

## Use cases

- Calculating the passive resistance of a sand backfill behind a concrete retaining wall.
- Estimating the lateral soil resistance for sheet pile wall design under uniform surcharge conditions.
- Performing quick geotechnical sensitivity analyses by varying the soil friction angle and unit weight.

## Frequently asked questions

### What is the formula used for the passive earth pressure coefficient?

The calculator uses Rankine's formula: K_p = tan²(45° + φ/2), where φ is the soil friction angle.

### Can I include a surcharge load in the calculation?

Yes, you can input an optional uniform surcharge (q) in kPa, which will be added to the lateral pressure distribution.

### Does this calculator account for wall friction?

No, Rankine's theory assumes a smooth vertical wall with no friction. The calculated K_p represents an upper bound, and designers typically apply a reduction factor.

### What types of soil is this calculator suitable for?

It is designed for cohesionless soils (such as sand or gravel) where cohesion is zero.

### What outputs does the calculator provide?

It outputs the passive earth pressure coefficient (K_p), the base pressure (p₀), the total lateral force (P), and the height of the resultant force.

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