Rankine Earth Pressure Calc
Active Earth Pressure
What Is a Geotechnical Retaining Wall Design Calculator?
A Geotechnical Retaining Wall Design Calculator is a digital tool that helps you quickly estimate:
- Active earth pressure coefficient (Ka)
- Total active earth pressure (Pa) acting on a retaining wall
- Based on Rankine’s earth pressure theory for cohesionless soil
Instead of manually going through formulas, conversions, and checks, the calculator allows you to:
- Enter a few key parameters such as:
- Wall height (H)
- Soil unit weight (γ)
- Soil friction angle (ϕ)
- Backfill slope angle (β)
- Get:
- Ka – coefficient of active earth pressure
- Pa – total active earth force per unit length of wall
This is especially useful at preliminary design stage, in classroom learning, and quick checks on site.
Why Earth Pressure Matters in Retaining Wall Design
A retaining wall is not only resisting the weight of soil. It must resist:
- Lateral earth pressure – the horizontal push of soil behind the wall
- Additional effects from:
- Surcharge loads
- Sloping backfills
- Groundwater and seepage
- Possible seismic loads (in advanced design)
Under active condition, the soil behind the wall expands slightly and mobilizes its shear strength, reducing lateral pressure to the active earth pressure state.
This active pressure is what Rankine’s theory is all about – and that’s exactly what your Rankine Earth Pressure Calculator computes.
If we underestimate this pressure → the wall can slide, overturn, or develop distress.
If we greatly overestimate it → we end up with an overly heavy and uneconomical wall.
So having a clear, quick, and consistent way to compute Ka and Pa is essential.
Rankine Earth Pressure Theory in Simple Words
Rankine’s earth pressure theory gives us a way to estimate:
- Active earth pressure (Pa) – when wall moves slightly away from backfill
- Passive earth pressure (Pp) – when wall moves towards the soil
Your calculator specifically focuses on active pressure for cohesionless backfill (no cohesion, only frictional soil).
Key Rankine assumptions include:
- Soil is homogeneous and isotropic
- Soil is dry or water pressure is treated separately
- Soil behaves as a Coulomb friction material with friction angle ϕ
- Wall back is vertical
- Backfill may be level or sloping at angle β
- No wall friction is considered (δ = 0 in Rankine model)
Under these conditions, the active earth pressure coefficient Ka is determined from ϕ and β. Once we know Ka, we can compute the total active force Pa on the wall.
Overview of Your Rankine Earth Pressure Calculator
Your Geotechnical Retaining Wall Design Calculator is a compact but powerful web-based tool that:
- Accepts wall geometry and soil properties
- Uses Rankine’s analytical expression for Ka
- Outputs Ka and Pa in clean, readable form
- Provides clear input error feedback to prevent incorrect usage
User Inputs
The calculator asks for:
- Wall Height (H)
- Height of the retained soil behind the wall
- Controls the magnitude of the earth pressure, since Pa ∝ H²
- Soil Unit Weight (γ)
- Bulk unit weight of the backfill soil
- Higher γ → heavier soil → greater lateral force
- Friction Angle (ϕ in degrees)
- Internal friction angle of the backfill soil
- Some typical values:
- Loose sand: ~28°–30°
- Dense sand: ~34°–40°
- Higher ϕ generally results in lower active pressure (smaller Ka)
- Backfill Slope (β in degrees)
- Slope of the backfill surface measured from the horizontal
- β = 0° → level backfill
- β > 0° → sloping backfill (rising behind the wall)
- Steeper backfill slopes increase the complexity and can increase active pressure.
The calculator also limits ϕ and checks that:
- β is not greater than or equal to ϕ
(for stability and validity of Rankine’s equation)
If β ≥ ϕ, the tool gently warns you that the combination is physically and analytically unsuitable.
What the Calculator Outputs
Once valid values are entered, the calculator shows:
Coefficient of Active Earth Pressure (Ka)
This is a dimensionless factor that tells you how much of the vertical stress becomes lateral stress under active conditions.
- Larger Ka → higher lateral pressure
- Smaller Ka → lower lateral pressure
For level backfill (β = 0), Rankine’s formula simplifies to:
Ka = (1 − sinϕ) / (1 + sinϕ)
For sloping backfill, your calculator uses the more general form implemented via trigonometric relationships, including cosβ and cosϕ.
Total Active Force (Pa)
The total active force per unit length of wall is computed as:
Pa = ½ × γ × H² × Ka
Key points about Pa:
- It is proportional to H² – doubling the wall height makes Pa four times larger.
- It is per unit length (e.g., per metre or per foot), depending on the units chosen for H and γ.
- The calculator clearly writes that units of Pa follow from:
Force units = (Unit weight × Height²) / 2
The tool also displays a note:
- Rankine theory is used
- The total active force acts parallel to the backfill slope (which is an important design detail)
These outputs are ideal for use in:
- Hand calculations of sliding and overturning
- Preliminary sizing of base width and stem thickness
- Conceptual and educational understanding of earth pressure behaviour
How the Internal Logic Works (Plain English Version)
Behind the friendly interface, the calculator carries out a series of steps:
- Reads inputs: H, γ, ϕ, β
- Checks that these are valid:
- All must be positive numbers
- ϕ must be realistic
- β must be less than ϕ
- If validation fails:
- The results panel turns into an error message display
- Ka and Pa are replaced with placeholders (
---) - A clear message explains what went wrong
- If validation passes:
- Converts angles from degrees to radians (because JavaScript uses radians for trigonometry)
- Applies the Rankine formula for Ka with sloping backfill:
- Uses cosβ, cosϕ, and a square root term involving these
- Calculates:
- Ka as a function of β and ϕ
- Pa = ½ × γ × H² × Ka
- Displays results:
- Ka: to three decimal places
- Pa: with comma separators and two decimal places (for readability)
- A note: “Rankine Theory used. Force acts parallel to backfill slope.”
- A Reset button:
- Clears inputs
- Resets backfill slope β to 0 (level backfill)
- Hides the result area until the next calculation
This behaviour makes the calculator feel both engineering-accurate and user-friendly.
How to Use This Calculator in Real Design Work
While this tool focuses on earth pressure calculation, that’s only one part of a complete retaining wall design.
After you obtain Ka and Pa, a typical design workflow includes:
- Determine resultant location
- For a simple triangular distribution, Pa acts at H/3 above the base.
- Check Sliding Stability
- Compare resisting forces (like base friction, passive resistance if used) to Pa.
- Ensure the factor of safety (FS) against sliding is acceptable.
- Check Overturning Stability
- Compare stabilizing moments (due to wall and soil weights) with overturning moments (due to Pa).
- Confirm acceptable FS against overturning.
- Check Bearing Pressure
- Evaluate contact pressure at the base.
- Ensure the maximum bearing pressure is below the allowable soil bearing capacity.
- Check Structural Design of Wall Components
- Design the stem, toe, and heel for bending and shear.
- Check reinforcement if it’s a reinforced concrete wall.
Your calculator is especially helpful at the early stage:
it gives a reliable estimate of how much soil pressure the wall must resist, so you can size the base and shape logically before doing detailed structural design.
Strengths and Limitations of the Rankine-Based Calculator
Strengths
- Quick and intuitive for students, site engineers, and designers
- Follows classical Rankine theory, widely taught and accepted
- Handles both level and sloping backfills
- Provides transparent and focused output: Ka and Pa
- Clear error messages prevent misuse
- Perfect for:
- Teaching sessions
- Concept checks
- Preliminary designs
- On-site comparison of alternative backfill arrangements
Limitations (What It Does Not Include)
To keep things clean and focused, the calculator intentionally does not include:
- Cohesive soils (c–ϕ soils)
- Wall friction (δ) or non-vertical wall faces
- Surcharge loads due to vehicles, buildings, or stockpiles
- Groundwater effects or seepage pressure
- Seismic earth pressure (Mononobe–Okabe, etc.)
These effects must be handled outside this Rankine tool, using more advanced analysis or code-based methods when needed.
Therefore, the calculator is best understood as a Rankine-based active earth pressure module within a broader retaining wall design process.
