Effective stress in soil: practice problems with step-by-step solutions

1What effective stress is

Soil is a skeleton of solid grains with small gaps between them, called pores. Below the water table the pores are full of water.

Every point in the ground carries the weight of everything above it. This load per square metre is the total stress σ, in kPa.

Part of that load is carried by the water in the pores, the pore water pressure u. The rest is carried by the grains pressing on each other.

The part carried by the grains is the effective stress:

This is Terzaghi’s principle. Strength and settlement of soil depend on σ′, because only grain-to-grain contact gives friction and compression.

Think of a wet sponge in a bucket of water. Water pressure all around it does not squeeze it. Only a load on the sponge itself does.

2Total stress from unit weights

The unit weight γ is the weight of one cubic metre of soil, in kN/m³. A soil band of thickness h adds γ·h to the total stress below it.

Soil above the water table is drained, so you use the dry unit weight γ_d. Below the water table the pores are full of water, so you use the heavier saturated unit weight γ_sat.

Typical values: dry sand 15 to 17 kN/m³, saturated sand 18 to 21 kN/m³, water 9.81 kN/m³.

Example: 2 m of dry sand with γ_d = 16 kN/m³ on top of 3 m of saturated sand with γ_sat = 20 kN/m³. At 5 m depth σ = 16·2 + 20·3 = 32 + 60 = 92 kPa.

3Pore water pressure

Water in the pores is connected, like water in a tank. So the pore water pressure grows with depth below the water table, and is zero at the water table itself.

Here z is the depth of the point, z_w the depth of the water table and γ_w = 9.81 kN/m³. Above the water table u = 0.

Only the depth below the water table counts. The soil type and the layers above do not change u.

4The method step by step

1. Mark the point X, its depth z and the depth of the water table z_w on the profile.
2. Split the soil above X into bands: each band has one unit weight. The water table and each layer boundary start a new band.
3. Total stress: add γ·h for every band. Use γ_d above the water table and γ_sat below it.
4. Pore water pressure: u = γ_w·(z − z_w), or 0 if X is above the water table.
5. Effective stress: σ′ = σ − u.

Write the units at every step. Unit weights in kN/m³ times depths in m give kPa.

5A worked example

Follow every step. Try to predict the next one before you open it.

6Common mistakes

• Measuring u from the ground surface. Use only the depth below the water table: z − z_w.
• Using γ_sat above the water table, or γ_d below it.
• Forgetting to subtract u and giving σ when σ′ is asked.
• In two layers, using one unit weight for both. Each layer has its own γ.
• Mixing up which value is asked: σ, u or σ′. Read the last line of the question again.

7How exam questions look

• Easy: one saturated sand layer with the water table at the surface. Then σ = γ_sat·z and u = γ_w·z.
• Medium: the water table is inside the layer. Use γ_d above it and γ_sat below it, as in the example.
• Hard: two layers, sand over clay, with X at the middle of the clay. Add a third band with the clay’s own γ_sat.

The question may ask for σ, for u or for σ′. The method is the same; you stop at the step that gives the asked value.