Worked problem

Footing Bearing Pressure with an Eccentric Load

Find maximum and minimum soil pressure under a 2.5 m × 2.0 m footing with an axial load and moment, and check the middle-third rule.

Civil & Structural · Last reviewed

Given

A rectangular footing is 2.5 m long in the direction of bending and 2.0 m wide. The service axial load at the base, including the footing weight, is P = 400 kN. A moment of M = 120 kN·m acts about the width axis. For this exercise the allowable soil bearing pressure is 150 kPa.

Find

The maximum and minimum contact pressures, whether the whole base stays in contact with the soil, and the result for a larger moment of 200 kN·m.

Solution

  1. Eccentricity e = M ÷ P = 120 ÷ 400 = 0.30 m.
  2. Kern limit L ÷ 6 = 2.5 ÷ 6 = 0.417 m. Since e < L ÷ 6, the resultant lies inside the middle third and the whole base stays in compression.
  3. Average pressure = P ÷ (B × L) = 400 ÷ (2.0 × 2.5) = 80 kPa.
  4. q(max) = 80 × (1 + 6e ÷ L) = 80 × (1 + 0.72) = 137.6 kPa.
  5. q(min) = 80 × (1 − 0.72) = 22.4 kPa.
  6. The maximum of 137.6 kPa is below the 150 kPa allowable value.

Answer

q(max) = 137.6 kPa | q(min) = 22.4 kPa | full contact | within 150 kPa

When the moment is larger: M = 200 kN·m

  1. e = 200 ÷ 400 = 0.50 m, which is greater than L ÷ 6 = 0.417 m. The linear formula would give a negative minimum pressure (soil cannot pull on the footing), so it no longer applies.
  2. The base lifts off on one side. With partial contact: q(max) = 2P ÷ [3B × (L ÷ 2 − e)] = 800 ÷ [3 × 2.0 × (1.25 − 0.50)] = 177.8 kPa.
  3. The contact length is 3 × (L ÷ 2 − e) = 2.25 m of the 2.5 m. The maximum pressure of 177.8 kPa exceeds the 150 kPa allowable value, so this case fails.

Common mistakes

  • Using q = P ÷ A ± M ÷ S when the eccentricity is outside the middle third. The formula then gives tension that the soil cannot supply.
  • Leaving the footing weight and the soil on top of it out of the axial load.
  • Measuring the eccentricity or the footing length in the wrong direction. Use the dimension in the plane of the moment.
  • Comparing service pressures with an allowable value from a different load combination, or factored loads with an allowable value.

Related tools and labs

Calculators, labs and simulators are part of the All-Access Pass.

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Questions

Why does the middle-third rule matter?

When the load resultant is inside the middle third of the base, the whole base is in compression. Outside it, part of the footing lifts off, the pressure on the rest rises, and the footing may rock or settle unevenly.

Is 150 kPa a typical value?

It is an exercise input. Real allowable pressures come from the geotechnical report.

What about overturning and sliding?

They are separate checks. This problem only covers bearing pressure under the footing.

Educational material. Examples use simplified, hypothetical inputs and are not a design, a construction document or professional engineering advice. Confirm requirements with the codes and standards adopted where the work is built and with a licensed engineer.

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