Biaxial Footing Contact Pressure Example

Problem Statement

A rigid rectangular footing is $B=10\ \mathrm{ft}$ wide and $L=12\ \mathrm{ft}$ long. At footing-base elevation, the simultaneous service reactions are:

• Vertical compression $P=400\ \mathrm{kip}$.
• Moment about the width axis $M_B=120\ \mathrm{kip\!\cdot\!ft}$.
• Moment about the length axis $M_L=80\ \mathrm{kip\!\cdot\!ft}$.

The allowable service bearing pressure is $5.0\ \mathrm{ksf}$. Calculate the eccentricities, combined kern ratio, and four corner pressures under the linear full-contact model.

Step 1: Average Pressure

The footing area is:

The average pressure is:

Average pressure alone does not show the effect of either moment.

Step 2: Load Eccentricities

Moment about the length axis shifts the resultant in the width direction. Moment about the width axis shifts it in the length direction:

Step 3: Combined Kern Check

For biaxial loading on a rectangular footing:

Because $0.27<1.0$, the resultant lies inside the biaxial kern and the full-contact linear solution remains compressive.

Step 4: Pressure Variations From Moment

The pressure variation along the footing length caused by $M_B$ is:

The variation along the width caused by $M_L$ is:

Step 5: Four Corner Pressures

Combine the average pressure and both gradients:

All corners remain in compression. The maximum-pressure utilization is:

Interpretation

The service-pressure check passes for the entered allowable value, and full contact is maintained. The pressure is not uniform: the maximum corner is about 27 percent above average, while the minimum is about 27 percent below average.

The engineer should still verify that the allowable pressure applies to maximum local pressure under this load case, evaluate settlement and rotation, and use compatible factored pressure distributions for structural design.

What Would Cause Partial Contact?

If moments increase while $P$, $B$, and $L$ remain fixed, the combined kern ratio approaches 1.0. At $\eta_k>1.0$, the full-contact equation predicts a negative corner pressure and a partial-contact analysis is required.

The threshold depends on both moments together. Passing separate middle-third checks in each direction does not necessarily pass the biaxial kern condition.

References And Further Reading

• FHWA GEC 6, Shallow Foundations.
• USACE EM 1110-1-1905, Bearing Capacity of Soils.
• Project-specific load combinations and geotechnical criteria.

FAQ

Why are moments evaluated at footing-base elevation?

Contact pressure equilibrium acts at the soil-contact plane. Horizontal shear acting through pedestal and footing depth can add overturning moment between the column reference and that plane.

Should absolute values be used in the pressure equation?

Use signed moments to identify which corner receives each increase or decrease. Absolute values are useful in the combined kern-ratio magnitude check.

Does 84.7 percent utilization mean the footing is 84.7 percent designed?

No. It describes one service bearing-pressure comparison. Settlement, sliding, uplift, concrete strength, reinforcement, and construction requirements remain.