Introduction
This calculator allows the user to analyse a rigid rectangular spread footing with piers for uniaxial or biaxial resultant eccentricities. The following checks are performed: overturning, sliding, uplift and soil bearing at the four corners of the footing.
❗This calculation has been written in accordance with ACI (American Concrete Institute) guidance.
Calculator
⬇️ Inputs
Refer to image above for the nomenclature and sign convention used in this calculator.
Footing Properties
Concrete unit weight, γc (kcf)
:0.15
Footing length, L
:8.0ft
Footing width, B
:5.0ft
Footing thickness, T
:2.0ftSoil Properties
Depth of soil, D
:2.0ft
Soil unit weight, γs (kcf)
:0.12
Coefficient of passive pressure, Kp
:3.0
Coefficient of friction, μ
:4.0Factor of Safety
Use the same factor of safety for all checks?
:No
Factor of safety
:1.5
FS (overturning)
:1.5
FS (sliding)
:1.5
FS (uplift)
:1.5
FS (bearing)
:1.50
Allowable bearing pressure, (kips-sf)
:150.0Loads
⬆️ Outputs
Summary of Applied Forces
Overturning Check
Sliding Check
Uplift Check
Bearing Check
Explanation
Rectangular footings, also known as spread footings, are structural elements which transfer load imposed by the superstructure to the soil underneath. Spread footings are typically wider than they are deep and are used to support columns, walls, or other vertical members.
The engineer must ensure that the bearing pressure does not exceed the soil bearing capacity. Distribution of the bearing pressure depends on the eccentricity of the loads - a concentric load result in even distribution, while an eccentric load leads to a greater pressure on one side than the other. Eccentric loads lead to overturning moments, which are significantly more dangerous than concentric loads as they cause rotation and differential settlement.
There are two types of eccentricities: uniaxial and biaxial. When the imposed loads on a footing produces a moment along only one axis (say, the x- or y-axis), it is said to have a uniaxial eccentricity. Loads which produce moments in both directions (x- and y-axis) have biaxial eccentricity.
For a footing subject to only one load, the resultant moment due to eccentricity is fairly easy to calculate by resolving it to the footing centroid:
When there are multiple loads on a footing (e.g. a strip or mat footing with multiple columns), the same approach is taken:
Loads which are eccentric in both x- and y-axis induce biaxial bending. Depending on the location of these, induced moments may be acting in opposing directions. If the moment in one direction is greater than the other, it leads to an uneven bearing pressure distribution. In such cases, these moments are said to be either overturning or resisting - the convention is up to the designer.
In the example above, P1 and P2 are acting against each other but since P2 has a greater eccentricity, the bearing distribution will be greater under it than under P1. The ratio between the resultant overturning and resisting moment is called the factor of safety (FoS). Different standards and codes recommended varying FoS values, generally greater than 1.5.
Bearing failure is the most common failure mode for footings. There are also other failure modes such as sliding and uplift but these are rare. Geotechnical investigations are conducted to ensure soil parameters (friction angle, cohesion, etc.) are adequate prior to construction and footings are almost always subject to compressive loads, hence no uplift.
Even more explanation
Time to get math-y.
Let's consider a footing with the following load configuration:
Based on the location of the loads, we notice the following:
- P1 and P2 acts against P3 and P4 about the x-axis
- P1 and P4 acts against P2 and P3 about the y-axis
This can be verified by calculating the moment due to each load relative to footing centroid:
If we assume positive moment to be overturning and negative to be resisting, then:
Now, we can also find the resultant eccentricity by summing the vertical loads and rearranging equations:
Why does this matter in practical design? A footing is most stable when it is in full compression i.e. the entire soffit of the footing is in bearing against the soil. Studies have shown that a footing is deemed to be in full compression if the resultant eccentricity is within an area within a sixth of its width/length from the centre. When the eccentric load is within the highlighted zone, the maximum and minimum bearing pressure can be calculated as:
If a footing is in partial compression or there is loss of contact with soil, there is a higher risk of failure due to overturning and the development of differential settlement. This occurs when the resultant eccentric load does not lie in the highlighted zone, leading to an asymmetric bearing pressure distribution:
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