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Dezi et. al (2010)
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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.
Rectangular Spread Footing Design Elevation & Plan


Calculation

Inputs

Refer to image above for the nomenclature and sign convention used in this calculator.

Footing Properties



γc, kcf
:0.15​



L
:8ft​



B
:5ft​



T
:2ft​


Soil Properties



D
:2ft​



γs kcf
:0.12​



Kp
:3​



μ
:4​


Factor 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​


Loads

Output

Summary of Applied Forces

Overturning Check

Resisting moments:


ΣMrx, ft-kips
:291.84ft kips​



ΣMry, ft-kips
:291.84ft kips​


Overturning moments:


ΣMox, ft-kips
:412.25ft kips​



ΣMoy, ft-kips
:-410.25ft kips​


Factors of safety:


FSx (overturning)
:0.71​



FSx (overturning) Safe/Fail?
:Fail​




FSy (overturning)
:0.71​



FSy (overturning) Safe/Fail?
:Fail​


Sliding Check

Resisting forces:


Ffx
:0.0kips​



Ffy
:0.0kips​


Pushing forces:


Fpx
:46.08kips​



Fpy
:46.08kips​


Factors of safety:


FSx (sliding)
:4.61​



FSx (sliding) Safe/Fail?
:Safe​




FSy (sliding)
:4.61​



FSy (sliding) Safe/Fail?
:Safe​


Uplift Check



Total downward load, ΣPz
:-76.71kip​



Total upward load, ΣPz
:100.0kip​


Factor of safety:


FS (uplift) (1)
:0.7671​



FS (uplift) Safe/Fail?
:Fail​



Bearing Check

Pressures at four corners:


P1 (kips-sf)
:0.00​



P2 (kips-sf)
:0.00​



P3 (kips-sf)
:0.00​



P4 (kips-sf)
:0.32​


Factor of safety:


Mobilised FS (bearing)
:0.0021067941878887​



FS (bearing) Safe/Fail?
:Fail​



Bearing Distribution Geometry



Biaxial eccentricity case
:Case 1​

Different cases of biaxial bearing pressure



Dist. x =
:12.6947885654948ft​



Dist. y =
:11.7721043972002ft​



Brg. Lx =
:4.06775765434247ft​



Brg. Ly =
:4.35356137131792ft​



%Brg. Area
:88.7979059480117sqft​



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.
Concentric load

Eccentric load


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.
Uniaxial eccentricity


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:

Moment about the x-axis:  Mx=P × eyMoment about the y-axis:  My=P × ex\text{Moment about the x-axis: }\space M_{x} = P\space\times\space e_y\\\text{Moment about the y-axis: }\space M_{y} = P\space\times\space e_x
When there are multiple loads on a footing (e.g. a strip or mat footing with multiple columns), the same approach is taken:

Resultant moment about the x-axis:  Mx=Σ(P × ey)Resultant moment about the y-axis:  My=Σ(P × ex)\text{Resultant moment about the x-axis: }\space M_{x} = \Sigma (P\space\times\space e_y)\\\text{Resultant moment about the y-axis: }\space M_{y} = \Sigma (P\space\times\space e_x)
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.
Example of a footing subject to biaxial bending from multiple loads

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.

Factor of safety=Resultant resisting momentResultant overturning moment=ΣMRΣMO\text{Factor of safety}=\frac{\text{Resultant resisting moment}}{\text{Resultant overturning moment}} = \frac{\Sigma M_R}{\Sigma M_O}
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:
  1. P1 and P2 acts against P3 and P4 about the x-axis
  2. 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:

Mx1=P1 × ey1Mx2=P2 × ey2Mx3=P3 × −ey3Mx4=P4 × −ey4M_{x1} = P_1\space\times\space e_{y1} \hspace{5cm} M_{x2} =P_2\space\times\space e_{y2} \\ M_{x3} = P_3\space\times\space -e_{y3} \hspace{5cm} M_{x4} = P_4\space\times\space -e_{y4}

My1=P1 × ex1My2=P2 × −ex2My3=P3 × −ex3My4=P4 × ex4M_{y1} = P_1\space\times\space e_{x1} \hspace{5cm} M_{y2} =P_2\space\times\space -e_{x2} \\ M_{y3} = P_3\space\times\space -e_{x3} \hspace{5cm} M_{y4} = P_4\space\times\space e_{x4}
If we assume positive moment to be overturning and negative to be resisting, then:

ΣMx=Mx1+Mx2+Mx3+Mx4ΣMy=My1+My2+My3+My4  Resultant overturning moment about x-axis: ΣMox=Mx1+Mx2Resultant overturning moment about y-axis: ΣMoy=My1+My3 Resultant resisting moment about x-axis: ΣMrx=Mx3+Mx4Resultant resisting moment about y-axis: ΣMry=My2+My4\Sigma M_x = M_{x1} + M_{x2} + M_{x3} + M_{x4}\\ \Sigma M_y = M_{y1} + M_{y2} + M_{y3} + M_{y4}\space \\ \space \\ \text{Resultant}\space\textbf{overturning}\space\text{moment about x-axis: } \Sigma M_{ox}=M_{x1}+M_{x2} \\ \text{Resultant}\space\textbf{overturning}\space\text{moment about y-axis: } \Sigma M_{oy}=M_{y1}+M_{y3} \\ \space \\ \text{Resultant}\space\textbf{resisting}\space\text{moment about x-axis: } \Sigma M_{rx}=M_{x3}+M_{x4} \\ \text{Resultant}\space\textbf{resisting}\space\text{moment about y-axis: } \Sigma M_{ry}=M_{y2}+M_{y4}
Now, we can also find the resultant eccentricity by summing the vertical loads and rearranging equations:

ΣP=P1+P2+P3+P4Resultant eccentricity from the x-axis: ex=ΣMxΣPResultant eccentricity from the y-axis: ey=ΣMyΣP\Sigma P = P_1 + P_2 +P_3 + P_4 \\ \text{Resultant eccentricity from the x-axis: } e_x = \frac{\Sigma M_x}{\Sigma P} \\ \text{Resultant eccentricity from the y-axis: } e_y = \frac{\Sigma M_y}{\Sigma P}

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:

Pmax=ΣPArea[1+6(exL)]Pmin=ΣPArea[1−6(exL)]P_{max} = \frac{\Sigma P}{Area} [1 + 6(\frac{e_x}{L})] \\ P_{min} = \frac{\Sigma P}{Area} [1 - 6(\frac{e_x}{L})]

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: