By: Javier Encinas, PE | October 9, 2019

ASDIP FOUNDATION is a structural engineering software for footing design. It includes the design of concrete spread footings based on the latest ACI 318 provisions. This article discusses the complexities inherent to the design of spread footings when subjected to a combination of vertical and horizontal loads, and biaxial bending.

For a simple spread footing with a vertical concentric compression load, the resulting bearing pressure is uniform and the shear and bending can be calculated easily by hand. If the column is at the corner of a building, then biaxial moments could be transferred to the footing. Additional moments are produced if the column is placed eccentrically on the footing, or due to the horizontal loads applied at the top of the pedestals.

The images below show the general case of a rectangular spread footing with an offset column subject to a vertical load, two horizontal loads, and two bending moments around the orthogonal axes.

## What makes biaxial bending so complex?

Since the bearing pressure distribution is the result of the combination of both the vertical load and the applied moments, it's more convenient to do the calculations in terms of the eccentricities ex = Mz/P and ez = Mx/P.

For biaxial bending, the maximum bearing pressure occurs at one corner of the footing, and the minimum pressure at the corner diagonally opposite. For small eccentricities, the entire footing will be under compression, and the bearing pressures may be calculated easily with the well known formula P/A + Mx/Sx + Mz/Sz, where the (+) signs may be (-) in order to calculate the minimum pressure (Type I in the images below). Note that this is valid only for eccentricities smaller than the kern = L/6 and W/6, with a straight interaction line between the two points. The kern is represented by the triangular area in the right image below.

As the eccentricities increase and fall outside the kern, the calculations become quite complex, mainly because the tensile resistance of the soil is negligible, and therefore a portion of the footing needs to be ignored in the analysis (Types II to IV). The first difficulty is then determining the shape and size of the remaining effective portion of the footing. Depending on the location of the resultant, this shape in plan view could vary from a full rectangle (Type I), through a pentagon (Type II), through a trapezoid (Type III), to a triangle (Type IV), as the blue areas shown above.

Obviously in a perfect world all footings would be Type I, but the reality is that in practice we may have existing footings or other obstructions in the area, or underground piping, or a property line nearby, or probably you need to check an existing footing for higher loads. In such cases the other footing types may be an option.

## How do you calculate the shears in the footing?

The shear is produced by the soil bearing pressure acting upwards on the footing. If the pressure is uniform, or if the entire footing is under compression, this force can be easily calculated as the volume of the parallelogram delimited by the critical section, the footing edges, and the bearing pressure.

Consider now a partial bearing diagram, such as the Type II, III or IV above, and trace the shear critical section at "d" from the column face. Try to visualize the resulting irregular shape formed in this scenario, as shown in the images below in lighter blue. You still need to calculate the volume of these irregular solids formed by inclined and vertical planes that look like stalactites, but now the calculation is substantially harder to accomplish.

ASDIP FOUNDATION uses an algorithm based on triple integrals of the type V = âˆ«âˆ«âˆ«dV to find the volume of these solids, and therefore the shear forces in the footing. The screen shots above show the factored shear forces of a typical footing under biaxial bending as a result of partial bearing. Note the different hatch representing the effective shear areas in both directions. Similar calculations are required to find the punching shear.

## Are the bending moments easier to calculate?

No, the calculation is actually even more difficult. The bearing diagram exerts an upward pressure against the footing, but this force acts through the centroid of the solid described above, and produces a moment with respect to the critical section. The footing bending moment is therefore the volume of the solid, multiplied by the centroid with respect to the face of the column.

The calculation of the centroid of this irregular body is therefore another required step. ASDIP FOUNDATION uses a series of triple integrals of the type È² = âˆ«âˆ«âˆ«xdV / V to calculate the centroid of the resulting irregular solids, and then it calculates the corresponding moments, as shown in the screen shots below in lighter blue.

Once the bending moments are found, the reinforcing steel may be designed per the concrete design theory. It should be noted that the bearing pressures are calculated using service loads, but both shears forces and bending moments must be calculated by applying the factored loads.

## Takeaway

The design of spread footings subject to biaxial bending may be cumbersome and time-consuming, particularly for partial bearing conditions. In such cases the calculation of bearing pressures, shears, and moments may become very complex. ASDIP FOUNDATION includes the design of spread footings, with multiple options to optimize the design easily.

For software usage, please read the blog post How to Design Spread Footings Using ASDIP FOUNDATION. For a footing design example, please see the blog post Spread Footing Real-Life Example Using ASDIP FOUNDATION.