Spread Footings Under Biaxial Bending: A Complex Design Subject
By: Javier Encinas, PE | February 7, 2016
ASDIP FOUNDATION includes the design of concrete spread footings. This structural engineering software is based on the latest ACI 318 provisions. This article covers the complexities inherent to the design of spread footings when subjected to a combination of vertical loads, horizontal loads, and biaxial bending.
For a simple spread footing with a vertical 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.
Why is biaxial bending so complex?
Since the bearing pressures are the result of the combination of both the vertical load and the applied moments, it’s more convenient to visualize this subject in terms of the eccentricity e = M/P.
For biaxial bending, the maximum bearing pressure will occur 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 figure below).
As the eccentricities increase and fall outside the kern, the computations 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. 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 triangle, through trapezoid, through pentagon, to a full rectangle.
What about 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. Try to visualize the resulting irregular shape formed in this scenario, as shown in the figures below in lighter blue. You still need to calculate the volume of this amorphous solid formed by inclined and vertical planes that resembles a stalactite, but now the calculation is substantially harder to accomplish.
ASDIP FOUNDATION uses an algorithm based on triple integrals of the type V = ∫∫∫dV to find the volume of these solids, and therefore the shear forces in the footing. The pictures below 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 for the punching shear.
Are the bending moments easier to calculate?
The short answer is no. The bearing diagram exerts an upward pressure against the footing, and this force acts through the centroid of the solid described before and produces a moment with respect to the critical section, which for bending is at the face of the column. The footing bending moment is therefore the volume of the solid times the centroid respective the face of the column.
The calculation of the centroid of the irregular body is therefore another required step. ASDIP FOUNDATION uses a series of triple integrals of the type Ȳ = ∫∫∫xdV / V to calculate the centroid of the resulting irregular solids, as shown in the figures 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.