Strip Foundations Optimization

Walls are structural elements widely used throughout the world. Particularly in seismic zones, they take on special relevance, as they are slender elements that provide great stiffness and strength to any structure. Strictly speaking, it is possible to build a wall with isolated foundations at a certain spacing; however, this would impose great additional stresses on the wall. Thus, it becomes imperative to create a new type of foundation for these continuous elements: strip foundations.

How do we analyze a strip foundation?

When we analyze an isolated foundation, relatively few parameters are necessary to correctly define both geometry and loads. This is why analytical solutions are viable. In strip foundations, some very simple cases can be studied analytically, which are quite far from the geometry of walls in practice. In this way, the best alternative for studying strip foundations is to model them with the finite element method (FEM).

Geometry of a strip foundation

Although a strip foundation can be used in an arrangement of columns, they are most commonly used to found walls. Thus, there are a series of typical geometries for strip foundations that respond to typical wall geometries in buildings, as shown in the following figure:

All these geometries can be defined from segments, which in turn are defined from a width and a succession of points. In practice, strip foundations can be even more complex, for example, a foundation system for a residential building.

Thus, the most flexible way to define the geometry of a strip foundation is through a polyline, which allows for the creation of different segments, each with its own independent width.

Finite element method

In simple terms, the finite element method consists of discretizing a large and complex problem into a series of small and simple problems, which are easy to solve independently. Basically, all structural analysis software has implemented some version of the finite element method, with greater or lesser efficiency. In this sense, finite element meshing is extremely important to achieve a balance between the precision of the results and calculation times.

In this case, each of the triangles in the mesh is a simple problem, which has 3 possible movements (i.e., degrees of freedom) at each of its vertices. In this way, determining the pressures and settlements of a single strip foundation becomes a mathematical and computational challenge of great dimensions.

Loads on a strip foundation

In the case of a strip foundation for a group of columns, each column corresponds to a load point for the foundation, which transmits 5 force/moment components ($F_x, F_y, F_z, M_x, M_y$). If the strip foundation is supporting a wall, the number of load points will depend on the discretization of the wall.

Strictly speaking, an adequate discretization is determined through trial and error, verifying that the stress/displacement results converge as a finer mesh is made. However, for a relatively regular building, whose walls are modeled with shell-type elements in commercial software, a discretization of $\approx 0.8\,m$ should work quite well.

Pressures on the bearing surface of strip foundations

The finite element method allows us to estimate the pressures on the bearing surface for the loads considered in the analysis. As a result, a pressure value is obtained for each node of the mesh, which is interpolated to generate a pressure color map. This directly allows us to identify which nodes have uplift (i.e., zero pressure) and thereby calculate the percentage of the compressed area of the foundation.

How to optimize a strip foundation?

When we talk about optimizing a foundation, we refer to minimizing the volume of concrete, as long as we continue to comply with the expected performance parameters of the foundations, as this tends to be a good indicator of the total project cost. Each engineer can define their own desired performance parameters, but these can usually be classified as:

• Maximum pressure: A reasonable and widely accepted design criterion is to ensure that at least 90% of the foundation area is under the allowable pressure ($\sigma_{adm}$), where

  $$\sigma_{adm} = \frac{q_u}{FS}$$

  corresponds to a conservative value provided in the geotechnical report to be sufficiently far from the soil's bearing capacity ($q_u$). This allows for ignoring highly localized stress peaks that the soil is naturally capable of redistributing.

• Minimum contact area: This parameter guarantees global stability against overturning. Since soil does not resist tension, any node with zero pressure indicates an uplift of the foundation. Generally, it is sought that the compressed area is $\geq 80\%$ for service combinations.

• Allowable settlements: Beyond strength, the foundation must limit vertical deformations so as not to compromise the integrity of partitions and finishes. Both absolute settlement (total sinking) and differential settlement (the difference in level between two points of the same strip foundation) must be verified. In complex wall foundations, the control of differential settlement is critical to avoid diagonal cracks in reinforced concrete elements, which are much more sensitive to angular distortion than frame structures.

The Foundaxis Secret

The analysis and optimization of strip foundations is not a trivial process. Finite element meshing, assembly of the stiffness matrix, solving matrix equations, and optimizing dimensions are highly complex processes, mathematically rigorous and demanding for any calculation engine. However, in Foundaxis, we achieve this automatically, accurately, and in just a few minutes.

In addition to some mathematical and computational tricks, at Foundaxis, we have implemented an efficient optimization algorithm for strip foundations. Basically, in each iteration of the analysis, we determine the most efficient way to grow a foundation, so we do not have to increase the dimensions of all segments at once. We identify which nodes exhibit excessive pressure or uplift, which segments they belong to, and how much these segments need to grow. Thus, we achieve optimal and reliable designs with fast results even for the most complex geometries.

 

Discover how strip foundations optimization is integrated into the complete Foundaxis workflow:

Foundation Design Software: A Complete Guide for Structural Engineers 2026