## 1 Introduction  
Due to its complexity, the interaction between raft foundations and soils is generally modeled with the help of linear elastic springs based on [PERSON]'s theory [1]. The spring constant is obtained from the modulus of subgrade reaction, which corresponds to the ratio of contact pressure to settlement at any given point, multiplied by the contributory area of the node it is connected to [2, 3, 4]. Bending moments, shear forces, and deflections on structural elements can only be computed when the modulus of subgrade reaction is accurately determined. Relatively high values of the subgrade reaction modulus lead to faulty design of structural components, while relatively low values result in a considerable increase in the required reinforcement [5, 6]. Contact pressure, which is required for the determination of the settlement and thus the subgrade reaction modulus for a raft foundation, depends on the relative stiffness of the soil-foundation system, so a distinction is theoretically made between infinitely rigid and infinitely flexible foundations [6, 7]. According to ACI committee [7], the behavior of a soil-foundation system can be estimated with the help of the rigidity factor \(K_{r}\), calculated using Equation (1). The system can be assumed rigid for a value of \(K_{r}>0.5\) and flexible for \(K_{r}<0.5\):  
\[K_{r}=\frac{E^{\prime}\cdot l_{b}}{E\cdot B^{3}} \tag{1}\]  
where \(E^{\prime}\) is [PERSON]'s modulus of the structure material, E is [PERSON]'s modulus of the soil, B is the foundation width, and \(\mathrm{I_{b}}\) is the moment of inertia of the structure per unit length at right angles to B. Ignoring the effect of the superstructure, DIN--Technical Report 130 [8] suggests Equation (2) to determine the relative stiffnesses of soil-foundation systems. Theoretically, a foundation behaves as infinitely rigid for K-values \(>>0\) and infinitely flexible for K-values \(\approx 0\):  
\[K=\frac{1}{12}\cdot\frac{E_{f}}{E}\cdot(\frac{d}{L})^{3} \tag{2}\]  
where \(\mathrm{E_{f}}\) is the Young's modulus of the foundation material, E is the Young's modulus of the soil, and d and L are the foundation thickness and length, respectively. According to [PERSON] and [PERSON] [9], the system can be considered rigid for K-values \(>0.1\), semi-rigid for \(0.01<\mathrm{K}<0.1\), semi-flexible for \(0.001<\mathrm{K}<0.01\), and flexible for K-values \(<0.001\):  
Further equations and limit values to estimate the foundation rigidity can be found in the literature [10; 11]. In an infinitely rigid foundation with a centrally acting axial load on an elastic half-space, the contact pressure q can be calculated using Equation (3) [12]:  
\[q=\frac{(4\cdot N)/(a\cdot b)}{\pi^{2}\cdot\left[\left[1-\left(\frac{x}{a/2} \right)^{2}\right]\cdot\left[1-\left(\frac{y}{b/2}\right)^{2}\right]\right]^{ 0.5}} \tag{3}\]  
where N is the axial load acting centrally, x and y are the coordinates of any point where the contact pressure q (x,y) is calculated, and a and b are foundation length and width, respectively. As shown in Figure 1a, infinitely large pressures appear at the edges of the foundation. However, these large pressures are reduced to finite values through the development of plastic soil deformations, which leads to an increase in the contact pressure at the central zone of the foundation. As a result, a uniform contact pressure corresponding to the ratio of the axial load to the foundation area N/A can be used for infinitely rigid foundations (Figure 1b). Assuming a straight-line distribution of soil pressure, the maximum and minimum soil pressures developing under eccentrically acting axial loads can be calculated from the formula provided in Figure 1c. Infinitely flexible foundations have no bending stiffness and thus can freely adapt to soil deformations, so contact pressures correspond to column and wall loads. However, infinitely rigid foundations cannot follow soil deformations due to their infinitely high bending stiffnesses.