## 1 Introduction  
The settlements of a raft foundation under vertical loads can be estimated using either analytical or numerical solutions. The analytical formula provides a relatively simple solution; however, it has a shortcoming when modeling variations of foundation settlements. For numerical solutions, one-dimensional (1D) beams on elastic foundation (i.e., the Winkler foundation) or two-dimensional (2D) plates underlain by a set of soil springs can be used to model the foundation. A three-dimensional (3D) structure-foundation-soil system can also be adopted as a more rigorous analysis. The relative stiffness (or rigidity) of the foundation is typically important in affecting the load-response mechanism of the foundation. Design engineers must know that the performance of a raft foundation can be influenced by the rigidity of that foundation. It has been generally noted that rigid foundations under vertical loads would result in more uniform settlements and variable contact pressures of the soils (i.e., soil reactions). On the contrary, flexible foundations can exert large differential settlements and more consistent contact pressures of the soils. To estimate foundation rigidity, the relative stiffness of a raft foundation (\(K_{r}\)) can be evaluated using the formula suggested by the ACI committee 336 report, DIN 4018 code,and IS 2950-1 code [1; 2; 3]. Ignoring the superstructure, the value of \(K_{r}\) can be simply computed as follows,  
\[K_{r}=\frac{E_{b}}{12E}\left(\frac{d}{L}\right)^{3}, \tag{1}\]  
where \(E_{b}\) is the Young's modulus of the concrete raft, \(E\) is the Young's modulus of the soil, \(d\) is the thickness of the raft, and \(L\) is the length of the raft. A rigid or flexible raft is indicated by \(K_{r}>0.5\) or \(K_{r}<0.5\), respectively. Alternatively, the effect of changes in the Poisson's ratios of the foundation and soil can be expressed by Equation (2) [4]. \[K_{r}=\frac{E_{b}\big{(}1-v^{2}\big{)}}{12E(1-v_{b}{}^{2})}\left(\frac{d}{L} \right)^{3}, \tag{2}\]  
where \(v_{b}\) and \(v\) are the Poisson's ratio of the concrete and the soil, respectively. Per the definition used by [PERSON] [5] in their finite element (FE) analysis, \(K_{r}<0.05\), \(0.05<K_{r}<5\), and \(K_{r}>5\) represent flexible, semi-rigid, and rigid foundations, respectively. The detailed expressions of Equation (2) can be found in [PERSON] and [PERSON] [6]. Moreover, [PERSON] [7] reported that \(K_{r}\) can be calculated following the Egyptian code [8]:  
\[K_{r}=\frac{E_{b}}{E}\left(\frac{d}{L}\right)^{3}. \tag{3}\]  
According to [PERSON] [7], \(K_{r}<0.005\), \(0.005<K_{r}<2\) and \(K_{r}>2\) can represent for flexible, semi-rigid, and rigid foundations, respectively. In calculating \(K_{r}\) values of the foundations, this study found that the Egyptian code is more stringent than others in satisfying the flexible foundation. When a 2D raft foundation is considered as a 1D beam and implemented with a vertically uniform load, the settlements and contact pressures (or the soil reactions) of the foundation are not only dependent of the rigidity of the foundation but also affected by the type of soil underneath the foundation. Figure 1 depicts qualitative plots of the foundation settlements and the contact pressure of the soil for a raft foundation subjected to a vertically uniform load. For granular soil such as sand, if the foundation is elastic (i.e., flexible), the settlement is larger at the edges and smaller at the center; corresponding stress of the soil reaction is relatively uniform. For rigid foundations in sand, uniform settlements can be expected and the stress of the soil reaction is larger at the center and negligible at the edges. For a flexible foundation on clayey soil with similar load, the settlements will be larger at the center and the stress of the soil reaction will tend to be uniform. Rigid foundations in clayey soil will result in uniform settlements, and the stress of the soil reaction is smaller at the center and higher at the edges.