Introduction

The presence of the structure on compressible subsoil causes settlements for the foundation and also for the structure itself. Values of settlements and settlement differences depend not only on the thickness of the compressible soil layer under the foundation, the value and distribution of structure loads, the foundation depth and contact pressure under the foundations but also on the flexural rigidity of the structure.

One of the properties that has a considerable influence on the development of settlement is the rigidity of the superstructure. The more rigid structure has more uniform settlement and conversely, structure that is more flexible has greatest difference in settlement. The entire structure can be defined as the three media: superstructure, foundation and soil. The analysis of the entire structure as one unit is very important to find the deformations and internal forces.

However, most of the practical analyses of structures neglect the interaction among the three media to avoid the three-dimensional analysis and modeling. The structure is designed on the assumption of non-displaceable supports while the foundation is designed on the assumption that there is no connection between columns. Such accurate analysis of the entire structure is extremely complex.

The early studies for consideration the effect of the superstructure were by *Meyerhof *(1953) who suggested an approximate method to evaluate the equivalent stiffness that includes the combined effect of the superstructure and the strip beam foundation. *Kany * (1959) gave the flexural rigidity of a multi-storey frame structure by an empirical formulae. Also, *Kany *(1977) analyzed the structure with foundation using a direct method. *Demeneghi * (1981) used the stiffness method in the structural analysis. *Panayotounakos/ Spyropoulos/ Prassianakis *(1987) presented an exact matrix solution for the static analysis of a multi-storey and multi-column rectangular plexus frame on an elastic foundation in the most general case of response and loading.

At the analysis of foundations with considering the superstructure stiffness, it is required to distinguish between the analysis for plane structures (two-dimensional analysis) and that for space structures (three-dimensional analysis). Further, it is required to distinguish between approximation methods with closed form equations (*Kany *(1974), *Meyerhof * (1953), *Sommer *(1972)) and refined methods such as conventional plane or space frame analysis (*Kany * (1976)), * Finite Elements *(*Meyer *(1977), *Ellner/ Kany * (1976), Zilch (1993), *Kany/ El Gendy * (2000)) or * Finite Differences *(*Bowles *(1974), *Deninger * (1964)).

In addition, many analytical methods are reported for analysis of the entire structure as one unit by using the finite element.

For examples:

*Haddadin *(1971) presented an explicit program for the analysis of the raft on Winkler's foundation including the effects of superstructure rigidity.

*Lee/ Browen *(1972) analyzed a plane frame on a two-dimensional foundation.

*Hain/ Lee *(1974) employed the finite element method to analyze the flexural behavior of a flexible raft foundation taking into account stiffness effect of a framed superstructure. They proposed the use of substructure techniques with finite element formulation to model space frame-raft-soil systems. The supporting soil was represented by either of two types of soil models (Winkler and half-space models).

*Poulos *(1975) formulated the interaction of superstructure and foundation by two sets of equations. The first set links the behavior of the structure and foundation in terms of the applied structural loads and the unknown foundation reactions. The second set links the behavior of the foundation and underlying soil in terms of the unknown foundation reactions.

*Mikhaiel *(1978) considered the effect of shear walls and floors rigidity on the foundation.

*Bobe/ Hertwig/ Seiffert * (1981) considered the plastic behavior of the soil with the effect of the superstructure.

*Lopes/ Gusmao *(1991) analyzed the symmetrical vertical loading with the effect of the superstructure.

*Jessberger/ Yuan/ Thaher/ Ming-bao * (1992) considered the effect of the superstructure in case of raft foundation on a group of piles.

*Zilch *(1993) proposed a method for interaction of superstructure and foundation via iteration.

*Kany/ El Gendy *(2000) proposed an iterative procedure to consider the effect of superstructure rigidity on the foundation. In the procedure, the stiffness of any substructure such as floor slab or foundation, connected by the columns can be represented by equivalent spring constants due to forces and moments at the connection nodes. Consequently the stiffness matrices of the slab floors, columns and foundation remain unaffected during the iteration process.

Description of problem

An application of the proposed iterative procedure is carried out to study the behavior of foundation resting on nonlinear soil medium with considering influence of the superstructure rigidity.

The previous example shown in Figures. (6.11) and (6.12) is also chosen here to show the analysis of structure on nonlinear soil medium with some modification to be a practical problem.

The floor is chosen to be a slab of 22 [cm] thickness resting on skew paneled beams. The slab carries a uniform load of 11.8 [kN/m^{2}]. Foundation is considered as a raft foundation with openings. The dimensions of paneled beams, columns and foundation are the same as those of the previous example.

Soil properties

Two different types of soil models are considered in this case-study:

i) Winkler’s model that represents the subsoil by isolated springs.

ii) Layered model that considers the subsoil continuum medium.

The foundation is resting on a soil layer of 10 [m], overlying a rigid base. The soil types are represented by the modulus of elasticity, * Es*, for layered model that yields modulus of subgrade reaction, *k*_{s}, for Winkler’s model. Table (6.7) shows two different soil types examined in this study according to the soil properties * E*_{s} and * k*_{s}. The two soil types are selected to represent weak and stiff soil. *Poisson’s* ratio is taken ν_{s }= 0.3 for the two soil types.