Introduction

In many practical cases, it becomes important to study the interaction of elastic or rigid foundations, which are constructed simultaneously. In this case, there will be interaction of foundations due to the overlapping of stresses through the soil medium, however the structures are not statically connected. The interaction of foundations will cause additional settlements under all foundations.

The conventional solution of this problem assumes that the contact pressure of the foundation is known and distributed linearly on the bottom of it. Accordingly, the soil settlements due to the system of foundations can be easily determined. This assumption may be correct for small foundations, but for big foundations, it is preferred to analysis the foundation as a plate resting on either elastic springs (Winkler’s model) or continuum model. In spite of the simplicity of the first model in application, one cannot consider the effect of neighboring foundations or the influence of additional exterior loads. Thus, because Winkler’s model is based on the contact pressure at any point on the bottom of the foundation is proportional to the deflection at that point, independent of the deflections at the other points. Representation of soil as Continuum model (methodes 4, 5, 6, 7 and 8) enables one to consider the effect of external loads.

The study of interaction between a foundation and another neighboring foundation or an external load has been considered by several authors. Stark (1990) presented an example for the interaction between two rafts. Kany (1972) presented an analysis of a system of rigid foundations. In addition, he presented a solution of system of foundations considering the rigidity of the superstructure using a direct method (Kany 1977). Recently, Kany / El Gendy (1997) and (1999) presented an analysis of system of elastic or rigid foundations on irregular subsoil model using an iterative procedure.

This section presents a general solution for the analysis of system of foundations, elastic or rigid, using the iterative procedure of Kany / El Gendy (1997) and (1999).

Description of the problem

Settlement joints are usually used in the foundation when the intensity of loads on it differs considerably from area to another. In such case, the foundation may be divided corresponding to its load intensity to avoid cracks. Settlement joint is constructed by making a complete separated joint in the foundation or a hinged joint. If the foundation has a separated joint, each part will settle independently but it will be interaction between parts of the foundation through the subsoil. In the other case of hinged joint, there will be transmission of shearing forces between connection parts.

This example is carried out to examine the interaction of two rafts considering settlement joint. Consider two equal square rafts I and II will be constructed side by side. Each raft has a side of 12 [m] and 0.5 [m] thickness. Raft I is subject to a uniform load of 400 [kN/m²], while raft II carries a uniform load of 200 [kN/m²].

Introduction

In many practical cases, it becomes important to study the interaction of elastic or rigid foundations, which are constructed simultaneously. In this case, there will be interaction of foundations due to the overlapping of stresses through the soil medium, however the structures are not statically connected. The interaction of foundations will cause additional settlements under all foundations.

The conventional solution of this problem assumes that the contact pressure of the foundation is known and distributed linearly on the bottom of it. Accordingly, the soil settlements due to the system of foundations can be easily determined. This assumption may be correct for small foundations, but for big foundations, it is preferred to analysis the foundation as a plate resting on either elastic springs (Winkler’s model) or continuum model. In spite of the simplicity of the first model in application, one cannot consider the effect of neighboring foundations or the influence of additional exterior loads. Thus, because Winkler’s model is based on the contact pressure at any point on the bottom of the foundation is proportional to the deflection at that point, independent of the deflections at the other points. Representation of soil as Continuum model (methodes 4, 5, 6, 7 and 8) enables one to consider the effect of external loads.

The study of interaction between a foundation and another neighboring foundation or an external load has been considered by several authors. Stark (1990) presented an example for the interaction between two rafts. Kany (1972) presented an analysis of a system of rigid foundations. In addition, he presented a solution of system of foundations considering the rigidity of the superstructure using a direct method (Kany 1977). Recently, Kany / El Gendy (1997) and (1999) presented an analysis of system of elastic or rigid foundations on irregular subsoil model using an iterative procedure.

This section presents a general solution for the analysis of system of foundations, elastic or rigid, using the iterative procedure of Kany / El Gendy (1997) and (1999).

Description of the problem

To illustrate the application of the iterative procedure of Kany / El Gendy (1997) for the interactive system of foundations, consider the system of two equal large circular rafts shown in the Figure. The rafts rest on a soil layer of thickness 15 [m]. Each raft has a diameter of 22 [m] and 0.65 [m] thickness. Loading on each raft consists of 24 column loads in which 16 columns loads have P₁ = 1250 [kN] and 8 column loads have P₂ = 1000 [kN]. The Young’s modulus of the raft material is E_b = 2.6×10 [kN/m ] and Poisson’s ratio is ν_b = 0.15 [-], while the soil values are E_s = 9500 [kN/m] and ν_s = 0 [-].

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 the problem

This example was carried out to show the influence of flexure rigidity of the superstructure on the settlements, contact pressures for a raft of high rise building. It is required to analysis a raft for the building shown in Figure (6.6) in three simplified sections. The building is a reinforced concrete skeleton structure consists of a cellar and 13 storeys. The floor height is 3 [m] while the bay width is 3.6 [m]. The number of bays is 18. The total building length is 66 [m] while the total width of the cellar basement is 17.55 [m]. The raft thickness is 1.2 [m]. In the following study the raft is analyzed considering subsoil behavior.

Also, a simplification estimation of the superstructure deformations is carried out. In the analysis, settlements and contact pressures are determined in which a comparison is carried out in four cases as:

i) For not stiffened raft

ii) For compound system raft-cellar

iii) For compound system raft-cellar-superstructure

iv) For completely rigid raft

The stiffness of the structure system parallel to the long axis can be determined from the data given in Figures (6.6) and (6.7).

Introduction

The foundation is considered as rigid, elastic or flexible, depends on the ratio between the rigidity of the foundation and the soil. The oldest work for the analysis of foundation rigidity is that of Borowicka (1939). He analyzed the problem of distribution of contact stress under uniformly loaded strip and circular rigid foundations resting on semi-infinite elastic mass.

After Borowicka’s analysis, many authors introduced formulae to find the foundation rigidity for plates resting on different subsoil models. For examples, Gorbunov/ Posadov (1959) introduced formula for an elastic solid medium. Cheung/ Zienkiewicz (1965) introduced formulae for Winkler springs and isotropic elastic half space model. Vlazov/ Leontiv (1966) introduced formula for a two-parameter elastic medium. A good review for those formulae may be found in Selvadurai (1979).

Lately, based on great number of comparative computations for the modulus of compressibility method, Graßhoff (1987) proposed various degrees of system rigidity between foundation and the soil until case of practical rigidity using Equation (5.2). The equation still used in many national standard specifications such as German standard (DIN 4018) and Egyptian Code of Practice (ECP 196-1995).

Description of the problem

Ribbed raft may be used for many structures have heavy loads or large spans, if a flat level for the first floor is not required. Consequently, concrete is reduced. Such structures are silos and elevated tanks. In spite of this type of foundation has many disadvantages if used in normally buildings, still used by many designers. Such disadvantages are the raft needs deep foundation level under the ground surface, fill material on the foundation to make a flat level and an additional slab on the fill material to construct the first floor. The use of the ribbed raft relates to the simplicity of analysis by hand calculations.

First, both of the two rafts with and without ribs are clearly saves and correct, but there is still a question, whose one of the two types is more rigid? To answer this question the following example is presented. Consider the foundation of an elevated tank may be designed for both types of foundations. The foundation has the dimensions of 20 [m] × 20 [m], transmits equal loads for all 25 columns, each of 1000 [kN]. The loads give average contact pressure on soil q_av = 62.5 [kN/m]. Columns are equally spaced, 4 [m] apart, in each direction as shown in the Figure.