This example is carried out to show the influence of irregular subsoil on the values of settlements, contact pressures and moments.

The analysis of the square raft is carried out by the two familiar types of soil models: Winkler’s and Continuum models for elastic foundations, besides the analysis of rigid raft on Continuum model, using the following three calculation methods:

Method 3         Variable modulus of subgrade reaction method

Method 7         Modulus of compressibility method

Method 8         Rigid raft on compressible subsoil

A square raft of 10 [m] side is subdivided into 144 square elements. The raft thickness is d = 0.4 [m].

The influence of irregularity of the subsoil material on the behavior of foundations is illustrated through the study of the differential settlements for a system of 9 footings. Consider the group of footings shown in Figure 2.29 and Table 2.3. Thickness of footings is d = 0.5 [m]. Unit weight of the footing is γf =25 [kN/m3]. Arrangement of footings and footing loads are shown in Figure 2.29a

Besides the possibility of studying the influence of neighboring structures on the foundation by the program ELPLA, the described algorithm of ELPLA can be used also for the calculation of settlements outside the foundation. This can be carried out through one of the following two ways:
i) Using a net for the foundation and the unloaded areas outside the foundation. Then, the rigidity of the unloaded areas can be eliminated by assuming very small thickness
ii) Using two independent nets, one for the foundation and the other for the unloaded areas outside the foundation as considered in this example
Figure 3.2 shows an irregular raft that has the contact area I with opening inside it. It is required to determine the settlements at the area II around the raft and at the opening of area III.

For the explanation of the influence of a neighboring building, the influence of a new building on an existing old one is examined in this example. Figure 3.6 shows plan and section of a new building II beside a similar old one I. The building I was constructed since long time, while the building II will be constructed close to the first one. The two buildings have the same construction geometry and loads. Also, every building is symmetrical about both x- and y-axes.

Figure 3.10 shows a raft of a building that consists of two rectangular parts, which are completely connected. The raft is 50 [cm] thick and has a foundation depth of 2.50 [m] under the ground surface. It is planned to construct a tunnel diagonally to the building axis. A primary estimation expects that the tunnel will cause a settlement trough of about 9 [m] width with a maximum lowering of 3 [cm] for the building ground. The settlement trough is plotted in Figure 3.10 as contour lines, running symmetrically to the tunnel axis. The influence of the settlement trough due to construction of the tunnel is considered in the analysis of the raft. The raft carries two equal column loads, each of P = 18000 [kN] and line loads of p = 300 [kN/m] from edge walls. The edge walls have 0.30 [m] breadth and 3 [m] height.

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 Figure 4.3. The rafts rest on a soil layer of 5 [m] thickness. 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 Eb = 2.6 × 107 [kN/m2] and Poisson’s ratio is νb = 0.15 [-], while the soil values are Es = 9500 [kN/m2] and νs = 0.0 [-].

To verify the iterative procedure of Kany/ El Gendy (1997) and evaluate its accuracy for interactive large system of rigid rafts, consider the example 2 in the User’s Guide of program STAPLA (Kany (1976)). The computed settlements obtained from the iterative procedure are compared with those of program STAPLA (Kany (1976)).

For a sewerage station, two isolated containers A and B were constructed simultaneously. Then, lately to extend the station another two isolated containers C and D would be constructed at the same area. Those two external C and D containers would provide an additional settlement on containers A and B.

It is required to assess the tilting of each container and the settlement considering the interaction between the containers through the subsoil at the end of construction. The tilting and settlement of the containers are main factors for designing the pipe connections.

4.2 Settlement behavior of four containers

Besides, the possibility of analysis of large foundation system with many elements by the procedure of Kany/ El Gendy (1997), the mesh of the rigid foundation can be generated in analog mode to the finite element mesh of the elastic foundation in one program. Comparing results from analysis of system of rigid rafts with those of elastic or flexible rafts with the same input data is possible. Subsequently the results of the three analyses are compared in an example.

In this example, the settlements of structures due to interactive analysis of system of rigid, elastic and flexible rafts are studied. This example is chosen from Graßhoff/ Kany (1997). Two large rafts and additional two external footings are constructed near each other.

4.3 Interaction of two rafts considering two additional footings

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. A 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 which will be constructed side by side. Each raft has a side of 12 [m] and 0.5 [m] thickness. Raft I is subjected to a uniform load of 400 [kN/m2], while raft II carries a uniform load of 200 [kN/m2].

4.4 Interaction of two square rafts constructed side by side

A swimming pool is supposed to be constructed at a river. The existing ground around the pool has to be increased up to a meter. The pool has dimensions of 25 [m] × 10 [m] and maximum water depth of 1.20 [m] as shown in Figure 4.32. The foundation level is 1.45 [m] under the ground surface. Slab and walls are reinforced concrete of concrete grade B 25 with thickness of 25 [cm] for slab and 20 [cm] for walls. It is divided into two independent parts through a joint at the pool middle. The filling material around the pool is non-cohesive soil (Figures 4.33 and 4.34). The filling is supposed to be carried out after finishing the pool.

In this example, it is required to study the following:
i) Influence of the joint on the settlements, contact pressures and internal forces of the pool slab and the pool walls in case of the pool is completely filled by water
ii) Influence of the ground rising by additional filling soil material at the southern part of the pool on the settlement

4.5 Analysis of a swimming pool

For comparison with complex foundation rigidity problems, no solution is yet available. Therefore, for judgment on the analysis of El Gendy (1998) to find the system rigidity of foundation, consider the simple example of raft foundation shown in Figure 5.2. The raft has dimensions of 12 [m] × 12 [m] and carries four symmetrical and equal loads, each of P = 9000 [kN]. The raft rests on a homogenous soil layer of thickness 20 [m]. Young’s modulus of the raft and soil materials is Eb = 2 × 107 [kN/m2] and Es = 10000 [kN/m2], respectively. Poisson’s ratio of the raft material is νb = 0.15.

5.1 Rigidity of a simple square raft

A general numerical example is carried out to show the applicability of system rigidity analysis, proposed by El Gendy (1998), to find the rigid thickness of rafts of any shape considering re-entrant corner and opening within the rafts.

In one case the raft carries many types of external loads; concentrated loads, distributed load, line load and moments in x- and y-direction as shown in Figure 5.9. The raft parameters are Young’s modulus Eb = 2 × 107 [kN/m2] and Poisson’s ratio νb = 0.25. Level of foundation is df = 2.7 [m].

5.2 Rigidity of irregular raft on irregular subsoil

Ribbed raft may be used for many structures with 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, it still is 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 save 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] and transmits equal loads for all 25 columns, each of 1000 [kN]. The loads give average contact pressure on soil qav = 62.5 [kN/m2]. Columns are equally spaced, 4 [m] apart, in each direction as shown in Figure 5.14.

5.3 Effect of girders on the raft rigidity

El Arabi/ El Gendy (2001) examined the structural analysis and design of the three common foundation systems: raft, grid and isolated footings. They carried out the examination to evaluate the different types of structural systems in order to decide the most suitable ones for a specific situation. Here, an example is chosen from the above study with some modifications. Consider the foundation system shown in Figure 5.22, which may be designed as raft or grid. The raft dimensions are 30.5 [m] × 30.5 [m] while the overall grid dimensions are 33.0 [m] × 33.0 [m], with a constant strip width in both directions. The foundation carries 49 column loads, which are equally spaced, 5.0 [m] apart, in each direction. Column loads and the arrangement of columns are shown also in Figure 5.22.

5.4 Comparison between raft and grid foundations

This example was carried out to show the influence of flexure rigidity of the superstructure on the settlements and contact pressures for a raft of high rise building.

It is required to analyze a raft for the building shown in Figure 6.6 in three simplified sections. The building is a reinforced concrete skeleton structure and 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, an 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

6.1 Analysis of a raft for a high rise building

To verify the iterative procedure and evaluate its accuracy, a five-storey building resting on foundation through 36 columns is considered. The building is composed of five bays in both x- and y-directions, each bay is 5.0 [m] span. The height of the first storey is 4.0 [m] while the height of the other storeys is 3 [m]. The typical floor of the five storeys is chosen to be skew paneled beams as shown in Figure 6.11. The dimensions and loads of floor beams are shown in Table 6.4. The foundation is a grid type with 0.5 [m] thickness and 2.5 [m] breadth, Figure 6.12. The columns are square cross sections, the column models and dimensions for each storey are shown in Table 6.5.
The building material is reinforced concrete and has the following properties:
Young’s modulus Eb = 3 × 107 [kN/m2]
Poisson’s ratio νb = 0.15 [-]
Shear modulus Gb = 1.3 × 107 [kN/m2]
The soil mass below the foundation is idealized as Winkler’s medium. The modulus of subgrade reaction of the soil ks is 40000 [kN/m3].

6.2 Verification of the iterative procedure

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/m2]. 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.

6.3 Analysis of structure on nonlinear soil medium

To verify the nonlinear analysis of the program ELPLA for Winkler’s soil model, the results of a square footing resting on elastic springs obtained through nonlinear analysis by Hasnien (1993) are compared with those obtained by the program ELPLA.

A flexible square footing of 0.12 [m] thickness with dimensions of 2 [m] × 2 [m] is considered as shown in Figure 7.4.

7.1 Verification of nonlinear analysis for Winkler’s model

For comparison with complex foundation shape, no analytical solution is yet available. Therefore, for judgment on the nonlinear analysis of foundations for simple assumption model, consider the rectangular foundation shown in Figure 7.5. The foundation has the length L = 8.0 [m] and the width B = 6.0 [m]. The foundation carries an eccentric load of N = 2000 [kN]. Both of the x-axis and y-axis are main axes, which intersect in the center of gravity s of the foundation area. The position of resultant N is defined by the ordinates x = ex and y = ey. Within the rectangle foundation area five zones are represented. It is found that the contact area and maximum corner pressure max qo depends on the position of the resultant N in these five zones (Irles/ Irles (1994)). In this example, the maximum corner pressure max qo is obtained using the program ELPLA for each zone and compared with other analytical salutations, which are available for rectangular foundation.

7.2 Rectangular foundation subjected to eccentric loading

Another example is considered to show the applicability of nonlinear analysis of foundations using the program ELPLA for simple assumption model to different foundation types. The results of nonlinear analysis for a circular raft calculated by Teng (1962) are compared with those obtained by the program ELPLA.

A circular raft of radius r = 5 [m] is considered as shown in Figure 7.7. The raft carries an eccentric load of N = 2000 [kN]. The position of the resultant N is defined by the ordinate e.

7.3 Circular foundation subjected to eccentric loading

One of the difficulties by applying the Continuum model to practical problems is the appearance of the high contact pressures at the raft edges, especially when the raft carries heavy loads. The appearance of plastic zones at the raft edges is related to the traditional mathematical soil models used in the analysis, which depend on the theory of elasticity. Therefore, an application example is carried out to show the applicability of the developed nonlinear analysis to redistribute the high contact pressures at the edges of both elastic and rigid rafts.

A rectangular raft with dimensions of 8 × 16 [m2] is chosen and subdivided into 512 square elements. Each element has a side of 0.5 [m] as shown in Figure 7.13. The raft carries a uniform load of 600 [kN/m2].

7.4 Elastoplastic analysis of a raft resting on Continuum medium