Abstract
In a combined piled raft foundation (CPRF) both raft and piles take their share of the total load applied. However, in practice, the contribution of a raft in taking load is usually ignored and the load is assumed to be supported on piles. This way of CPRF becomes excessively conservative and uneconomical. To economize the design, relative load sharing of raft and piles in CPRF has to be found. In this connection, different simplified methods have been developed, each one with some limitations. In this study, three simplified methods have been applied to two cases of pile-raft systems. The methods include Randolph, Poulos-Davis-Randolph, and modified Poulos-Davis-Randolph. The first case is a hypothetical case consisting of a 12 m × 12 m raft supported on a square group of nine piles. The second case study is an actual eight-story building to be constructed in Peshawar, Pakistan. The building is supported on a pile-raft system, with raft resting on very soft clay underlain by dense sand. The two case studies are also modelled in the finite element program PLAXIS 3D for comparison. The results of all the simplified methods are comparable with PLAXIS 3D. However, the Randolph method is much closer to PLAXIS 3D for the two cases studied. Furthermore, it is also shown that piles in a piled raft system can be used as “stress reducers” as well as a “settlement reducers.” Additionally, the effect of interaction factors is also evaluated with the s/d ratio as well as with varying soil stiffness. It was concluded that ignoring these factors leads to a very unsafe design of the pile-raft system.
1. Introduction
CPRF system consists of three foundation-bearing elements: piles, raft, and supporting soil. The serviceability and stability of a structure are ensured by transferring the superstructure load to the soil through the foundation system by one of these three components of the foundation system. Piles are used to carry out heavy structure load to deeper bearing strata while in the case of rafts, the loads are solely resisted by raft foundation. The third case which is very effective in carrying out heavy structural loads is the combined foundation system in which both piles and raft contribute in resisting lateral and vertical loads. The foundation system is a very complex structure leading to very problematic and complex phenomena and hence various researchers concluded different theories. Conventionally it was assumed that there is no contribution of the raft in resisting loads [1]. However, the raft has direct contact with soil and hence contributes to a significant resistance of loads leading to a too conservative approach. Reference [2] carried out research on clays subjected to huge structural loads produce stresses. They concluded that, in this case, some stresses are resisted by raft and remaining by pile creep state of loading.
The thickness of raft, length and dimension of pile, location of piles in raft, the soil properties, and stiffness of pile and raft play a crucial role in the analysis and design of CPRF foundation [3]. Applied load from the superstructure is transferred to the soil through these foundations bearing elements raft and pile by taking into account interaction between these foundations bearing elements. There are mainly two types of interaction which are pile-to-pile interaction and pile-to-raft interaction. These interaction factors are highly dependent upon the elastic modulus of soil, s/d ratio of piles, and length of piles. Ignoring these interactions will lead to unsafe design and will underestimate settlement and bending moments in raft [4]. Many researchers focused on the seismic response of pile-raft foundation systems considering seismic soil-structure interaction idealizing superstructure as a single degree of freedom system [5]. For proper load sharing between pile-raft components, consideration of interaction factors between pile-raft components is of utmost necessity to be taken in the design phase of the foundation system [6]. Similarly, for piled raft foundation design, [7] addressed a few very important issues. The first point he stressed is the limit states considered in the design of pile-rafts which are moment loads, vertical and horizontal loads, moments of the raft, differential settlements, and shear forces. The following minimum criteria must be followed at the preliminary design stage:(i)Loading capacity (ultimate).(ii)Moments and shear forces along with the raft and piles.(iii)Maximum and differential settlement.
The following procedures must be followed in light of the aforementioned philosophies.(i)Feasibility study of the provision of piled raft system and the number of piles in the preliminary phase.(ii)Relevant features and behavior of the pile-raft system.(iii)For piles numbers, configuration and number of piles, designed charts should be reviewed.
Every element in a piled raft system has its advantages. The raft can contribute to taking some percentage of load and can reduce substantially differential settlement depending upon raft thickness. Piles can be used as “stress reducers” as well as a “settlement reducers” [8, 9] in addition to an increase in the bearing capacity of the soil. Taking into account this contribution of both elements, piled raft foundation can be made very economical in terms of several piles, length of piles, and reinforcement in the raft.
Researchers have put their efforts to solve the problem of load sharing and analysis of CPRF; for example, [8, 10, 11] have proposed some simplified methods involving several simplifications. According to [12], piles contribute to 50–80% of the total superstructure load while the rest is resisted by raft. Another researcher [13] found about the contribution of the raft as 30–60% of the total load depending on the underneath soil, length of the piles, and spacing between piles. They concluded that this percentage decreases as the spacing between them decreases and the length of the pile increases. They also found that, for serviceability and bearing capacity problems in the case of soft clays, piled raft system is highly recommended. According to [14], rafts contribute to building loads up to 50%. Reference [15] reported that an average of 36% load is carried out by raft in a pile-raft system. Other approximate methods like “strip on springs” are proposed by [16] in which raft is represented by strip and piles as springs and “plate on springs” method is proposed by [17] in which raft is represented by plate and piles as springs. Finite Elements software like SAP2000, method proposed by [18], and PLAXIS 3D foundation [19] can provide a good option for numerical analysis. FEM is used to obtain an approximate solution to various nonlinear engineering problems. Numerical methods are the best approach in which the finite element method is the most popular one. Initially, this method was used as a powerful tool for the problems relating to structural engineering as an extension of matrix methods but later on, it is considered as the best approach in other fields of engineering like soil mechanics, fluid mechanics and rock mechanics, etc. [20]. Continuum in this method has the distribution of elements which is further divided into several nodes which is further subdivided into the degree of freedom.
For structure design of raft in a piled raft system, there are several simplified approaches, which include method in [8] and Winkler model for piled raft foundation system (WMPR) method [5]. These methods can provide reasonable results for preliminary analysis of rafts in a piled raft system.
1.1. Randolph Method (1994)
In this method response of equivalent raft with a single pile is used to predict the response of the whole CPRF system. Using stiffness of raft in isolation and pile group stiffness, combined piled raft stiffness can be calculated usingand in this equation, Kpr is piled raft stiffness, Kp is the pile group stiffness, kr is only raft stiffness, and arp is the pile-raft interaction factor and can be calculated fromwhere ζ = ln(rm/rp), = {0.25 + ξ [2.5 ρ (1 − υ) − 0.25] ∗ L}, ξ = Esl/Esb, ρ = Esav/Esl, rc = average radius of pile cap which is equal to total area of raft divided by number of piles, = maximum radius of influence of raft, ro = radius of the pile, L = length of pile, Esl = soil Young’s modulus at pile tip level, Esb = soil Young’s modulus below pile tip at bearing stratum, and Esav = average soil Young’s modulus along the pile shaft.
The load taking percentage between piles and raft is calculated by the following;
“Pr” is the percentage of load taken by raft, “Pt” is the total load applied, and Pp is the load taken by piles. Equation (4) can be used to find an average settlement of the combined piled raft foundation:where Papp is the load applied on the CPRF system and Kpr is the combined piled raft system.
Using this method, the average settlement of the whole CPRF system and the percentage load taken by piles and raft can be estimated. To incorporate the variation of stiffness along with the depth of soil, Randolph includes stiffness of soil along the pile shaft, at pile head, and at pile tip. Randolph does not consider the flexibility of raft and strength properties of soil (cohesion, friction angle).
1.2. Poulos-Davis-Randolph Method (2001)
Trilinear settlement curve is used to establish this method. Using this method, find a point (load “P1”) at which pile capacity is fully mobilized and the ultimate capacity “Pu” of CPRF. Using stiffness of pile group “Kp” and stiffness of raft “Kr” combined piled raft stiffness can be calculated “Kpr.” Using this method, load sharing percent between piles and raft can be estimated. Pile-raft interaction factor and stiffness of raft can be calculated from Randolph method and elastic theory, respectively [14].
The combined piled raft stiffness can be calculated from equation (1). The load taking percentage between piles and raft is calculated by equation (3). “Pr” is the percentage of load taken by raft, “Pt” is the total load applied, and Pp is the load taken by piles. Equation (4) can be used to find an average settlement of the combined piled raft foundation.
Consider this assumption that a point can be found which is actually a load level at which full mobilization of pile capacity will occur “P1” fromwhere “Pu” is the ultimate load capacity of the piles in the group and X = percentage of the total load taken by the raft.
After this load level, raft stiffness “Kr” of the CPRF system will act alone, and this will detain up till the ultimate load capacity “Pu” of the piled raft foundation system is reached. At this stage, the curve of the load-settlement relationship becomes parallel to the settlement axis.
1.3. Modified Poulos-Davis-Randolph Method
For this method, a hyperbolic settlement curve for piled raft system is used instead of a trilinear settlement curve. Also, in this method, the stiffness of raft and piles will change according to the applied load. Using this method, an allowable settlement for CPRF can be determined and load level at which piled raft system gets nonlinear and subsequently nonlinear behavior of piled raft system can be captured. To avoid the plastic behavior of the piled raft system, it is focused on not exceeding the nonlinear load level.
The load extent at which the CPRF system gets nonlinear is denoted by . SA indicates the allowable settlement of CPRF system. To eliminate plastic behavior of the CPRF system and excessive overall settlement, it is always aimed in the design phase not to surpass the point . The load can be found using the equation given below:where “” is pile group ultimate load level and “” is a percentage of load taken by piles.
CPRF stiffness can be found as shown in equation (7) below:where X = , “Kr” is secant stiffness of raft, and “Kp” is the secant stiffness of the pile group. “” load proportion carried by piles can be found out bywhere .
Pile group and raft alone secant stiffness are calculated by equations (9) and (10):Kp, Kr are pile group and raft secant stiffness, respectively. Kri, Kpi are initial stiffness of the raft and pile group, respectively. Rfr, Rfp are hyperbolic factors for raft and pile group, respectively. , are load taken by raft and piles, respectively. , are ultimate capacity of the raft and pile group, respectively.
It is recommended to use 0.75 and 0.5 for Rfr and Rfp, respectively, according to Poulos [6].
Piles and raft taken load from the applied load are given by equations (11) and (12):
For two different condition, the overall displacement of the CPRF system can be calculated as follows by equations (13) and (14):
Up to :
After ,
An iterative procedure is involved in this method because of the changing of the secant stiffness and load sharing percentage at different load extents. Due to this reason, for first calculation value of is assumed (recommended is 0.85).
1.4. Burland Method
As one of the simplified approaches for the geotechnical and structural design of piled raft systems, this approach is applicable only when piles are developing their full geotechnical capacity. In this method, the load is calculated corresponding to an allowable settlement of raft out of total load. The remaining excess load is assumed to be carried by piles. Bending moments in a raft of piled raft system can be calculated using a reduced column load “Qr” which is given bywhere “Q” is total column load and “Pus” is the pile geotechnical capacity without the factor of safety.
1.5. Winkler Model for Piled Raft System (WMPR)
WMPR method is based on the Randolph method. Adjusted stiffness for piles and raft can be calculated based on settlement results of the Randolph method and assigned as Winkler Springs in any FEM software. Pile-pile and pile-raft interaction factors were incorporated using Randolph (1994) equations and Poulos (2000) equations while finding this stiffness. Bending moments in the raft of a piled raft system can be calculated using this simplified approach. The flow diagram for the WMPR method is shown in Figure 1.
2. Methodology
For this research, the abovementioned methods were applied to two case studies. The first case study is an assumed ideal piled raft of 12 m × 12 m supported on a square group of nine piles while the second case study is an actual piled raft of dimensions 32 m × 83 m with 132 piles underneath an eight-story building to be constructed in Peshawar, Pakistan. Both case studies are also analyzed using FEM software PLAXIS 3D. FEM PLAXIS 3D analysis is used to get better insight into the percentage distribution of load between pile-raft components and settlement analysis. PLAXIS 3D Foundation is a high-performance 3-dimensional FEM software package. It can do nonlinear, static, and dynamic analysis for a large spectrum of engineering problems. PLAXIS 3D can take into account interaction factors by itself subjected to accurate modelling like interface elements and soil constitutive model. To model the structural behavior of the combined piled raft system, special types of elements are used. Plate elements are used for the raft. Plate elements consist of six-nodded triangular elements with six degrees of freedom at each node. Embedded piles which are model like beam elements can be used to model pile response. The main advantage of using embedded pile for modelling pile behavior is its ability to take interaction effects from the surrounding soil using skin resistance and base resistance. Embedded piles can be modelled within the soil in any direction. Using PLAXIS 3D, someone can easily find settlements of raft and piles, bending moments in raft and piles, and percentage of load carried by piles and raft if input parameters, interface elements, and constitutive model are provided accurately to PLAXIS 3D. Lateral load analysis of the CPRF system can be done easily in PLAXIS 3D.
In this case, the constitutive model is used as Mohr-Coulomb for soil modelling due to its simplicity and commercial usage. Raft and piles are modelled as a plate and beam element, respectively. In 2nd case study, two layers of the soil are modelled in which the top layer is modelled as clay and the bottom layer as sand.
Also, analysis of unpiled raft is performed to show usage of piles as “Stress Reducers” and “Settlement Reducers.”
2.1. Case Study (1)
This study includes the 3D finite element analysis of pile-raft systems, carried out in PLAXIS 3D in comparison with the abovementioned techniques. In this case, an ideal pile-raft foundation system is selected with 12 m × 12 m raft having nine piles. Figure 2(a) shows a typical layout of the model with a cross-section in Figure 2(b). The properties of soil, raft, and pile are mentioned in Tables 1–3. These properties are ideal and selected based on experience. Figure 2(c) shows a finite element meshed model. For pile-raft ad soil-pile interfaces, interfacial elements are defined and for all elements, fine mesh is used.
(a)
(b)
(c)
2.1.1. Analysis Results of Case Study 1
The results of the analysis are shown in Table 4. Also, in this case, all methods showed that more than 20 percent load is taken by raft supported on the soil. A trilinear settlement curve was drawn for Poulos-Davis-Randolph method as shown below in Figure 3. The load at which pile capacity is fully mobilized (P1) is calculated as 30 MN and the ultimate capacity “Pu” of piled raft foundation is 60 MN.
From the above comparative study, it can be seen that Randolph and modified Poulos-Davis-Randolph approaches are almost close to the results of PLAXIS 3D in the case of percentage load taking, showing its viability in the analysis of piled raft system.
2.2. Case Study (2)
For the second case study, piled raft of an eight-story building that was to be constructed in Peshawar is selected. Raft dimension was 32 m × 83 m having 0.76 m thickness with 132 piles underneath. The diameter of piles is 0.6 m with an embedded length of 25 m. The total weight of the building was 242 MN including self-weight of raft. The soil, pile, and raft properties are discussed below. An architectural 3D view of the proposed constructed building is shown in Figure 4. The raft layout and position of piles are shown in Figure 5. The geotechnical profile of the proposed site is also shown in Figure 6. A 15 m soft clayey soil layer is present at the top, which makes the foundation design challenging. PLAXIS 3D models are shown in Figures 7(a) and 7(b).
(a)
(b)
2.2.1. Parameter Properties
(1) Soil Properties. Geotechnical investigation of this site was performed by drilling two boreholes of depth of 45 m and 33 m. SPT, Shelby, and split spoon samples, moisture content, Atterberg limits, sieve analysis, and unconfined compression test were also performed. Using this data, elastic Young modulus of soil for soil layers is calculated using SPT correlations mentioned in [22]. Soil elastic modulus for clayey soil is calculated as 30 MPa and 55 MPa for sandy soil. Poisson ratio is selected as 0.4 for both layers.
(2) Raft and Pile Properties. Raft and pile properties were based on a concrete strength of 27.6 MPa. The modulus of elasticity for both elements was selected as 24.83 × 106 kPa based on the concrete strength. A value of 0.2 was assumed as a Poisson ratio. Unit weight of concrete is 23.5 kN/m3. The thickness of the raft was 0.7 m. The pile length and diameter selected are 25 m and 0.61 m, respectively, based on the geotechnical requirements.
2.2.2. Analysis Results of Case Study 2
By analyzing the piled raft using different methods in comparison with PLAXIS 3D, the analysis result is tabulated in Table 5.
In this case, the above-tabulated results show that the Randolph approach is showing converging results with PLAXIS 3D in both load taking and settlement while the other two methods show somehow deviation in this case.
3. Piles as a “Stress and Settlement Reducers”
Analysis was performed for an isolated raft to show piles as “settlement reducers” and “stress reducers.” Settlement of piled raft with piles and without piles is shown in Table 6. Usage of piles as “stress reducers” was shown by taking two cross-sections of bending moment from PLAXIS 3D analysis, one on a two-story side while the other on an eight-story side as shown in Figures 8 and 9, respectively.
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(b)
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(b)
Table 6 shows that provision of piles in pile-raft system decreases settlement up to 2 times only in the raft. Additionally, differential settlement is also largely reduced as shown in Table 6. Similarly, the results of bending moments show that piles cause a high reduction in both positive and negative bending moments illustrated in Figures 8 and 9.
4. Interaction Factors
Interaction factors between the elements of piled raft system are very important during the analysis of piled raft system because ignoring interaction factors will lead to unsafe design. In this study, interaction factors are calculated for specific cases using Randolph formulation for pile-raft interaction and the concept of Poulos was used to find out pile-pile interaction factor.
4.1. Pile-Raft Interaction Factor
Using the formulation of Randolph shown in equation (2), pile-soil-raft interaction can be calculated for any spacing of piles as well for any raft dimensions. Using parameters of Table 7, the pile-soil-raft interaction curve has been drawn for various s/d ratios. The number of piles is 9, spaced uniformly around the raft dimension.
Using the Randolph procedure, a curve is drawn shown in Figure 10 for pile-raft interaction factor with different s/d ratios.
It can be concluded from the curve drawn that as pile spacing increases or the diameter of pile decreases, the interaction of pile on raft decreases.
4.2. Pile-Soil-Pile Interaction Factor
The concept of Poulos has been used to find interaction factors using FEM software PLAXIS 3D. Poulos suggested to first find the settlement of a single pile and then find an additional settlement of the same pile due to adjacent loaded pile such that load applied is the same on both piles. It can be written as
In PLAXIS 3D, two piles with different s/d can be model for finding the additional settlement caused by the adjacent pile and consequently the pile-soil-pile interaction using equation (16). Approximately pile-soil-pile interaction can be incorporated in piled raft stiffness by multiplying the single pile stiffness with square root of number of piles [23]. A curve for pile-soil-pile interaction has been drawn which is shown in Figure 11 using parameters of Table 7. The calculation has been performed for two values of soil young’s modulus which are 40 MPa and 70 MPa. Effect of varying soil young’s modulus is also in Figure 11.
It is clearly shown from the figure that there is an inverse relation between pile spacing and pile-soil-pile interaction factor. It can also be concluded from the above figure that as soil young’s modulus increases, interaction factors decrease and the trend is continuous for all points selected for this study.
4.3. Effect of Ignoring Interaction Factor
A study has been conducted to find an effect of ignoring one interaction factor or both interaction factors on pile-raft stiffness. Randolph method of analysis has been selected for this procedure. The parameters which have been selected for this study are shown below in Table 8.
After performing calculations, using the Randolph method the differences in the stiffness by ignoring one or both interaction factors are tabled below in Table 9.
This is clearly shown from the table above that, by ignoring interaction factors, stiffness of piled raft foundation is overestimated. This overestimation of stiffness of piled raft foundation will lead to an unsafe design which is shown in the following study.
4.4. Effect of Stiffness on Bending Moment and Settlement
Another study was conducted where the effect of combined piled raft stiffness had been studied concerning bending moment in raft and settlement of the foundation system. The values selected are just randomly because the primary objective was to find the effect of increasing stiffness on bending moment and settlement of the piled raft system. Raft dimensions selected are 10 × 6 m, the thickness of raft is 0.5 m, and load applied is 12 MPa at center of the raft. The values of combined piled raft stiffness can be calculated using the Randolph method from equation (1) and are tabled below in Table 10.
The results are plotted against the stiffness and are shown below in Figure 12 for maximum positive bending moment at center of the raft and Figure 13 for total settlement. The maximum value of bending moment was considered against each stiffness value and absolute total settlement was taken at the middle of raft for each stiffness value.
It can be concluded from the figures drawn above that overestimation of the combined piled raft stiffness will lead to unsafe design, because both bending moment and settlement are decreasing with an increase in stiffness values. It means, by ignoring interaction factors, you are overestimating the stiffness which will lead to underestimation of bending moment in the raft as well as overall settlement of the piled raft system and it is also observed clearly from Table 9.
5. Conclusions
The analysis conducted shows that, in both case studies results, as compared to other methods, Randolph method gives results that are in close agreement to the numerical approach. So, it is concluded that, in the case of preliminary analysis of piled raft design, the Randolph method is recommended if someone does not have access to numerical FEM software. Piles in addition to geotechnical advantage can also be used to reduce the state of stress in raft and settlement of piled raft foundations. Hence, piled raft foundation can be made very economical by incorporating these aspects in piled raft design. From an interaction factor point of view, the following conclusions are made:(i)By ignoring interaction factors, stiffness of the combined piled raft foundation is overestimated.(ii)Overestimation of stiffness will cause underestimation of bending moment and settlement of piled raft foundation.(iii)It means ignoring interaction factors will lead to unsafe design.(iv)It can be also concluded that pile-soil-raft interaction is affecting stiffness more than pile-soil-pile interaction.
Data Availability
All the data used in this research work are present within the article and are allowed to anyone for using and verification purposes.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.