This paper has to be cited as: E. Díaz,Tomás R (2014) Revisiting the effect of foundation embedment on elastic settlement: A new approach. Computers and Geotechnics 62, 283–292. The final publication is available at Elsevier via
REVISITING THE EFFECT OF FOUNDATION EMBEDMENT ON ELASTIC SETTLEMENT: A NEW APPROACH
Abstract
The effect of foundation embedment on settlement calculation is a widely researched topic in which there is no scientific consensus regarding the magnitude of settlement reduction. In this paper, a non-linear three dimensional Finite Element analysis has been performed with the aim of evaluating the aforementioned effect. For this purpose, 1800 models were run considering different variables, such as the depth and dimensions of the foundation and the Young´s modulus and Poisson’s ratio of the soil. The settlements from models with foundations at surface level and at depth were then compared and the relationship between them established. The statistical analysis of this data allowed two new expressions, with a mean maximum error of 1.80 %, for the embedment influence factor of a foundation to be proposed and these to be compared with commonly used corrections. The proposed equations were validated by comparing the settlements calculated with the proposed influence factors and the true settlements measured in several real foundations. From the comprehensive study of all modelled cases, an improved approach, when compared to those proposed by other authors, for the calculation of the true elastic settlements of an embedded foundation is proposed.
Keywords: foundations; soil/structure interaction; settlement; elasticity; finite-element modelling; statistical analysis.
1
1. Introduction
Engineers must consider settlements when designing foundations, in order to avoid excessive, unacceptable vertical deformations. Therefore, it is important to calculate foundation settlements as accurately as possible, considering the contact pressure at the base of the foundation. Foundations are usually buried at depth in order to increase the bearing capacity of soil, reduce settlements in the foundation or provide protection against frost, desiccation and erosion. Multiple methods exist for calculating foundation settlements, although due to their ease of use compared with accuracy, the most commonly used in practice are analytical methods based on elastic theory and influence factors.
The general expression for calculating the elastic settlement of a foundation with a uniform net contact pressure distribution (qnet) resting on an elastic, homogeneous and isotropic medium is [1]:
[1]
where S is the settlement of the foundation, B is the foundation width, E is the Young´s modulus of the soil, ν is the Poisson’s ratio of the soil and I is the displacement influence factor.
The displacement influence factor (I) from eq. (1) mainly depends on the foundation shape [1]. However, this factor is also related to the relative foundation-soil stiffness, the existence of incompressible layers at a certain depth, the embedment of the foundation, the variation of E with depth (Gibson’s profile), etc. [1]. Therefore, this factor modifies the value of the calculated settlement according to the conditions of the foundation to provide a more accurate settlement calculation. Among all displacement influence factors considered by Poulos and David [2], Milovic [3] and Mayne and Poulos [1], one of the most controversial is that related to the depth of the foundation (IE).
The first proposed method for quantitatively evaluating the effect of foundation embedment on settlement magnitude was proposed by Fox [4], who studied the relationship between the settlement of a foundation resting at a certain depth and the settlement of the same foundation at the surface. To this end a constant value of 0.5 was considered with a uniformly loaded flexible foundation, applying Mindlin [5] solution. The relationship between both settlements (i.e. the displacement influence factor, IE), obtained for different L/B ratios, was plotted in a graph. The results show that for a given L/B ratio, the greater the embedment depth (D), the lower the displacement influence factor, which varies from 1.0 (for zero embedment depth) to 0.5 (for an infinite embedment depth). The main drawback of this method is that it is only valid for values equal to 0.5. Janbu et al. [6] obtained similar results considering circular and square foundations, resting at different depths. Nishida [7] studied the influence of the depth of a circular foundation considering different values, and concluded that for greater depths, the obtained displacement influence factors are similar to those proposed by Fox [4] for a value equal to 0.5. Schmertmann [8] used cone penetration tests (CPT) to propose an expression considering the embedment effect, concluding that, as Fox [4] and Janbu et al. [6] had previously asserted, the displacement influence factor is lesser than 0.5. Burland [9] revised the IE values proposed by Fox [4] and Janbu et al. [6], by calculating the settlements of circular foundations by means of the FEM. It was concluded that the effect of the foundation depth is lesser than that considered by the aforementioned authors (mainly for cases in which the embedment is high), and proposed a new expression for calculating IE. Butterfield and Banerjee [10], developed a numerical model based on the Mindlin [5] solution for different depths and a rigid circular foundation embedded in an elastic half-space, recommending IE values even lower than those proposed by Fox [4]. Christian and Carrier [11], also performed a detailed analysis of this problem, concluding that the IE values proposed by Fox [4] and Janbu et al. [6] are essentially the same and must be reviewed because they provide unsafe values which underestimate settlements. These authors proposed a chart to consider the existence of a non-deformable layer at a certain depth (depth correction factors, proposed by Giroud [12]) and the effect of the foundation embedment (embedment correction factors proposed by Burland [9]). However, they do not explicitly consider . Pells and Turner [13], reproduced Burland’s calculations [9], concluding that his proposed values are inaccurate. They proposed charts, were derived from FEM analysis, for obtaining IE, considering both rigid and flexible foundations. Yamaguchi [14], proposed an expression which considers soil deformability, the net contact pressure of the foundation and the bearing capacity of the soil. Due to the high number of variables considered by this author the expression is not easy to apply. Bowles [15] advised against using the formula proposed by Fox [4], as the author considered this approach to give unsafe values. Christian and Carrier [16] reaffirmed their previous conclusions [11] in response to the comments made by Bowles [15] about their method. Johnson et al. [17] and Kausel and Ushijima [18] analysed circular and strip foundations with vertical sides in contact with the ground. Gazetas et al. [19] applied a parametric boundary element method and proposed an analytical expression for calculating vertical elastic settlements of rigid arbitrarily shaped foundations embedded in a homogeneous and elastic half-space with a maximum error of 20%. Mei and Xu [20] presented an analytical solution for flexible foundations resting on an elastic half-space, based on the study of the most crucial parameters of Groth and Chapman [21]. Currently, although more reliable models which provide more realistic approximations to the problem (e.g. [14]) exist, geotechnical engineers often use traditional methods such as those proposed by Fox [4], Janbu et al. [6] or Burland [9]. For example, Mayne and Poulos [1], who proposed one of the most complete and widely used formula for the calculation of settlements, recommend the use of the IE values proposed by Burland [9].
Consequently, there is an evident controversy in the use of this displacement influence factor and there is no consensus in the magnitude of settlement reduction due to the effect of foundation embedment. As explained previously, solutions exist in which authors do not consider or E, propose algorithms which are difficult to apply, or propose solutions that are only valid for circular and flexible foundations without considering the shape of the foundation, etc. Additionally, due to the fact that this is a problem which involves soil-structure interaction, it cannot be satisfactorily solved by using analytical solutions based on the theory of elasticity [14].
Therefore, the main aim of this investigation is the study of this question, considering all parameters relevant to the problem (including soil-structure interaction), thereby improving upon the limitations of the existing models and proposing a new method.
In the present study the relationship between the settlement of a foundation at a given depth (Semb) and the settlement of the same foundation at surface level (Ssur) is analysed, in order to obtain the displacement influence factor (IE). For this purpose, different models have been developed, varying the key parameters for solving the problem using ANSYS+CIVILFEM v.11 software. This software allows the modelling of settlements suffered by a foundation by means of the FEM.
Additionally, the paper provides analytical expressions for an accurate consideration of the effect of the foundation depth on the calculated settlement. The study will be focused on the most common range of embedment/width (D/B) ratios, which in practice vary from 0 to 1.
The paper is organized as follows: section 2 describes the effects of the foundation embedment on the calculation of settlements, section 3 details the model and the geotechnical and geometric parameters adopted in the study and examines the most relevant results obtained from the analysis, section 4 presents the proposed analytical formulae for considering the effect of embedment on the calculation of settlement, section 5 shows a validation of the proposed formulae through the analysis of several case studies, and section 6 presents the main conclusions.
2. Effect of embedment on foundation settlement.
The embedment of a foundation has five different effects on the magnitude of vertical settlement [14, 19, 20]: The floating foundation effect, the Gibson effect, the trench effect, the side-wall contact effect and the retaining wall effect. The explanation of these effects is as follows:
-Reduction of the net pressure (qnet) transmitted by the foundation (floating foundation effect), that is related to the stress discharge (q0)effect due to excavation (Figure 1b) which can be calculated as:
[2]
where is the gross pressure transmitted to the subsoil by a foundation, is the soil unit weight and D is the depth of the foundation base.
-Deformation modulus usually increases with depth [22, 23] (Figure 1b). Consequently, foundations at greater depth transmit pressure to a stiffer soil than if they were resting on the surface, and as such show less settlement in the same loading conditions.
-The third factor to be considered is the so called trench effect (Figure 1c). This effect refers to the fact that in a homogeneous half-space, the settlement of a foundation placed at the bottom of a trench is smaller than that of the same foundation resting on the ground surface. This effect is illustrated in Figure 1c and is due to the normal () and shear () stresses from the overlying soil, which restrict the displacement of the horizontal plane in contact with the foundation base, thereby reducing the settlements. In the case of a foundation at surface level (Figure 1a) the deformation of the aforementioned plane is not influenced by any external stresses, and consequently the expected settlements are greater. Erden [24] experimentally observed that this effect takes place even in cases in which non lateral contact between the foundation and the soil exists, and that it is dependent on the stress level acting over the foundation base (i.e. the pressure in the surrounding soil at the foundation base level due to soil overburden, q0).
-The fourth effect is due to the lateral contact between the foundation and the soil (sidewall contact effect; Figure 1d). When a foundation’s vertical faces are in contact with the adjacent soil, a portion of the applied load is transmitted to the ground as shear stresses () acting along the lateral faces of the foundation. The reduction of the load transmitted by the foundation with sidewall contact compared to the same foundation without lateral contact also implies a reduction in settlement.
-The last factor is called the retaining wall effect [20], and is caused by the deformation of the trench walls towards the excavated area due to being limited by the lateral faces of the foundation, which has the positive effect of reducing vertical deformations (Figure 1e).
Figure 1. Illustration of (a) a surface foundation with a stress-free surface (q0=0), (b) reduction of net pressure (qnet) transmitted to the soil and increase in Young's modulus (E(z)) with depth (Gibson profile) effects, (c) trench effect, (d) sidewall effect, and (e) retaining wall effect (based on [14, 19, 20]).
The effect of the foundation embedment on settlement magnitude of settlement is due to the aforementioned causes. Thus, it is difficult to provide a satisfactory explanation of the problem through the application of numerical methods based on the theory of elasticity [14, 19]. In order to take into account these five effects, it is necessary to use a tool which is able to consider nonlinear soil-structure behaviour, solve a three-dimensional problem, include a “Gibson profile”, take into account the existing natural horizontal and vertical stress level at the foundation plane before excavation using the command ~TIS from CivilFEM and the deformation restrictions on foundation planes at depth due to the pressure caused by overburden.
3. Settlement modelling
The model used in this investigation for the study of the effect of foundation embedment is described in detail in section 3.1. Additionally, the main results of the modelling are shown in subsection 3.2.
3.1. Geometry and parameters
A nonlinear three-dimensional model was created using ANSYS+CIVILFEM version 11 [25-27] in order to establish a relationship between the settlement from a foundation buried at depth (Semb) and a surface foundation (Ssur) resting over the same soil (i.e. with the same geotechnical properties and with a variable E value with depth as shown in Figure 1b and later described in detail in this section). This was expressed through the IE value, defined as follows:
[3]
The soil and the foundation were meshed using 20-node quadratic solid elements called SOLID95, with three degrees of freedom per node (i.e. translation in the nodal x, y, and z directions). These elements were selected because they allow the modelling of soil-foundation contacts and irregular shapes (e.g. the foundation corners) without a loss of accuracy.
The soil-foundation contact was represented by means of a surface-surface contact model using TARGE170 and CONTA174 elements that accurately represent the reality of the contact [28]. These are elements without thickness or stiffness that are defined by means of the nodes from both areas (soil and foundation), which are initially at the same position.
TARGE170 elements were employed for the lateral faces and the base of the foundation and CONTA174 elements for the soil in contact with the foundation, interacting with TARGE170 elements. For modelling the friction between the foundation and the soil, a keyoption for CONTA174 elements allows the friction coefficient between materials to be considered.
The foundation was considered by ANSYS to be made of a concrete type material in accordance with Eurocode 2 [29], with a compressive strength of 25 MPa (C25/30 [29]), because this is one of the most commonly used concretes in practice. A centred vertical load was considered in the modelling. The foundation was considered to be rigid with a rough interface with the soil, due to the material being reinforced concrete. Note that this rough interface has been simulated using a Mohr-Coulomb law in the soil-footing contact. Consequently, horizontal displacement can only occur when the maximum shear force given by the Mohr-Coulomb law is exceeded. This shear force is proportional to the normal force and to the friction angle. In this case, a friction angle equals to 2/3 of the friction angle of the soil has been adopted as recommended by
[30].
From a geometric point of view, different width-length-embedment combinations were considered and compared with foundations with the same geometry at surface level.
Because the estimation of settlements is performed by applying formulae based on the theory of elasticity containing E and only, a surface Young´s modulus of soil has been considered (E0) and this value has been linearly increased with depth according to the vertical effective stress level considering an increase rate of 300 kN/m2 per meter of depth. This can be considered a conservative value for settlement calculation[31, 32].
After the initial stress state is calculated, the foundation is loaded with the net contact pressure.
Due to the symmetrical nature of the problem, only half of the model was defined, and symmetry boundary conditions in the middle plane were considered, which represented a significant resource optimization.
Regarding the model meshing, mesh refinement was performed in the area affected by the foundation pressure bulb. For this purpose rectangular prismatic elements were used, with a variable size depending on the zone of the model, considering 0.1 times the width of the foundation in the area of the foundation bulb and gradually increasing this value to 0.5 times the width of the foundation at the model boundaries (Figure 2).
The lower boundary, whose vertical displacements were impeded, was modified as a function of the dimensions of the foundation, varying from 20 to 25 times the width of the foundation (Figure 2).
The lateral borders of the model were positioned at a distance of between 6 and 12 times the foundation width from the edge of the foundation, to avoid modifying the stress distribution (Figure 2). Horizontal displacements were impeded on these lateral borders.
The validity of the mesh size and the model borders was verified through an optimization study. In this study, several analyses were performed considering different geometric configurations of the foundation used in this study, gradually reducing the size of the mesh and increasing the limits of the model to find the values for which the calculated settlements in consecutive models were constant.
Once the mesh selection was optimized, the model was verified. For this purpose, the settlements obtained from the FEM models and from Schleicher´s manual methods [33] were compared considering a foundation resting at surface level (i.e. not affected by embedment effects). From the comparison of both sets of data, it may be observed that the results provided by both methods were very similar and the differences irrelevant, providing a mean error of 4.65% (1.56%). Consequently, it may be concluded that the selected mesh sizes and model boundaries are valid for the purpose of this research.