Experimental–Numerical Investigation of Embedment Effect on Foundation Behavior Under Vertical Loading

Abstract

Embedded foundations have different behavioral characteristics in comparison to surface ones. In present study, to evaluate the behavioral aspects of five strip foundations with constant width and different embedment depths in sandy soil under monotonic vertical loading, experimental and numerical finite element (FE) modelings were implemented. Based on experimental results, the growth of foundation depth-to-width ratio from 0 to 1.2 caused the increase in ultimate bearing capacity and vertical stiffness for 33% and 23%, respectively. Moreover, regarding to the observed damages, an attempt was made to present an innovative categorization for dominant failure mode as a function of foundation embedment depth. The result of numerical framework designed in accordance with real material and interfaces properties showed a good agreement with experimental behavioral curves as well as ultimate bearing capacity and vertical stiffness. Achieved from FE modeling, the adequate dimensions for underlying soil model were suggested to eliminate any effect of boundary conditions on foundation vertical loading results. Finally, the validity of a common theoretical expression in literature for anticipating vertical stiffness of embedded foundation was explored. It was concluded that for considered domain of foundation embedment depth, the increase rate of vertical stiffness vs. foundation embedment curve achieved through theoretical expression was more than experimental and numerical results.

1 Introduction

Using embedded foundations is usual for some structures such as buildings with basements, bridges with caissons or embedded shafts, radio masts and antennas. For these structures, behavioral characteristics of embedded foundations differ from those of shallow foundations. One of the most important characteristics is ultimate bearing capacity of foundation. The importance of ultimate bearing capacity is due to the need for safety margin to prevent complete failure of foundation as a criterion of foundation design. Moreover, ultimate bearing capacity controls the total settlement experienced by the foundation and superstructure. In addition, the ratio of ultimate bearing capacity to vertical service stress under the foundation shows the dominant performance of foundation in the seismic excitation. In another word, this ratio known as safety factor against vertical loading (FS_V), determines the foundation behavior in uplift or sinking modes during lateral loading [1,2,3,4]. Vertical stiffness per unit area is another behavioral feature of the foundation named foundation subgrade modulus. Vertical stiffness of foundation is significant to control the foundation settlement in the soil linear behavioral phase. In literature, some researchers [5, 6] have focused on this parameter and presented expressions to calculate it. Accordingly, some research works in literature on bearing capacity and vertical stiffness of embedded foundations are introduced as below.

Terzaghi, Meyerhof, Hansen and Sun et al. [7,8,9,10] presented some theories to estimate ultimate bearing capacity of shallow and embedded foundations under vertical loading. The basis of the theories is algebraic relationships to attain ultimate vertical load which is presented as a function of soil strength parameters, soil density and foundation geometry. In these theories, the effect of embedment on ultimate bearing capacity has been considered through depth semi-empirical coefficients causing the ultimate bearing capacity of embedded foundation to increase compared to that for the shallow foundation with the same geometric dimensions. In spite of the wide application of this traditional method in foundation engineering, the results obtained from the mentioned method are quite approximate because of its semi-empirical nature.

In an advanced point of view, some researchers [11,12,13,14] considered failure state for foundations under the load combination including axial force, horizontal force and bending moment. These research fields led to present an envelope surface in Cartesian coordinates, named foundation bearing capacity surface. Accordingly, when the foundation load combination state meets the abovementioned surface, the failure of foundation is mobilized.

Some studies in literature were allocated to analytical evaluation of ultimate bearing capacity of foundations located on layered soil. Haghbin [15] presented an analytical method to determine the ultimate bearing capacity of strip foundations with/without embedment depth in the layered soil. This procedure considered ultimate bearing capacity in accordance with foundation underlying soil as well as virtual retaining walls. Determination of soil failure surface depth, consideration of second layer effects on foundation ultimate bearing capacity and investigation of underlying soil improvement on vertical response of foundation were such highlights of the mentioned research. Using concept of soil characteristic stress line, Lotfalizadeh and Kamalian [16] suggested a novel algorithm to calculate ultimate bearing capacity of strip surface foundation on the two-layer sandy soil. Based on relative strength of soil layers, they emphasized on factors controlling foundation ultimate bearing capacity. Moreover, in a generalized point of view, some researchers [17, 18] focused on the performance of various foundations on layered soil under the combined loading condition.

Based on two- and three-dimensional finite element limit analyses, Salgado et al. [19] achieved bearing capacities of rough strip, square, circular and rectangular foundations. Accordingly, they proposed accurate functions for shape and depth factors for foundation on clay. Regarding plasticity theorems of Hill [20], Drucker et al. [21] and Drucker and Prager [22], they stated that the presented results reduce the uncertainties in accordance with the bearing capacity equation.

Gajan et al. [1] carried out a series of horizontal and vertical push tests on embedded foundations. They considered foundation ultimate bearing capacity and vertical stiffness in their study. For cyclic vertical loading, they concluded that the stiffness of loading–unloading phase is greater than that for pure loading branch.

Patra et al. [23] investigated the ultimate bearing capacity of the embedded foundations in the granular soil—with and without geogrid reinforcement—through an experimental program. The results determined that with the increase in embedment depth, ultimate bearing capacity grows dramatically. Moreover, they stated that for small-scale experimental tests, scaling has influence on the results intensively.

Benmebarek et al. [24] assessed the undrained bearing capacity of embedded circular foundations via the finite difference method. They focused on the effects of different base and side foundation roughness conditions on foundation ultimate bearing capacity. These researchers declared that the type of procedure used for modeling the foundation side interface conditions influences result discrepancies. Finally, they concluded that practical formulae of depth factors are conservative for circular foundations.

Despite performing several experimental tests by the researchers to determine ultimate bearing capacity of embedded foundations, there seems to be an important need for more empirical observations to monitor the embedded foundations behavior under vertical load and recognize the failure mechanism of mentioned foundations. Moreover, there are some deficiencies related to the empirical–theoretical expressions presented in literature used to estimate ultimate bearing capacity of embedded foundations. As an example, Terzaghi’s expression ignores the shear strength of foundation surcharge soil and entails in approximate results [7]. Moreover, having experimental tests on embedded foundations, vertical stiffness per area in small-scale models will be recognized and compared. However, some expressions are presented in literature and codes to estimate embedded foundation stiffness [5, 6, 25, 26].

Accordingly, in the present study, to expand the empirical dataset, five experimental tests on ultimate bearing capacity of strip foundations with different embedment depths in sandy soil under pure vertical loading were carried out. The embedded foundations consisted of base slab and retaining walls (the latter has almost been neglected in previous works in literature) to provide a condition similar to the building basement foundations. In the following, based on observed damages, a categorization of failure patterns was implemented depending on foundation embedment depth. Then, referring to behavioral curves (in terms of vertical stress versus foundation settlement), ultimate bearing capacity (q_u) and vertical stiffness of foundation per unit area (k_z) were obtained for each test. In addition, defining mechanical characteristics of soil and foundation, finite element (FE) modeling of embedded foundations in soil was implemented via ABAQUS software [27]. Accordingly, the monotonic behavioral curves achieved through experimental and numerical modelings were compared and interpreted in both linear and nonlinear states (e.g., vertical stiffness and ultimate bearing capacity). Furthermore, the sufficient distances of embedded foundation under vertical loading from rigid boundary walls were suggested through several analyses to exclude unwanted boundary effects. Moreover, an effort was made to investigate the validity and applicability of a classic theoretical expression in literature for estimation of foundations vertical stiffness in testing domain of foundation embedment.

2 Experimental Methodology

In foundation engineering, experiments are carried out to simulate complicated systems under controlled conditions. Accordingly, it is convenient to choose required scaled models for simulative specimens with regard to the laboratory facilities, experimental targets and required precision. Using small-scale model—if the main characteristics of prototype are preserved based on precision—reflects an acceptable quantitative and especially qualitative understanding of foundation behavior. Accordingly, in this study, the height, width and length of the model foundations were decided by considering the geometry of strip-embedded foundation in practice. By assuming the width and the length of embedded prototype foundations as 3 and 17.55 m, respectively, the width and length of model were designed conveniently to be 100 and 585 mm, representing a linear scale factor of 1/30.

To extract the foundation behavioral curve under vertical loading and determine bearing capacity of surface and embedded foundations, static tests have been performed on small-scale models at Physical Modeling Center at University of Tehran. The physical models were constructed in a 1.46 m (length) × 0.6 m (width) × 0.9 m (height) soil container with the maximum vertical payload of 4 tons. One long side of the container was made of plexiglass with a thickness of 20 mm. Contact paper was attached to the inside of another long side of the box to reduce friction between the edge of the model foundation and the box. Moreover, container consists of four more sheets (three remained walls and bottom) made of Teflon with a thickness of 20 mm. Steel angle profiles in companion with steel box shape stiffeners were used to brace the outside of the container test box to control deformations during soil compaction and bearing capacity tests. Experiments were implemented for strip-embedded foundations in plane-strain condition.

2.1 Testing Materials

2.1.1 Soil

Firuzkooh 161 sand with 50% relative density (D_r = 50%) was used in all ultimate bearing experiments by a raining technique. Firuzkooh is a county near Tehran, the capital of Iran with different types of soils and mines. The Firuzkooh 161 sand gradation curve is similar to that of Toyora sand [28]. It is a fine, uniformly graded sand with the mean grain size D₅₀ = 0.27 mm. The mentioned sand was used by some researchers in the past experiments [29,30,31,32]. An acceptable stableness of soil was achieved through adding moisture of 5% and silt value of 10%. The soil properties were provided by the authors through distinct experiments. The specific gravity and maximum and minimum void ratios were G_s = 2.66, e_max = 0.815, and e_min = 0.534, respectively. At 50% relative density, the void ratio (e) and the unit weight of the soil (γ_wet) were 0.675 and 16.36 kN/m³, respectively. Table 1 summarizes the soil mechanical properties.

Table 1 Soil properties

In Table 1, c, ϕ and γ_d are soil average cohesion, effective average friction angle of soil and soil dry density, respectively. It is noted that e_max, e_min and γ_d were determined through experiments corresponding to the standards ASTM D-4253 [33] and ASTM D-4254 [34], respectively. G_s was attained by experiment conforming ASTM-854 [35].

Aimed to estimate the soil strength parameters (φ and c) (effective average soil friction angle and average soil cohesion corresponding to the stress level of interest for the desired D_r), a series of vertical pushover tests were performed on different foundation models. Thereby, φ and c were back-calculated using the classical expressions for ultimate bearing capacity [8, 23, 36].

2.1.2 Foundation

The foundation was modeled with a rigid aluminum section consisting of two longitudinal retaining walls, base strip slab and two transverse stiffeners. The foundation has the dimensions of 585 mm (length) × 100 mm (width) × 275 mm (height) as depicted in Fig. 1. Accordingly, dimensional properties of foundations are introduced in Table 2.

Fig. 1

Strip-embedded model foundation: a 3D view; b elevation and plan view

Table 2 Dimensional properties of foundations

As presented in the next sections, D is the only variable property of specimens in different tests. It possessed values of 0, 30, 60, 90 and 120 mm corresponding to 0B, 0.3B, 0.6B, 0.9B and 1.2B, respectively.

Regarding the application of foundation for probable tests with deeper embedment in future, the height of 270 mm for foundation was chosen. However, in the current study, experiments have been carried out for foundation with the maximum embedment of 120 mm because of limitation related to the maximum vertical payload. Retaining walls have a thickness of 15 mm as well as transverse stiffeners. Also, the thickness of bottom slab is 25 mm. The reason for using transverse stiffeners is to avoid any distortion of longitudinal retaining walls. Modeling longitudinal walls approximately considers the real condition of buildings with basement in which the foundation contacts with the surrounding soil through lateral retaining walls. Accordingly, the friction between retaining walls and the surrounding soil which raises the ultimate bearing capacity of embedded foundation will be considered in modeling. In addition, the use of retaining walls boosts the foundation section rigidity. It is important because in the subsequent experiments, the Terzaghi’s hypothesis will be used. The rigidity of strip foundation section is controlled through the expressions presented in valid references [7, 25, 37, 38].

Sand was glued to the sides and base of the model foundation to simulate a rough sand-coated surface. In addition, the edges of the model foundation were lightly coated with sponge to isolate foundation from long side walls of container. The dimensions of the container are much larger than the dimensions of the foundation model, which eliminates any boundary effect during the slow monotonic vertical loading. Specifications of foundation model are summarized in Table 3.

Table 3 Foundation model properties

It is noted that the material Al-6061 used in the current experiments had a density of 2700 kg/m³ and elasticity modulus of 70 GPa. In addition to the introduced foundation elements, an aluminum block with dimensions of 200 × 70 × 70 mm³ was leaned to the center of base slab and screwed to two lateral retaining walls. The aim for use of the block is to set a parallel loading path through retaining walls and base slab into surrounding soil. Moreover, it translates the loading exertion point to shear center of U-shape section of foundation. It is noted that for surface foundation (D/B = 0) stiffening was implemented just through an aluminum block added to the top of strip foundation.

2.2 Locating Foundation in Adequate Distance from Container Boundaries and Providing Plain-Strain Condition

2.2.1 Soil Depth Under Base Slab

In foundation ultimate bearing tests, the location of bottom boundary of container beneath the foundation is important to avoid any effect of the rigid boundary on the bearing capacity. Cerato and Lutenegger [39] considered influence of the thickness of soil under the foundation on the ultimate bearing capacity of foundation. They concluded that for H/B > 3, the thickness of soil does not affect the ultimate bearing capacity (H is the thickness of soil under the foundation and B is the width of strip foundation). In the present study, for foundations with the width of 100 mm, a soil thickness of 300 mm under the foundation was provided and consequently, a ratio of H/B = 3 was obtained.

2.2.2 Distance of Foundation Edge and Container Boundary Walls

The distance between boundary rigid walls of container and the edge of the foundation should possess a sufficient value to avoid any impact on the foundation bearing capacity. Leshchinsky and Marcozzy [40] during their experiments understood that failure slip footprint locates on soil surface in the range of $2.3 < l /B < 4.8$ in which l is the distance between the edge of foundation and the nearest rigid wall of container and B is the strip foundation width. With regard to the selected foundation width (B = 100 mm) in the current study, the mentioned distance, l was equal to 680 mm and subsequently, $l /B = 680 /100 = 6.8.$ However, this ratio has a significant difference with the upper limit of the range introduced by Leshchinsky and Marcozzy [40]. This safety margin is because of embedment of foundation that causes the failure slip footprint to develop more than that for shallow foundation. In other words, for settlement of embedded foundations without any rotation, the failure wedge intersects the soil surface for a distance about $\cot \;(45 - \phi /2) \times D$ closer to the rigid wall of container in comparison to the base slab level. Figure 2 shows a schematic view of the embedded foundation in soil container and defines distance from foundation edge to the rigid wall as well as experimental setup and instrumentation.

Fig. 2

Experimental setup and instrumentation for vertical pushover test

2.2.3 Providing Plane-Strain Condition for Foundation Bearing Tests

To provide a strip foundation condition, the aspect ratio of foundation should be regarded. Investigating the ultimate bearing capacity of strip foundations on reinforced sand, Azzam and Nasr [41] considered the length-to-width ratio of 2 for foundations. Shin and Dos [42], in studying bearing capacity of foundations on sand reinforced by geogrid, introduced a value of 2.6 for foundation dimensional ratio in plan. Leshchinsky and Marcozzy [40] used the length-to-width ratio of 4 for strip foundations to examine the Terzaghi’s [7] foundation bearing capacity expression. Regarding the introduced dimensional ratio for the strip foundations in literature; a length-to-width ratio of $L /B = 585 /100 = 5.85$ was chosen for present experiments which satisfied the strip foundation condition (L and B are the length and width of strip foundation, respectively).

2.3 Testing Procedure

To implement foundation vertical loading tests, sand was compacted in the container box in the 30-mm-thick layers. Compaction was achieved using a flat-bottomed hammer. Accuracy of sand placement and consistency of the relative sand density were checked during compaction by placing small cans of known volumes at different locations. Colored sand was also poured around and underneath the foundation to indicate the post-test ground deformation. The model foundation was placed at desired D/B values (D is the embedment depth and B is foundation width). Load was applied to the model foundation using an electric gear-controlled piston. The loading shaft is perfectly perpendicular to the foundation plan. According to Fig. 2, the loading shaft is in contact with foundation through a rounded device situated at the center of foundation; such an arrangement produced a hinge that allowed the foundation to rotate freely as it approached failure and eliminated any potential moment transfer from loading fixture. The load and corresponding foundation settlement were measured using a load cell and a linear variable differential transducer (LVDT) gauges, respectively. The LVDT gauge was located at the center of foundation width to capture the settlement of foundation’s midpoint. Both load cell and LVDT were connected to an eight-channel data-logger. It is noted that the test for each foundation configuration was repeated to ensure the uniformity of behavior and remove any random factor.

2.4 Loading Protocol

Load was exerted vertically to the center of the foundation section. Since an objective of current research was to attain the foundation nonlinear behavior, loading procedure was implemented in the displacement-controlled state. Accordingly, vertical incremental displacement was applied to the foundation, monotonically. Since the static behavior was considered, the loading speed was maintained at 0.075 cm/min. This loading rate was so close to those regarded by Cerato and Luetenegger [39] and Zhu et al. [43] in related experimental research. The loading process was continued until the behavioral curve reached a constant and stable stress state. Figure 3 shows the loading protocol utilized in current experiments.

Fig. 3

Loading protocol

3 Experimental Test Characteristics

Since the goal of the present study is to investigate the embedment effect on the foundation behavioral characteristics such as ultimate bearing capacity and vertical stiffness under vertical loading, the tests were performed on five foundations with different embedment depths. Experimental test characteristics are summarized in Table 4.

Table 4 Experimental specimen characteristics

For the specimens naming, “F” stands for “Foundation” and its subsequent index shows the number of each test. Also, “Emb” abbreviates “Embedment” and the subsequent number demonstrates the embedded depth (in mm). Five foundation-soil specimens are shown in Fig. 4. Accordingly, the undeformed configurations were demonstrated as well as foundations sections, loading shaft and compacted layers of soil.

Fig. 4

Tests configurations: a F₁-Emb₀; b F₂-Emb₃₀; c F₃-Emb₆₀; d F₄-Emb₉₀; e F₅-Emb₁₂₀

4 Experimental Results

Figure 5 shows the foundation and its underneath soil deformations for all specimens as well as soil slip surfaces in a Cartesian coordinate system to quantify deflections. To draw these figures, the undeformed states of provided soil layers (drawn parallel lines on plexiglass wall of container), were compared to deformed location of soil layers (according to deformation of colored sand alignments during loading process). The protraction of soil slip surface was achieved through connecting broken points along the colored sand line for each layer. As an example, the abovementioned broken points and the consequence slip surface are shown in Fig. 6 for a typical specimen (F₄-Emb₉₀). Since all models experimented in laboratory were categorized as shallowly embedded foundations, they resembled in some behavioral aspects under vertical loading. For example, all specimens rotated continually in companion with the settlement. Mentioned rotation caused asymmetric soil failure slip formation for such specimens. The asymmetry of soil failure slip surface is apparent for all specimens through considering the horizontal distance between foundation and intersection of soil slip protraction and soil surface, named d_slip.

Fig. 5

Trace of slip surface for: a F₁-Emb₀; b F₂-Emb₃₀; c F₃-Emb₆₀; d F₄-Emb₉₀; e F₅-Emb₁₂₀

Fig. 6

a Broken points of colored sand lines at soil failure state; b connecting broken points to achieve soil slip surface for specimen F₄-Emb₉₀

Moreover, as a consequence of rotation, the contact between adjacent soil and retaining walls deteriorated partially for embedded foundations. Another fact is that, for all specimens, the slip surface footprint on the top of soil became closer to the foundation body in two ends in comparison to the middle as a result of friction between soil and its container rigid walls. This observation is shown in Fig. 7 for a typical specimen (F₃-Emb₆₀). More or less, some differences in behaviors under vertical loading originated from embedment depth were observed. For example, for shallower specimens (D/B = 0 and 0.3), a mesh of longitudinal and transverse cracking pattern formed on soil surface in spite of foundations with deeper embedment as a result of extreme tension in the soil surface. In addition, for named specimens, failure happened by the type known as “general shear failure” (Fig. 5a, b). Another observation for all tests was swelling phenomenon on soil surface (at least in one side of foundation) as a consequence of soil slip surface formation. The variation trend of swelling with foundation embedment depth will be interpreted in the upcoming section.

Fig. 7

Soil slip footprint on the top of soil for a typical specimen (F₃-Emb₆₀)

For deeper embedded foundations (D/B = 0.6–1.2), slip surface formation was postponed until the creation of an obvious longitudinal crack on soil, parallel to the foundation length. This fact was observed apparently for F₄-Emb₉₀ and F₅-Emb₁₂₀ (Fig. 8). As a matter of fact, the embedded foundation mobilized a mass of adjacent soil as settled and consequently, one or two longitudinal cracks formed around the foundation. However, it was followed by the formation of slip failure surfaces for large settlements beyond the settlement corresponding to ultimate bearing capacity. Accordingly, the mentioned localized damages dominated the vertical behavior, subsequently, followed by the formation of soil slip surfaces. This type of failure is similar to the so called “local shear failure”. In other words, with the increase in foundation depth, the mentioned failure type became more apparent. For embedded foundations F₄-Emb₉₀ and F₅-Emb₁₂₀, some special damages such as mentioned longitudinal cracking, local settlement and separation of foundation retaining walls from adjacent soil were concentrated in a localized damage zone (Fig. 5d, e). The concentration of damages in localized damage zone avoided creation of significant swelling of soil surface near the foundation. Particularly, the occurrence of soil slip surface was a secondary failure mechanism in mentioned model foundations.

Fig. 8

Formation of longitudinal crack in soil surface: a F₄-Emb₉₀; b F₅-Emb₁₂₀

5 Discussion on the Experimental Results

In the current section, it is tried to discuss and quantify the observations and outputs of the experiments. As declared before, however, the theoretical distances from the foundation body were regarded to eliminate any effect of boundary conditions on the results, and the position of soil slip surface relative to the rigid boundary walls experienced in the tests can represent an understanding of real circumstances. According to Fig. 9a, increase in embedment ratio causes a growing trend on d_slip. As depicted in Fig. 9b, variation of soil swelling values with foundation embedment depth has approximately irregular scattering pattern. Although the linear trendline fitted to scattering shows a descending tendency, it seems the experimental dataset is not enough to reflect a robust rule for swelling variation with embedment depth. Another fact is that for foundations with deeper embedment, plastic strain develops in depth for larger zone instead of concentration near soil surface. In better words, it is expected that local shear failure highlighted with increasing of embedment ratio did not mobilize significant swelling. It is noted that increase in embedment ratio does not show a regular and significant change in foundation rotation at the time of failure. This finding confirms this fact that the rotation of foundation would be resulted in accordance with the stochastic factors such as soil media heterogeneity.

Fig. 9

Variation in slip surface position: a horizontal extension; b maximum soil surface swelling

In addition to the abovementioned observations, in this section, behavioral curves of embedded foundations under vertical loading are extracted and presented in terms of average applied vertical stress (q) under the foundation base slab and foundation center settlement (δ). As demonstrated in Fig. 10, for all specimens, the behavioral curve begins with a linear phase, then reaches a peak of strength and finally terminates after passing a plateau state. Since the ultimate bearing capacity of embedded foundation (q_u) is considered, there is an attempt to find a criterion for extracting q_u from vertical behavioral curve. Finding an appropriate method to determine q_u is important because the main conclusions of this study are based on this quantity. Moreover, it is noted that the current study was defined along research series named evaluating nonlinear interaction of soil-embedded foundation-structure (e.g., rocking, uplifting and sinking phenomenon) under seismic loading. Accordingly, necessary quantities such as foundation moment capacity (M_cap) and vertical factor of safety (FS_V) would be a function of q_u.

Fig. 10

Behavioral curves for all embedded model foundations (a–e are corresponding to D/B of 0–1.2, respectively)

There are several methods in literature for estimating q_u from experimental vertical behavioral curve. The commonplace ones are introduced as below:

• 1. 0.1B method (B is width of strip foundation) or selection of foundation stress corresponding to settlement value of 0.1B (Briaud and Jeanjean [44]).

• 2. Tangent intersection method (Trautmann and Kulhawy [45]).

• 3. Log–log method (De beer [46]).

• 4. Hyperbolic method.

It is noted that, in contrast to methods (2) and (3), 0.1B and hyperbolic methods make use of the total stress-settlement curve to anticipate ultimate bearing capacity. Focusing on 0.1B method, Briaud [47] claimed that for ratio of $S /B \approx 0.1$ (S is foundation settlement) it can be ensured that the soil strain reaches to its ultimate value. Achieving q_u from 0.1B method and comparing the normalized vertical behavior curves for foundations with different depths and similar widths, Lutenegger and Adams [48] concluded that the chosen method results in a single unifying concept which may be used to interpret the behavior of foundations properly. Moreover, this method has an advantage and it is that even when the soil failure point is not recognizable, the ultimate bearing capacity can be explored easily. Accordingly, since in current study, foundations with different embedment depths and analogous widths were considered, the 0.1B method was used. In addition, some researchers [39, 48] have used this method in relevant fields for foundation behavioral assessment.

For specimens F₁-Emb₀, F₂-Emb₃₀, F₃-Emb₆₀, F₄-Emb₉₀ and F₅-Emb₁₂₀, ultimate bearing capacities are equal to 3.640, 3.897, 4.518, 4.630 and 4.830 kg/cm², respectively. Accordingly, for these specimens with growth in embedment depth, the ultimate bearing capacity increases (Fig. 11). This result is in agreement with the findings in literature [7, 8].

Fig. 11

Variation of ultimate bearing capacity of model foundations with embedment depth ratio

In Fig. 11, q_u is ultimate bearing capacity of embedded foundation and D/B is the ratio of embedment depth to the strip-embedded foundation width. It seems that the increasing rate of the ultimate bearing capacity for foundations with D/B in range of 0–0.6 is slightly more than one for foundation with D/B of 0.6–1.2. It confirms that with increase in embedment ratio, the increasing rate of the ultimate bearing capacity tends to decrease. However, this change in the rate of ultimate bearing growth is smooth and ultimate bearing capacity of embedded foundations can roughly be considered a linear function of embedment ratio in the experimental range.

Now referring to vertical performance, it is tried to explain the behavioral curve of model foundations according to the observed damages. For model foundation without any embedment (F₁-Emb₀), the peak of strength is concurrent with the creation of the first failure slip surface. Afterwards, the strength diminishes gradually until approaches to a limit value.

For specimens with embedment depths, the behavioral curves experience a plateau state in maximum level of strength. The formation of soil slip surface occurred when the behavioral curve fell in mentioned plateau portion. The light uneven plateau state in behavioral curve can be assigned to the unavoidable errors that arose in test condition.

The point arises here is the effect of container boundary walls on the ultimate bearing capacity of embedded foundations. Since the probable effect of boundary walls appears at high strains of soil, it seems that the mentioned effects should be significant in higher settlements. As stated before, the ultimate bearing capacity is extracted from the foundation vertical behavioral curve at the stress corresponding to a settlement of δ = B/10. This point on the foundation behavioral curve is located approximately in constant phase and so far before the failure slip surface (failure wedge) formation instance. In better words, when the ultimate bearing capacity of foundation is extracted, the container boundary walls do not interfere yet. In addition, formation of first failure wedge for all experiments without incident to the container walls confirms the negligible effect of walls on results.

Another parameter extracted from the behavioral curve and explained in current section is the foundation vertical stiffness per unit area. Since in traditional foundation design procedure, the consideration of foundation vertical stiffness in linear phase is significant (i.e., vertical low-strain stiffness), in present study, the initial tangent of vertical behavioral curve in low settlement amplitudes was attained [49]. Accordingly, this method for determining initial stiffness of foundation would minimize any nonlinearity phenomenon (such as material and geometry nonlinearities). Also by this manner of determining stiffness, the possibility of comparing experimental values of stiffness with those calculated from theoretical formulation in literature would be provided. Figure 12 depicts the variation of foundation vertical stiffness versus embedment ratio. It is inferred that vertical stiffness increases with growth in embedment ratio gradually. This finding confirms the empirical relationships for embedded foundations suggested by Pais and Kausel [5], and Gazetas [6]. The vertical stiffness grows with embedment ratio in range of D/B = 0 to D/B = 0.9 nearly about 22.7%. However, the foundation stiffness decreases slightly with increase in embedment ratio from D/B = 0.9 to D/B = 1.2. It seems that more experiments are needed for the determination of embedment effect on the vertical stiffness for deeper foundations.

Fig. 12

Variation of foundation vertical stiffness with embedment ratio

6 Numerical Methodology

The calibrated numerical models can be used to anticipate the ultimate bearing capacity of embedded foundations. Although various numerical modeling approaches have been presented for behavioral estimation of embedded foundations [50,51,52], the comparison between numerical and experimental results in model scale has been scarce in literature. One of the important advantages of numerical modeling is feasibility for performing the parametric study on the behavioral characteristics without such difficulties and costs.

To implement the current numerical analysis in companion to the experimental program, the framework of finite elements modeling of embedded foundations in ABAQUS is explained in this paper. Accordingly, five experimented model foundations were considered for numerical modeling.

The modeled U-shape sections of strip-embedded foundations consisted of two side walls and base slab; while the surface foundation just included a simple slab. As noted in Sect. 2.1.2, the foundation section was rigid; therefore, a Constraint in type of Rigid Body was attributed to foundations. For numerical modeling of soil, properties defined in Sect. 2.1.1 were used. Moreover, soil elastic characteristics, i.e., elasticity modulus and Poisson ratio, defined for finite elements modeling were 15,600,000 Pa and 0.3, respectively. Mohr–Coulomb [53] was chosen as the soil plastic behavior and its parameters were those defined and modified in previous sections. In addition to in situ geostatic condition, the soil weight was defined as Body Force for soil. Moreover, for each contact interface between foundation body and its surrounding soil, two types of interaction properties were defined: normal behavior (Hard Contact) and tangential behavior (Penalty Friction Formulation with friction coefficient of 0.47). The loading on bottom of foundation (on base slab) was in displacement phase and continued until the settlement yielded to the limit value corresponding to the ultimate bearing capacity (δ = B/10). To avoid any sudden deformation of soil, the mentioned settlement was exerted to the foundation in a period of 100 s. Consequently, the analysis type was Dynamic Explicit. Also, four-noded meshes were chosen for soil in type of plane-strain quadrilateral reduced integration elements (CPE4R). The size of mesh was selected from 30 mm at far boundaries to 10 mm adjacent to the embedded foundation. It is remarked that boundary condition for two sides of model was defined through fixing the nodes for translational degree of freedom in just horizontal direction. For the bottom of model, both translational degrees of freedom in horizontal and vertical directions for nodes were fixed. Figure 13 shows the typical model view consisting of soil layer and model foundation (for D/B = 0.6).

Fig. 13

The geometry and meshing for an embedded foundation (D/B = 0.6) implemented in FE analysis

7 Numerical Modeling Results

Regarding the importance of boundary condition in finite element analysis and the its probable effects on the analysis output such as ultimate bearing capacity, two analyzing series were carried out on the model soil length (horizontal distance between two vertical side boundaries, L) and also the soil depth beneath the foundation (H). The reference model for the sensitivity analysis was F₅-Emb₁₂₀ with the model soil length of 1460 mm and the soil depth under base slab of 300 mm in experimental program. The selection of F₅-Emb₁₂₀ was because of its probable sensitivity to the boundaries in comparison to other ones. For each of two sensitivity analysis types, nine numerical models were selected. The first type was relevant to the model soil length which controls distance between the foundation edge and the boundary rigid wall. With this regard, the value of L was considered to be in the range of 3B–21B (300–2100 mm). The variation of q_u versus the mentioned distance is depicted in Fig. 14a. Referring to this figure, it is approved that for reference specimen, with choosing a model soil length of at least 900 mm (9B), the undesirable effects of boundary walls on the foundation ultimate bearing capacity are negligible.

Fig. 14

The effects of boundaries distances to the foundation body on the ultimate bearing capacity: a distance between foundation edge and container boundary wall; b soil depth under the foundation

The second type of sensitivity analyses belongs to the effect of soil depth under the base slab on the ultimate bearing capacity of foundations. This depth has changed from 0.75B to 5B (75–500 mm) in analyses. Figure 14b shows the variation of q_u versus the soil depth under the foundation base slab (H). According to figure, by having a depth of at least 2.5B (250 mm) under the foundation, it would be ensured that this depth does not affect the ultimate bearing capacity.

Carrying out the sensitivity analyses, it was confirmed that the boundary condition implemented in the experimental program could be used in numerical analysis for five specimens without the effects of rigid boundary walls on the bearing capacity of foundations (for pure vertical loading). Accordingly, in numerical modeling, the width of soil media was considered as 1460 mm and the depth of soil under the foundation equals 300 mm. It is noted that because of model configuration and loading symmetry, just half space of soil media and foundation was modeled. By implementing finite elements analysis, vertical behavioral curve was achieved and compared with experimental one for each of model foundations (Fig. 15). Apparently for all numerical responses, there is an about linear phase followed by an approximately plateau nonlinear response; therefore, the comparison of behavioral parameters such as stiffness and ultimate bearing capacity between experimental and numerical performances would be possible. Accordingly, the estimated and real vertical stiffness values are approximately in agreement for model foundations, F₁-Emb₀, F₄-Emb₉₀ and F₅-Emb₁₂₀. For the other two, F₂-Emb₃₀ and F₃-Emb₆₀, however, experimental and numerical behaviors do not match in linear phase; two curves become approximately parallel after beginning portion of behavior. By modifying the mentioned beginning portion of experimental curves originating from possible test imperfections, it is expected that appropriate closeness of experimental and numerical curves is achieved. As declared before, in numerical modeling, the maximum settlement of 0.1B was exerted to foundation base slab; so the experimental and numerical curves are comparable up to the mentioned settlement. It is noted that for all specimens, the numerical behavior curve reaches to its nonlinear (approximately plateau) phase before touching settlement value of 0.1B. For specimens F₁-Emb₀, F₂-Emb₃₀ and F₃-Emb₆₀, it is evident that the nonlinear numerical behavioral curve initiates at lower settlement than experimental one. This comparison reflects the difference between ideal elastoplastic soil model and soil real response (e.g., gradual plastification inherence of soil grains). For two specimens with the deepest embedment, F₄-Emb₉₀ and F₅-Emb₁₂₀, a good matching is evident between experimental and numerical responses in nonlinear phase. Moreover, for two mentioned model foundations, the experimental behavioral curve has the form of bilinear and resembles numerical performance. It can be stated that the presented numerical modeling has more capability to anticipate vertical behavior of deeply embedded foundations in comparison with shallower ones. Moreover, it seems that for specimen F₁-Emb₀ which has apparent softening branch after peak of strength, the numerical modeling could not anticipate post-peak response properly.

Fig. 15

The comparison between experimental and numerical behavioral curves for five tested model foundations under vertical loading

8 Discussion on Numerical Results

Now, experimental, numerical and theoretical [5] values of initial vertical stiffness for five model foundations are compared (Fig. 16). Approximately, for each specimen, the results of stiffness achieved from all three methods are close together. However, some minor differences are apparent. For example, it seems that numerical values of stiffness are about more than those of experimental and theoretical methods. This observation confirms this fact that finite elements modeling provides stiffer results than real condition; moreover, possibly in experiments, soil behavior did not follow idealized elastoplastic model and its gradual plastification initiated at smaller strains than assumed ones. It is noted that an effort was made to correct the beginning part of experimental behavior curve for some specimens related to testing instrumental imperfections. Ignoring the differences between stiffness values, it seems that the general tendency of numerical and experimental variation of stiffness is analogous (see attributed trendlines in Fig. 16). On the other hand, it is apparent that variation of theoretical stiffness vs. D/B ratio has steeper tangent than two others. The comparison shows that the theoretical relationship reflects a reasonable approximation of the stiffness values based on empirical arguments. The theoretical expression follows the results of the experimental tests for studied domain of foundation embedment with exception of the result for the specimen F₅-Emb₁₂₀. In fact, the achieved value for the stiffness of specimen F₅-Emb₁₂₀ was already remarked before, as the one that does not follow the tendency (see Fig. 12). Therefore, the Pais and Kausel [5] formulation might be considered for practical purposes to estimate the stiffness, especially in experimented domain of embedment depth. Although, it is expected that for foundation embedment depth greater than experimental ratios, the theoretical values of vertical stiffness exceed those of experimental and numerical modelings.

Fig. 16

The comparison between experimental, numerical and theoretical estimations of foundation vertical stiffness values

To compare the vertical stiffness of embedded foundation with that of surface foundation, Fig. 17 is presented. In this figure, $\eta$ is the stiffness ratio defined as follows:

$$\eta = k_{\text{z,embedded}} /k_{\text{z,surface}} .$$

(1)

Fig. 17

Comparison of stiffness ratio resulted from experimental and numerical modelings with those proposed by Pais and Kausel [5]

In which k_z,embedded and k_z,surface are vertical stiffness per area corresponding to the embedded and surface foundations, respectively. In addition to test results, the vertical stiffness ratio for numerical modeling and theoretical expression (Pais and Kausel) is presented in Fig. 17. Apparently from Fig. 17, the maximum stiffness ratios are 1.23, 1.24 and 1.59 for experimental, numerical and theoretical results, respectively. It seems that stiffness ratios are roughly close to each other for experimental and numerical modelings; however, for most specimens, the numerical value is more than the experimental one. Totally, it would be concluded that for numerical modeling, the stiffness ratio tends to a limiting value of about 1.24. For stiffness ratios obtained from theoretical expression [5], the result is different. Accordingly, it seems that the slope of trendline fitted to variation of stiffness values achieved from mentioned theory is more than those of experimental and numerical results.

Figure 18 shows the comparison of ultimate bearing capacities resulted through experimental and numerical modelings. Apparently, the numerical model (FE) is capable of anticipating the bearing capacity of embedded foundations for those with D/B in the range of 0.6–1.2. However, it seems that for embedded foundations, FE reaches to higher value than experimental results for about 6%, averagely. One of reasons is that the rotation of foundation happened during experimental tests, especially for embedded foundations, reduced the axial bearing capacity of foundation; while in numerical analysis just the vertical settlement of foundation was considered in a symmetric loading condition. Moreover, the real strength parameters of soil were back-calculated through Terzaghi’s. The mentioned theory ignores the shear strength of soil above the foundation base slab. But in the numerical analysis not only the strength parameters of soil achieved from Terzaghi’s theory were used, but also the soil above the foundation base slab was modeled and subsequently its corresponding shear strength was considered. Accordingly, it is common to reach the higher value of ultimate bearing capacity of embedded foundation from FE in comparison to experiments. Closeness of numerical and experimental results of ultimate bearing capacities relates to this fact that soil strength parameters (i.e., friction angle and cohesion) used in numerical analyses were back-calculated from experimental results of ultimate bearing capacity tests. In this regard, the empirical inherence of input parameters fed in numerical modeling takes into account some uncertain factors such as soil heterogeneity, instrumental imperfections and foundation rotation indirectly in numerical responses. In addition, with increase in foundation embedment depth, and consequently increase in underlying soil effective stress at foundation level, the sensitivity of strength parameters is reduced and their applicability becomes more reliable and nearer to reality in comparison to surface foundations. For foundation without embedment, the numerical analysis underestimates ultimate bearing capacity for about 12%. It can be inferred as a result of assumption of soil frictional angel. Obviously, the numerical framework used an average amount of soil frictional angel achieved from back-calculation. On the other hand, in surface foundation model under very low effective stress, the value of soil friction angel is significantly sensitive to the confining stress and more than the value assumed in analysis. Accordingly, the predicted value for q_u through the numerical analysis is less than that from the experiment for surface foundation.

Fig. 18

Comparison of variation of ultimate bearing capacity with embedment depth between numerical and experimental modeling

Figure 19 demonstrates the deformed shape of soil near the embedded foundation at the settlement of δ = B/10 for two representative model foundations introduced in the experimental program. For each specimen, the displacement resultant (U) was extracted and shown as arrow symbols. Although the numerical model is not capable of capturing the rotation of foundation as a result of soil heterogeneity and instrumental deficiency, the convexity of soil which is adjacent to the foundation is in agreement with all experimental results. Apparently from Fig. 19, with increase in embedment ratio, the swelling zone of soil develops toward depth and sides. However, it seems that with growth in embedment ratio, the magnitude of soil surface swelling decreases. For foundations with D/B of 0.9 and 1.2, the displacement arrows touch the side boundary and they are forced to have a vertical direction. However, because the mentioned arrows around the boundary possess the small values, their influence on the results are negligible.

Fig. 19

The resultant deformation (U in m) of soil in FE analysis at the foundation settlement of δ = B/10: a F₂-Emb₃₀; b F₄-Emb₉₀

Another fact arising from the finite elements modeling is that the swelling of soil is beyond the rigid boundary walls and, therefore, the numerical modeling cannot monitor the meeting of failure wedge and rigid boundary walls.

Figure 20 shows the variation of principal in-plane plastic strain in the soil media for foundations F₂-Emb₃₀ and F₄-Emb₉₀ specimens. From Fig. 20, especially for F₂-Emb₃₀, the formation of three distinct zones is obvious: a triangular zone directly under the foundation, a radial zone, and a Rankine passive zone reminding the three zones assumed by Terzaghi. It is evident that with increase in embedment ratio, the triangular zone is stretched to the bottom boundary. In other words, the maximum shear strain forms in deeper location from the foundation base slab.

Fig. 20

The variation of maximum plastic strain of soil in FE analysis at the foundation settlement of δ = B/10: a F₂-Emb₃₀; b F₄-Emb₉₀

9 Conclusion

In the present study, an effort was made to investigate such behavioral parameters of embedded foundations including ultimate bearing capacity and vertical stiffness through experimental and numerical methods. For both programs, five embedded foundations with D/B of 0, 0.3, 0.6, 0.9 and 1.2 were studied. Foundations were strip, rigid and in plane-strain conditions (analogous to the Terzaghi’s assumption). Accordingly, the following results were obtained:

• 1. All tested specimens rotated continually in companion to the settlement. Mentioned rotation caused asymmetric soil failure slip formation for such specimens.

• 2. For shallower foundations (F₁-Emb₀ and F₂-Emb₃₀), failure happened by the type known as “general shear failure”.

• 3. For deeper embedded foundations, F₄-Emb₉₀ and F₅-Emb₁₂₀, some special damages such as longitudinal cracking, local settlement and separation of foundation retaining walls from adjacent soil happened in a localized damage zone.

• 4. Maximum swellings of soil surface did not follow a regular variation pattern with growth in embedment depth.

• 5. For experimental and numerical investigations, with growth in embedment depth, the ultimate bearing capacity increases. This result is in agreement with the findings in literature.

• 6. The slope of the q_u-D/B curve for foundations with D/B in range of 0–0.6 is slightly more than curve slope for foundation with D/B of 0.6–1.2. It confirms that with increase in embedment, the increasing rate of the ultimate bearing capacity tends to a limiting value.

• 7. For model foundation without any embedment (F₁-Emb₀), the peak of strength is concurrent with the creation of the first failure slip surface. Then, the strength decreases gradually until approaches to a limit value.

• 8. For embedded model foundations, the behavioral curves experience a plateau state in maximum level of strength. The formation of soil slip surface occurred when the behavioral curve fell in mentioned plateau portion.

• 9. Performing the sensitivity analysis via FE analysis, it is concluded that for a model soil length of at least 9B, the undesirable effects of boundary walls on the foundation ultimate bearing capacity are eliminated. In addition, having a soil depth of at least 2.5B under the foundation, removes the unwanted effect of bottom boundary on the modeling results.

• 10. Since the proposed relationship by Pais and Kausel could predict the vertical stiffness variation in considered domain of embedment except F₅-Emb₁₂₀, it might be considered for possible practical purposes; however, it seems that the slope of trendline fitted to variation of stiffness values achieved from mentioned theory is more than those of experimental and numerical results.

• 11. The maximum stiffness ratios are 1.23, 1.24 and 1.59 for experimental, numerical and theoretical results, respectively, in considered range of embedment depth. It seems that more experiments are needed for the determination of embedment effect on the vertical stiffness for deeper foundations.

Results of the present study led to generalize existing experimental database on vertical behavior of embedded foundations. In spite of some limitations, variation tendencies of behavioral components (i.e., foundation vertical stiffness and ultimate bearing capacity) presented useful interpretations having appropriate origins in literature. However, it should be noted that current research results were limited to the surface and embedded foundations with maximum embedment ratio of 1.2 lying on silty sand. Moreover, providing monotonic vertical loading circumstances and 1 g testing condition were other specifications of experimental and numerical results.