Effect of Footing Shape on the Rocking Behavior of Shallow Foundations

Abstract

Sources such as wind or severe seismic activity often exert extreme lateral loading onto the shallow foundations supporting high-rise structures such as bridge piers, buildings, shear walls, and wind turbine towers. Such loading conditions may cause the foundation to exhibit nonlinear responses such as uplift and bearing capacity mobilization of the supporting soil (i.e., rocking behavior). Previous numerical and experimental studies suggest that while such inelastic behaviors may engender residual deformations in the soil–foundation system, they offer potential benefits to the overall integrity of structures through dissipating energy and reducing inertia forces transmitted to the superstructure, thereby limiting seismic demand on structural elements. This study investigates the effect of footing shape on the rocking performance of shallow foundations in different subgrade densities and initial vertical factor of safety (FSv). To this end, a series of reduced-scale slow cyclic tests under 1 g condition were conducted using a single degree of freedom (SDOF) structure model. The performance of different footing shapes was studied in terms of moment capacity, recentering ratio, rocking stiffness, damping ratio, and settlement. For three foundations with different length-to-width ratios, the results indicate that increasing the safety factor and length-to-width ratio leads to thinner, S-shaped moment–rotation curves, mainly owing to the enhanced recentering capability and the P-δ effect. Moreover, across all foundation types, the repetition of a limited loading cycles with consistent rotation amplitude does not cause stiffness degradation or moment capacity reduction.

1. Introduction

The field of earthquake engineering has undergone a paradigm shift in recent decades, catalyzed by the devastating impact of strong earthquakes around the globe. The conventional design aims to mitigate seismic damages by permitting ductility-controlled inelastic behavior of the superstructure. This method, predominantly reliant on structural elements for energy dissipation, intentionally limits significant nonlinear behavior in the soil–foundation system [1,2]. However, emerging insights challenge this philosophy, suggesting it may lead to more expensive and, in some instances, more vulnerable structures [3].

Innovative techniques have emerged to control seismic responses by altering earthquake input motion characteristics. These include using isolating synthetic liners in the soil mass to reduce earthquake acceleration at the soil surface and incorporating rubber soil mixtures under foundations for energy dissipation [4,5,6]. The concept of rocking isolation in foundation design is also gaining traction as an effective method to reduce seismic forces on structures. This approach invites nonlinear behavior during intense ground motions to the soil–foundation system by intentionally under-designing it, thus allowing energy dissipation through foundation uplift and transient mobilization of bearing capacity failure [7,8,9].

Contrary to the traditional viewpoint, the soil–foundation system presents untapped potential in seismic force mitigation. Its capacity for geometric nonlinearity (like foundation uplift and sliding) and material inelasticity (such as bearing capacity mobilization) can significantly influence the inertia forces and energy transmission to the superstructure [7]. This aspect of soil–foundation interaction, particularly in the form of foundation rocking, has been substantiated through extensive research, including historical evidence and case histories [10,11], in situ field tests [12,13,14,15,16], reduced scale experiments under 1 g condition [1,2,17,18,19], centrifuge tests [20,21,22,23], and numerical modeling [7,24,25,26].

To date, methods to incorporate rocking and uplift into the design and modeling of shallow foundations have been employed for a range of structure types, including buildings [27,28,29,30,31], bridges [32,33], and more recently wind turbine towers [34,35]. In this regard, Kourkoulis et al. [31] explored the application of the rocking isolation concept to frames with spread footings. They utilized nonlinear finite element analyses to identify potential limitations of the concept. Additionally, they conducted a parametric study to assess its effectiveness in more intricate structures, using a two-bay frame as an illustrative example. According to their findings, the rocking-isolated two-bay frame exhibited satisfactory performance, successfully withstanding all analyzed earthquake scenarios with minimal distortion to the columns. Nouri et al. [36] investigated the effectiveness of employing rocking foundations for a three-story building and demonstrated that steel frames with rocking foundations on over-consolidated clay exhibited improved levels of story drift.

Deng et al. [33] proposed a direct displacement-based design methodology for the seismic design of rocking shallow foundations for bridges under earthquake loading, in which a multilinear model capturing the nonlinear moment–rotation behavior, as well as a novel empirical relationship linking initial rotational stiffness to moment capacity, is introduced. Agalianos et al. [37] constructed a three-dimensional finite element model encompassing an entire bridge, along with its supporting foundations and soil. The outcomes of dynamic analyses indicated that rocking foundations exhibit higher displacement capacities, greater resistance to toppling, and reduced residual tilting.

Ajaei and El Naggar [34] investigated the incorporation of the rocking foundation concept in repurposing shallow wind turbine foundations to accommodate higher-capacity turbines. Their investigation focused on the feasibility of replacing a 5 MW wind turbine with larger capacity turbines, considering seismic and aerodynamic loads. The study revealed consistent trends in the increase in maximum gapping and tilting with higher turbine power. Notably, they found that circumventing the no-uplift regulation under normal operational conditions enables the installation of an 8 MW wind turbine on a foundation initially designed for 5 MW without compromising safety or serviceability requirements. Furthermore, Wang and Ishihara [38] conducted a comparative analysis of two Winkler models to study the effects of foundation uplift on the wind turbines seismic loads, one incorporating the ability to simulate foundation uplift and the other lacking this capability. Their findings illustrated that incorporating foundation uplift in the modeling process led to a reduction in the tower base moment but an increase in the bending moment within the foundation cross-section. In this context, some of the most widely employed wind turbine support structure codes and guidelines provide insights into the potential occurrence of foundation uplift in specific loading cases. According to DNVGL-ST-0126, the foundation should withstand repeated gapping cycles without adverse effects, ensuring the stress state at the interface maintains soil plane shape stability beneath the foundation throughout its operational lifespan. In this respect, DNV permits the foundation to gapping to the center of gravity of the bottom area of the foundation for some load cases [39]. Furthermore, IEC 61400-6 advocates a zero-gapping criterion to keep the foundation in full contact with the soil in certain load cases, preventing the risk of progressive or sudden degradation of soil capacity or stiffness. However, it permits gapping between the soil and foundation under specific conditions, provided certain requirements are met [40].

Despite an expanding knowledge base and increasing experimental evidence, the utilization of foundation rocking as a seismic isolation mechanism for enhancing structural seismic performance is still viewed with skepticism. Primary concerns include the potential for excessive rotation and settlement of the foundation, the risk of the structure tipping over, and the inherent challenges in precisely predicting the performance of rocking foundations, especially in the face of uncertainties in soil properties and earthquake loading. These concerns collectively hinder the widespread adoption of foundation rocking as a design approach for mitigating seismic forces and ductility demands imposed on structures [41,42,43].

A broader understanding of the influential factors on the performance of rocking shallow foundations, as well as the corresponding soil deformation mechanisms, is necessary at this stage for the dependable, widespread use of such foundations in practice. In this respect, foundation shape is one of the most important factors influencing the rocking behavior of soil–foundation systems, and thus far, limited studies have been conducted in this regard [44]. Therefore, the present study is designed to investigate the shape effects on the rocking performance of shallow foundations in varying subgrade relative densities and initial vertical factors of safety in a series of 1 g, slow cyclic loading experiments. The performance of rectangular (loaded in the direction of weak and strong axes), circular, and square foundations in terms of moment capacity, damping ratio, settlement response, rotational stiffness, and recentering ratio were explored systematically in this study. Furthermore, footing deformation mechanisms for each foundation shape and different subgrade conditions were investigated and used to identify major sources of undesirable deformation in the soil–foundation system.

2. Physical Model Description and Testing Program

The current study methodically examines the rocking behavior of a rigid single-degree-of-freedom (SDOF) structure placed on dry sand subjected to slow cyclic lateral loading. This section details the various components of the physical model, outlines the test preparation procedure, describes the loading protocol, and presents the test program.

2.1. The Testing Tank, Foundation, and Structure Model

A soil container with dimensions of 126 × 80 × 75 cm (length × width × height) with rigid walls was utilized to model the foundation subgrade. The structure, modeled as a single-degree-of-freedom (SDOF) system, comprises a footing, a column, and a concentrated mass, all interconnected with rigid joints to limit deformations to the soil–foundation system only. The concentrated mass is afforded by steel plates positioned on two platforms on either side of the column at a height of 45 cm above the foundation base. Steel footings of circular, square, and rectangular shapes, all with equivalent areas and a thickness of 2 cm, were designed and used. The foundations are made of St37-2 steel with a Young’s modulus of 210 GPa and a yield strength of 235 MPa. The column is a steel box profile, 8 cm wide and 2 mm thick. Considering the stress level in the experiments and the dimensions of the column and foundation, a rigid response is anticipated from both. To simulate the friction between a real concrete foundation and its subgrade, an abrasive sheet (sandpaper) was attached to the footing’s bottom. The friction angle between the sand and this abrasive sheet was determined through direct shear tests in a modified shear box, measuring 33.6° and 35.3° for soil relative densities of 35% and 65%, respectively.

2.2. Subgrade Properties and Preparation Procedure

Utilizing Firouzkouh dry silica sand (code D1) for subgrade preparation, the sand raining technique was employed to establish a uniform subgrade with the desired density. The grain size distribution and key parameters of the Firouzkouh sand are represented in Figure 1. The e_min and e_max were determined based on the procedures recommended by ASTM D-4253 and ASTM D-4254 [45,46]. The raining height and rate can be adjusted to achieve the target density in the sand raining technique. Notably, the sand raining hopper covered the entire container area, ensuring a uniform density across the surface. For loose and dense subgrades, relative densities of 35% and 65% were considered, respectively, corresponding to dry unit weights of 15.4 and 16.1 kN/m³. These specific relative densities were achieved by implementing raining heights of 10 and 60 cm. To assess the uniformity of the relative density across the subgrade tank surface using the sand raining method, the relative density of various points on the tank surface was examined. To this end, three containers, positioned at different locations on the subgrade tank, were employed to conduct the tests. During these tests, the sand raining procedure was executed to fill each of the designated containers. Subsequently, the containers, each with predetermined volumes, were weighed. Utilizing the properties outlined in Figure 1, the relative density at each location on the subgrade tank was then calculated. Detailed information on the relative density at three distinct points on the tank surface, corresponding to specified raining heights, is provided in

Table 1. Relative densities for two respective raining heights.

Test No.	Raining Height (cm)	Container No.	Container Mass (gr)	Container Vol. (cm3)	Full Container Weight (gr)	Unit Weight (gr/cm3)	Void Ratio (e)	Relative Density (%)	Ave. Relative Density (%)
1	10	1	76.4	641	1061	1.53	0.727	33	35
		2	77.1	641	1065	1.54	0.721	35
		3	73.2	637	1055	1.54	0.721	35
2	60	1	76.4	641	1110	1.61	0.645	65
		2	77.1	641	1112	1.61	0.643	65	65
		3	73.2	637	1100	1.60	0.647	64

. The results affirm the efficacy of the raining system adopted in delivering uniform density throughout the entire container surface. Additional insights into the sand raining technique used in this study can be found in Asli et al. [1].

2.3. Vertical and Horizontal Loading Systems

To measure the bearing capacity of foundations with different shapes resting on different subgrade conditions, bearing capacity tests were performed before conducting the rocking tests. A pneumatic vertical loading system was employed to measure the bearing capacity of various foundation shapes on different subgrade densities. Figure 2 illustrates different components of the vertical loading system, including a pneumatic jack, a load cell, a loading shaft, and a linear variable differential transformer (LVDT).

Table 2. Details of bearing capacity tests conducted.

Test ID	Footing Shape	Subgrade Relative Density (%)	Footing Dimension, B × L (cm × cm)	Bearing Capacity (kg-f)	Bearing Capacity (kPa)
BC-Sq-D	Square	65%	15 × 15	405	176
BC-Circ-D	Circular	65%	16.9	376	164
BC-Rec-D	Rectangular	65%	22.5 × 10	365	159
BC-Sq-L	Square	35%	15 × 15	280	122
BC-Circ-L	Circular	35%	16.9	255	111
BC-Rec-L	Rectangular	35%	22.5 × 10	243	106

outlines the specifications of the bearing capacity tests. In this study, the load-bearing capacity of a square footing with a width of 15 cm, a rectangular footing with dimensions of 10 × 22.5 cm, and a circular footing with a diameter of 16.9 cm were determined using the vertical loading system. It is noteworthy that the vertical loading system was used solely to determine the ultimate load-bearing capacity of the foundations. The vertical load in lateral loading tests is provided by the mass of the structure and the concentrated mass attached to it, not by the vertical loading jack.

The naming of the experiments indicated in Table 2 consists of three parts. The first part, ‘BC’, stands for bearing capacity test, and this section remains constant in all relevant tests. The second part indicates the shape of the footing: ‘Sq’ for square, ‘Rec’ for rectangular, and ‘Circ’ for circular. The third part represents the relative density of the bed, with ‘D’ representing dense subgrade and ‘L’ representing loose subgrade. The outcomes of the bearing capacity tests on square, circular, and rectangular footings placed on loose and dense subgrade conditions are presented in Figure 3. This study defines the ultimate bearing capacity as the load at which the load–settlement curve peaks or, in the absence of a peak, the load at which the vertical stiffness‘s rate of change becomes constant [2,23,47]. Table 2 includes the corresponding values at these points.

Furthermore, Figure 4a,b display the horizontal loading system, along with the structural model. The structure’s center of mass is horizontally displaced through a displacement control actuator, which is supported by the tank wall. As shown in Figure 4a and Figure 5, the actuator is linked to the structure through a combination of shear joints and pin connections, enabling unrestricted rotation and settlement of the structure. The actuator’s height is adjustable, ensuring precise application of displacement to the pin connection. Additionally, the vertical load corresponding to different FSv values is generated by the weight of the structure model, encompassing both lumped mass and structural mass.

A load cell is positioned between the actuator and the shear joint to measure the load transmitted to the structure. Vertical displacements, or settlements, are monitored using an LVDT directly placed on the foundation. To track the rotation of the structure model, a rotary encoder is attached to the pin connection. All mentioned instruments are linked to a data logger, which consistently records and transmits the necessary data to a computer.

2.4. Loading Protocol

In this test series, the structure model was subjected to slow periodic displacements utilizing a lateral loading system under 1 g condition. Previous studies investigated the rocking behavior of foundations using one of the four following loading conditions:

• 1 and 2—Reduced scale experiments under 1 g condition with dynamic loading (shake table) or slow cyclic loading
• 3 and 4—Centrifuge testing (Ng acceleration) under dynamic loading (shake table) or slow cyclic loading

Among the mentioned methods, centrifuge condition and shake table testing stand out for their respective advantages. Centrifuge testing reduces scale effects and generates stress levels closer to real conditions, while shake table tests offer a more accurate simulation of seismic excitations and inertial loads on structures. These methods are more suitable for studying the influence of various factors on stresses and deformations in both the structure and subgrade soil. Slow cyclic loading, on the other hand, has its own set of advantages in studying the rocking behavior of soil–foundation systems. Slow cyclic loading produces more transparent and better-organized moment–rotation and settlement–rotation diagrams as opposed to dynamic loading, facilitating the observation and interpretation of behavioral mechanisms in detail. Additionally, in slow cyclic loading, soil mass deformations result solely from foundation displacements. In dynamic loading, however, soil mass behavior and deformations are predominantly influenced by inertia from imposed acceleration, complicating result interpretation.

The reliability of the 1 g condition, slow cyclic loading in the study of rocking foundations has been addressed in previous studies. Kokkali et al. [48] conducted a comprehensive study of the rocking behavior of foundations under both 1 g and centrifuge conditions. Their findings revealed a high degree of correlation, particularly in aspects like stiffness degradation, energy dissipation, and cyclic moment overstrength. Furthermore, they noted remarkable similarities in the moment–rotation and settlement–rotation diagrams, as well as in the rotational stiffness, between the 1 g test results and those from centrifuge testing. Furthermore, Gajan et al. [49] revealed that the moment–rotation backbone curve (the maximum moment values of each cycle), obtained from cyclic loading when the load is exerted at slow rates, complies well with that of dynamic loading test results. They concluded that the performance of the soil–foundation system, subjected to slow cyclic loading, can be used to anticipate its performance under dynamic loading. This similarity in response, however, can be expected only for rigid structures [50].

Accordingly, the earthquake load simulation was performed by applying lateral load packages in the form of displacement to the structure’s center of mass. As shown in Figure 6, the loading pattern consists of three phases with amplitudes of 2.25 mm, 9 mm, and 36 mm (equivalent to rotations of 0.5%, 2%, and 8%, respectively), representing minor, moderate, and significant deformations in the soil environment. Each loading phase includes three similar cycles. Repeating more cycles in a specific phase (with constant rotational amplitude) does not significantly alter the behavior of the foundation; therefore, further repetition of loading cycles in a phase was disregarded. To reduce the effects of the structure’s inertial force during cyclic loading, a loading speed of 1 mm per second was considered, resulting in a total loading duration of 567 s for each test.

2.5. Test Program

The purpose of this study is to investigate the rocking behavior of foundations with different geometry and FSv values, and to this end, different subgrade relative densities (35%, referred to as loose subgrade, and 65%, referred to as dense subgrade) and different structural mass (70 kg and 120 kg) were considered to achieve various FSv values. Footing shape effects were systematically studied by considering four footing shapes, namely rectangular footings loaded in the direction of weak and strong axes (referred to as weak-axis rectangular footing and strong-axis rectangular footing, respectively), circular, and square footings. With respect to the mentioned variables, eight tests with FSv values ranging from 2 to 5.8 were conducted, the names and specifications of which are tabulated in

Table 3. Details of the horizontal loading tests conducted.

Test ID	Footing Shape	Subgrade Relative Density (%)	Structural Mass (kg)	$\mathbf{F}{\mathbf{S}}_{\mathbf{v}}$	Footing Static Stress (kPa)
LL-RecW-L	Rectangular/Weak Axis	65%	70	5.2	30
LL-Sq-L	Square	65%	70	5.8	30
LL-Circ-L	Circular	65%	70	5.4	30
LL-RecS-L	Rectangular/Strong Axis	65%	70	5.2	30
LL-RecW-H	Rectangular/Weak Axis	35%	120	2.0	52
LL-Sq-H	Square	35%	120	2.3	52
LL-Circ-H	Circular	35%	120	2.1	52
LL-RecS-H	Rectangular/Strong Axis	35%	120	2.0	52

. The name of each experiment includes three parts: the first part, ‘LL’, stands for lateral loading; the second part indicates the shape of the footing (Sq: square, RecW: weak-axis rectangular footing, RecS: strong-axis rectangular footing, and Circ: circular); and in the third part ‘H’ represents heavily loaded foundations (high structural weight and loose subgrade, hence small FSv) and ‘L’ denotes lightly loaded foundations (low structural weight and dense subgrade, hence, large FSv).

3. Results and Discussion

As previously highlighted, the initial vertical factor of safety significantly impacts the rocking behavior of foundations. In line with this, this study conducted a comparative analysis between two distinct experimental groups. The first group includes experiments with large FSv values (lightly loaded foundations: dense subgrade and a low structural mass), and the second group consists of experiments with small FSv values (heavily loaded foundations: loose subgrade and a high structural mass). The moment–rotation and settlement–rotation diagrams of these experiments for different foundation shapes are presented in Figure 7 and Figure 8. Following this, an in-depth examination was conducted on the performance of different footing shapes, focusing on aspects such as their moment capacity, self-centering ability, rotational stiffness, damping ratio, and settlement.

3.1. Foundation Moment Capacity

The comparison of diagrams related to the experiments with large and small FSv values in Figure 7 and Figure 8 shows that with the increase in the safety factor of the foundation, its moment capacity also increases. The general equation for the moment capacity of a foundation is presented in Equation (1). The normalized moment capacity (on which the diagrams of this study are based) was also obtained by dividing the moment capacity of the foundation by the overturning moment of a rigid foundation on a rigid base (M_r = P × L/2) using Equation (2):

$${\mathrm{M}}_{\mathrm{ult}}=\mathrm{P}\times \frac{\mathrm{L}}{2}\left[1-\frac{{\mathrm{A}}_{\mathrm{c}}}{\mathrm{A}}\right]$$

(1)

$$\frac{\mathrm{M}}{\mathrm{Mr}}=1-\frac{{\mathrm{A}}_{\mathrm{c}}}{\mathrm{A}}$$

(2)

In the aforementioned equation, ‘P’ denotes the structure’s weight, and ‘L’ refers to the foundation’s dimension in the direction of the horizontal load. The term ‘Ac’ equals the required area for supporting the structure’s weight during rocking motions, while ‘A’ represents the foundation’s total area. The ratio of Ac to A is inversely proportional to the foundation’s static FSv. Hence, it is predictable that an increase in the foundation’s normalized moment capacity correlates with a rise in the safety factor (which corresponds to a reduction in the Ac/A).

As seen in Figure 7 and Figure 8, the moment capacity of foundations in the second phase of loading increased compared with the first phase in all experiments conducted. This implies that the foundation’s moment capacity was not entirely mobilized in the first phase due to minimal rotational displacement; that is, the foundation did not reach its minimum soil contact area (Ac). In foundations with large FSv, this occurs in the second phase; consequently, there is no significant increase in moment capacity values when transitioning from the second to the third phase. Moreover, based on the results presented in Figure 7, it can be observed that the repetition of loading cycles within a single phase does not significantly impact the foundation’s moment capacity.

On the other hand, in foundations with small FSv (Figure 8), the transition from the second phase to the third phase not only leads to an increase in the moment capacity of foundations but also the repetition of the loading cycle in a specific phase leads to an increase in the moment capacity. This increase in the moment capacity of the foundation results from two simultaneous occurrences. First, due to the low subgrade density and high structural mass, the sand mass undergoes densification during the loading and unloading quarter cycles; hence, the moment capacity increases with increased subgrade density. Furthermore, with respect to the low footing factor of safety, significant settlement materializes during cyclic loading, resulting in an increased embedment depth of the foundation and ultimately leading to an increase in both the safety factor and the moment capacity of the foundation.

An important point to note is that repeating loading cycles within a specific amplitude of rotation does not lead to a reduction in the moment capacity of a rocking footing, and in some cases, an increase in strength may materialize. This is noteworthy as a common concern with seismic load-bearing elements and earthquake energy absorbers is their strength degradation in repeated loading cycles. However, it should be noted that this behavior is observed for dry sand subjected to a small number of cycles in this study, and this should be examined for other soil conditions, such as clayey soil or unsaturated sand.

In examining the effect of the footing shape on the moment capacity considering the FSv, it is observed that the moment capacity of foundations generally increases with an increase in the length-to-width ratio (L/W) of the footing. For instance, from Figure 7 and Figure 8, it can be inferred that LL-RecS-L and LL-RecS-H (FSv = 5.2 and 2, respectively) with L/W = 2.25 have greater moment capacity compared with the LL-Sq-L and LL-Sq-H (FSv = 5.8 and 2.3, respectively) with L/W = 1. The same trend is applicable to the LL-RecW-L and LL-RecW-H. Furthermore, circular footings showed inferior moment capacity compared with square footings. This may be due to the square footing’s greater second moment of area compared with the circular footings [51].

3.2. Shape of the Moment–Rotation Diagrams

The inspection of moment–rotation hysteresis loops provides insights into the general rocking foundation behaviors. Figure 7 and Figure 8 demonstrate that in large amplitudes of rotation in experiments with large FSv, the loops are S-shaped and slender. In contrast, tests with small FSv produce fat, elliptic moment–rotation diagrams. These shape differences arise from variations in the soil–foundation contact area and the degree of toe (loaded side) penetration into the subgrade during rocking. Moradi et al. [52] studied the deformation mechanisms beneath rocking foundations using the PIV technique. They found that foundations with large FSv have a smaller contact area with the soil during loading cycles, resulting in a notable uplift on the heel side (unloaded side). Conversely, foundations with small FSv establish larger soil contact area and undergo significant settlement and wedge deformation. This causes the footing to sink into the soil (a sinking mechanism) and exhibit minimal uplift on the unloaded side of the foundation (Figure 9).

When observing the beginning of the unloading quarter cycles, particularly in the third phase, there is a behavioral distinction in the moment–rotation diagram of foundations with small and large FSvs, impacting whether the diagrams become S-shaped or elliptical. In foundations with large FSvs, the force gradually decreases until it reaches zero at around half to two-thirds of the maximum rotation, while in foundations with small FSvs, the force drops quickly and reaches zero at a rotation greater than two-thirds (and sometimes around 90%) of the maximum rotation. This difference in behavior demonstrates that in foundations with large FSvs, less force is needed to move the foundation toward zero rotation due to the smaller contact area with the subgrade and the positive P-δ effect, creating a self-centering effect. Conversely, in foundations with small FSvs, due to the large foundation–subgrade contact area and significant sinking into the soil, the P-δ effect is negligible, requiring force to be applied from the beginning of unloading to return the foundation to its initial equilibrium (zero rotation). These variations in the unloading path of foundations lead to the creation of S-shaped or elliptical loops.

In examining how the shape of a footing influences the characteristics of moment–rotation diagrams, it was observed that an increase in the footing’s length-to-width ratio results in slenderer, S-shaped diagrams. This effect is attributable to the foundations’ recentering capacity and the P- δ effect (see Figure 10). Essentially, a higher length-to-width ratio in footings increases the distance between the structure’s center of mass and the soil–foundation contact surface’s center (or, in other words, the δ value). This increased distance increases the moment caused by the P-δ effect, which facilitates the recentering of the structure back to its original equilibrium. Consequently, this leads to the formation of slenderer, more S-shaped hysteresis loops in the moment–rotation diagrams.

3.3. Self-Centering

Recentering of a structural system is its ability to minimize residual displacements during loading and unloading cycles. The preferred structural systems are those that, in addition to dissipating energy, have the least residual displacement after loading. In Figure 11, a simplified diagram of the moment–rotation behavior of a rocking foundation during cyclic loading (based on Deng et al. [33]) is shown. Based on Figure 11, the recentering ratio is defined as follows:

$${\mathrm{R}}_{\mathrm{c}}=1-\frac{\mathrm{z}}{\mathrm{d}}$$

(3)

where d denotes the maximum cycle rotation, and z represents the residual rotation (at zero moment). A rocking foundation on a rigid base has zero residual rotation, yielding a recentering ratio of one. Conversely, a foundation with a safety factor of one shows no uplift in rocking motions, resulting in a recentering ratio of zero.

In Figure 12, the recentering ratios for footings of different shapes are presented, categorized by foundations with large and small FSvs. As expected from the previous section’s explanations, foundations with large FSvs exhibit significantly greater recentering ratios than those with smaller FSvs. Additionally, based on the result of three tested foundations with the length-to-width ratio of 0.44, 1, and 2.25, it was observed that an increase in the length-to-width ratio of the foundation increases its recentering ratio, a phenomenon explained in the previous section. It can also be concluded that the recentering ratios of the circular and square footings are quite similar to one another, with the circular footings having slightly greater recentering ratios considering their smaller FSv values.

3.4. Rotational Stiffness

Rotational stiffness is an important parameter in analyzing the behavior of rocking foundations. Figure 13 shows a diagram of the foundation’s moment variation and corresponding rotation for a loading cycle. Based on this figure, the rotational stiffness is equal to the slope of the line connecting the maximum and minimum points of the moment–rotation hysteresis loop using the following equation.

$${\mathrm{K}}_{\mathsf{\theta }}=\frac{2{\mathrm{M}}_{\max }}{2{\mathsf{\theta }}_{\max }}$$

(4)

Figure 14 shows the rocking stiffness of shallow foundations of varying shapes. The findings indicate a reduction in stiffness as the amplitude of rotation increases. This is mainly due to the reduction in the soil–foundation contact area and the non-proportional increase in the moment with foundation rotation. As highlighted by Gajan et al. [49], at small strains, significant moment capacity is mobilized and, consequently, the rotational stiffness is high. Afterward, the system undergoes softening in larger strains, and the rotational stiffness degrades.

As expected, foundations with large FSvs (lightly loaded foundations) exhibit greater stiffness than those with lower values. Moreover, increased length-to-width ratios enhance rocking stiffness, with strong-axis rectangular footings displaying maximal stiffness during each loading phase. Furthermore, it can be noted that circular and square footings have relatively close rocking stiffness, with square footings having slightly greater values.

3.5. Energy Dissipation

As previously discussed, significant energy dissipation of rocking foundations is a key advantage of these foundations over conventional foundation design methods. Based on Figure 13, the foundation’s damping is determined by a ratio of the energy absorbed in one hysteresis loop to the elastic energy, calculated using the following equations.

$${\mathsf{\Delta }\mathrm{E}}_{\mathrm{elastic} }=\frac{1}{2}{\mathrm{K}}_{\mathsf{\theta }}*{\left(\frac{\left|2{\mathsf{\theta }}_{\max }\right|}{2}\right)}^2$$

(5)

$$\mathsf{\zeta }=\frac{{\mathsf{\Delta }\mathrm{E}}_{\mathrm{cycle} }}{4{\mathsf{\pi }}^{*}{\mathsf{\Delta }\mathrm{E}}_{\mathrm{elastic} }}$$

(6)

Figure 15 presents the damping ratio results for foundations of various shapes, categorized by large and small FSvs. The second cycle of each phase was used to compute the damping ratio. It is evident that with a decrease in the safety factor, the damping ratio increases. This increase is due to more significant plastic deformations and the development of more pronounced failure mechanisms in footings with smaller FSvs (see Figure 9). Additionally, this behavior is observed with an increase in rotation from the first phase to the third phase. As foundation rotation and, consequently, soil deformation under the foundation increases, the overall damping ratio of the foundation also increases.

In examining the effect of foundation shape, it was noted that the weak-axis rectangular footing exhibits the highest damping. This conclusion is also inferred from comparing moment–rotation hysteresis loops in Figure 7 and Figure 8, in which the hysteresis loops for weak-axis rectangular footings are wide and elliptical, and as the length-to-width ratio increases, the hysteresis loops become slim and S-shaped, indicating a reduction in the energy dissipation. As previously discussed, this behavior is due to the greater unloading stiffness and lower recentering properties of foundations with a small length-to-width ratio compared with those with a larger length-to-width ratio. Furthermore, the results presented in Figure 15 show that the square footing exhibits greater damping ratios compared with circular footings.

It is worth noticing that damping is significant across all tests conducted with different safety factors and footing shapes. For instance, the lowest damping value, shown in Figure 15 for the first loading phase of the lightly loaded, strong-axis rectangular footing, is 22%—a considerable amount.

3.6. Examination of the Settlement–Rotation Results

The primary concern with implementing the rocking isolation design approach is the potential for permanent deformations, such as settlement and rotation. This concern often hinders the adoption of the rocking isolation design approach, with significant foundation settlement being a major issue. However, the main objective of this approach is to prevent extreme structural damages or collapse during earthquakes beyond the design level, exemplified by the 1995 Kobe earthquake’s destruction of the Fukae bridge piers in Japan. Settlements of even several centimeters in a foundation are, therefore, viewed as favorable outcomes compared with total structural failure. To address these concerns, researchers developed various ground improvement techniques to mitigate foundation settlement and rotation caused by rocking movements.

The settlement–rotation diagrams in Figure 7 and Figure 8 show that with an increase in the foundation’s safety factor, the settlement values decrease. Additionally, the uplift behavior of foundations in the loading quarter cycles is more evident in foundations with larger safety factors. As the safety factor increases, soil densification mechanism, formation of soil failure wedges, and soil escape from beneath the foundation become limited. Consequently, the foundation settlement values decrease during the loading and unloading quarter cycles.

Figure 16 illustrates the settlement trend of shallow foundations with various shapes at the end of each loading cycle. The data show an increase in settlement as the footing’s length-to-width ratio rises. Specifically, strong-axis rectangular footings, having the largest length-to-width ratio, experience the most settlement. This trend is consistent for both categories of foundations with small and large safety factors. Furthermore, as can be seen in Figure 16, circular footings have a slightly better settlement response compared with square footings.

To understand the reasons behind the considerable settlement of footings with large length-to-width ratios, it is essential to examine soil displacement mechanisms during rocking motions. Figure 17 presents the following three primary soil mass deformation mechanisms under a rocking foundation: (I) the densification mechanism that increases soil density under the foundation during loading and unloading quarter cycles; (II) the wedge deformation mechanism, causing soil displacement outward from the foundation, creating soil heave in the direction of loading; and (III) the scoop deformation mechanism, which pushes the soil from the loaded side toward the gap created under the unloaded side of the foundation (creating heave in the unloaded side). The main reason for foundation settlement during rocking motions is the escape of soil from underneath the foundation as a result of the wedge deformation mechanism and, to a lesser degree, soil compaction beneath the foundation. Even when fully developed, the scoop deformation mechanism does not notably contribute to the overall settlement of the foundation. Nevertheless, due to its considerable soil displacement and the creation of extensive slip surfaces, the scoop deformation mechanism is capable of absorbing a significant amount of energy during these rocking motions.

As depicted in Figure 18, the critical contact area of a strong-axis rectangular footing is relatively square-shaped (rectangular contact area with a b/Lc ratio of near 1—Figure 18b), while the critical contact area of a weak-axis rectangular footing is elongated and strip-shaped (rectangular contact area with a high b/Lc ratio—Figure 18a). In this study, the b/Lc ratio of strong-axis rectangular foundations in loose and dense subgrade conditions is 1 and 1.5, respectively. Further, this ratio is about 3.1 and 5.1 for weak-axis rectangular foundations resting atop loose and dense subgrades, respectively.

In strong-axis rectangular footings, due to the shape of the soil–foundation contact area, a small amount of soil in contact with the foundation returns under the footing (by scoop mechanism), and most of the soil in contact with the foundation is pushed outward by the wedge deformation mechanism. However, in weak-axis rectangular footings, most of the soil is displaced under the foundation (by the scoop mechanism), and a relatively insignificant amount of soil is pushed outward in the direction of loading. In other words, only a small amount of the soil under the foundation is pushed outward. This difference in the amount of soil displaced outward in different footing shapes is the main cause of settlement differences due to rocking motions.

The described mechanisms in relation to the subgrade soil displacements during rocking motions can be correlated with heave around the foundation at the end of loading cycles. As can be seen in Figure 19a, most of the soil displacement in the weak-axis rectangular footing has occurred toward the right of the foundation. The accumulated soil mass on the right side of the footing is related to the displacement mechanisms described earlier (heave only on one side). In contrast, in the strong-axis rectangular foundation (Figure 19b), it can be seen that soil displacement and heave occur on all sides of the foundation (in the direction and perpendicular to the direction of loading). These lateral displacements, as previously stated, are the main cause of significant settlements in strong-axis rectangular footings. It is notable that in the case of square and circular footings, these displacements occur all around the foundation.

4. Conclusions

The rocking isolation design concept is an emerging approach in the seismic design of foundations that promotes the exhibition of nonlinear behaviors such as uplift and bearing capacity mobilization in the soil–foundation system. Despite a fair amount of research conducted on the behavior of rocking foundations, there are still concerns regarding the practical use of this approach, mainly due to a lack of precise understanding of the rocking behavior of the soil–foundation system under various soil and foundation conditions. A precise understanding of the effect of foundation shape on its rocking behavior and deformation mechanisms can enhance our understanding and, hence, facilitate the reliable application of such a design concept. Considering this, in the current research, the rocking behavior of foundations of various shapes with different vertical factors of safety resting on different subgrade conditions was investigated using physical modeling under 1 g condition and slow cyclic loading.

Based on the results obtained, the following conclusions can be drawn:

• The rocking stiffness of all foundation types tested decreases as the amplitude of rotation increases. Regarding the effect of foundation shape on its rocking stiffness, it was observed from the three length-to-width ratios studied that the rocking stiffness increases as the length-to-width ratio of the foundations increases. Circular and square foundations also showed relatively close rocking stiffness, with square foundations showing slightly greater values.
• In all foundation types, the repetition of limited loading cycles (three cycles) with similar amplitude of rotation does not lead to stiffness degradation and moment capacity reduction. In fact, in foundations with a small FSv, the repetition of loading cycles with similar amplitude of rotation can lead to an increase in the moment capacity due to the soil densification and an increase in the foundation embedment depth (as a result of foundation settlement).
• The findings revealed that foundation settlement is significantly influenced by the length-to-width ratio. Foundations with a high ratio (strong-axis rectangular foundations) form a square-shaped critical contact surface during rocking, leading to soil escaping from three sides and, hence, greater settlements. In contrast, weak-axis rectangular foundations form a strip-shaped contact surface, returning almost half of the displaced soil under the foundation through the scoop deformation mechanism, leading to comparatively smaller settlements. The results also indicate that circular foundations have relatively better settlement response compared with square foundations.
• Increasing the safety factor of the foundation leads to a decrease in the damping of the soil–foundation system. The damping ratio of the foundation increases with the increase in the foundation’s amplitude of rotation and, in general, foundations with smaller length-to-width ratios exhibit higher damping ratios. Square foundations exhibited greater damping ratios compared with circular foundations in this study.
• With increasing the safety factor and length-to-width ratio of the foundation, the moment–rotation curves become thinner and S-shaped, which can be justified by the foundation’s recentering capability and the P- δ effect.
• The recentering ratio, which indicates the system’s ability to return to its initial position, increases with an increase in the safety factor of the foundation. Furthermore, this ratio increases as the foundation’s length-to-width ratio increases. Comparing the results of square and circular foundations, it can be concluded that they have almost similar recentering ratios, with circular foundations having slightly greater recentering capability.