Combined loading apparatus for monotonic and cyclic planar loading of foundations Open Access

$D$

pile diameter (prototype scale)

$d$

diameter of the strong box

$FS$

safety factor of a single pile under axial loading

$fc$

unconfined compressive strength of concrete

$fys$

yield strength of steel reinforcement

$H$

horizontal load at the foundation

$Hl$

load of the lower actuator (compression or tension)

$Hu$

load of the upper actuator (compression or tension)

$h$

lever arm of single actuator (pushover configuration)

$hl$

lever arm lower actuator

$hu$

lever arm upper actuator

$l$

depth of the strong box

$M$

overturning moment at the foundation (reference point)

$N$

target scale of the centrifuge tests

$s$

spacing of transversal reinforcement (prototype scale)

$u$

horizontal displacement

$ul$

displacement of the lower actuator (contraction or extension)

$uu$

displacement of the upper actuator (contraction or extension)

$V$

vertical load at the foundation

$w$

settlement

$θ$

rotation

The design of foundations under general planar loading (combined vertical, $V$⁠, lateral, $H$⁠, and moment, $M$⁠) is essential for a variety of engineering applications. Offshore installations subjected to environmental (wind and wave) loading, or foundation systems subjected to seismic loading at earthquake-prone regions are characteristic such examples. Foundation design historically treated combined loading within the framework of the $3N$ bearing capacity formulae (Terzaghi, 1943). Modification factors were introduced to extend the basic solution (failure under vertical loading) to more complex load cases. Inclination factors were used to account for the effect of lateral loading, while the overturning moment was treated as an eccentric vertical load, reducing the foundation width to an effective one (Meyerhof, 1953). This approach is implemented in many modern design codes and is still widely applied in engineering practice.

A decisive turning point on foundation bearing capacity can be traced back to the late $1970s$⁠, with the introduction of the first 3D failure envelopes for shallow foundations. Figure 1 presents some of the first such interaction domains proposed by Ticof (1977), based on a series of scaled-down tests of model footings in dry sand, under 1g conditions. Using a simple experimental assembly, consisting of a single actuator, a variety of inclined and eccentric load combinations were studied. Connecting the obtained points of this 1g test campaign, led to the well-known today parabolic failure envelopes for shallow foundations.

In the following decades, the increasing demand from the offshore industry fuelled a variety of such studies. The convenient representation of the ultimate bearing capacity in the form of interaction diagrams motivated the development of different apparatuses either for 1g (Martin, 1994) or for centrifuge testing (e.g. Dean et al., 1997; Zhang et al., 2013). A comprehensive overview of the available test setups is provided by Zhang et al. (2013), and some indicative examples of existing developments along with their main functionality are illustrated in Figure 2. Gottardi et al. (1999) present a detailed overview of the different techniques available to experimentally derive failure envelopes. Application of straight displacement paths, or of straight load paths, can be employed to reach the failure locus at a single point, while swipe tests (extending the ‘side-swipe’ introduced by Tan, 1990) are often employed to track the yield surface, thus requiring a limited number of tests to derive the full 3D failure envelope.

Today, 3D failure envelopes are available for almost any foundation type, being the state-of-practice for safety factor-based foundation design under $VHM$ loading, especially for offshore applications. Moreover, failure envelopes are a fundamental element of the more general plasticity-based macro-element modelling approach, which is commonly applied for performance-based foundation design. Although many of the solutions available today have also been derived analytically or numerically, their significant evolution in the past decades was undoubtedly driven by experimental studies that exploited the advancements in the field of physical modelling. The developed experimental setups allowed for extensive and repeatable testing, thus allowing for experimental verification of novel design concepts and approaches.

Recognising the need to experimentally support any new design concept, a combined loading apparatus capable of performing swipe testing was developed at ETHZ (Figure 2(d)) and used in the drum centrifuge (Springman et al., 2001) of the centre. Combining three electromechanical actuators and a number of mechanical parts, along with hinged connections (materialised by means of roller bearings), the developed setup could successfully apply any desired displacement path (combinations of settlement, lateral translation, and rotation) at any reference point (Sakellariadis et al., 2022).

All setups presented in Figure 2, including the authors’ previous work, share the same functionality: they apply combinations of planar lateral translation and rotation, either by restricting or prescribing the vertical displacement of the foundation. For certain geotechnical applications, the sinking or uplifting behaviour of the foundation is a desired output, while often the vertical load is a key input parameter of the studied problem. With those two aspects in mind, the setup presented in Figure 2(d) was modified, maintaining only the two horizontal actuators and adjusting their end condition from hinged to fixed. Placing the pair of actuators at a certain spacing allows to apply any desired combination of lateral translation, $u$⁠, and rotation, $θ.$ The latter is possible utilising a rigid column, which transmits the loads from the actuators to the foundation level, being connected to the actuators by means of sliding hinges. Figure 3 provides an illustration of the drum centrifuge test setup, highlighting its key attributes, along with a snapshot during testing. This setup was successfully employed to study a 2 × 1 pile group at 100g, subjected to both monotonic and cyclic loading. The campaign explored both rocking (overturning moment-dominated) and shear (lateral load-dominated) regions of the response, leading to the development of experimental failure envelopes, corresponding to distinguishable, load path-dependent, failure modes. Indicative results are shown in Figure 3.

This paper introduces the latest upgraded version of such an experimental setup, developed for the new beam centrifuge of the ETHZ Geotechnical Centrifuge Center (GCC). Having proof-tested the main concept of the assembly in the (smaller) drum centrifuge, the new apparatus offers significantly increased capacity and enhanced robustness. Furthermore, several technical limitations identified along the implementation of the previous setup are now discussed and tackled. The key concept of the new experimental setup is outlined, and its main capabilities are illustrated through selected examples and indicative preliminary test results.

The previously described test series (Figure 3) were performed making use of relatively small models at 1:100 scale with respect to the prototype problem. This was dictated by space limitations in the drum centrifuge, related to both the size of the strong boxes that were fitted in the drum channel, and the unavoidable (due to space limitations) small size of the actuators, that limited the range of applicable forces. The maximum lateral loading capacity of 3 $kN$ (2 × 1.5 $kN$⁠) corresponds to 30 $MN$ at prototype scale when the testing is performed at 100g, while it would reduce to merely 7.5 $MN$ if testing was performed at 50g.

Careful evaluation of the drum centrifuge test setup revealed certain additional limitations, calling for enhancement. The first issue is related to the establishment of fully fixed end conditions for each actuator, due to the small but non-negligible compliance of the support system. The second refers to the (limited) bending of the actuators caused by their self-weight, when subjected to the increased gravity field of the geotechnical centrifuge. The latter was found to partially affect the control precision of the two actuators, calling for a closed-loop control regulation system that was successfully developed and employed for the drum centrifuge test campaign. Both issues are directly related to the space limitations of the drum centrifuge, and can therefore be addressed when developing such an experimental setup for the much larger beam centrifuge. The new test apparatus is developed to study foundations under monotonic and cyclic combined planar loading, exploiting the considerably larger capabilities of the newly installed beam centrifuge of the ETHZ GCC (Figure 4).

Without the previous space limitations of the drum centrifuge, the new experimental setup aims at testing foundation systems at lower g-levels (target scale 1:50), being however designed and proof-tested for up to 100g. The reduced g-level allows for increased modelling details, unavoidably resulting to higher force demands. With a capacity of 7 $kN$ and maximum stroke of 100 mm, the selected stepper motor actuators (FESTO ESBF-BS-63-100-5P) satisfy the $N2$ scaling requirements, allowing for flexibility in testing at lower g-levels. Via a state-of-the-art controlling system, either load- or displacement-controlled testing is possible with displacement accuracy ± 0.015 mm and velocity from 0.01 to 270 mm/s.

The increased forces are also calling for a rigid support system, ensuring stability with minimum deformability. This is achieved by adding four stiffeners between the fixed-end supports of the two actuators at two different adjustable heights, selected to coincide with the axis of each actuator (Figure 5). Simple L-sections made of steel are used for this purpose, serving an additional dual role: (i) they allow mounting of the pair of laser sensors required to measure displacements and rotations of the rigid loading column; and (ii) they act as tracks, supporting rollers connected to the end-part of the actuator arms. Special attention was paid to the design of this part (rollers on tracks) to ensure the establishment of a linear slider condition, that is capable of undertaking the increased self-weight of the actuator arm without any undesired bending. The configuration was designed and constructed in-house at ETHZ (Figure 5). A key part of this configuration is the sliding-hinged connections between the actuators and the rigid column, materialised by roller bearings and a slot of marginally larger width, but with sufficient length (Figure 5). The latter allows the foundation to freely settle during spin-up, without developing undesired negative skin friction. During combined loading, the foundation can freely settle or uplift, depending on the loading conditions, without altering the vertical load which remains constant.

Depending on the requirements of each test series, the developed experimental setup can be set either to load- or displacement-controlled mode. The concept of the apparatus is schematically illustrated in Figure 6. The two actuators are set in different distances from the reference point of the foundation, namely, $hu$ and $hl$⁠, for the upper and the lower actuator respectively. The actuators are connected to the foundations via a rigid column and appropriate sliding hinges. Thanks to the selected connection, no bending moment or shear is transmitted to the actuators, which experience only compressive of tensile loads (⁠$Hu$⁠, $Hl$⁠) as they extend or contract by a measured amount: $uu$⁠, $ul$ (the subscripts correspond to the position of each actuator, namely, upper and lower). The applied forces and the resulting displacements are measured by two load cells, fixed to the actuator arms, and a system of laser sensors, respectively.

By setting a certain target load to each actuator and adjusting their spacing, it is possible to apply any $M-H$ load path at the foundation level:

Similarly, setting certain target displacement to the actuators, any $u-θ$ combination is possible:

The configuration can be easily adjusted to conduct conventional pushover testing, simply by disconnecting the one actuator (Figure 6(b)). In this case, even when the actuator is set to displacement-controlled mode, the foundation is subjected to a certain load path, $M-H$⁠, controlled by the lever arm, $h$⁠, between the actuator and foundation reference point:

Besides its significantly increased load capacity, a key advantage of the new apparatus is its versatility and ease of conversion from one experimental setup to another. In both test configurations (combined loading or pushover loading) the target constant vertical load is provided by the adjustable mass at the top of the rigid loading column. Throughout the test, the foundation is allowed to settle or uplift freely, which is attributed to the sliding hinged connection between the actuators and the rigid loading column.

Certain limitations of the developed setup are highlighted and discussed. The developed assembly has been designed to apply one-directional planar loading. Applying multidirectional loading (in- and out-of-plane loading combinations), which may be important for certain foundation systems, is not possible with the proposed setup. Another key limitation of the setup at its current state is the fact that it can only be used to study wished-in-place foundations. A future upgrade is planned as illustrated in Figure 7, and is briefly discussed herein. The loading frame, currently used for 1g-jacking purposes, is designed to fit in the combined loading apparatus. Thanks to the sliding hinged connection of the actuators to the loading beam, the test foundation can be jacked in-flight. The vertical actuator can then be retracted, and subsequently the foundation can be subjected to combined loading, allowing free vertical movement. Special care and adaptations are essential so that the vertical actuator provides the capacity and stroke needed for each target installation.

The new experimental setup was proof-tested in two recent test series related to: (a) lateral loading of a single reinforced concrete (RC) pile; and (b) pushover loading of a light-weight structure on a block foundation. A brief overview of the employed model preparation technique is given herein. Both test campaigns used steel strong boxes of diameter $d$ = 0.75 $m$ and depth $l$ = 0.75 $m$⁠. To ensure uniform and repeatable soil model preparation, a curtain-type sand pluviation system was employed (Figure 8(a)). The latter was calibrated for the target relative density, $Dr$ = 85%, by maintaining a constant drop height and varying the sand flow and passage velocity. The achieved soil density was verified after each test using pots, carefully placed at different depths of the soil model, at adequate distance from the model foundation. All tests were conducted using fine poorly graded Perth Sand. A detailed characterisation of the sand can be found in Sakellariadis et al. (2022) and Sakellariadis and Anastasopoulos (2024).

Focusing on the first test series, the installation of the model piles employed the same actuator used in the combined loading assembly, supported by a rigid frame (Figure 8(b)). The adopted slow 1g monotonic jacking is expected to cause limited disturbance, and therefore the model piles can be assumed to behave as wished-in-place (Madabhushi, 2017). Once the soil pluviation is completed and the model foundation is installed, the loading assembly is mounted on the strong box using a custom support base-frame (Figure 8(c)). In the pile testing series, the connection of the model pile to the rigid loading column is established via a ring with a slot and four screws along its periphery. Once model preparation is completed, the strong box with the loading device atop is transported using a crane, and mounted to the centrifuge basket.

The newly developed combined loading apparatus was proof-tested revisiting a classic problem of foundation engineering: lateral loading of a single reinforced concrete (RC), long, fixed-head pile in cohesionless soil. The zero-rotation condition at the pile head is established by coordinating the two actuators to move simultaneously, applying exactly the same lateral displacement. The latter can be achieved through a regulation system of the two actuators, preferably via closed-control loops, on the basis of an external measurement (e.g. laser sensors).

A key feature of the studied problem is that the ultimate limit state is controlled by the structural failure of the RC pile in the form of two plastic hinges, the first one at the pile-cap fixity and the second at a certain depth (Broms, 1964). In this regard, selecting an appropriate modelling technique to reproduce the non-linear response of the prototype RC pile within the environment of a geotechnical centrifuge, is a crucial factor. Simplified techniques are typically employed: the model piles are made of aluminium tubes, with their properties selected to match the bending stiffness of the prototype RC section, and epoxy-sand coating to achieve rough pile–soil interface conditions. It is well recognised that this technique comes with certain limitations: the bending moment capacity is overestimated and is independent of the level of axial load (e.g. Anastasopoulos et al., 2015; Knappett et al., 2011; Loli et al., 2014; Madabhushi, 2017; Sakellariadis and Anastasopoulos, 2024; Trombetta et al., 2013).

These limitations can be overcome by employing more advanced state-of-the-art physical modelling techniques. Matching both the bending stiffness and bending moment capacity would require the introduction of mechanical fuses (Trombetta et al., 2013), artificial plastic hinges (Anastasopoulos et al., 2015) or miniature RC (Knappett et al., 2011; Loli et al., 2014). Inspired by the work of Knappett et al. (2011) and Loli et al. (2014), miniature RC piles were produced and tested at the component level (Figure 9). Deviating from the original work of Knappett et al. (2011), the production process proposed by Del Giudice et al. (2022) was followed. A mixture of cement, water and fine poorly graded Perth sand acting as scaled aggregate is poured inside a 3D-printed split casing, where the 3D-printed steel reinforcement cage is already securely placed. When the mixture has hardened (after 7 days) the casing is removed, revealing the miniature RC pile. The latter is then wrapped with plastic foil until concrete curing and hardening is mostly completed (28 days). Additive steel manufacturing was outsourced (ECOPARTS AG), while many different mixtures were tested to ensure repeatability and achieve the target concrete properties.

Certain aspects of this modelling technique require future investigation to fully demonstrate its capabilities and limitations. For instance, the use of scaled aggregates may lead to an overestimation of the concrete tensile and shear resistance, potentially resulting in a small but non-negligible overestimation of the plastic hinge capacity. With respect to reinforcement, the use of 3D-printed steel cages could introduce material anisotropy, both in terms of strength and stiffness. Despite these aspects, Del Giudice et al. (2022) demonstrated that overall, this technique reproduces the key features of RC response under both monotonic and cyclic loading.

The produced miniature RC piles target prototype RC piles of diameter $D$ = 1 $m$ with unconfined concrete compressive strength $fc$ = 35 $MPa$⁠. The target yield strength of the steel reinforcement is $fys$ = 500 $MPa$⁠, whereas a typical longitudinal reinforcement ratio $ρs$ = 1% is assumed, realised with 12 bars of nominal diameter Ø30. The transversal hoops assume the same steel quality, nominal diameter Ø15 and spacing $s$ = 0.1 $m$⁠. Figure 9 (left) depicts the 3D-printed split casing employed to cast the model RC piles, the 3D-printed steel reinforcement cage, and the final hardened miniature RC pile produced at 1:50 scale. Extensive component-level characterisation of the model RC piles was performed thereafter, by means of conventional four-point bending, cantilever, and unconfined compression testing. Indicative snapshots from such characterisation tests are depicted in Figure 9 (middle and right).

Key results of five fixed-head lateral load centrifuge tests at the target 50g-level (using the pile tip as reference) are comparatively assessed in Figures 10 and 11 in terms of lateral load–horizontal displacement $(H- u)$ response (presented in model scale). The first four tests are monotonic, while the fourth test is slow cyclic. The first test employs the simplified aluminium-coated pile technique (designed to match the target RC bending stiffness at zero axial loading), whereas the four other tests are conducted with the miniature RC model piles. A key scope of this study is to demonstrate the advantages of employing the detailed RC modelling technique, compared to the simplified approach. All tests aimed to reach the ultimate state with full development of plastic hinges. Given this scope and considering the quasi-brittle response of miniature RC, no instrumentation (e.g. strain-gauges) was installed along the pile, and the documented response is limited to the pile-head reactions.

As shown in Figure 10(a), the two modelling approaches yield similar lateral stiffness, indicating successful stiffness-matching for the aluminium pile. However, the latter is exhibiting a significantly higher capacity as opposed to the RC pile. This is in accord with the literature, being a well-recognised unavoidable limitation of the simplified modelling approach. As expected, a considerably higher ductility of the aluminium pile is also clearly observed. The two piles reach their capacity at a similar horizontal displacement of ≈7 mm (marginally higher in the case of the aluminium pile), but the displacement needed to exceed their available ductility is 65% higher for the aluminium pile (≈25 mm) compared to the RC piles (≈15 mm). The monotonic testing of the miniature RC pile was performed twice to assess repeatability, yielding overall very similar load–displacement curves (solid vs. dashed black line in Figure 10(b)).

An additional important limitation of the simplified technique is related to its inability to reproduce the axial load dependency of bending moment capacity and stiffness, characteristic of RC sections. The latter is typically depicted by means of $M - V$ interaction diagrams of parabolic shape. In contrast to the simplified method, the detailed RC modelling technique successfully reproduces this feature of RC response. In this test series, two different levels of axial loading are examined: (a) zero vertical load $(V = 0)$⁠; and (b) excessive vertical load, corresponds to a vertical safety factor $FS$= 1.5. As clearly depicted in Figure 10(b), the increase of axial load results in an increase of the ultimate lateral resistance as well. The latter is consistent with the compressive axial loads examined in this study, which correspond to the increasing branch of the $M - V$ parabola. Moreover, the lateral stiffness of the heavily loaded pile is evidently higher, which is also attributed to the axial load dependency of RC stiffness.

The last test refers to slow cyclic lateral loading of a model RC pile, with increasing displacement amplitude under zero axial load. The corresponding $H$ – $u$ response is presented in Figure 10, along with the comparison to the ‘backbone’ curve resulting from the previously discussed monotonic test. Excellent agreement is observed between the two tests (monotonic vs. cyclic), further confirming the repeatability of the proposed experimental setup. The cyclic test results are shown in more detail in Figure 11 (corresponding to raw data without any correction or filtering). Four cycles of increasing amplitude were applied: 1–10 mm. As shown in Figures 11(a) and 11(b), gapping is observed in the derived loops. This consistent trend should not be related to RC or soil behaviour, but is rather attributed to the clearance of the sliding-hinged connections which needs to be subtracted from the response. The results indicate progressive stiffness degradation, while softening is observed in the last cycle of higher amplitude. The ductility is exceeded at 15 mm, similar to the monotonic test, which is attributed to the dense transverse reinforcement.

Post-testing snapshots of the deformed model (aluminium and RC) piles from the four conducted centrifuge tests are collected in Figure 12. The characteristic double-hinged failure mode postulated by Broms (1964) for fixed-head long piles is evident in all cases. Nevertheless, the depth of the second plastic hinge differs between the tests, being deeper in the case of the aluminium pile. The latter is consistent with the higher lateral resistance, due to the inability of the simplified approach to match the stiffness and the bending moment capacity of the prototype. A noticeable difference (although smaller) can also be observed between the RC piles with varying axial load, being consistent with their lateral resistance. When closely examining the RC plastic hinges, concrete cracking and spalling of the tensile side and crashing of the compressed zone is observed, qualitatively confirming the anticipated RC failure mechanism.

Overall, the presented preliminary centrifuge campaign reproduced fundamental features of RC pile behaviour, such as axial load dependency of bending moment capacity and stiffness, and stiffness degradation under cyclic loading. These observations were made possible thanks to the detailed modelling of the piles, which contrary to simplified modelling techniques, allows for realistic development of plastic hinges. More tests of different reinforcement detailing and soil conditions (e.g. loose sand, stiff clay), are essential to fully explore the capabilities of this technique. With respect to the combined loading apparatus, its performance was found to be reliable and robust, as further verified by external laser sensor readings.

The tests presented so far utilised the two actuators under displacement-controlled mode: regulating and forcing equal extension-contraction between the two actuators, lateral translation was imposed under zero rotation condition. Similarly, with appropriate regulation of the two actuators a load-controlled test is possible with the same setup. Nevertheless, there are several cases of foundations that resist specific combinations of lateral load and overturning moment, known a priori. In such cases the use of a single actuator to apply pushover-loading of the foundation system is often advantageous, since the control requirements are less complicated. Interestingly, the pushover setup applies a straight load path (constant ratio of overturning moment to lateral load; controlled by the position of the actuator) through a displacement-controlled application. The latter allows recording of softening response as a result of progressive failure. This is not possible with load-controlled applications, where catastrophic collapse is observed once the ultimate state is reached.

This section presents the first application of this adaptation of the proposed setup by Arnold et al. (2024) to apply pushover loading of a light-weight structure on a block foundation at 30g. The vertical position of the single actuator was adjusted accordingly to correspond to the target lever arm with respect to the ground surface reference level (Figure 13). Lateral loading was applied by the actuator, connected to the rigid column via. a roller bearing, to ensure that only compressive loads are transmitted without parasitic moment or shear. The applied load and the resulting lateral displacement were measured by a load cell and laser sensors, respectively. The roller supports were maintained to undertake the self-weight of the actuator arm, minimising undesired bending. Indicative snapshots are shown in Figure 13 (Arnold et al., 2024).

Building on past experience from developing experimental setups for the ETHZ drum centrifuge, this paper introduced a newly developed combined loading apparatus for the larger beam centrifuge of the ETHZ GCC. The experimental setup comprises a pair of fixed-end actuators, connected through sliding hinges to a rigid loading column atop of the foundation system. The key technical aspects of the assembly were outlined, emphasising key improvements with respect to the overall rigidity of the loading system and the support of the actuators. Using roller bearing supports, undesired bending of the actuator arms under their self-weight in the adverse conditions of the geotechnical centrifuge is significantly limited.

The key capabilities of the new apparatus were demonstrated for two different preliminary test campaigns. The full combined loading assembly was employed to study single long RC piles subjected to fixed-head lateral loading. The key advantages of detailed modelling of RC piles using miniature reinforced concrete, as opposed to simpler modelling techniques (bending stiffness-matched aluminium tubes) were illustrated through selected preliminary results. Moreover, displacement-controlled lateral loading of the RC pile under zero rotation at the pile head was successfully reproduced, verified by external laser sensor measurements. Finally, the key features of a simpler version of the loading assembly, employing a single actuator for pushover testing, were outlined with reference to a second test series (Arnold et al., 2024) on a block foundation supporting a light-weight structure.

Although the proposed configuration was proof-tested for two characteristic foundation types, it is practically applicable to any foundation system. Having demonstrated its reliability and robustness, the developed assembly can be employed to apply both monotonic and cyclic combined loading, either under displacement- or load-controlled conditions. In addition to its significantly increased load capacity and rigidity (compared to the drum centrifuge experimental setup), a key advantage of the newly developed apparatus is its versatility and ease of conversion from one experimental setup to another, as demonstrated through the two examples outlined in this paper.