## 2 Materials and Methods
### Experimental Setup and Test Program  
In the model experiments, a 1000 mm \(\times\) 750 mm \(\times\) 750 mm\({}^{3}\) (height \(\times\) width \(\times\) thickness) square-section sand tank was used, as shown in Figure 6. The frame of the test tank was made of steel profiles, and 8 mm thick tempered glass was used on the front and back surfaces. The side surfaces and base were made of 3 mm-thick steel. The test tank was designed to be rigid to constitute plane strain conditions. At the same time, lateral deformations were prevented by placing steel profiles on the side walls of the tank, which were resistant to loading. Vibrating equipment compacts the soil, increases its bearing capacity, and reduces soil settlement [53]. In the model experiments, the Makita HM0870C (Makita Co., Nagoya, Japan) model vibration device with an energy of 12 joules at 33 Hz was used to obtain a very loose density. (Dr = \(\leq\)10%), medium- dense soil (Dr = \(\leq\)40%), and dense soil (Dr = \(\geq\)65-90%). The foundation brief names, sizes, and relative densities of the model tests are listed in Table 2. The sand was poured into the test tank in 10 cm-thick layers using the pluviation method and compacted in a controlled manner until the determined relative density ratios were reached using a vibration device (Table 3). The sand pluviation technique [54] was used to control the height and flow rate of the sand, ensuring the smoothness and compactness of the fill. After the compaction process was completed, the foundations were placed, and the leveling of the soil surface was considered (Figure 7b). The load was applied vertically to the center of the foundation using a hydraulic jack at a loading speed of 1.5 mm/min. Settlement measurements were performed using two Opkon brand LVDTs (Linear Variable Differential Transformers) placed at both corners of the foundation. To measure the additional lateral stress increases in the soil, five FSG model soil pressure transducers with a capacity of 200 kPa were installed on the tank wall at 100 mm intervals, as shown in Figure 7a. The data received as digital signals were then transmitted to a computer for storage and analysis. During the experiments, the data were  
Figure 6: Model test system: (**a**) 3D schematic sketch, (**b**) cross-section of test tank. recorded using a data logger (Figure 7c). Figure 7d illustrates a noticeable increase in the settlement at the foundation and heaving around it as it approached its ultimate bearing capacity. The analog signals from the sensors (pressure transducers, load ring, and LVDTs) were amplified by voltage amplifiers and then taken as numerical values at one data point per second with an analog-to-digital converter. The loading tests were completed when the foundation reached its ultimate bearing capacity. (Figure 7e). At this stage, the foundations were loaded slowly and continuously using a hydraulic loading system until they reached their ultimate bearing capacity. When the loading tests were completed, general shear failure was observed in medium-dense (40%) and dense (65% and 90%) soils, as shown in Figure 7e,f. To determine the effects of the foundation size and relative density on the bearing capacity and lateral stress changes, 12 distinct model tests were carried out.