## 1 Introduction  
With the rapid development of China's economy and the acceleration of urbanization, traffic congestion has become an increasingly severe problem, causing significant inconvenience to residents during travel. The construction of urban underground railways has emerged as an important measure to address these issues. However, the layout of urban underground pipelines is intricate and complex, and tunnel excavation or foundation pit excavation must often occur close to existing pipelines [1, 2]. This increases the complexity of underground engineering and places higher demands on deformation control during the project.  
Numerical simulation has been widely used to predict the deformation and disturbance caused by underground engineering projects such as tunnels and foundation pits [3, 4]. When finite element software is used for modeling, the choice of soil constitutive model has a significant impact on the prediction results.  
The traditional Mohr\(-\)Coulomb constitutive model is an ideal elastoplastic model that does not account for the correlation between soil stiffness, stress, and stress path. Generally, it is considered to effectively reflect the failure stress state of soil and is widely used in engineering. However, this model does not account for soil unloading and rebound behaviors, resulting in calculation outcomes that are often overly conservative and unsuitable for engineering projects with stringent deformation-control requirements.  
[PERSON] and [PERSON] [5] defined the small-strain range for soil as being between 0.0001% and 0.1%. Numerous engineering practices have shown [6; 7; 8; 9] that a considerable portion of the soil surrounding underground structures is in a small-strain state under working loads. For instance, [PERSON] [7] analyzed the excavation deformation of a deep foundation pit in the UK and found that the deformation of the soil was within 0.03%. Figure 1[9] shows the strain range of the soil under various engineering conditions. The stiffness of soil under small-strain conditions exhibits a high degree of nonlinearity, and the true stiffness value of soil under small-strain conditions is significantly higher than the nominal elastic stiffness value obtained from conventional experiments. The HSS (hardening soil model with small-strain stiffness) model is a high-order elastoplastic hyperbolic model that considers the increased stiffness of soil during unloading, comprehensively considers the small-strain characteristics of soil, and can more accurately describe the plastic characteristics and small-strain conditions of soil. After years of research, this constitutive model has matured and is widely applied in engineering practice [10; 11; 12; 13; 14]. After practical engineering verification [11; 15; 16; 17], compared with simple models such as M\(-\)C, the prediction results of the HSS model were found to be closer to the measured values in engineering applications.  
The difficulties and key point in using the HSS model for analysis and calculation lie in determining soil parameters. Currently, extensive relevant research has been conducted in only a few areas, such as Shanghai [18; 19], while there are few instances of using the HSS constitutive model to study underground engineering in the loess region of Xi'an. In practical engineering, the HSS model parameters for loess are often based on empirical data from other regions, which significantly affects the accuracy of model predictions. Against this backdrop, this paper focuses on the newly constructed subway tunnel in Xi'an, and the research combines indoor tests and back-analysis methods to recommend HSS parameter values suitable for the loess region of Xi'an.