## 1 Introduction  
The pre-stressed high-strength concrete (PHC) pile offers high bearing capacity, a wide application range, good driving resistance, and fast construction speed. Widely utilized in high-rise buildings, bridges, ports, and other industries [1; 2; 3]. PHC piles have become increasingly important for ensuring safety through reliable construction and thorough quality control. Although standards vary by country, conducting dynamic load tests on all piles remains impractical. Instead, these tests occur on a select few to estimate bearing capacity while managing the remaining piles through penetration depth. Ensuring that all piles share the same conditions is impossible, making it problematic to manage piles based solely on penetration depth. Two major issues must be addressed for reliable pile management. First, the method of measuring penetration depth during the final driving requires improvement. Traditionally, laborers manually measure penetration depth and rebound values using pen and paper [4; 5]. Although simple and widely used, this method is prone to inaccuracies unless performed flawlessly by experienced laborers in ideal conditions. Additionally, important information may be omitted, and records can be manipulated, complicating construction management and safety assurance. Recently developed non-contact penetration measurement devices and sensors help mitigate these issues [6; 7]. While these advancements have improved accuracy and reliability in measuring penetration depth, further enhancements in the measurement process contribute to better pile management. The second issue involves reliance on on-site penetration depth values for construction and quality management. Penetration depth is influenced by various factors such as hammer height, ram weight, and pile type, making it difficult to ensure reliable pile quality using penetration depth alone. It is crucial to consider penetration depth and hammer energy through driving formulas to address this issue, as this approach can significantly improve reliability. However, current practices often fail to integrate these factors comprehensively, leaving room for improvement in pile quality management. Various researchers have conducted studies from different perspectives to improve the accuracy and applicability of pile driving formulas. For instance, [PERSON] improved the traditional Hiley formula by considering the time effects, using high-strain dynamic tests to propose a method for predicting the bearing capacity of piles more accurately [8]. Similarly, [PERSON] presented a new analytical solution for vibratory pile driving rates, considering vibratory creep to better estimate the pile embedding rate [9]. [PERSON] proposed improved wave equation analysis methods and load and resistance factor design (LRFD) recommendations for piles in soil-based intermediate geomaterials, introducing new damping parameters to enhance the reliability of pile resistance predictions [10]. Also, [PERSON] incorporated the setup effect, where the bearing capacity of driven piles increases over time, into the reliability analysis to improve the prediction accuracy and economic efficiency of pile designs [11]. [PERSON] suggested improvements to the commonly used Hiley kinetic equation, which could result in more accurate evaluations of pile bearing capacity [12]. [PERSON] used four different formulas, including the Hiley formula, to estimate pile bearing capacity and found that the Hiley formula tended to underestimate these values [13]. [PERSON] compared the predicted geotechnical pile bearing capacities using the Hiley formula and the high-strain method. Their findings indicated that the values derived from the Hiley formula were consistently lower than those obtained from the high-strain method and static load tests [14]. This paper aims to enhance the reliability of quality management for driven piles without dynamic load tests by reviewing and improving existing driving formulas, especially for PHC piles.