## 1 Introduction  
Thick hard roofs (THRs) in coalfaces have unique breaking characteristics, including a large roof overhang area, difficulties related to spontaneous caving, concentrated mining-induced stresses, and a significant dynamic load upon breaking [1, 2, 3]. In recent years, the shallow coal deposits of many coal mines have been depleted, and mining depths are increasing incessantly. Strata pressure behaviors (SPBs) due to THRs have become increasingly frequent in coalfaces. This is especially true in the coal mines in central and eastern parts of China (e.g., the Huaibei Mining Area) and northwest China (e.g., the Huijirt Mining Area), where mining depths are continuously increasing, resulting in strong SPBs in THRs and inducing rock bursts and support-crushing accidents [4]. Studying the deformation and failure characteristics of THRs in coalfaces is important to achieve safe and efficient mining. Progress has been made in understanding the breaking law and control technology of THRs in coalfaces. [PERSON] et al. [5] utilized the medium-thickness plate theory to theoretically analyze the first and periodic weighting of a thick main roof. [PERSON] et al. [6] studied the height of a damaged zone in a hard roof using several methods and performed field tests and validation. [PERSON] et al. [7] investigated the weakening effect caused by the advanced fracturing of THRs on SPBs in the Shendong Mining Area. [PERSON] et al. [8] examined the weighting length of hard rock strata with significant thickness variations in the close distance and explained the different weighting phenomenon of a coalface. [PERSON] et al. [9] studied the fracture evolution characteristics at different loading stages for thick hard sandstone roof specimens. [PERSON] et al. [10] investigated the critical conditions for the instability of THRs, as well as the instability pattern, influence factors of instability, and mechanism of instability. [PERSON] et al. [11] analyzed the dynamic evolution law of vertical displacement in caving ultra-thick hard roofs and established a relationship between the mining-induced stress distribution and roof movement. [PERSON] et al. [12] explored the judgment criteria for rock bursts in a coalface under high-position hard rock conditions. [PERSON] et al. [13] derived a method for determining the position and size of a fractured THR under different supporting boundary conditions. [PERSON] et al. [14] studied the factors that influence fracture propagation in the thick hard limestone roof of the Taiyuan Formation and performed field tests and validation. [PERSON] et al. [15] studied the fracture development characteristics, SPBs characteristics, and broken roof control technology for hard thick quartz sandstone. From an energy perspective, [PERSON] et al. [16] revealed the characteristics that influence strong SPBs due to the breaking and instability of THRs. [PERSON] et al. [17] proposed the precursory characteristics and mechanism of faults that induce rock bursts in the presence of ultra-thick hard roofs. The above studies offer important theoretical and technical support for safe mining under THRs. However, few articles have been published concerning the deformation and breaking characteristics of THRs of upward mining coalfaces (UMCs). Some researchers have only established a simple mechanical model based on one theory alone, which does not conform to the reality of underground mining conditions. There is an urgent need for further investigation in this respect. Here, a mechanical model of the bearing capacity of a THR in an UMC is built based on bending theories for beams with single generalized displacement and the elastic foundation beam theory.