## 1 Introduction  
The forefoot longitudinal bending stiffness (FLBS) of a shoe's sole is an important variable affecting running economy and lower-limb biomechanics [1]. Increased FLBS has been shown to improve running economy by 2.8% to 4.2% [2, 3, 4]. Biomechanically, increased FLBS reduces the dorsiflexion angle [5], slows the dorsiflexion rate [6], and decreases negative work [7] of the metatarsophalangeal joint. Additionally, it shifts the point of application of ground reaction forces further forward, increasing the runner's mechanical advantage [8]. The three-point bending test is the most commonly used method for quantifying the FLBS of shoe soles in research [9]. As a key technique, the three-point bending test is frequently used to determine the bending stiffness, ultimate strength [10], and fatigue strength [11] of materials such as plastics [12], metals [13], and fiber-reinforced composites [14]. In the field of research on carbon-plated running shoes, the three-point bending test is utilized to quantify the FLBS around the metatarsophalangeal joint [9]. Despite simplifying the complex loading conditions of running by applying a unidirectional static load--whereas running involves dynamic, multidirectional forces and complex sole deformation--the test still provides valuable data on sole performance, such as the impact of the FLBS on running. Different stiffness levels of shoe soles impact the foot, causing changes in the kinematics and dynamics of the metatarsophalangeal and ankle joints [5; 6; 7]. These changes then affect running economy. Currently, there is no explicit standard for testing the FLBS of shoes using the three-point bending test [1], and the test procedures vary across various studies. A common pre-testing step is to position the loading head at the location corresponding to the metatarsophalangeal joint on the shoe sole, while also aligning it directly above the center of two supporting beams spaced 80 mm apart [15; 16; 17]. Differences in testing procedures mainly involve the loading rate and displacement range [18; 19] of the loading head [20]. Considering that the dimensions of the three-point bending fixture can also affect the comparability of test results across different studies, and in order to facilitate simulation, it is necessary to clarify whether varying loading rates will influence stiffness data under the same three-point bending fixture and displacement range. The finite element method (FEM) is an effective approach for simulating and analyzing the mechanical properties of materials. Previous studies have integrated three-point bending mechanical tests with finite element simulations to analyze the mechanical responses of structures such as composite fiber tubes [14], femurs [21], and trabecular bone [22]. In the research field of carbon-plated running shoes, some studies have established a 'foot-shoe-plate' model to simulate running characteristics and analyze the effects of different carbon plate structures on the stress and strain of foot bones and plantar soft tissues [23; 24]. However, these studies do not consider the anisotropic material properties of carbon plate layup designs, nor do they test the stiffness of different carbon plates coupled with the shoe sole. In a typical carbon-plated running shoe design process, the bending stiffness of the shoe or carbon plate is first obtained through mechanical testing, and then the shoe and carbon plate are integrated to produce the final running shoe product. Subsequent motion testing is conducted to evaluate the impact of carbon plates with varying bending stiffness on shoe performance. This process is costly, time-consuming, and slow to provide feedback. Therefore, it is essential to use FEM simulations of composite materials to analyze the three-point bending behavior of the shoe sole, the carbon plate, and the 'plate-sole' system, which can reduce trial-and-error costs in the early design stages and shorten the product development cycle.