Calculations of the optimised corrugated structure utilised in high heels

Introduction
Raising in the height of a shoe heel increases the pressure under the forefoot area, which leads to medical problems (Eshraghi et al., Citation2019). It also interferes with the human natural movement (Cronin, Citation2014). The conventional high heels can be more comfortable if they are being designed with an advanced technology. The wall thickness and the angle of the connected walls are important factors. The geometry of the structure has a direct effect on the stiffness and the flexibility and comfortability.
Purpose of the study
The purpose of the study is to obtain the optimized corrugated structure regarding the thickness of the walls and the angle of the arcs, which leads to the better force distribution.
Method
The high heel shoes was designed and manufactured. Size 38, heel height 7 cm. The gait data including plantar pressure and force in the static and dynamic position were obtained. The volunteer was 69 kg with no foot deformity. The Research Ethics Committee of the School of Engineering, Design and Physical sciences of Brunel University London approved the conduct of the study. The corrugated structure was designed and analyzed with Abaqus software, which was used to simulate the high heel forefoot structure in a static position while uniform load was applied from the top. The average pressure on the forefoot was obtained from RSscan device data. The structure’s stiffness with different thicknesses were measured and compared to get the optimum thickness by FEA. The structure is shown in Figure 1, which is located under the forefoot in rows with the width of 1 cm each. The material was Thermoplastic polyurethane (TPU) with the 34.4 Mpa module of elasticity. The optimum stiffness was found theoretically and with FEA.
Figure 1. a is the structure and Θ is the degree of the arc, r is the circle radius, b is the FEA model with the pressure analysis.

Figure 1. a is the structure and Θ is the degree of the arc, r is the circle radius, b is the FEA model with the pressure analysis.

The structure perceived as two arcs over each other and each one assumed as one spring therefore two springs attached and the system was analyzed based on the subsequent order. The spring constant was achieved by the formula 1. K
was found based on arc’s angle. In the EquationEquation 1

, C is the constant coefficient of TPU and the result is the coefficient of elasticity of the spring (the arc of the structure). The optimum wall thickness of the arc and the angle were assumed to be unknown and they were achieved by following formulas which was done to get the best stiffness possible. Two arcs of the circle were used for designing the structure and Θ shown in the Figure 1 has an effect on the stiffness and flexibility.
K(Θ)=ΘC4−−−√
(1)
d=rΘ−2r sin Θ2=r[Θ−2 sin Θ2]
(2)
K=ΘC4−−−√ ×[ 2r(1−cos Θ2)+h0 ](r[Θ−2sin Θ2])
(3)
The displacement of the structure was obtained according to the formula 2. The final stiffness of the structure was achieved by EquationEquation (3)

in which h0
is the initial height of the structure.

Results and discussion
The optimum thickness was gained through FEA and showed to be 1 mm with the less strain on the structure. Furthermore, the optimum wall thickness and arc degree was found through the formulations and showed to be 0.66 mm and 87.76˚. Due to the limitations of making the wall thickness of less than 1 mm in FEA the results are contradicted.

With the high heel’s narrow and pointed forefoot area, keeping the size standards was important when implementing the structure. Hollow shaped structure was more flexible so the arc thickness had an effect on flexibility. In addition, the Θ degree of the arcs was a factor affecting the pressure distribution. The paper hypothesis included two defined unknowns such as the optimum Θ degree and the arc thickness, which were calculated and compared with FEA results.

Conclusion
The calculations were more reliable as there was a limitation in FEA such as considering the walking pace and applied pressure on the forefoot. The optimum arc angle and thickness were obtained which they had direct effect on comfortability and pressure distribution. The 0.66 mm wall thickness and 87.76˚ arc angle were the optimum numbers for the structure to reduce forefoot plantar pressure.