Errors of approximate analytical solutions for pressurized toroidal shells

This dataset is presented for the paper "Curved pipes subjected to mechanochemical corrosion under pressure: Analytical and numerical estimates of the lifetime". AUTHORS: A. Ilyin, Y. Pronina. Saint-Petersburg State University Corresponding E-MAIL: [e-mail] A linearly elastic toroidal shell with mean toroidal radius R, inner radius ri and outer radius ro loaded with internal pi and external po pressures (pi ≥ 0 and po ≥ 0) is considered. The dataset contains the values of the relative error in circumferential stresses calculated by the approximate analytical "thin-shell" (eq.(8) of this paper) and "thick-shell" (eq.(9)-(10) of this paper) solutions compared to Finite element modeling (FEM) results, for the inner and outer surfaces at the torus intrados. For the "thin-shell" solution the graphical representation of the current dataset is presented in Figure 7(a) for the inner surface and in Figure 7(b) for the outer surface of the toroidal shell. For the "thick-shell" solution the graphical representation of the current dataset is presented in Figure 8(a) and Figure 8(b) for the inner and outer surfaces, correspondingly. The values of the error are calculated using the following formula: Error = (S^{an} - S^{FEM})/S^{FEM} * 100%where:S^{an} -- circumferential stress obtained by the approximate analytical solution (eq.(8) - eq.(10)),S^{FEM} -- circumferential stress obtained by the Finite element modeling (FEM). NOTATIONS: ro -- outer radius of the toroidal shell;ri -- inner radius of the toroidal shell;R -- mean radius of the toroidal shell;h -- relative thickness of the toroidal shell ( h = (ro - ri) / 2 );Error_Inner_Thin -- relative error of the "thin-shell” solution (eq. 8) compared to the FEM for the inner surface (Figure 7 (a)), in percent;Error_Outer_Thin -- relative error of the "thin-shell” solution (eq. 8) compared to the FEM for the outer surface (Figure 7 (b)), in percent;Error_Inner_Thick -- relative error of the modified Kornecki solution (eq. 10) compared to the FEM for the inner surface (Figure 8 (a)), in percent;Error_Outer_Thick -- relative error of the modified Kornecki solution (eq. 9) compared to the FEM for the outer surface (Figure 8 (b)), in percent.本数据集配套于论文《受压工况下承受机械化学腐蚀的弯管：寿命的解析与数值估算》（"Curved pipes subjected to mechanochemical corrosion under pressure: Analytical and numerical estimates of the lifetime"）。 作者：A. Ilyin、Y. Pronina，所属单位：圣彼得堡国立大学 通讯邮箱：[e-mail] 本研究考虑一平均环向半径为$R$、内半径为$r_i$、外半径为$r_o$的线弹性环壳（toroidal shell），承受内压$p_i$与外压$p_o$作用（$p_i geq 0$且$p_o geq 0$）。本数据集包含两种近似解析解——“薄壳”解（对应本文式(8)）与“厚壳”解（对应本文式(9)-(10)）——计算得到的环向应力（circumferential stresses），与有限元建模（Finite element modeling, FEM）结果相比的相对误差，取值覆盖环壳内、外表面的拱腹区域。其中，“薄壳”解的本数据集可视化结果分别对应图7(a)（内表面）与图7(b)（外表面）；“厚壳”解的可视化结果则分别对应图8(a)与图8(b)，依次对应环壳内、外表面。 相对误差采用如下公式计算： $$ ext{Error} = frac{S^{ ext{an}} - S^{ ext{FEM}}}{S^{ ext{FEM}}} imes 100\%$$ 其中： $S^{ ext{an}}$为近似解析解（式(8)-式(10)）得到的环向应力； $S^{ ext{FEM}}$为有限元建模（FEM）得到的环向应力。 符号说明： $r_o$：环壳外半径； $r_i$：环壳内半径； $R$：环壳平均半径； $h$：环壳相对厚度（$h = frac{r_o - r_i}{2}$）； `Error_Inner_Thin`：“薄壳”解（式8）针对内表面的相对误差（对应图7(a)），单位为百分比； `Error_Outer_Thin`：“薄壳”解（式8）针对外表面的相对误差（对应图7(b)），单位为百分比； `Error_Inner_Thick`：改进的Kornecki解（式10）针对内表面的相对误差（对应图8(a)），单位为百分比； `Error_Outer_Thick`：改进的Kornecki解（式9）针对外表面的相对误差（对应图8(b)），单位为百分比。