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 应用科技  2020, Vol. 47 Issue (3): 11-16  DOI: 10.11991/yykj.201911025 0

### 引用本文

ZHANG Haichen, TONG Lili, WANG Lijun, et al. Theoretical model analysis of unbonded flexible riser under axisymmetric load[J]. Applied Science and Technology, 2020, 47(3): 11-16. DOI: 10.11991/yykj.201911025.

### 文章历史

Theoretical model analysis of unbonded flexible riser under axisymmetric load
ZHANG Haichen, TONG Lili, WANG Lijun, SHEN Yichong, LIU Hao
College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: The structure of unbonded flexible riser is complex, so it is very important to build a high-efficiency mechanical model. Based on the principle of minimum potential energy, the stiffness matrix of flexible riser under axisymmetric load is derived in this paper. By adding the additional function of normal contact between layers, the matrix of riser contact of is obtained, and the global stiffness matrix of flexible riser is obtained by adding the two matrices. Based on this model, the penalty parameters are discussed, and the mechanical responses of flexible risers under tensile and torsional loads are analyzed and compared with numerical simulation and experimental results. The results show that the larger the penalty parameter, the higher the calculation accuracy; but if the penalty parameter is too high, the equation will be ill conditioned and the result will be inaccurate; the theoretical model can correctly reflect the mechanical characteristics of the unbonded flexible riser under the axial tension and torque, which provides a certain reference for the initial stage of engineering design.
Keywords: unbonded flexible riser    principle of minimum potential energy    stiffness matrix    interlayer contact    theoretical model    numerical model    penalty parameter    axial tension    torque

1 理论模型

1)柔性管具有较大的轴向和径向刚度，所以各层变形均为小变形；

2)模型中的材料视为线弹性且各向同性；

3)柔性管各层的轴向变换量和绕轴向扭转角度相同。

1.1 螺旋层刚度矩阵

1.1.1 螺旋铠装层条带的轴向和径向应变

 ${\varepsilon _1}{\rm{ = co}}{{\rm{s}}^2}\alpha \cdot \frac{{{\mu _Z}}}{L} + {\sin ^2}\alpha \cdot \frac{{{\mu _R}}}{R} + R\sin \alpha \cos \alpha \cdot \frac{{{\mu _\theta }}}{L}$

 ${\varepsilon _2} = \frac{{\Delta t}}{t}$

 ${\sigma _1} = \frac{E}{{1 - {\nu ^2}}}\left( {{\varepsilon _1} + \nu {\varepsilon _2}} \right)$
 ${\sigma _2} = \frac{E}{{1 - {\nu ^2}}}\left( {{\varepsilon _2} + \nu {\varepsilon _1}} \right)$

 ${U_1} = \frac{1}{2}\int {_v\left( {{\sigma _1} {\varepsilon _1} + {\sigma _2} {\varepsilon _2}} \right)} {\rm{d}}v$

 ${U_{\rm{e}}} = {P_{\rm{I}}}\Delta {V_{\rm{I}}} - {P_{\rm{O}}}\Delta {V_{\rm{O}}} + F{\mu _Z} + T{\mu _\theta }$

1.1.2 螺旋层条带的弯曲和扭转

 ${U_2} = \frac{1}{2}\int {_v\left( {E{I_{\rm{b}}} \cdot \Delta k_{\rm{b}}^2 + GJ \cdot \Delta {\tau ^2}} \right)} {\rm{d}}s$

 $\varPi = n U - {U_{\rm{e}}}$

 $\delta \varPi = 0$

 $\left[ {\begin{array}{*{20}{c}} {{K_{11}}}&{{K_{12}}}&{{K_{13}}}&{{K_{14}}} \\ {{K_{21}}}&{{K_{22}}}&{{K_{23}}}&{{K_{24}}} \\ {{K_{31}}}&{{K_{32}}}&{{K_{33}}}&{{K_{34}}} \\ {{K_{41}}}&{{K_{42}}}&{{K_{43}}}&{{K_{44}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{X_Z}} \\ {{X_R}} \\ {{X_t}} \\ {{X_\theta }} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {F + {\rm{{\text{π}}}} {P_{\rm{I}}}R_{\rm{I}}^2 - {\rm{{\text{π}}}} {P_{\rm{O}}}R_{\rm{O}}^2} \\ {2{\rm{{\text{π}}}} {P_{\rm{I}}}R_{\rm{I}}^2 - 2{\rm{{\text{π}}}} {P_{\rm{O}}}R_{\rm{O}}^2} \\ { - {\rm{{\text{π}}}} {P_{\rm{I}}}{R_{\rm{I}}}t - {\rm{{\text{π}}}} {P_{\rm{O}}}{R_{\rm{O}}}t} \\ T \end{array}} \right]$
1.2 圆柱层刚度矩阵

 ${\varepsilon _1} = \frac{{{\mu _{ Z}}}}{L};{\varepsilon _2} = \frac{{\partial {\mu _R}}}{{\partial R}};{\varepsilon _3} = \frac{{{\mu _R}}}{R};{\gamma _{12}} = R\frac{{{\mu _\theta }}}{L}$

 $\left\{ {\begin{array}{*{20}{l}} {{\sigma _1} = \lambda \left( {{\varepsilon _1} + {\varepsilon _2} + {\varepsilon _3}} \right) + 2G{\varepsilon _1}} \\ {{\sigma _2} = \lambda \left( {{\varepsilon _1} + {\varepsilon _2} + {\varepsilon _3}} \right) + 2G{\varepsilon _2}} \\ {{\sigma _3} = \lambda \left( {{\varepsilon _1} + {\varepsilon _2} + {\varepsilon _3}} \right) + 2G{\varepsilon _3}} \\ {{\tau _{12}} = G{\gamma _{12}}} \end{array}} \right.$

 $U = \frac{1}{2}\int {_v\left( {{\sigma _1} {\varepsilon _1} + {\sigma _2} {\varepsilon _2} + {\sigma _3} {\varepsilon _3} + {\tau _{12}}{\gamma _{12}}} \right)} {\rm{d}}v$

 $\left[ {\begin{array}{*{20}{c}} {{K_{11}}}&{{K_{12}}}&{{K_{13}}}&{{K_{14}}} \\ {{K_{21}}}&{{K_{22}}}&{{K_{23}}}&{{K_{24}}} \\ {{K_{31}}}&{{K_{32}}}&{{K_{33}}}&{{K_{34}}} \\ {{K_{41}}}&{{K_{42}}}&{{K_{43}}}&{{K_{44}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{X_Z}} \\ {{X_{{\varphi _{\rm{{\rm I}}}}}}} \\ {{X_{_{{\varphi _{\rm{O}}}}}}} \\ {{X_\theta }} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {F + {\rm{{\text{π}}}} {P_{\rm{I}}}R_{\rm{I}}^2 - {\rm{{\text{π}}}} {P_{\rm{O}}}R_{\rm{O}}^2} \\ {2{\rm{{\text{π}}}} {P_{\rm{I}}}R_{\rm{I}}^2} \\ { - 2{\rm{{\text{π}}}} {P_{\rm{O}}}R_{\rm{O}}^2} \\ T \end{array}} \right]$

1.3 接触附加矩阵

 $\Delta {R_{n + 1}} = \Delta {R_n} + \frac{1}{2} \left( {\Delta {t_{n + 1}} + \Delta {t_n}} \right) + {g_i}$

 ${\varPi _{{\rm{cp}}}} = \sum\limits_{j = 1}^{m - 1} {\left[ {{\omega_u}{{\left( {{\mu _{j{R_{\rm{O}}}}} - {\mu _{\left( {j + 1} \right){R_{\rm{I}}}}}} \right)}^2} + {\omega_P}{{\left( {{P_{j{\rm{O}}}} - {P_{\left( {j + 1} \right){\rm{I}}}}} \right)}^2}} \right]}$ (1)

 $\varPi = {\varPi _{\rm{s}}} + {\varPi _{\rm{cp}}}$

 ${{{k}}_{\rm{cp}}} = {{w}} \cdot {{{k}}_{\rm{cpo}}}$

${{{k}}_{{\rm{cpo}}}}$ 中除以下系数外均为0： ${k_{33}} = 2R_{1{\rm{o}}}^2$ ${k_{43}} =$ ${k_{34}} = - 2{R_{1{\rm{o}}}}{R_{2{\rm{I}}}}$ ${k_{44}} = 2R_{2{\rm{I}}}^2$ ${k_{55}} = 2R_{2{\rm{o}}}^2$ ${k_{65}} = {k_{56}} = - 2{R_{2{\rm{o}}}}{R_{3{\rm{I}}}}$ ${k_{66}} = 2R_{3{\rm{I}}}^2$ ${k_{77}} = 2R_{3{\rm{o}}}^2$ ${k_{87}} = {k_{78}} = - 2{R_{3{\rm{o}}}}{R_{4{\rm{I}}}}$ ${k_{88}} = 2R_{4{\rm{I}}}^2$ ${k_{99}} =$ $2R_{4{\rm{o}}}^2$ ${k_{109}} = {k_{910}} = - 2{R_{4{\rm{o}}}}{R_{5{\rm{I}}}}$ ${k_{1010}} = {2R_{5{\rm{I}}}^2}$ ${k_{1110}} = {k_{1011}} = 2{R_{5{\rm{I}}}}{t_5}$ ${k_{1111}} = 2t_5^2$ ${k_{1211}} = {k_{1112}} = - 2R{R_{5{\rm{I}}}}{t_5}$ ${k_{1212}} = 2R_{6{\rm{I}}}^2$ ${k_{1313}} = 2R_{6{\rm{o}}}^2$ ${k_{1413}} =$ ${k_{1314}} = - 2{R_{6{\rm{o}}}}{R_{7{\rm{I}}}}$ ${k_{1414}} = 2R_{7{\rm{I}}}^2$ ${k_{1513}} = {k_{1315}} = 2{R_{7{\rm{I}}}}{t_7}$ ${k_{1515}} = 2t_7^2$ ${k_{1615}} = {k_{1516}} = 2{R_{8{\rm{I}}}}{t_7}$ ${k_{1616}} = 2R_{8{\rm{I}}}^2$

1.4 建立刚度矩阵

 ${{{\varDelta}} _{\rm{s}}} = \left[ {{X_Z}\;\;X_R^1\;\;X_\theta ^1\;\;X_{\varphi {\rm{I}}}^2\;\;X_{\varphi {\rm{o}}}^2\;\;X_R^3\;\;X_\theta ^3\;\;X_{\varphi {\rm{I}}}^4\;\;X_{\varphi {\rm{O}}}^4 \cdots X_{\varphi {\rm{I}}}^m\;\;X_{\varphi {\rm{O}}}^m\;\;{X_\theta }} \right]$

 ${{K}} = {{k}} + {{{k}}_{\rm{cp}}}$

 ${{K}}{{{\varDelta}} _{\rm{s}}} = {{{F}}_{\rm{S}}}$

${{{F}}_{\rm{S}}}$ 为外力向量：

 ${{{F}}_{\rm{S}}} = \left[ {\begin{array}{*{20}{c}} {{F_Z}}&0&0&{{P_{\rm{I}}}}&0&0& \cdots &{}&0&{{P_{\rm{O}}}}&N \end{array}} \right]$

2 有限元模型 2.1 立管基本参数

2.2 模型与网格

2.3 分析步和边界条件

3 计算结果分析 3.1 罚参数计算

3.2 拉伸工况

3.3 扭转工况