﻿ 深海高强度钢环肋圆柱壳单位容积重量优化设计
 舰船科学技术  2018, Vol. 40 Issue (7): 11-16 PDF

Weight to volume optimization design of deep-sea ring-stiffened cylindrical high-strength steel shell
YU Jun, OUYANG Lv-wei, LI Yan-qing, LU Bo
China Ship Scientific Research Center, Wuxi 214082, China
Abstract: Weight per unit volume is an important indicator of undersea pressure hull with light weight design. The lower weight per unit volume of the structure can have, the more goods and staff it can load, the safer it can be for manipulation and floating. Based on this method, Isight, a multidisciplinary optimization software, is usedto reach the optimized weight per unit volume of ring-stiffened cylindrical shell, which is made of high strength steel. Both multi-island genetic algorithm (MIGA) and Sequential Quadratic Programming method (NLPQL) is adopted to search optimization results. Based on optimization results, a fitting curveand formula of weight per unit volume and maximum working pressure, as well as a fitting curveandformula of radius-thickness ratio and maximum working pressure, is put forward for ring-stiffened cylindrical shell made of 600MPa level high strength steel. These formulas can set foundations foroptimal design and assessment of deep-sea ring-stiffened cylindrical shell.
Key words: weight per unit volume     ring-stiffened cylindrical shell     optimization design     high strength steel
0 引　言

1 环肋圆柱壳无量纲化的强度稳定性计算方法

 $u = \frac{{\sqrt[4]{{3(1 - {v^2})}}}}{2}\frac{l}{{\sqrt {Rt} }}{\text{，}}$ (1)
 $\beta = \frac{{lt}}{A} = \frac{{lt}}{{(h{t_1} + b{t_2})}}{\text{。}}$ (2)

 ${P_j} = K{P_{jx}}{\text{，}}$ (3)

1.1 环肋圆柱壳无量纲化的强度计算方法

 $\gamma = 0.5\sqrt {3(1 - {v^2})} {P_j}/E(R/t){\text{，}}$ (4)

 ${u_1} = u\sqrt {1 - \gamma } {\text{，}}$ (5)
 ${u_2} = u\sqrt {1 + \gamma } {\text{，}}$ (6)
 ${F_5} = {u_2}sh2{u_1} + {u_1}sh2{u_2}{\text{，}}$ (7)
 ${F_1} = \sqrt {1 - {\gamma ^2}} (ch2{u_1} - \cos 2{u_2})/F_5{\text{，}}$ (8)
 ${F_4} = 2(1 - 0.5v)({u_1}ch{u_1}\sin {u_2} + {u_2}sh{u_1}\cos {u_2})/{F_5}{\text{，}}$ (9)
 ${K_2} = 1 - {F_4}/(1 + \beta {F_1}){\text{，}}$ (10)
 ${K_1} = 0.5 + {F_2}/(1 + \beta {F_1}){\text{，}}$ (11)
 ${K_f} = (1 - v/2)\beta {F_1}/(1 + \beta {F_1}){\text{，}}$ (12)

 ${\sigma _1} = {K_1}{P_j}(R/t)/{\sigma _s}{\text{，}}$ (13)

 $\sigma _{_2}^0 = {K_2}{P_j}(R/t)/{\sigma _s}{\text{，}}$ (14)

 ${\sigma _f} = {K_f}{P_j}(R/t)/{\sigma _s}{\text{。}}$ (15)
1.2 环肋圆柱壳无量纲化的稳定性计算方法

 ${P_{e1}} = \left\{ \begin{array}{l}0.6E/{(R/t)^2}/(u - 0.37){\text{，}}\;\;\;\;u \geqslant 1{\text{，}}\\1.21E{(R/t)^2} {\text{，}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0 < u < 1{\text{，}}\end{array} \right.$ (16)
 ${\sigma _{e1}} = {P_{e1}}R/t{\text{，}}$ (17)
 ${\tau _1} = {\sigma _{e1}}/{\sigma _s}{\text{。}}$ (18)

 {C_{s1}} = \left\{ \begin{aligned}{l}0.044\,42\tau _1^5 - 0.476\,5\tau _1^4 + 1.985\tau _1^3 - 3.905\tau _1^2 +\\ 3.214{\tau _1} + 0.084\,22,\;\;0.75 \leqslant {\tau _1} \leqslant 3{\text{,}}\\1{\text{,}}\;\;\;\;\;\;\;\;\;\;0 < {\tau _1} < 0.75{\text{。}}\end{aligned} \right. (19)

 ${C_{g1}} = \left\{ \begin{array}{l}1{\text{，}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0 < u \leqslant 1{\text{，}}\\0.75{\text{，}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;1 < u \leqslant 2{\text{，}}\\1.25 - 0.25u{\text{，}}\;\;\;\;\;\;\;\;\;u > 2{\text{。}}\end{array} \right.$ (20)

 ${P_{cr1}} = {C_{g1}}{C_{s1}}{P_{e1}}{\text{，}}$ (21)
 $\alpha = \pi /(L/R){\text{，}}$ (22)

 $b/l = \frac{{0.5(b/t)\sqrt[4]{{3(1 - {v^2})}}}}{{u(R/t)}}{\text{，}}$ (23)
 ${t_1}/l = \frac{{0.5{t_1}/{t_0}\sqrt[4]{{3(1 - {v^2})}}}}{{u(R/t)}}{\text{。}}$ (24)

 \begin{align}{y_0}/t = \frac{\begin{array}{l}(b/t)({t_2}/t)(0.5{t_2}/t + h/t + 0.5) + \\(h/t)({t_1}/t)(0.5h/t + 0.5)\end{array}}{{(h/t)({t_1}/t) + (b/t)({t_2}/t)}}{\text{，}}\end{align} (25)
 $\begin{split}&I/{R^3}l = \displaystyle\frac{{{I_0} + {{({y_0} + 0.5t)}^2}ltA/(lt + A) + l{t^3}/12}}{{{R^3}l}}=\\&\displaystyle\frac{b}{{12l}}\displaystyle\frac{{{{({t_2}/t)}^3}}}{{{{(R/t)}^3}}} + \displaystyle\frac{{{t_1}}}{{12l}}\displaystyle\frac{{{{(h/t)}^3}}}{{{{(R/t)}^3}}} + \displaystyle\frac{{{{({y_0}/t + 0.5)}^2}}}{{(1 + \beta ){{(R/t)}^3}}} + \displaystyle\frac{1}{{12{{(R/t)}^3}}}{\text{。}}\!\!\!\end{split}$ (26)

 $\begin{array}{l}U = \displaystyle\frac{{0.535{t^3}L}}{{I/l\sqrt {R(t + A/l)} }}=\\\;\;\; \!\! \displaystyle\frac{{0.535\sqrt {R/t} L/R}}{{[\displaystyle\frac{b}{{12l}}{{(\displaystyle\frac{{{t_2}}}{t})}^3} \!\!+\!\! \displaystyle\frac{{{t_1}}}{{12l}}{{(\displaystyle\frac{h}{t})}^3} \!\!+\!\! {{(\displaystyle\frac{{{y_0}}}{t} \!+\! 0.5)}^2}/(1 \!+\! \beta ) \!+\! 1/12]\sqrt {1 \!\!+\!\! 1/\beta } }}{\text{。}}\end{array}$ (27)

 ${P_{e2}} = \displaystyle\frac{E}{{{n^2} - 1 + 0.5{\alpha ^2}}}[\displaystyle\frac{{{\alpha ^4}}}{{{{({\alpha ^2} + {n^2})}^2}\displaystyle\frac{R}{t}}} + \displaystyle\frac{I}{{{R^3}l}}{({n^2} - 1)^2}]{\text{，}}$ (28)

$n$ 为使欧拉压力取最小值的周向失稳波数。

 ${\tau _2} = \frac{{{\sigma _2}}}{{{\sigma _s}}} = \frac{{0.95{P_{e2}}R}}{{(t + A/l)}} = \frac{{0.95{P_{e2}}(R/t)\beta }}{{(1 + \beta ){\sigma _s}}}{\text{。}}$ (29)

 ${C_{s2}} = \left\{ \begin{split}(0.04442\tau _2^5 - 0.4765\tau _2^4 + 1.985\tau _2^3 - 3.905\tau _2^2 + \\ 3.214{\tau _2} + 0.08422) \times 0.9{\text{，}}\;\;\;0.75 \leqslant {\tau _2} \leqslant 3{\text{，}}\\0.9{\text{，}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0 < {\tau _2} < 0.75{\text{。}}\!\!\!\end{split} \right.$ (30)
 ${C_{g2}} = \left\{ \begin{array}{l}1 - 0.14U{\text{，}}\;\;\;\;\;\;\;\;0 < U < 1.2{\text{，}}\\0.83{\text{，}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;U \geqslant 1.2{\text{。}}\end{array} \right.$ (31)

 ${P_{cr2}} = {C_{g2}}{C_{s2}}{P_{e2}}{\text{。}}$ (32)
2 优化模型和优化算法 2.1 设计变量

 $\begin{array}{l}0 < R/t < 150{\text{，}}1 < b/t < 5{\text{，}}0 < \beta < 5{\text{，}}2 < h/t < 15{\text{，}}\\0 < {t_1}/t < 1{\text{，}}0 < {t_2}/t < 3{\text{，}}0 < u < 3{\text{。}}\end{array}$
2.2 目标函数

 $\min f = \min \{ 2\delta \left( {1 + \frac{1}{\beta }} \right)/(R/t)\}{\text{，}}$ (33)

2.3 约束条件

1）强度约束

 $\sigma _2^0 \leqslant {k_1}{\sigma _s} {\text{或}} {g_1} = \sigma _2^0 - {k_1}{\sigma _s} \leqslant 0{\text{，}}$ (34)

 ${\sigma _1} \leqslant 1.15{\sigma _s} {\text{或}} {g_2} = {\sigma _1} - 1.15{\sigma _s} \leqslant 0{\text{，}}$ (35)

 ${\sigma _f} \leqslant 0.6{\sigma _s} {\text{或}} {g_3} = {\sigma _f} - 0.6{\sigma _s} \leqslant 0{\text{。}}$ (36)

2）稳定性约束

 ${P_{cr1}} \geqslant {P_j} {\text{或}}{g_4} = {P_j} - {P_{cr1}} \leqslant 0{\text{，}}$ (37)

 ${P_{cr2}} \geqslant 1.2{P_j} {\text{或}} {g_5} = 1.2{P_j} - {P_{cr2}} \leqslant 0{\text{，}}$ (38)

3）几何约束

 $\frac{h}{R} \leqslant 0.12 {\text{或}} {g_6} = \frac{h}{R} - 0.12 \leqslant 0{\text{。}}$ (39)
2.4 优化算法

2.5 优化过程

 图 1 环肋圆柱壳确定性优化流程图 Fig. 1 Deterministic optimization process of ring-stiffened cylindrical shell
3 600 MPa级高强度钢单位容积重量优化结果

 图 2 拉丁方设计（左）和最优拉丁方设计（右） Fig. 2 Latin square design (left) and optimal Latin square design (right)

 图 3 $f$ 与自变量的复相关系数 Fig. 3 Multiple correlation coefficient between $f$ and independent variable

 图 4 ${R_0}$ （ $R/t$ ）与自变量复相关系数 Fig. 4 Multiple correlation coefficient between ${R_0}$ ( $R/t$ ) and independent variable

 图 5 $L/R$ 与自变量复相关系数 Fig. 5 Multiple correlation coefficient between $L/R$ and independent variable

 图 6 ${P_{jx}}$ 与自变量复相关系数 Fig. 6 Multiple correlation coefficient between ${P_{jx}}$ and independent variable

 $f = 0.0516{P_{jx}} + 0.0067{\text{。}}$ (40)

${R^2} = 0.9995$ 与1接近，说明拟合精度较高，该公式可以用于600 MPa高强度钢环肋圆柱壳最小的单位容积重量计算。

 ${(\frac{R}{t})_{opt}} = 363.02P_{jx}^{ - 1.002}{\text{。}}$ (41)

${R^2} = 0.9961$ 与1接近，说明拟合精度较高，该公式可以用于600 MPa级高强度钢环肋圆柱壳最小单位容积重量情况下 ${\left(\displaystyle\frac{R}{t}\right)_{opt}}$ 计算。

 图 7 600 MPa级高强度钢环肋圆柱壳 ${f_{\min }}$ 与关系 Fig. 7 Relation between ${f_{\min }}$ and ${P_{jx}}$ of the 600 MPa ring-stiffened cylindrical high-strength steel shell

 图 8 600 MPa级高强度钢环肋圆柱壳 ${\left(\displaystyle\frac{R}{t}\right)_{opt}}$ 与 ${P_{jx}}$ 关系 Fig. 8 Relation betweenan ${\left(\displaystyle\frac{R}{t}\right)_{opt}}$ d ${P_{jx}}$ of the 600 MPa ring-stiffened cylindrical high-strength steel shell
4 结　语

1）本文依据GJB/Z 21A-2001潜艇结构设计计算方法建立了600 MPa级高强度钢环肋圆柱壳强度和稳定性无量纲的计算方法，该方法可用于环肋圆柱壳强度和稳定性校核和优化设计。

2）在上述计算方法的基础上对环肋圆柱壳单位体积重量进行优化，通过多岛遗传算法（MIGA）整体寻优和序列二次规划法（NLPQL）局部寻优，得到了在不同的圆柱壳长度半径比、不同的最大工作压力下，600 MPa级高强度钢环肋圆柱壳最小的单位容积重量和对应的最优的半径厚度比，在圆柱壳长度半径比为1，2，3，4情况下分别绘制了600 MPa级、高强度环肋圆柱壳最大工作压力与最小的单位容积重量关系曲线，最大工作压力与最优的半径厚度比关系曲线，并证明了最小的单位容积重量、最优的半径厚度比仅和材料、最大工作压力有关，和圆柱壳长度半径比关系不大。

3）根据上述优化结果，提出了计算600 MPa级高强度环肋圆柱壳最小的单位容积重量、最优的半径厚度比的拟合公式，该公式可以用于评估高强度钢环肋圆柱壳单位容积重量是否最优，可以计算单位容积重量最优情况下的半径厚度比，指导深海耐压环肋圆柱壳优化设计与评估。

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