﻿ 侧向载荷作用下的T型加筋板梁柱屈曲载荷-端缩曲线修正
 舰船科学技术  2021, Vol. 43 Issue (7): 19-26    DOI: 10.3404/j.issn.1672-7649.2021.07.005 PDF

1. 上海交通大学 海洋工程国家重点实验室，上海 200240;
2. 上海交通大学 高新船舶与深海开发装备协同创新中心，上海 200240

A correction method of Load-end shortening curves for T-stiffened plate beam column buckling under lateral load
CHEN Yu-zhe1,2, WANG De-yu1,2
1. State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, Shanghai 200240, China;
2. The Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China
Abstract: The load–end shortening curve of stiffened plate element is an important factor affecting the accuracy of Smith method, however, the effect of lateral load has not been considered in the load-end shortening curve specified by HCSR. To consider the effect of lateral load on the load-end shortening curve of stiffened plate element and broaden the scope of application of Smith method, the ultimate strength of 192 T-stiffened plate elements with beam column buckling failure mode under longitudinal compression load and lateral load is calculated by nonlinear finite element method. Through regression analysis, the load-end shortening curve correction formula of T-stiffened plate element is obtained. The critical stress and strain of 8 T-stiffened plates are calculated by the correction formula and the nonlinear finite element method, respectively. The relative error of results is less than 10%, which verifies the validity of the correction formula.
0 引　言

Paik等[4]研究了加筋板在轴向载荷、面内弯矩以及侧向载荷联合作用下的极限强度，并将计算结果和试验结果进行了对比，证实了提出公式的有效性。Shanmugam[5]对受单轴压缩和侧向载荷联合作用下的加筋板极限强度进行了模型试验研究和数值计算，得出对于此联合作用下的加筋板，板柔度系数 $\ \beta$ 的增大会引起加筋板极限承载能力下降的结论。

 $\frac{{{\sigma _u}}}{{{\sigma _{Yeq}}}}{\rm{ = }}\frac{1}{{{\beta ^{{Z_1}}} \cdot \sqrt {1.0 + {\lambda ^{{Z_2}}}} }}{\text，}$ (1)
 $\frac{{{\sigma _u}}}{{{\sigma _{Yeq}}}}{\rm{ = }}\frac{1}{{{\beta ^{0.28}} \cdot \sqrt {1.0 + {\lambda ^{3.2}}} }}{\text。}$ (2)

 $\beta {\rm{ = }}\frac{b}{{{t_p}}}\sqrt {\frac{{{\sigma _{Yeq}}}}{E}}{\text，}$ (3)

 $\lambda {\rm{ = }}\frac{a}{{\pi r}}\sqrt {\frac{{{\sigma _{Yeq}}}}{E}}{\text。}$ (4)

 $r = \sqrt {\frac{I}{{b{t_p} + {b_f}{t_f} + {h_w}{t_w}}}} {\text，}$ (5)
 $\begin{split} I \!=& \frac{{b{t_p}^3}}{{12}} \!+\! b{t_p}{\left( {{z_0} \!-\! \frac{{{t_p}}}{2}} \right)^2} \!+\! \frac{{{h_w}^3{t_w}}}{{12}} \!+\! {h_w}{t_w}{\left( {{z_0} \!-\! \frac{{{h_w}}}{2} \!-\! {t_p}} \right)^2} \!+ \\ &\frac{{{b_f}{t_f}^3}}{{12}} \!+\! {b_f}{t_f}{\left( {{t_p} \!+\! {h_w} \!+\! \frac{{{t_f}}}{2} \!-\! {z_0}} \right)^2}{\text，} \end{split}$ (6)
 ${z_0} = \frac{{0.5bt_p^2 + {h_w}{t_w}({t_p} + 0.5{h_w}) + {b_f}{t_f}({t_p} + {h_w} + 0.5{t_f})}}{{b{t_p} + {b_f}{t_f} + {h_w}{t_w}}}{\text。}$ (7)

 $\frac{{{\sigma _{xu}}}}{{{\sigma _{Yeq}}}} = \frac{1}{{\sqrt {\left( \begin{gathered} {X_0} + {X_1}\lambda + {X_2}\beta + {X_3}\lambda \beta + {X_4}{\lambda ^2} + {X_5}{\beta ^2} + \\ {X_6}{\lambda ^2}{\beta ^2} + {X_7}{\lambda ^3} + {X_8}{\beta ^3} + {X_9}{\lambda ^3}{\beta ^3} + {X_{10}}{\lambda ^4} \\ \end{gathered} \right)} }} \leqslant \frac{1}{{{\lambda ^2}}}{\text。}$ (8)

1 计算模型介绍 1.1 模型参数

 ${\sigma _{{\rm{Y}}eq}}{\rm{ = }}\frac{{{\sigma _{{\rm{Y}}p}}b{t_p} + {\sigma _{{\rm{Y}}s}}({b_f}{t_f} + {h_w}{t_w})}}{{b{t_p} + {b_f}{t_f} + {h_w}{t_w}}}{\text。}$ (9)

Do [9]统计出船体加筋板的长宽比通常在5～6之间，因此在设计模型参数时，取加筋板长宽比为5。Zhang[6]统计了12个油船和10个散货船的设计数据，总结出典型船体板柔度系数及梁柱柔度系数的分布范围，其中板柔度系数多分布在1～2.5之间，而梁柱柔度系数多分布在0.25～0.95之间。本文通过调整加筋板几何参数，得到4种不同的板柔度系数：1.1731，1.4664，1.9552，2.3234，以及分布在[0.3067，1.0240]之间的一系列梁柱柔度系数，与船体加筋板统计得到的分布范围基本一致。

1.2 载荷与边界条件

 图 1 加筋板节点编号示意图 Fig. 1 Diagram of stiffened plate node number

1.3 初始缺陷

 ${w_{op}} = {A_0}\sin \left( {\frac{{m\text{π} x}}{a}} \right)\sin \left( {\frac{{\text{π} y}}{b}} \right){\rm{ + }}{B_0}\sin \left( {\frac{{\text{π} x}}{a}} \right)\sin \left( {\frac{{\text{π} y}}{b}} \right){\text，}$ (10)
 ${w_{oc}} = {B_0}\sin \left( {\frac{{\text{π} x}}{a}} \right)\sin \left( {\frac{{\text{π} y}}{b}} \right){\text，}$ (11)
 ${w_{os}} = \frac{{{C_0}z}}{{{h_w}}}\sin \left( {\frac{{\text{π} x}}{a}} \right){\text。}$ (12)

 图 2 加筋板初始变形示意图（变形放大10倍） Fig. 2 Diagram of initial distortion of stiffened plate model （Deformation magnified 10 times）
2 载荷-端缩曲线修正 2.1 有限元法计算结果

 图 3 不同侧向载荷作用下无因次化临界应力有限元法计算结果 Fig. 3 Dimensionless critical stress results of stiffened plate model under different lateral loads by finite element method

 图 4 不同侧向载荷作用下无因次化临界应变有限元法计算结果 Fig. 4 Dimensionless critical strain results of stiffened plate model under different lateral loads by finite element method
2.2 拟合公式

 图 5 σu/σu0关于λ拟合公式示意图 Fig. 5 Diagram of the fitting formula about σu/σu0 and λ

 图 6 εu/εu0关于λ拟合公式示意图 Fig. 6 Diagram of the fitting formula about εu/εu0 and λ

 $\frac{{{\sigma _u}}}{{{\sigma _{u0}}}}\left( \lambda \right) = {p_1}{\lambda ^3} + {p_2}{\lambda ^2} + {p_3}\lambda + {p_4}{\text，}$ (13)
 $\frac{{{\varepsilon _u}}}{{{\varepsilon _{u0}}}}\left( \lambda \right) = {p_5}{\lambda ^3} + {p_6}{\lambda ^2} + {p_7}\lambda + {p_8}{\text。}$ (14)

 图 7 临界应力修正公式系数p1～p4关于β和LP拟合结果 Fig. 7 The fitting results of coefficients of the critical stress correction formula about β and LP

 图 8 临界应变修正公式系数p5～p8关于β和LP拟合结果 Fig. 8 The fitting results of coefficients of the critical strain correction formula about β and LP

 $\begin{split} {p_i}(\beta ,LP) = &{p_{00}} + {p_{10}} \cdot \beta + {p_{01}} \cdot LP + {p_{20}} \cdot {\beta ^2} +\\ &{p_{11}} \cdot \beta \cdot LP + {p_{02}} \cdot L{P^2} + {p_{30}} \cdot {\beta ^3} + \\ &{p_{21}} \cdot {\beta ^2} \cdot LP +{p_{12}} \cdot \beta \cdot L{P^2} + {p_{03}} \cdot L{P^3} {\text。} \end{split}$ (15)
2.3 T型加筋板梁柱屈曲载荷-端缩曲线修正公式

 ${\sigma _{CR1}}{\rm{ = }}\varPhi {\sigma _{C1}}\frac{{{A_s} + 10{b_{E1}}{t_p}}}{{{A_s} + 10{b_E}{t_p}}}{\text。}$ (16)

 $\varPhi {\rm{ = }}\left\{ {\begin{array}{*{20}{c}} {{\rm{ - 1}}},&{\varepsilon < - 1}{\text，}\\ \varepsilon ,&{{\rm{ - 1}} \leqslant \varepsilon \leqslant {\rm{1}}}{\text，}\\ {\rm{1}},&{\varepsilon > 1}{\text。} \end{array}} \right.$ (17)

$\varepsilon$ 为相对应变：

 $\varepsilon = \frac{{{\varepsilon _E}}}{{{\varepsilon _Y}}}{\text。}$ (18)

${\varepsilon _E}$ 为单元应变，根据单元自身中和轴位置、船体横剖面中和轴位置和曲率计算得到； ${\varepsilon _Y}$ 为单元达到屈服应力时的应变：

 ${\varepsilon _Y} = \frac{{{\sigma _{Yeq}}}}{E}{\text。}$ (19)

 ${\sigma }_{C1}=\left\{\begin{array}{cc}\dfrac{{\sigma }_{E1}}{\varepsilon },& {\sigma }_{E1}\leqslant \dfrac{{\sigma }_{Yeq}}{2}\varepsilon {\text，} \\ {\sigma }_{Yeq}\left(1-\dfrac{{\sigma }_{Yeq}\varepsilon }{4{\sigma }_{E1}}\right),& {\sigma }_{E1}>\dfrac{{\sigma }_{Yeq}}{2}\varepsilon {\text。}\end{array}\right.$ (20)

 ${\sigma _{E1}} = {{\text{π}} ^2}E\frac{{{I_E}}}{{{A_E}{l^2}}}{10^{ - 4}}{\text。}$ (21)

 ${b_{E1}} = \left\{ {\begin{array}{*{20}{c}} {\dfrac{b}{{{\beta _E}}}},&{{\beta _E} > 1.0} {\text，}\\ b,&{{\beta _E} \leqslant 1.0} {\text，} \end{array}} \right. \;\; {\beta _E} = {10^3}\dfrac{b}{{{t_p}}}\sqrt {\frac{{\varepsilon {\sigma _{Yp}}}}{E}}{\text。}$ (22)

 ${b_E} = \left\{ {\begin{array}{*{20}{c}} {\left( {\dfrac{{2.25}}{{{\beta _E}}} - \dfrac{{1.25}}{{\beta _E^2}}} \right)b},&{{\beta _E} > 1.25}{\text，} \\ b,&{{\beta _E} \leqslant 1.25} {\text。} \end{array}} \right.$ (23)

 ${\sigma _{CR1}}{\rm{(}}\varepsilon '){\rm{ = }}{f_1}(LP,\beta ,\lambda )\Phi {\sigma _{C1}}\frac{{{A_s} + 10{b_{E1}}{t_p}}}{{{A_s} + 10{b_E}{t_p}}}{\text。}$ (24)

 $\begin{split} {p_i}(\beta ,LP) = &{p_{00}} + {p_{10}} \cdot \beta + {p_{01}} \cdot LP + {p_{20}} \cdot {\beta ^2} + \\ &{p_{11}} \cdot \beta \cdot LP + {p_{02}} \cdot L{P^2} + {p_{30}} \cdot {\beta ^3} + \\ &{p_{21}} \cdot {\beta ^2} \cdot LP +{p_{12}} \cdot \beta \cdot L{P^2} + {p_{03}} \cdot L{P^3} {\text。} \end{split}$ (25)
2.4 算例验证

3 结　语

1）侧向载荷对T型加筋板梁柱屈曲状态下的载荷-端缩曲线具有显著影响，较大的侧向载荷（ $LP <$ $0.15$ MPa）可使柔度系数较大（ $\ \beta > 1.95,\lambda > 0.6$ ）的加筋板极限承载能力下降40%以上。

2）对于柔度系数较小（ $\ \beta < 1.45,\lambda < 0.5$ ）的T型加筋板，由于无侧向载荷时加筋板的弯曲方向和侧向载荷的方向相反，加筋板的临界端缩量随侧向载荷的增加呈现先增大后减小的趋势，拐点通常出现在LP = 0.05 MPa～0.1 MPa范围内；当 $\ \beta$ $\lambda$ 增大时，拐点对应的侧向载荷值减小。

3）当 $\ \beta \leqslant 1.45,\lambda \leqslant 0.4,LP \leqslant 0.1MPa$ 时，侧向载荷对T型加筋板梁柱屈曲状态下的临界应力影响较小，在5%以内。

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