结构极限与安定分析的数值方法研究及其工程应用
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摘要
当前极限与安定性研究的一个重要课题是研究应用的策略，寻找切实可
行的计算方法，使极限与安定性理论得以在工程实际中应用。本文深入、系
统研究了结构极限与安定分析的数值理论和计算方法，并运用于带体积型缺
陷压力容器的安全评估中。
本文发展了三维结构极限上限分析的数值方法．基于凸分析和非光滑分
析理论，通过采用罚一对偶方法，成功地解决了三维极限上限分析中的塑性
不可压问题，建立了三维结构上限分析的有限元规划格式，并给出了相应的
无搜索优化迭代算法，克服了目标函数非线性非光滑所导致的数值困难。
本文发展了轴对称结构极限下限分析的一种有限元计算方法。根据下限
定理，通过引入P泛数，并采用应力函数法构造平衡应力场，建立了极限
下限分析的有限元数学规划格式。采用修正的Newton—Raphson迭代算法
求解了非线性规划问题。
针对两种放松的机动安定准则，建立了相应的三维机动安定规划格式，
提出了各自的数值或解析求解方法。根据凸分析和对偶理论，建立了三维机
动安定分析的有限元规划格式，采用一套直接迭代算法求解，每一内部迭代
相当于求解一个相关的弹性问题。
根据Melan定理，采用杂交应力元构造自平衡应力场，使用单元内屈
服条件平均化的思想，建立了静力安定分析的有限元非线性规划格式。通过
采用弹塑性分区降维迭代法，克服了安定分析中的维数障碍，使得求解大规
模非线性规划问题成为可能。
利用本文所提出的数值方法，对带体积型缺陷压力容器进行了数值极限
与安定分析。研究了各种类型体积型缺陷对受压容器的失效模式及极限与安
定载荷的影响，给出了一系列极限与安定载荷计算曲线与拟合公式。本文的
计算结果为带体积型缺陷压力容器的安全评估提供了理论依据。
At present an important subject on the limit and shakedown analyses is
to study the strategies of applications, search for efficient and feasible
computational methods so that the limit and shakedown theories can find
their applications in engineering practice. In this paper, the numerical theo-
ries and computational methods for limit and shakedown analyses of struc-
tures are systematically studied and applied to the safety assessment of the
pressure vessels with volume defects.
A numerical method is developed for the upper bound limit analysis of
3-D structures. Based on the theories of the convex analysis and the
nonsmooth analysis, the penalty-duality method is used to deal with the
plastic incompressibility condition. The limit analysis of 3-D structures is
formulated as a discrete nonlinear mathematical programming problem with
equality constraints by means of finite element technique. An optimal direct
iterative algorithm is given to solve this formulation.
A finite element technique using the definition of the P-norm is devel-
oped for calculating lower bounds of the limit load multiplier for
axisymmetric structures which obey the von Mises yield criterion. Based on
the lower bound limit theorem, a finite element mathematical programming
formulation is established by applying the stress function method to con-
struct the equilibrated stress field, and solved by the modified
Newton-Raphson iterative algorithm.
For two relaxed kinematic shakedown criteria, the corresponding 3-D
numerical kinematic formulations are established and the respective solution
approaches are presented. Based on the convex analysis and duality theorem,
a kinematic shakedown formulation for 3-D problems is derived. A finite el-
ement discretization of the kinematic extremum problem is considered. A di-
rect iterative algorithm is employed to solve the discretized system. Every
step of the inner iteration is equivalent to solving a relevant elastic problem.
According to Melan's theorem, a nonlinear finite element programming
formulation for static shakedown analysis is established by using hybrid
stress finite element to construct the self-equilibrated stress field with the
yield criterion of the mean. In order to avoid the dimension obstacle of large
scale nonlinear programming, a new technique, e.g. elastic-plastic dimension
reduction-based iteration approach is proposed and suitably applied to the
shakedown analysis of the pressure vessels with volume defects.
In light of the proposed methods, numerical limit and shakedown ana-
lyses are performed for the pressure vessels with volume defects. The effects
of various shapes and sizes of volume defects on the limit and shakedown
loads are investigated and evaluated. Two kinds of typical failure modes cor-
responding to different dimensions of volume defects are studied. The re-
search has lead to a series of computational curves as well as the fit formulae
for the limit and shakedown loads. The calculated results can provide theo-
retical foundations for the safety assessment of the pressure vessels with vol-
ume defects.
Ph. D. Candidate Liu Yinghua (Solid Mechanics)
Directed by Professor Xu Bingye &
Professor Cen Zhangzhi
行的计算方法，使极限与安定性理论得以在工程实际中应用。本文深入、系
统研究了结构极限与安定分析的数值理论和计算方法，并运用于带体积型缺
陷压力容器的安全评估中。
本文发展了三维结构极限上限分析的数值方法．基于凸分析和非光滑分
析理论，通过采用罚一对偶方法，成功地解决了三维极限上限分析中的塑性
不可压问题，建立了三维结构上限分析的有限元规划格式，并给出了相应的
无搜索优化迭代算法，克服了目标函数非线性非光滑所导致的数值困难。
本文发展了轴对称结构极限下限分析的一种有限元计算方法。根据下限
定理，通过引入P泛数，并采用应力函数法构造平衡应力场，建立了极限
下限分析的有限元数学规划格式。采用修正的Newton—Raphson迭代算法
求解了非线性规划问题。
针对两种放松的机动安定准则，建立了相应的三维机动安定规划格式，
提出了各自的数值或解析求解方法。根据凸分析和对偶理论，建立了三维机
动安定分析的有限元规划格式，采用一套直接迭代算法求解，每一内部迭代
相当于求解一个相关的弹性问题。
根据Melan定理，采用杂交应力元构造自平衡应力场，使用单元内屈
服条件平均化的思想，建立了静力安定分析的有限元非线性规划格式。通过
采用弹塑性分区降维迭代法，克服了安定分析中的维数障碍，使得求解大规
模非线性规划问题成为可能。
利用本文所提出的数值方法，对带体积型缺陷压力容器进行了数值极限
与安定分析。研究了各种类型体积型缺陷对受压容器的失效模式及极限与安
定载荷的影响，给出了一系列极限与安定载荷计算曲线与拟合公式。本文的
计算结果为带体积型缺陷压力容器的安全评估提供了理论依据。
At present an important subject on the limit and shakedown analyses is
to study the strategies of applications, search for efficient and feasible
computational methods so that the limit and shakedown theories can find
their applications in engineering practice. In this paper, the numerical theo-
ries and computational methods for limit and shakedown analyses of struc-
tures are systematically studied and applied to the safety assessment of the
pressure vessels with volume defects.
A numerical method is developed for the upper bound limit analysis of
3-D structures. Based on the theories of the convex analysis and the
nonsmooth analysis, the penalty-duality method is used to deal with the
plastic incompressibility condition. The limit analysis of 3-D structures is
formulated as a discrete nonlinear mathematical programming problem with
equality constraints by means of finite element technique. An optimal direct
iterative algorithm is given to solve this formulation.
A finite element technique using the definition of the P-norm is devel-
oped for calculating lower bounds of the limit load multiplier for
axisymmetric structures which obey the von Mises yield criterion. Based on
the lower bound limit theorem, a finite element mathematical programming
formulation is established by applying the stress function method to con-
struct the equilibrated stress field, and solved by the modified
Newton-Raphson iterative algorithm.
For two relaxed kinematic shakedown criteria, the corresponding 3-D
numerical kinematic formulations are established and the respective solution
approaches are presented. Based on the convex analysis and duality theorem,
a kinematic shakedown formulation for 3-D problems is derived. A finite el-
ement discretization of the kinematic extremum problem is considered. A di-
rect iterative algorithm is employed to solve the discretized system. Every
step of the inner iteration is equivalent to solving a relevant elastic problem.
According to Melan's theorem, a nonlinear finite element programming
formulation for static shakedown analysis is established by using hybrid
stress finite element to construct the self-equilibrated stress field with the
yield criterion of the mean. In order to avoid the dimension obstacle of large
scale nonlinear programming, a new technique, e.g. elastic-plastic dimension
reduction-based iteration approach is proposed and suitably applied to the
shakedown analysis of the pressure vessels with volume defects.
In light of the proposed methods, numerical limit and shakedown ana-
lyses are performed for the pressure vessels with volume defects. The effects
of various shapes and sizes of volume defects on the limit and shakedown
loads are investigated and evaluated. Two kinds of typical failure modes cor-
responding to different dimensions of volume defects are studied. The re-
search has lead to a series of computational curves as well as the fit formulae
for the limit and shakedown loads. The calculated results can provide theo-
retical foundations for the safety assessment of the pressure vessels with vol-
ume defects.
Ph. D. Candidate Liu Yinghua (Solid Mechanics)
Directed by Professor Xu Bingye &
Professor Cen Zhangzhi
引文
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