开孔板结构应力及稳定性半解析分析方法研究

发布时间：2018-05-23 07:19

本文选题：开孔板结构 + 平面应力　；参考：《华中科技大学》2014年博士论文

【摘要】：含有开孔的各种板结构广泛地应用于工程结构中。由于开孔的存在,其应力集中问题和稳定性问题是结构设计中需要关注的两个重要问题。因此,对开孔板结构的应力及稳定性问题进行研究具有重要的理论意义和实用价值。本文提出了一种新的求解含有任意开孔形状有限平板的平面应力分布问题和弯曲应力分布问题的半解析方法；并在此基础上,提出了一种新的求解开孔平板和开孔加筋板稳定性问题的半解析方法。论文主要的研究工作包括以下几个方面：首先,对含任意开孔形状的开孔有限平板在面内均布载荷作用下的应力分布问题,提出了一种基于平面问题复变函数方法的半解析方法,即平面应力问题的应力函数重构法。这个方法第一步是考虑含开孔无限大板的情况,通过保角映射变换将在物理平面内(z平面)的开孔及开孔外部的无限大区域映射到映射平面内(ζ平面)的单位圆及单位圆外部的无限大区域。通过Cauchy积分求解得到开孔无限大板在ζ平面内的应力函数；第二步是再考虑开孔有限板的情况,在ζ平面内,将求得的开孔无限大板应力函数的两组特征项进行扩展,重新构造得到开孔有限板在ζ平面内的应力函数；第三步通过最小二乘边界配置法来确定应力函数未知的待定系数,最终求得开孔有限板的整个应力场。其次,对该方法进行验证及参数分析。针对七种不同开孔形状的无限大板和有限板,采用该方法进行平面应力分布计算,并与有限元ANSYS计算结果和已有解析结果进行对比验证。结果表明,开孔足够大时,平面应力问题的无限大板理论将不适用,而本文所提出的应力函数重构法计算简便、适用性强、计算精度高,可适用于具有不同开孔形状的无限板和有限板。采用该方法研究了几种不同开孔形状的无限板和有限板分别在单向拉伸、双向拉伸、剪切载荷作用下,开孔大小对孔边应力分布和应力集中的影响。并针对矩形开孔的应力分布及应力集中,进一步分析了开孔大小、开孔角度、矩形板边长比的影响。接着,针对开孔板弯曲应力分布问题,基于薄板小挠度弯曲理论的复变函数方法,提出了一种新的求解含有任意开孔形状有限板弯曲应力分布问题的半解析方法,即弯曲应力问题的应力函数重构法。针对4种不同开孔形状的无限板和有限板,采用该方法进行弯曲应力分布计算,并与有限元ANSYS计算结果和已有解析结果进行对比。计算结果表明,该方法适用性强、计算简便、计算精度高,可适用于具有不同开孔形状的无限大板和有限板。针对一对边具有均布弯矩作用的4种开孔形状的无限大板和有限板,采用该方法研究了开孔大小对孔边应力分布和应力集中的影响。然后,针对含有矩形开孔的受压矩形平板总体稳定性进行了半解析分析。先采用前面提出的求解开孔板平面应力分布问题的应力函数重构法精确地求解开孔板的内力分布。再采用区域分解法求出含矩形开孔的矩形平板挠曲面函数,该挠曲面函数不仅满足开孔板的外边界条件,还满足开孔板的内边界条件(矩形孔边的边界条件)；基于求得的内力分布和挠曲面函数,通过能量法求出其临界屈曲载荷,并与有限元ANSYS计算结果进行对比。计算结果表明,该方法计算精度高。采用该方法讨论了在四边简支、四边固支、承载边简支无载边自由、承载边固支无载边自由四种典型边界条件下,开孔大小对其稳定性的影响。最后,针对含有矩形开孔的受压加筋板总体稳定性进行了半解析分析。将加筋板结构简化处理为板和梁的组合。与求解开孔平板稳定性问题相似,将采用应力函数重构法求得的开孔平板的内力分布作为开孔加筋板铺板的内力分布；采用区域分解法求得满足含矩形开孔加筋板的内外边界条件的挠曲函数；并进一步求得板和梁的弯曲应变能及外力功。再通过能量法求出其临界屈曲载荷,并与有限元ANSYS结果进行对比。计算结果表明,针对几种典型开孔加筋板形式,当开孔大小在不超过某个值的范围内,计算结果误差较小。在不同边界条件下,讨论了开孔大小对其稳定性的影响。本文的研究成果对开孔板结构的理论研究和工程设计具有一定的参考价值。
[Abstract]:The problems of stress concentration and stability are two important issues that need to be paid attention to in structural design due to the existence of openings . Therefore , it has important theoretical significance and practical value to study the stress and stability of open - plate structures .

In this paper , a new semi - analytic method for solving the problem of plane stress distribution and bending stress distribution is presented .
Based on this , a new semi - analytical method for solving the stability problem of open - hole plate and open - hole stiffened plate is presented . The main research work includes the following aspects :

First , a semi - analytic method based on the plane problem recurrence function method is presented , which is the stress function reconstruction method of plane stress problem . The first step of this method is to consider the stress function reconstruction method of plane stress problem . The first step of this method is to take into account the condition of infinite plate with open hole . The first step of this method is to take into account the infinite area outside the unit circle and the unit circle in the mapping plane through conformal mapping transformation . The stress function of the open pore infinite plate in the zeta plane is obtained by Cauchy integral solution .
the second step is to expand the two groups of characteristic items of the open - hole infinite plate stress function , and reconstruct the stress function of the open - hole finite plate in the zeta plane ;
and the third step determines the undetermined coefficient unknown to the stress function by the least square boundary collocation method , and finally obtains the whole stress field of the open - hole finite plate .

In this paper , the method is used to calculate the stress distribution and stress concentration of seven kinds of infinite plates and limited plates with different opening shapes . The results show that the stress distribution and stress concentration of the rectangular openings can be applied to the stress distribution and stress concentration of the rectangular openings , and the influence of the opening size , the opening angle and the ratio of the edge length of the rectangular plate is further analyzed .

This paper presents a new semi - analytical method for solving the problem of bending stress distribution of open - hole plate , which is based on the theory of bending stress of thin plate . A new method for solving bending stress distribution with arbitrary open - hole shape is presented . The results show that the method has strong applicability , simple calculation and high calculation precision . It can be applied to the infinite plate and limited plate with different open - hole shapes .

In this paper , a semi - analytical analysis of the overall stability of a rectangular flat plate with rectangular openings is carried out firstly . The internal force distribution of the perforated plate is solved by the method of stress function reconstruction , which is based on the problem of plane stress distribution of the perforated plate .
Based on the obtained internal force distribution and deflection surface function , the critical buckling load is calculated by the energy method , and compared with the calculation result of finite element ANSYS . The results show that the method has high calculation precision .

Finally , a semi - analytical analysis of the overall stability of the stiffened plate with rectangular openings is carried out . The stiffened plate structure is simplified into a combination of plates and beams . The internal force distribution of the open - hole plate obtained by the stress function reconstruction method is used as the internal force distribution of the open - hole stiffened plate .
The deflection function of the inner and outer boundary conditions of the stiffened plate with rectangular openings is obtained by using the regional decomposition method .
The bending strain energy and external force work of the plate and beam are obtained . The critical buckling load is calculated by the energy method and compared with the result of finite element ANSYS . The results show that the error is less when the opening size is within the range of no more than a certain value , and the influence of the opening size on its stability is discussed under different boundary conditions .

The research results of this paper have some reference value to the theoretical research and engineering design of open - hole plate structure .
【学位授予单位】：华中科技大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TU311

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