求解弹性地基上自由矩形中厚板弯曲问题的辛-叠加方法  

A Symplectic Superposition Method for Bending Problems of Free-Edge Rectangular Thick Plates Resting on Elastic Foundations

作　　者：李锐 田宇 郑新然 王博[1] LI Rui;TIAN Yu;ZHENG Xinran;WANG Bo(Department of Engineering Mechanic;International Research Center for Computational Mechanics,Dalian University of Technolog;State Key Laboratory of Structural Analysis for Industrial Equipment（Dalian University of Technology）,Dalian,Liaoning 115024,P.R.China)

机构地区：[1]大连理工大学工程力学系国际计算力学研究中心工业装备结构分析国家重点实验室(大连理工大学),辽宁大连116024

出　　处：《应用数学和力学》2018年第8期875-891,共17页Applied Mathematics and Mechanics

基　　金：国家重点基础研究发展计划(973计划)(2014CB049000);中国科协青年人才托举工程项目(2015QNRC001);国家自然科学基金(11302038)~~

摘　　要：基于近年来提出的辛-叠加方法,解析求解了弹性地基上自由矩形中厚板的弯曲问题.首先将原问题拆分为3类子问题,在Hamilton体系下,运用辛几何方法推导出子问题对应的弹性地基上对边滑支矩形板弯曲问题的辛解析解;以此为基础,通过叠加法思想,求出弹性地基上四边自由矩形中厚板的弯曲解.与半逆法等传统解析方法相比,辛-叠加方法兼备了辛方法理性和叠加法规律性的优点,在求解过程中不需要预先假定解的形式,而是由弹性力学基本方程出发,经过逐步严格推导获得解析解,因而大大拓展了可求解问题的范围,成为一种求解以矩形板问题为代表的弹性力学高阶偏微分方程复杂边值问题的有效解析方法.Based on the symplectic superposition method proposed in recent years,the bending problems of free-edge rectangular thick plates resting on elastic foundations were analytically solved. The original problem was split into 3 subproblems corresponding to the bending problems of rectangular thick plates with 2 opposite edges slidingly clamped and resting on elastic foundations,which were solved with the symplectic geometry method. The analytic solution of the original problem was then obtained through superposition. Compared to the conventional analytic approaches such as the semi-inverse method,the symplectic superposition method has the advantages of both rationality of the symplectic method and regularity of the superposition method. The solution procedure starts from the basic equations of elasticity,and a rigorous derivation yields the analytic solutions,thus extending the scope of problems to be solved. The present method can serve as an effective analytic approach to complex boundary value problems of high-order partial differential equations in elasticity,as represented by the rectangular plate problems.

关 键 词：辛-叠加方法 弹性地基 中厚板 弯曲 

分 类 号：O302[理学—力学]