【作者】 杨帅；

【导师】 陈朝晖；

【作者基本信息】 重庆大学 ， 土木工程， 2020， 硕士

【摘要】 薄膜和薄壳结构,被广泛应用于飞机、风力发电机等机翼以及大跨建筑结构中。膜结构面外刚度很小几乎可以忽略,而薄壳结构的基本几何特性是厚度方向的尺寸远小于另外两个方向的尺寸,面内薄膜张力效应良好而面外抗弯刚度较弱。荷载作用下,膜和薄壳结构容易产生大变形,具有明显的几何非线性特性。传统有限元方法通常基于连续介质力学的大变形理论,从推导虚应变能与荷载势能开始,引入形函数,并通过数值积分得到相应的弹性刚度矩阵和几何刚度矩阵,最后采用增量迭代法确定荷载—位移路径。但上述方法中非线性的虚应变能与荷载势能的推导非常繁琐,且由于形函数的近似性,所得几何刚度矩阵不尽准确。为此,本文采用了刚体准则思想分析薄膜和薄壳结构的大位移大转动问题。基于更新的拉格朗日格式的增量迭代法,根据刚体准则思想,初始受力平衡的单元在经历刚体位移后,其单元结点力方向随单元发生转动而大小不变,单元仍保持平衡。在常见几何非线性问题中,可认为刚体位移占了单元位移的主要部分,而弹性变形为次要部分。采用更新的拉格朗日格式的增量迭代法,每一荷载增量步,将单元的变形视为刚体运动与自然变形两部分,其中刚体位移及其对应结点力采用刚体准则处理;弹性变形采用小变形线性化理论处理。利用刚体准则处理单元结点力经历刚体位移的效应,大大简化了单元结点力计算过程。本文基于刚体准则的基本思想,构建了一类新型三角形刚体准则壳单元和三角形刚体准则膜单元。三角形刚体准则空间壳单元的几何刚度矩阵由空间梁单元的几何刚度矩阵推导得到,弹性刚度矩阵由考虑膜作用的平面混合单元的刚度矩阵和考虑弯曲作用的混合应力模型单元的刚度矩阵组合得到,该三角形刚体准则空间壳单元可容易退化为三角形刚体准则平面壳单元。三角形刚体准则膜单元由三根空间杆件组成铰接三角形,并在中间张拉薄膜而成,杆件的材料与薄膜相同,其几何刚度矩阵由杆单元的几何刚度矩阵推导得到,弹性刚度矩阵则由平面应力单元构成。该方法几何刚度矩阵推导简单,无需引入对单元大变形的人为假定,增量迭代计算过程充分考虑刚体准则,通过对若干膜和壳结构经典算例的分析及与已有方法的比较,验证了所建单元与方法的准确性以及计算效率,基于刚体准则建立的上述单元将在相关工程结构的非线性分析中发挥其概念清晰、计算高效的优势。更多还原

【Abstract】 Membrane and thin shells are very important structural types,and they have been widely used in the wings of space shuttles,wind turbines,and large-span building structures.A basic characteristic of the thin shell structure is that the dimensions in the thickness direction are much smaller than those in the other two directions,and have good in-plane film tension effects and out-of-plane bending stiffness.The out-of-plane stiffness of the membrane structure is very small,even without the out-of-plane stiffness.Therefore,under the action of load,the thin shell and membrane structure tend to produce very large deformation,which has obvious geometric nonlinear characteristics.Based on a large-deflection theory of continuum mechanics,such as non-linear analysis,using the FEM on shell and membrane structures usually begins by deriving linear virtual strain energy and non-linear virtual strain energy.Then,shape functions of displacements are introduced and the corresponding elastic stiffness matrix and geometric stiffness matrix can be obtained by using numerical integration.Finally,uses of the incremental-iteration procedure can determine the path of load-deflection equilibrium of structures.However,the derivation of nonlinear virtual strain energy and load potential energy in the conventional finite element method is very tedious,and the resulting geometric stiffness matrix is not accurate due to the approximation of the shape function.In view of the limitations of the element displacement and deformation description in the geometric nonlinear analysis method mentioned above,the rigid body rule is adopted to analyze the large displacement and large rotation of the structure in this paper.According to the rigid body rule,when an initial stressed element undergoes rigid body displacement,the directions of the nodal forces at the element will change with the rigid rotation of the element while the magnitude of the nodal forces will remain unchanged in order to keep the element in balance.In common geometric nonlinear problems,the rigid body displacement accounts for the main part of the element displacement.The rigid body rule can be used to clearly deal with the effect of the element force undergoing the rigid body displacement,which greatly simplifies the calculation process of the element node force.The physical concept of nonlinear analysis process based on rigid body rule is clear and the calculation process is simple,which overcome the disadvantages of traditional methods.Based on the basic idea of rigid body rule,this paper constructs triangular shell elements and triangular membrane elements.The geometric stiffness matrix of the triangular space shell element is derived from the geometric stiffness matrix of the space beam element.the elastic stiffness matrix adopted for the triangular space shell element is constructed as the composition of Cook’s plane hybrid element for membrane actions and the hybrid stress model(HSM)of Batoz et al.for bending actions.Space shell elements can easily degenerate into triangular planar shell elements.This membrane element is composed of three bars to form a pin-joint triangle with a triangle film stretched inside.The material of the bars is assumed the same as that of the film.The geometric stiffness matrix of the proposed membrane element is derived by the geometric stiffness matrix of spatial bar element which is rigid body rule qualified and the elastic stiffness matrix is composed of plane stress elements.By rooting the rigid body rule into the UL incremental-iteration method,the effect of rigid body rotation is fully considered in each stage of analysis and the residual effects of natural deformation is treated by the linearization.The derivation of the geometric stiffness matrix of the proposed membrane element is quite clear without introducing any assumption on the large deformation of the membrane element and can be easily degenerated into a plane membrane element.The accuracy and efficiency of the present shell element and membrane element as well as the rigid-body-rule-rooted nonlinear analysis method is proved by the analyzing of several typical space membrane structures and by the comparison with the results of other methods.更多还原