四节点矩形弹簧元及其特性研究

弹簧元法是一种将单元离散为一系列弹簧的数值计算方法。不同的单元具有不同的离散方式，确定相应的离散弹簧的刚度系数表达式是弹簧元法的关键。将四节点矩形单元离散为6个基本弹簧，每个基本弹簧包括法向弹簧和切向弹簧两个派生弹簧，并用泊松弹簧和纯剪弹簧描述单元的泊松效应和剪切效应，用有限元的单元刚度矩阵标定各弹簧的刚度系数，实现了一种四节点矩形弹簧元的构造形式。该单元的同类弹簧具有相同的表达形式。法向与切向弹簧的刚度表达式中分别含有法向和切向弹簧刚度待定系数。通过改变待定系数的值可使该单元分别对应于有限元的常应变、双线性及Wilson非协调单元。将上述弹簧元方法进行理论推导，并应用于基于连续介质的离散单元法（CDEM）的核心计算进行简单算例验证，证明了提出方法的正确性。通过以上研究发现，四节点矩形弹簧元有以下特点：对于相同问题，不同单元有不同的计算精度；对于梁弯曲问题，应用该单元可显著提高离散单元法的求解精度；改变待定系数的值，可得到更高或者更低精度的单元。

Study of four-node rectangular spring element and its properties

The spring element method (SEM) is a numerical method that uses a spring system to describe an element. Different elements can be described as different spring systems; and the definition of the spring stiffness expressions in the systems is the key point of the spring element method. The four-node rectangular element is described by 6 basic springs, each of which contains two derived springs: normal spring and tangential spring. Poisson spring and pure shear spring are used to describe Poisson and shear effects of the element. Thus a four-node rectangular spring element is presented. Compared with the element stiffness matrix of finite element method, the stiffness expression of each spring is obtained. Springs of the same kind have the same expressions. The stiffness expressions of the normal and tangential springs have corresponding coefficients to be decided. By varying the coefficients in the stiffness expressions of springs, expressions of constant strain, bilinear or Wilson incompatible finite element are achieved by this element. The accuracy of the SEM is verified by theoretical derivation; and this method is applied to the continuum-based discrete element method (CDEM) for case verification. The features of the four-node rectangular spring element are as follows. Different accuracies can be found in different elements. This element can significantly improve the accuracy of the bending problem of beam. Elements with different accuracies can be achieved by using different coefficients.