(2.5.2-1)

没有考虑非协调模式的单元内任意的坐标x、y和平动位移u、v使用形函数表示如下：

                                                                                                                                   (2.5.2-2)

三节点单元的形函数

                                                                                                       (2.5.2-3)

四边形单元的形函数

                                                                                 (2.5.2-4)

节点位移u与应变ε使用几何转换矩阵Bi表示如下：

                                                                                                                                                                   (2.5.2-5)

转换矩阵Bi可以使用形函数的微分表示如下：

                                                                                                                                             (2.5.2-6)

与面内变形相关的单元刚度矩阵如下：

                                                                                                                                           (2.5.2-7)

式中：

  t 厚度；

Ae 面积。

各向同性材料的应力与应变的本构矩D如下：

                                                                                                                                          (2.5.2-8)

线性分析时，四节点单元使用非协调模式，除了节点位移以外会有下列其它自由度的位移：

                                                                                                                                                (2.5.2-9)

考虑非协调模式的节点坐标x、y和平动位移u、v如下：

                                                                                                                                        (2.5.2-10)

非协调模式的形函数如下：

                                                                                                                                                                          (2.5.2-11)

应变ε使用节点位移和非协调模式表示如下：

                                                                                                                                                             (2.5.2-12)

与非协调模式相关的转换矩阵Ba如下：

                                                                                                                               (2.5.2-13)

使用转换矩阵Bi和Ba计算面内刚度矩阵可得下面四个刚度矩阵：

                                                                                                                                                           (2.5.2-14)

四个刚度矩阵的关系如下：

                                                                                                                               (2.5.2-15)

使用静力凝聚(static condensation)方法消除非协调模式的刚度：

                                                                                                                           (2.5.2-16)

在计算与非协调模式相关的转换矩阵Ba时，微分中使用自然坐标系(natural coordinate)相对于原点(ξ=η=0)的雅可比转换。非协调模式可以模拟如图2.5.3所示的面内弯曲变形，所以可提高单元性能。

图2.5.2  非协调模式的形状(弯曲)