基于第二类四边形面积坐标的弹性力学纯弯解及MacNeal局限定理的破解

为了提高有限元的性能,弹性力学的解析解(齐次方程的通解)常常可用作有限元的试探函数。然而单元自由度数与完备的直角坐标解析解个数并不匹配,不完备的试函数会导致单元有方向依赖性。利用新型局部自然坐标——第二类四边形面积坐标QACM-II(S,T),给出了平面问题对应任意方向纯弯曲状态的应力函数解析解,即S3和T3的线性组合,并推导出了这两组应力函数对应的应力、应变和位移解析解。之后,利用QACM-II表示的解析解构造了非对称的平面4节点8自由度单元USQ4,该单元可以同时通过常应力/应变分片检验和纯弯测试,从而破解了MacNeal局限定理对平面低阶单元的限制。

Analytical solutions in terms of the quadrilateral area coordinates for pure bending state and the finite element model overcoming the limitation of MacNeal's theorem

In order to improve the performance of finite element models,analytical solutions of problems in theory of elasticity (the general solutions of the homogeneous equations )were often used as the element trial functions.However,the number of the element DOFs usually does not match the number of the complete analytical solutions,and such incomplete trial functions may lead to direction dependence of the finite elements.In this paper,a new local natural coordinate method,i.e.,the second form of the quadrilateral area coordinate method QACM-II (S,T),was employed to formulate the analytical solutions (the linear combination of S3 and T3 )of the Airy stress function for pure bending state along arbitrary directions.And corresponding analytical solutions for stresses,strains or displacements were also derived out.Then,by utilizing above solutions,a new unsymmetric 4-node,8-DOF plane quadrilate-ral element,denoted by USQ4,was successfully created.The new element can pass both the constant strain/stress patch test and the pure bending test,which means that the limitation defined by the MacNeal’s theorem is overcome.