基于修正形函数的Euler-Bernoulli开口裂纹梁单元刚度矩阵

Element stiffness matrix of Euler-Bernoulli open crack beam based on modified shape function

带裂纹参数的单元刚度矩阵是裂纹构件动力计算及裂纹损伤识别的基础。现有研究主要通过裂纹梁截面变化表征裂纹对单元刚度的影响,而裂纹主要影响单元的应力应变分布。针对两结点四自由度Euler-Bernoulli开口裂纹梁单元,在三次形函数基础上,采用阶跃函数考虑裂纹的影响,叠加线性函数对三次形函数进行修正,提出含裂纹参数的新形函数,再结合虚位移原理得到Euler-Bernoulli裂纹梁单元刚度矩阵。仿真算例表明:裂纹深度比小于0.5时,用形函数计算的挠度值与有限元结果比较,相对误差最大为1.714%,用裂纹梁单元刚度矩阵计算的一阶固有频率误差最大为0.936%。新的形函数能准确描述裂纹单元应力应变分布,裂纹梁单元刚度矩阵能用于结构静动力分析,为考虑裂纹对单元刚度的影响提供了新的研究思路。

Element stiffness matrix with crack parameters is the basis of dynamic calculation and crack damage identification of cracked components.The existing studies characterize effects of crack on element stiffness mainly through variation of crack beam cross-section,while crack mainly affects element stress-strain distribution.Here,aiming at Euler-Bernoulli open crack beam element with 2 nodes and 4-DOF,based on the cubic shape function,the step function was used to consider effects of crack,linear functionswere superimposed to modify the cubic shape function,and a new shape function with crack parameters was proposed.Then,combined with the principle of virtual displacement,the element stiffness matrix of Euler-Bernoulli cracked beam was obtained.Simulation examples showed that when the crackdepth ratio is less than 0.5,compared with the finite element results,the maximum relative error of structure deflection calculated with the proposed shape function is 1.714%,the maximum error of the first-order natural frequency of structure calculated with the obtained element stiffness matrix of cracked beam here is 0.936%;the proposed new shape function can accurately describe stress-strain distribution of crack element,the obtained crack beam element stiffness matrix can be used for structural static and dynamic analysis,and the study results can provide a new study idea for considering effects of crack on element stiffness.