Study On Element Stiffness Model Of Euler-bernoulli Cracked Beam

Cracks is one of the common forms of damage in civil engineering structures,and cracks will lead to the reduction of structural stiffness,load-bearing capacity and durability,and affect the normal operation of the structure.Therefore,it is of great theoretical significance and engineering value to establish an accurate crack model and to study the hydrostatic behavior of cracked beams to ensure the safety of beam structures and to avoid the failure of structures and their members.In this paper,the stiffness models of two types of cracks are studied and investigated for open cracked beams and breathing cracked beams.The specific research work is as follows.(1)Based on the local flexibility theory,the analytical solution of the deflection of a planar Euler-Bernoulli open cracked beam deflection is obtained by directly attaching the crack-induced deflection change to the displacement field of the crack-free beam.Based on this displacement field,the shape function of the Euler-Bernoulli open-cracked beam element is obtained.By combining the modified shape function and virtual displacement principle,the element stiffness matrix of the planar Euler-Bernoulli cracked beam are derived.The results of the study show that the modified shape function fits the element displacement field in agreement with the finite element analysis results.The open-cracked element stiffness model established in this paper can reflect the influence of cracks on the inherent frequency of cantilever open-cracked beams,and the calculated results are in good agreement with the literature test results.(2)The variation of contact displacement and stress on the crack surface of cantilever beam with three shapes of ideal crack,U-shaped crack and V-shaped crack and different crack width are analyzed to study the breathing behavior of the cracked beam.The relationship between the three parameters of relative displacement of neutral layer junction,relative turning angle of crack surface and representative value of vertical displacement of cantilever beam and respiration state is explored.Through numerical simulation,the breathing time-varying process of crack opening and closing of cantilever cracks of ideal crack,U-shaped crack and V-shaped crack under load is obtained.The relative displacement of the neutral layer junction,the relative rotation angle of the crack surface and the vertical displacement of the cantilever beam can approximate the breathing state of the ideal crack model,but they cannot directly approximate the breathing state of the U-shaped crack and the V-shaped crack.(3)An improved cosine stiffness model is proposed based on the characteristics of V-shaped cracked beams in different stages of complete opening,partial contact and complete closure.The influence of crack location and crack depth on the nonlinear characteristics of the improved cosine stiffness model is discussed.The research results show that the structural dynamic characteristics of the breathing crack model in this paper are consistent with the existing research results and finite element calculation results.Under the action of single-frequency interference force,the model proposed in this paper can excite obvious super-harmonic and sub-harmonic features,and the deeper the crack,the greater the impact on the stiffness of the improved cosine model,and the more obvious the structural nonlinearity.The closer the crack is to the free end,the smaller the effect on the stiffness of the improved cosine model,and the more obvious the linear characteristic of the structure.