Study On Variational Asymptotic Equivalent Model And Multi-scale Modeling Of Corrugated Structures

The corrugated structure shows excellent structural performance because of its special geometric structure,so it is widely used in wall panel,retaining wall,tunnel and bridge,et al.There are many achievements in the theory of corrugated plate,but additional assumptions make the displacement field incompatible with the stress field.It is logically reasonable to use mature finite element software to analyze such structure by meshing all the details of the microstructure,,but this is not a practical and effective method,because it requires too many degrees of freedom（i.e.computational cost）to capture the micro behavior.Therefore,this paper makes full use of the characteristics that the size of a single corrugation is much smaller than the geometric size of the whole corrugated structure,establishes the equivalent model of the corrugated structure by using the variational asymptotic method,and deduces the expression of the equivalent stiffness and the recovery relationship of local fields.The main research contents and conclusions are as follows:（1）The corrugated structure specimen is printed by 3D printing technology,and the three-point bending test is carried out by using electronic universal testing machine and digital image correlation（DIC）related equipment.The test results show that the static deformation results of uni-periodic trapezoidal corrugated plate（TCP）,bi-periodic TCP and corrugated web beam predicted by the equivalent model were consistent with the corresponding three-dimensional model and DIC results.The free vibration and buckling analysis results of corrugated structures predicted by the equivalent model and the corresponding three-dimensional model were also highly consistent.This verified the accuracy of the equivalent model for corrugated structure.The calculation efficiency of the equivalent model and three-dimensional model was compared.The results show that the calculation time of equivalent model was significantly shorter than that of three-dimensional model,which due to the significant reduction of the number of elements and nodes.On the premise of ensuring sufficient engineering accuracy,the equivalent model can greatly improve the calculation efficiency.In addition,the equivalent model has more advantages than the three-dimensional model in the definition of contact,boundary and load.（2）The effects of geometric parameters（height,wall thickness and angle）of uni-periodic TCP on the equivalent stiffness are studied.The results show that in order to maximize A₁₁or D₁₁,uni-periodic TCP with high height,small angle and large wall thickness should be selected;for A₁₃,A₃₃ or D₁₃,the uni-periodic TCP with small height,large angle and large wall thickness should be selected.The effects of geometric parameters（height,wall thickness and angle）of bi-periodic TCP on the equivalent stiffness are studied.The results show that the equivalent stiffness decreases with the increase of angle,especially when the angle was less than 120°,the equivalent stiffness decreases sharply with the increase of angle.When the angle was greater than 120°,the decrease rate of equivalent stiffness with the increase of angle gradually tends to zero,that is,the equivalent stiffness of bi-periodic TCP was no longer sensitive to the change of angle.The equivalent stiffness terms of all bi-periodic TCP increase with the increase of thickness,among which D₁₁ and D₃₃ increase the most with the increase of thickness.The equivalent stiffness of bi-periodic TCP with single concave convex was less than that with double concave convex.The stiffness components B₂₂,D₁₁ and D₂₂ of bi-periodic TCP with single concave convex increase with the increase of angle,especially D₂₂increases most obviously with the increase of angle.（3）The applicability of one-dimensional equivalent model to geometric nonlinear analysis is investigated.The results show that when considering geometric nonlinearity,the central axis deflection of cantilever beam with corrugated web under free-end load was less than the corresponding linear analysis deflection.With the increase of the load,the difference increases gradually,and the closer it is to the free-end of the cantilever beam,the greater the difference.The influence of different height-to-span ratio H/L on the accuracy of the one-dimensional equivalent model of cantilever beam with corrugated web is studied.The results show that the central axis deflection of the one-dimensional equivalent model was consistent with the three-dimensional model.The natural frequencies of cantilever beam with corrugated webs with different height-to-span ratios calculated by the one-dimensional equivalent model were consistent with the corresponding three-dimensional model.The fundamental frequency of the cantilever beam with corrugated web under preload was less than that without preload,and the difference increased with the decrease of height-to-span ratio（H/L）.The influence of aspect ratio（H/W）of cantilever beam with corrugated web on the equivalent stiffness is studied.The results show that the equivalent stiffness of cantilever beam with corrugated web decreased with the increase of H/W,and C₄₄decreased the most with the increase of H/W than C₁₁,C₂₂ and C₃₃,that is,C₄₄ was the most sensitive to H/W.The natural frequency of cantilever beam with corrugated web decreased with the increase of H/W,but the decrease range was small.