Perturbation Solution Of Large Deformation Problem Of End-Clamped Beams Based On Cylindrical Bending Model

The large elastic deformation problem of flexible materials and the corresponding structures has been the focus of scholars’ attention.Because of its strong geometric nonlinearity,it is difficult to establish governing equations of the problem.In structural engineering,both plates and beams are two basic bending members,and it is found in existed works that when the deformation is small,the internal response of the plates and beams will be dominated by the bending effect,supplemented by the membrane force;but the opposite will be true for the large deformation;when the bending effect disappears completely and the membrane force becomes the dominant internal response,the bending problem of plates and beams will gradually evolve into the tensile problem of the corresponding membranes and cables,respectively.At present,there are a lot of research works on beams and plates,but it seems that they are separate and unrelated.In addition,the combination of the researches of beams and plates is rarely reported.A phenomenon which cannot be ignored is that when a rectangular thin plate is under cylindrical bending,its deformation characteristic is very similar to that of the beam.This fact suggests that we can study the problem of beams from the perspective of the problem of plates;and at the same time,it’s also helpful to clarify the transformation relationship from the problem of plates and beams to the problem of membranes and cables when the bending stiffness gradually disappears.In this thesis,a new method was developed to study the large deformation problem of end-clamped beams by using the cylindrical bending model of thin plates.To this end,first of all,we gave up the small-rotation-angle assumption commonly adopted in the Von Kármán large deflection theory of thin plates,and then derived the equation of equilibrium and consistency equation of large deformation problem of thin plates,respectively.By introducing the stress function to simplify the number of stress components,we finally obtained governing equation expressed in terms of the deflection component and stress function.Based on the cylindrical bending model,we further simplified the governing equation,and employed the perturbation method to obtain the perturbation solution of the large deformation problem of the rectangular thin plates,in which the central deflection was taken as the perturbation parameter.This solution obtained is also a perturbation solution of the large deformation problem of the endclamped beams(which belongs to a plane strain problem in the theory of elasticity).In addition,by ignoring the flexural rigidity,the large deformation perturbation solution of the rectangular membranes was obtained without the small-rotation-angle hypothesis,which is also a perturbation solution of the problem of flexible cables.So far,we have obtained the all solutions of the plates and beams problem and the corresponding membranes and cables problem.Moreover,the influences of each parameter in the analytical expression on the solution as well as the influence of small-rotation-angle hypothesis on the solution were discussed in detail.There are the following important conclusions: first,for the cylindrical bending model,the membrane force will be positively associated with the square of the depth-span ratio and the mid-span deflection.And with the increase of depth-span ratio and mid-span deflection,the deflection curves of the end-clamped beams will tend to be gentle near the supports and steeper near the mid-span;second,when the ratio of the deflection of the plate(or beam)to the plate thickness(or the beam depth)is less than 1.5,the influence of the small-rotation-angle hypothesis on membrane force will be small,even negligible.However,when the ratio is greater than 1.5,the influence of the smallrotation-angle hypothesis will gradually become larger and increase with the increase of mid-span deflection and depth-span ratio.The innovative work of this thesis is mainly reflected in: the governing equations for large elastic deformation problem of rectangular thin plates without small-rotation-angle hypothesis were derived,for the first time.The result shows that,comparing with the classical Von Kármán large deflection equation,the second-order derivative terms of displacement in the equilibrium equation all originate from the curvature of thin plate.Second,the application of the mechanical model,namely,the cylindrical bending model with the combination of the vanished flexural rigidity,enables us to connect exactly the problem of plates and membranes in the two-dimensional state with the problem of beams and cables in the one-dimensional state.The results obtained in this thesis can provide a theoretical support for the analysis and design of flexible materials and structures.