| Beam structures are widely used in bridges,domes,space structures,marine platforms,aerospace and other fields because of their high structural load-bearing efficiency.Since the nonlinearity of the deformation of most materials in practical engineering problems,more complex and accurate mechanical models are needed to provide theoretical instructions in the future.However,the real performance curves of materials cannot be obtained by ignoring the physical nonlinearity to simplify the calculation.Considering the nonlinear deformation equations of materials,analytical solutions are difficult to be obtained.It is of great scientific significance and practical value to solve such nonlinear problems by numerical methods.The traditional numerical methods have difficulties in solving such problems because of complicated calculation process,low accuracy and technology blockage.The wavelet numerical method proposed in the previous work of our group shows good calculation accuracy and efficiency due to its excellent smoothness,orthogonality and compact support of the basis functions,and has great potential in solving nonlinear problems.The wavelet integral collocation method,with high accuracy and efficiency for solving nonlinear problems,is applied to solve the large deflection bending deformation problem of Ramberg-Osgood intrinsic beam.The feasibility of the algorithm is verified by numerical examples.The main work and conclusions of the paper are as follows:The Ramberg-Osgood constitutive relation(hereafter referred to as the R-O constitutive relation)cannot be directly applied to the moment-stress equation because of its special form of stress-strain relationship.In order to study the large deflection bending deformation behavior of the beams with R-O constitutive relation.Firstly,we investigate the inverse of the R-O constitutive relation,and explore the calculation method of the nonlinear coefficient n and its influence on the variation of the stress-strain relationship in the R-O constitutive relation.It is found that the curves corresponding to different nonlinear coefficients n all pass through the same point(stress σ=400MPa,strain ε=0.00394174).When n increases gradually from 1,the stress-strain relationship in the R-O constitutive relation gradually changes from a straight line to a curve with similar logarithmic form.When n tends ∞,the stress-strain relationship degenerates to an ideal elastic-plastic model.By deriving the moment-curvature relations for four types of sections,namely,double-flange section,rectangular section,thin-walled tube,and solid circle under the R-O constitutive relation,the approximation equations to all types of sections are presented.It can be found that the nonlinear moment-curvature equations for beams with R-O constitutive relation are approximated to the R-O constitutive relation.And the equilibrium equation that satisfies the constitutive relation was derived.In this paper,a set of strongly nonlinear differential equations with boundary conditions is developed to describe the large-deflection bending of beams with R-O constitutive relation,which is solved numerically by applying the wavelet integral collocation method.The main idea is to discretize the independent variable,namely the second derivative of the rotation angle by the wavelet approximation formula and the multiple integration of the scaling function,and to obtain the discretization curvature and rotation angle based on the governing equation and boundary conditions.Consequently,these variables are substituted into the governing equation to complete the discretization of the equation,realizing the discretization of the equation by the wavelet integral collocation method.Finally,numerical examples were used to demonstrate advantages of the wavelet integral collocation method,verifying its accuracy and computational efficiency.For rectangular section beams and thin-walled tube section beams,we compute the numerical results by finite element method with the number of 16625 and 8896 elements respectively as exact solutions.It can be found that for two types of cross section beams,the wavelet integral collocation method only uses 64 nodes to obtain more accurate results than the Runge-Kutta method under a total step of 256.The wavelet numerical solution under 64 nodes is very close to the finite element solution under the selected large number of grids,which undoubtedly reflects the computational accuracy and efficiency of the wavelet integral collocation method.In addition,it can be found that the wavelet integral collocation method can also avoid the ill conditioned problem of equations caused by large differences between the in-plane and out of plane scales of elements when calculating thin-walled tube beams problems. |
