| Elastic beams are widely used in many fields,such as bridge engineering,automobile manufacturing,marine and wharf,civil engineering and aircraft manufacturing.The design of strength and stiffness of elastic beams is an important issue,because it is directly related to the application safety of beams.For the elastic beam,the strength calculation and stiffness analysis are related to the calculation of stress and deformation respectively,and the stress and deformation are interrelated and corresponding to each other.Therefore,based on the literature,this paper takes the deformation of the elastic beam as the main analysis object to study and explore a more effective calculation method in theory.The classical calculation method of elastic beam deformation(deflection)is the differential equation method of section bending moment and second-order deflection.In addition,there are three bending moment method,force method,displacement method and finite element method based on the application of energy method.These methods are widely used in the analysis of deformation and stress calculation of statically indeterminate beam and statically indeterminate beam,The displacement method and the finite element method are numerical methods;The three moment method,force method and energy method all involve the analysis of moment and the deduction of complex energy expression,which is inconvenient for the occasion of bearing various loads;In addition,these methods are only applicable to the static analysis of the beam,and the dynamic deformation calculation of the beam needs further consideration.For this reason,this paper puts forward two analysis methods,one is algebraic polynomial method,the other is algebraic polynomial method;One is infinite series method.The algebraic polynomial method makes use of the relationship between the fourth-order deflection differential and the set degree of the distributed load.On the basis of expressing the concentrated load(force and couple)as the formally distributed load by using the generalized function,and with the aid of the superposition principle,the deflection expression of the beam with undetermined coefficient can be assumed in advance,Then,the exact linear algebraic equations with unknown coefficients are obtained by using the definite solution conditions of the problem.Finally,the solution of the problem is obtained;The infinite series method is to find out the suitable orthogonal function system according to the constraint characteristics of the beam end.The deflection of the beam is assumed to be the expansion of the orthogonal function system,and the undetermined coefficients contained in it can be determined by the intermediate constraint conditions in the problem,For statically indeterminate beams,the infinite algebraic equations with undetermined coefficients can be obtained by means of the principle of minimum potential energy.The common feature of the two methods is that they completely avoid the complicated process of determining bending moment by section method and the complicated process of calculating deflection by many piecewise integrals;In addition,the infinite series method is also a very effective method for the calculation of the steady-state dynamic deformation of beams.The results of the example given in this paper are compared with those of the traditional method to verify the feasibility of this method. |
