دانلود رایگان مقاله خم و کمانش گرادیان فشار غیر محلی پرتو الاستیک

Abstract

Featured by the two material length parameters in the nonlocal strain gradient theory, it is still unknown that what are the boundary conditions of nonlocal strain gradient beams, since the equations of motion and boundary conditions of these beam models appear in the same form as those of the classical ones. Based on the weighted residual approaches, this paper provides the boundary value problems of Euler–Bernoulli beams within the framework of the nonlocal strain gradient theory in conjunction with the von Kármán nonlinear geometric relation. The closed-form solutions for bending and buckling loads of nonlocal strain gradient beams are obtained. Numerical results show that the higher-order boundary conditions have no effect on the static bending deflection of beams for the cases studied. However, the higher-order boundary conditions and the material length parameters have a significant effect on the buckling loads. Finally, when the two material length parameters are the same, the buckling loads can not always reduce to the classical solutions, the findings of which violate our expectations. The results provided in this work are expected to be helpful for the applications of this theory to the analysis of engineering structures.

6. Conclusions

Based on the nonlocal strain gradient theory, the governing equations of motion of nonlinear Euler–Bernoulli beams are derived. Then, these equations are used to determine all possible (lower-order and higher-order) boundary conditions by the WRAs. These derivations, on the one hand, allow us to derive the boundary conditions for nonlocal beams, and on the other hand, they enable us to obtain the boundary conditions for the strain gradient beams. The bending deflections of a nonlocal strain gradient beam subjected to the distributed load are analytically obtained. Meanwhile, the boundary value problems of the buckling behaviors of nonlocal strain gradient beams are studied. In conclusion, the main findings of this work are summarized as follows:  Reformulation of the boundary value problems of nonlinear Euler–Bernoulli beams (see Eqs. (20), (21) and (31)) within the framework of the nonlocal strain gradient theory.  The bending deflections of nonlocal strain gradient beams subjected to the distributed load are found to be independent of the choices of higher-order boundary conditions. The stiffening and softening behaviors of beams are observed by comparing the effect of the two material length parameters.  The buckling loads are affected by the choices of the higherorder boundary conditions. When the two material length parameters are the same, the buckling loads of nonlocal strain gradient beams are not always the same; this conclusion is not the same as those reported in the literature. The authors believe that by utilizing the WRAs, the present work can be extended to the studies of dynamic boundary value problems for rods, beams, plates and shells. As a result, further work is needed to provide WRAs as a useful tool for the engineering structures such as functionally graded materials and piezoelectric materials.