用非局部欧勒-伯诺利束理论对螺旋单壁碳纳米管静态反应的精确分析解决方案

The deflection of a simply supported beam that carries a central point load can be analyzed using structural mechanics principles, particularly by applying Castigliano's theorem. This theorem relates the displacement at the load application point to the partial derivatives of the strain energy in the structure. The simply supported beam with a point load at its center has symmetric reaction forces at the supports, each bearing half of the load. The bending moment at any point along the beam is...

The structural behavior of beams under distributed loads is critical for engineering analysis, which focuses on predicting how beams bend and react under such conditions. Different types of beams (e.g., cantilever, supported, or overhanging) behave differently under distributed load conditions.
For all beams, the analysis of the beam's reaction to distributed loads begins by understanding the relationship between a beam's load and the resulting shear forces and bending moments. Initially, this...

In curved beams, unlike straight beams, the stress distribution across the cross-section is not uniform due to the beam's curvature. This non-uniformity arises because the neutral axis, where stress is zero, does not align with the centroid of the section. In a curved beam, the strain varies along the section as a function of the distance from the neutral axis.
Consider the curved member described in the previous lesson. According to Hooke's law, which relates stress to strain within the...

The moment-area method is an analytical tool used in structural engineering to determine the slope and deflection of beams under various loads. Consider a cantilever with a concentrated load and moment at the free end. The first step is constructing a free-body diagram to calculate the reactions at the fixed end. Next, the bending moment diagram is plotted to visualize how the bending moment varies along the beam's length, focusing on points where the bending moment equals zero.
The M/EI...

A cantilever beam with a rectangular cross-section under distributed and point loads experiences shearing stresses. The analysis begins by identifying the loads acting on the beam. Then, the reactions at the beam's fixed end are calculated using equilibrium equations. The vertical reaction is a combination of the distributed and point loads, while the moment reaction is the sum of their moments. The shear force distribution along the beam, resulting from these loads, is established by creating...

In analyzing a thin-walled hollow shaft subjected to torsional loading, a segment with width dx is isolated for examination. Despite its equilibrium state, this segment faces torsional shearing forces at its ends. These forces are quantitatively described by the product of the longitudinal shearing stress on the segment's minor surface and the area of this surface, leading to the concept of shear flow. This shear flow is consistent throughout the structure, indicating a uniform distribution of...