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Introduction
Published in Ansel C. Ugural, Youngjin Chung, Errol A. Ugural, Mechanical Engineering Design, 2020
Ansel C. Ugural, Youngjin Chung, Errol A. Ugural
Note that solutions based on the mechanics of materials give average stresses at a cross-section. Since, at concentrated forces and abrupt changes in a cross-section, irregular local stresses (and strains) arise, only at a distance about equal to the depth of the member from such disturbances are the stresses in agreement with the mechanics of materials. This is due to Saint-Venant’s Principle: the stress of a member at points away from points of load application may be obtained on the basis of a statically equivalent loading system; that is, the manner of a force’s application on stresses is significant only in the vicinity of the region where the force is applied. This is also valid for the disturbances caused by the changes in the cross-section. The mechanics of materials approach is therefore best suited for relatively slender members.
Introduction
Published in Ansel C. Ugural, Youngjin Chung, Errol A. Ugural, MECHANICAL DESIGN of Machine Components, 2018
Ansel C. Ugural, Youngjin Chung, Errol A. Ugural
Note that solutions based on the mechanics of materials give average stresses at a section. Since, at concentrated forces and abrupt changes in cross section, irregular local stresses (and strains) arise, only at distance about equal to the depth of the member from such disturbances are the stresses in agreement with the mechanics of materials. This is due to Saint-Venant’s Principle: the stress of a member at points away from points of load application may be obtained on the basis of a statically equivalent loading system; that is, the manner of force application on stresses is significant only in the vicinity of the region where the force is applied. This is also valid for the disturbances caused by the changes in the cross section. The mechanics of materials approach is therefore best suited for relatively slender members.
Viscoelastic stress and deformation analysis
Published in Roderic S. Lakes, Viscoelastic Solids, 2017
Saint Venant’s principle states that a localized self-equilibrated load system produces stresses which decay with distance more rapidly than stresses due to forces and moments. It is applicable in many situations of interest in engineering. There are some counterexamples (see, e.g., Fung [5.2.3]). Consider a sandwich panel with rigid face sheets and an elastic material of Poisson’s ratio ν sandwiched between them. For Poisson’s ratios in the vicinity of 0.5, stresses applied to the end will decay [5.6.3] with distance z as σ(z) ∝ e–yz. The decay rate is
Exact static axisymmetric solutions of thick functionally graded cylindrical shells with general boundary conditions
Published in Mechanics of Advanced Materials and Structures, 2022
Wenfeng Hu, Tao Xu, Jinsheng Feng, Lei Shi, Jun Zhu, Jianyou Feng
Figure 7 shows the displacements and stresses of the FGM cylindrical single shell ( ) along the axial direction under various boundary conditions. It clearly reflects that the influence of boundary conditions and span-to-radius ratios on the static responses of the FGM cylindrical shells is significant. As Saint-Venant principle points out, when the span-to-radius ratio equals 5, the end effect only influences in the end region, and the displacement w and stresses gradually tend to be a horizontal line at the mid-span area. As known, the values of the horizontal lines for CC conditions are the results of the plane strain problem in the rθ plane. Meanwhile, because the free end and the simply supported end lack the constraint along the axial direction x, the values of the horizontal lines for CS and CF conditions are the results of the plane stress problem, hence the lines are almost coincident. When the span-to-radius ratio equals 1, the influence of the end effect is obvious in the entire length of the short shell. Therefore, the present solution can accurately describe the end effects which are ignored by Saint-Venant principle.
Identify the distribution of 2D residual stresses around notches based on the Willis-form equations
Published in Inverse Problems in Science and Engineering, 2021
Zhuyou Hu, Jianing Xie, Jinlong Zhao, Yixiao Sun, Zhihai Xiang
According to the Saint Venant’s principle, the influence of loads is limited to very local regions. Therefore, the deformation due to cutting the specimen can be reasonably calculated by certain analytical equations, in which the loads are taken as the released RS to be measured. Since the elastic solution is unique, the deformation is uniquely determined by the equivalent Remote Virtual Loads (RVL) added on the configuration after cutting. This means the released RS can be regarded as the RVL applied to the cut region. This equivalence is the theoretical basis of many RS identification methods, e.g. the widely used hole-drilling method [22] and the crack compliance method [23]. Although the RS near the cut are re-distributed after cutting, the change of remote RS can be neglected according to the Saint Venant’s principle.