The study of the perturbation of normal stresses in the rod of the I-section in the zone of application of the load

The study of the implementation of the Saint-Venant principle for restrained rods of the I-section exposed to various loads at its free end is carried out. When using the program complex LIRA SAPR are defined zones of disturbance of normal stresses. 1 Research In previously published works [1, 2] authors investigated the applicability of the SaintVenant principle for rods of the rectangular cross-section. It was found that the attenuation of stress perturbations, both normal and tangential, occurs approximately at a distance of one transverse dimension. In this work, the task was to investigate the behavior of normal stresses in local areas near the loads applied at the free end of the rigidly clamped rod of the I-section (Figure 1) length of 4.8 m. Finite element mesh 5x5 cm was used in calculations. Fig. 1. Cross-section of the rod In calculation No. 1, a vertical force F = 1000 kN was applied to the free end of the rod (Figure 2). * Corresponding author: Alexglebov@bk.ru © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). MATEC Web of Conferences 251, 04025 (2018) https://doi.org/10.1051/matecconf/201825104025 IPICSE-2018


Research
In previously published works [1,2] authors investigated the applicability of the Saint-Venant principle for rods of the rectangular cross-section. It was found that the attenuation of stress perturbations, both normal and tangential, occurs approximately at a distance of one transverse dimension. In this work, the task was to investigate the behavior of normal stresses in local areas near the loads applied at the free end of the rigidly clamped rod of the I-section ( Figure 1) length of 4.8 m. Finite element mesh 5x5 cm was used in calculations.   disturbance zone was 1.1b for this calculation, where b is the cross-sectional vertical dimension.
In spite of the fact that the I-beam is a single section represented by three rectangular sheets connected together, the disturbance of the stressed state can be considered separately for each element. The stressed state of the web and the bottom shelf of the I-beam (the most remote place from the point of application of the load in the cross section) was considered in this calculation.
Numerical experiments have shown that the attenuation of the stress state disturbance for a web practically coincides with the results obtained for a rod of rectangular cross section [1]. For the bottom flange, we have the results shown in Figure 4.  As can be seen from the diagrams of normal stresses, in the region of the interface between the web and the flange, the diagram does not have a constant character, but forms a local region of stress heterogeneity along the width of the flange, which decays along the entire length of the flange.
In the calculation of No. 2a, the rod was loaded with a centrally applied tensile force F = 1000 kN. Figure 6 shows the normal stress field for this loading case.  Based on numerical experiments to verify the applicability of the Saint-Venant principle, it was found that, for a given type of loading, the damping of the disturbance of normal stresses occurs at a distance of more than one transverse dimension of the rod. Considering the stress state of the elements separately, it can be seen that a similar phenomenon is observed both in the flange and in the web plate of the rod. Based on the results of the previous article [1], for centrally stretched rods of rectangular cross section, the damping of the perturbation of normal stresses occurred at a distance of the order of 0.92b, where b is the transverse dimension of the rod. Figure 8 shows the diagrams of normal stresses for calculating No. 2a in characteristic cross-sections. As can be seen from the diagrams of normal stresses along the sections (Figure 8), the damping of the normal stress disturbance occurs at a distance of 85 cm from the free end, which corresponds, approximately, to 1.2b, where b is the transverse dimension of the rod. However, if you apply a pair tensile load F/2 = 500 kN (calculation No 2b) along the edges of the free end (Figure 9), then the disturbance of the stress state quickly decays at a distance of approximately 0.5b from the load application area.   As can be seen from the diagrams of normal stresses, damping occurs faster than in case No. 2a, at a distance of about 0.5b from the free end.
The perturbation of normal stresses in the web decays at a farther distance than in the flange. Consequently, we can conclude that redistribution of stresses is most effective if the load is applied to the flanges of the I-beam, thus causing a decrease in the zone of disturbance of the stress-strain state along the length of the rod. In the case of a rectangular cross section [1], the difference in the damping zones of a stressed state under the action of a concentrated force or two forces is small (0.95b for lumped and 0.92b for a pair, respectively). This fact is confirmed by the calculation of No.3 (Figure 11), where the rod is influenced by an eccentrically applied force F = 1000 kN at the free end. We also studied the following types of beam loading.
-A pair of horizontal forces applied at close range (10 cm) at the end of the rod.
-Two vertical forces at the free end.
-The vertical force at the free end at the center of gravity of the I-beam Calculations showed that the best convergence with the analytical solution is the last calculation, since, obviously, it corresponds most closely to the analytical representation of beam bending in the rod setting, where the load is applied to the axis of the rod. The attenuation of the perturbation with an allowable error of 5% occurred already at a distance of 25 cm from the force application zone, which corresponds to 0.3b, where b is the crosssectional dimension.

Conclusions
1. The linear distribution of normal stresses on the axis of symmetry in the crosssectional height for the stepped profile, adopted in the strength of materials, is generally valid for places remote from the load application zone.
2. For some types of loading, the damping of the perturbation of normal stresses was performed at a distance of 1.2b (where b is the transverse dimension of the rod).