This study investigated elastic stress distribution in rotating cylinders made of Titanium and Bronze material analytically. Von Mises yield criterion is considered in the plane strain reference interval. Elastic ranges were determined analytically. As a result, the obtained findings were shared with the literature in graphics. At the end of the study, it was determined that the stresses occurring in the titanium material cylinder were higher than the bronze disc, and the pressures were inversely proportional to the increase in the grading parameter in the cylinders.
Research Article
Investigation of Stresses Occurring in Rotating Cylinders
https://doi.org/10.21203/rs.3.rs-2631535/v1
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Elastic stress analysis is an essential concept for engineering. In this regard, it is even more critical to determine the appropriate material belonging to the cylinder referenced. When literature research is carried out, in general, In this study, it is taken as a reference that the material properties change only in the radial direction. The analytical solution is made for a long hollow cylinder with rotating Boron carbide (B4C) material. The results obtained were shared with the literature. As a result, the importance of stresses in selecting materials for cylinders was revealed. It is possible to come across many studies about rotating disks and deleted ones in the literature. Only the automatic selection of materials is essential in the design and model of building materials. Previously, a literature review was conducted; in one study, an Elastic Analysis was performed to Rotate a Functionally Graded Annular Disk, the Exponentially Changing Profile and Properties of which were investigated. At the end of the study, Considering the proposed FGM disc under the boundary conditions FF, CC, CF and FF, it was determined that the radial displacement and circumferential stress along the discs are more significant under the boundary conditions FF. Another radial displacement under the CC boundary condition and the radial stress under the FC boundary condition were found to be small along the discs [1].
In a different study, the displacement and stress distribution of a functionally graded rotating disk subjected to variable angular velocities was mechanically studied [2]. In additional research, the mechanical behaviour of rotating cylindrical pressure vessels with functional grade materials was investigated. The results obtained were determined graphically by comparing them with the literature [3]. In other studies, the elastic stress behaviour of exponential FGM discs exposed to internal and external pressure and the mechanical and thermal stresses formed in the hollow cylinder with functional grade were determined [4–5].
In another study, thermal stress distributions formed in a brake disc/pad subjected to friction force over time were investigated. At the end of the study, it was determined that different friction force profiles, depending on time, cause tensile stresses on the friction surface of the brake disc. In addition, it has been found that tensile stresses occurring in the circumferential direction lead to thermal microcracks [6]. The stresses occurring in the rotating discs were investigated and the results were shared with the literature [7].
The materials of the cylinder to be analyzed were selected from the appropriate material. The mechanical properties of the cylinder materials are given in Table 1 below.
| Modulus of elasticity | Density | Cylinder Inner half diameter | Cylinder Outer semi-diameter |
|---|---|---|---|
| 120 GPa | 4510 kg/m3 | 60 mm | 120 mm |
| 96 Gpa | 8800 kg/m3 | 60 mm | 120 mm |
2.1. Stress analysis in cylinders
Two-dimensional equilibrium equation in cylindrical coordinates [8];
$$\frac{\text{d}\left({{\sigma }}_{\text{r}}\right)}{\text{d}\text{r}}-\frac{1}{\text{r}}\frac{\left({\text{d}{\tau }}_{\text{r}{\theta }}\right)}{\text{d}{\theta }}+\frac{\left({{\sigma }}_{\text{r}}-{{\sigma }}_{{\theta }}\right)}{\text{r}}+\text{R}=0 \left(1\right)$$
$$\frac{{\text{d}{\tau }}_{\text{r}{\theta }}}{\text{d}\text{r}}-\frac{1}{\text{r}}\frac{\left({\text{d}{\tau }}_{{\theta }}\right)}{\text{d}{\theta }}+\frac{\left({2{\tau }}_{\text{r}{\theta }}\right)}{\text{r}}+\text{R}=0 \left(2\right)$$
$$\frac{\text{d}\left({{\sigma }}_{\text{r}}\right)}{\text{d}\text{r}}-\frac{\left({{\sigma }}_{\text{r}}-{{\sigma }}_{{\theta }}\right)}{\text{r}}+{\rho }{\left(\text{r}\right){\omega }}^{2}{\text{r}}^{2}=0 \left(3\right)$$
Stresses in the radial and tangential directions;
$${{\epsilon }}_{\text{r}}=\frac{1}{\text{E}\left(\text{r}\right)}{[{\sigma }}_{\text{r}-\text{V}}{({\sigma }}_{{\theta }}{+{\sigma }}_{\text{Z}}\left)\right] \left(4\right)$$
$${{\epsilon }}_{{\theta }}=\frac{1}{\text{E}\left(\text{r}\right)}{[{\sigma }}_{{\theta }-\text{V}}{({\sigma }}_{\text{r}}{+{\sigma }}_{\text{Z}}\left)\right] \left(5\right)$$
$${{\epsilon }}_{\text{z}}=\frac{1}{\text{E}\left(\text{r}\right)}{[{\sigma }}_{\text{z}-\text{V}}{({\sigma }}_{\text{r}}{+{\sigma }}_{{\theta }}\left)\right] \left(6\right)$$
\({{\epsilon }}_{\text{z}}=0\) , Eq. 7 is obtained as follows.
$${{\sigma }}_{\text{z}=\text{V}}{({\sigma }}_{\text{r}}{+{\sigma }}_{{\theta }}\left) \right(7)$$
If \({{\sigma }}_{\text{z}}\) obtained above is put in their place, then;
$${{\epsilon }}_{\text{r}}=\frac{1+\text{v}}{\text{E}\left(\text{r}\right)}\left[\right(1-\text{v}\left){({\sigma }}_{\text{r}}{-\text{v}{\sigma }}_{{\theta }}\right)\left] \right(8)$$
$${{\epsilon }}_{{\theta }}=\frac{1+\text{v}}{\text{E}\left(\text{r}\right)}\left[\right(1-\text{v}\left){({\sigma }}_{{\theta }}{-\text{v}{\sigma }}_{\text{r}}\right)\left] \right(9)$$
For the connection between stress strain
$${{\epsilon }}_{\text{r}}=\frac{\text{d}\text{u}}{\text{d}\text{r} } \left(10\right)$$
$${{\epsilon }}_{\text{r}}=\frac{\text{u}}{\text{r} } \left(11\right)$$
For the stress analysis equation in rotating cylinders;
\({\text{r}}^{2}\frac{{\text{d}}^{2}F}{\text{d}{\text{r}}^{2}}+\text{r}\left[1-\text{r}\frac{{\text{E}}^{{\prime }}\left(r\right)}{\text{E}\left(\text{r}\right)}\frac{\text{d}F}{\text{d}\text{r}}\right]+\left[\text{v}\left(\text{r}\right)\frac{{\text{E}}^{{\prime }}\left(r\right)}{\text{E}\left(\text{r}\right)}-1\right]\text{F}={\rho }{\left(\text{r}\right){\omega }}^{2}{\text{r}}^{3}[\text{r} \frac{{\text{E}}^{{\prime }}\left(r\right)}{\text{E}\left(\text{r}\right)}-\frac{{{\rho }}^{{\prime }}\left(\text{r}\right)}{{\rho }\left(\text{r}\right)}-3--\frac{\text{v}}{1-\text{v} }\) (12)]
In Eq. 12, the elastistie modulus and density vary according to the functions.
$$\text{E}\left(\text{r}\right)={\text{E}}_{0}\frac{{\text{r}}^{\text{n}}}{{r}_{0}^{n}} \left(13\right)$$
$${\rho }\left(\text{r}\right)={{\rho }}_{0}\frac{{\text{r}}^{{\gamma }}}{{r}_{0}^{{\gamma }}} \left(14\right)$$
if r = et transformation is performed, \({\text{E}}_{0}\),\(\)modulus of elasticity, ,\({{\rho }}_{0}\) density reference value, n and γ are optional constants.
$${\text{r}}^{2}\frac{{\text{d}}^{2}F}{\text{d}{\text{t}}^{2}}-\text{n}\frac{\text{d}F}{\text{d}\text{t}}+\left(\text{n}\text{v}-1\right)\text{F}={[r}_{0}{{\omega }}^{2}\left[n-{\gamma }-3-\frac{\text{v}}{1-\text{v}}\right]/\left({r}_{0}^{{\gamma }}\right){]e}^{\left(3+{\gamma }\right)\text{t}}$$
15
For the stress function;
$$\text{F}={\text{C}}_{1}{\text{r}}^{(\text{n}+\text{k})/2}+{\text{C}}_{2}{\text{r}}^{(\text{n}-\text{k})/2}+\text{A}{\text{r}}^{(3+{\gamma } )} \left(16\right)$$
Here, for the;
$$\text{A}=-\frac{{{\rho }}_{0}{{\omega }}^{2}}{{r}_{0}^{{\gamma }}\left[\right(3+{{\gamma })}^{2}-n\left(3+{\gamma }\right)+\text{v}\text{n}-1]}\left[n- {\gamma }-3-\frac{1}{1-v}\right] \left(17\right)$$
For radial and tangential stress values;
$${{\sigma }}_{\text{r}}={\text{C}}_{1}{\text{r}}^{(\text{n}+\text{k}-2)/2}+{\text{C}}_{2}{\text{r}}^{(\text{n}-\text{k}-2)/2}+\text{A}{\text{r}}^{(2+{\gamma })} \left(18\right)$$
$${{\sigma }}_{{\theta } }\left(\text{M}\text{P}\text{a}\right)=\frac{\text{n}+\text{k}}{2}{\text{C}}_{1}{\text{r}}^{(\text{n}+\text{k}-2)/2}+\frac{\text{n}-\text{k}}{2}{\text{C}}_{2}{\text{r}}^{(\text{n}-\text{k}-2)/2}+\left(3+{\gamma }\right)\text{A}{\text{r}}^{(2+{\gamma })}+{\rho }\left(\text{r}\right){{\omega }}^{2}{\text{r}}^{2}$$
19
C1 and C2 are integral constants. For boundary conditions;
C1 = A[-\({r}_{i}^{({\gamma }+2)}{r}_{0}^{(\text{n}-\text{k}-2)/2}+{r}_{i}^{(\text{n}-\text{k}-2)/2}]{r}_{0}^{({\gamma }+2)}\) (20)
$${\text{C}}_{1}=A\left[\frac{{r}_{i}^{\left({\gamma }+2\right)}{r}_{0}^{\frac{\text{n}-\text{k}-2}{2}}+{r}_{0}^{\left({\gamma }+2\right)}{r}_{i}^{\frac{\text{n}-\text{k}-2}{2}}}{{r}_{i}^{\frac{\text{n}+\text{k}-2}{2}}{r}_{0}^{\frac{\text{n}-\text{k}-2}{2}}-{r}_{0}^{\frac{\text{n}+\text{k}-2}{2}}{r}_{i}^{\frac{\text{n}-\text{k}-2}{2}}}\right] \left(21\right)$$
$${\text{C}}_{2}=A\left[\frac{-{r}_{0}^{\left({\gamma }+2\right)}{r}_{İ}^{\frac{\text{n}+\text{k}-2}{2}}+{r}_{İ}^{\left({\gamma }+2\right)}{r}_{0}^{\frac{\text{n}+\text{k}-2}{2}}}{{r}_{i}^{\frac{\text{n}+\text{k}-2}{2}}{r}_{0}^{\frac{\text{n}-\text{k}-2}{2}}-{r}_{0}^{\frac{\text{n}+\text{k}-2}{2}}{r}_{i}^{\frac{\text{n}-\text{k}-2}{2}}}\right] \left(22\right)$$
The inner half diameter of the modelled cylinder is 60 mm, and the outer half diameter is 120 mm. One of the cylinders was selected as Titanium material. The material of another cylinder was deemed appropriate to be selected as Bronze. At the end of the literature review, it was understood that titanium and bronze material cylinders can be used today. Radial, tangential, axial and equivalent stress distributions have been formed in cylinders consisting of two hollow cylinders exposed to angular velocity during rotation. In Fig. 1 and Fig. 2 below, it is determined that the stress distributions are zero in the inner and outer parts of the cylinder. It has been determined that the stress is inversely proportional to the increase in the rating parameter values. The maximum radial stress value was determined to be in the middle region closer to the inner part of the cylinder. It has been observed that radial stresses occur in the form of tensile stress. Tangential stresses show a decreasing distribution from the inner to the outer part of the cylinder, as can be seen in Fig. 3 and Fig. 4 below. It has been determined that the most regular distribution occurs at the enormous value of the rating parameter. It is determined that the most significant tangential stress occurs for the smallest value of the rating parameter on the inner surface of the cylinder.
Radial, tangential, and axial stresses formed in a cylinder consisting of hollow Titanium and Bronze materials exposed to an angular velocity of 50 rad/sec are shown in Fig. 1 and Fig. 6 above. Radial stresses are minimal in the inner part of the cylinder and increase towards the outer part. The tangential stresses are determined as maximum in the inner part of the cylinder and minimum in the outer parts.
In Table 2, stress values of bronze and titanium material cylinders exposed to different angular velocities are given.
Table 2. Stresses occurring in the cylinder
In Fig. 1 below, radial stresses occurring in a titanium material cylinder rotating at an angular velocity of 50 rad/sec are given.
In Fig. 2 below, radial stresses occurring in a bronze material cylinder rotating at an angular velocity of 50 rad/sec are given.
In Fig. 3 below, radial stresses occurring in a titanium material cylinder rotating at an angular velocity of 50 rad/sec are given.
In Fig. 4 below, the tangential stresses occurring in a titanium material cylinder rotating at an angular velocity of 50 rad/sec are given.
In Fig. 5 below, axial stresses occurring in a titanium material cylinder rotating at an angular velocity of 50 rad/sec are given.
In Fig. 6 below, axial stresses occurring in a bronze material cylinder rotating at an angular velocity of 50rad/sec are given.
This study investigated the stresses occurring in rotating hollow cylinders with Titanium and Bronze materials. As a result of research, Radial stresses were determined as zero in the innermost and outermost parts of the cylinders.
It has been determined that the stresses are inversely proportional to the increase of the rating parameter in the cylinders.
The tangential stresses occurring in the interior of the titanium cylinder are approximately 4.14% more than the stresses occurring in the Bronze cylinder when n=-1 and w = 50 rad/sec;
The tangential stresses occurring on the outside of the titanium cylinder are approximately 4.39% more than the stresses occurring in the Bronze cylinder when n=-1 and w = 50 rad/sec;
The tangential stresses occurring in the interior of the titanium cylinder are approximately 5.17% more than the stresses occurring in the Bronze cylinder when n=-0.5 and w = 50 rad/sec;
The tangential stresses that occur on the outside of the titanium cylinder are approximately 6.38% more than the stresses that occur in the Bronze cylinder when n=-0.5 and w = 50 rad/sec;
The tangential stresses occurring on the outside of the titanium cylinder are approximately 4.28% more than the stresses occurring in the Bronze cylinder when n=-0.5 and w = 50 rad/sec;
The tangential stresses occurring in the interior of the titanium cylinder are approximately 4.76% more than the stresses occurring in the Bronze cylinder when n=-1 and w = 50 rad/sec;
The tangential stresses occurring on the outside of the titanium cylinder are approximately 3.35% more than the stresses occurring in the Bronze cylinder when n=-1 and w = 50 rad/sec;
The tangential stresses occurring in the interior of the titanium cylinder are approximately 4.74% more than the stresses occurring in the Bronze cylinder when n=-0.5 and w = 50 rad/sec;
It has been determined that the maximum radial stress value occurs in the middle regions closer to the inner part of the cylinders. This stress is more significant in the Titanium cylinder than in the Bronze material.
Conflict of interest The authors declare that there are no conficts of interest and funding regarding the publication of this article.
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