**2. Analytical design of swaging circle fitting**

The swaging circle fitting principle basically relies on mechanical-elastic deformation of the fitting with the external pressure load and also subjecting the inner connection pipe to elastic–plastic deformation.

*How Impact the Design of Aluminum Swaging Circle Fitting on the Sealing for Piping… DOI: http://dx.doi.org/10.5772/intechopen.99938*

#### **Figure 4.**

*Swaging principle with swaging circle fitting (sleeve) [13].*

**Figure 4** shows the principle of elastic–plastic deformation applied to the inner pipe and itself by applying pressure to the swaging circle fitting.

In general, stresses occur in the circumferential and radial directions in a pipe loaded under an externally applied pressure. Its radial stress is equal to the applied pressure on the pressure surface and 0 at the other inner surface. Ratio of pipe inner radius to thickness is in the thin pipe class ri*=*s>5, calculations are made according to the middle radius and very small variation in thickness can be ignored [17]. Based on this acceptance, the centrifugal tension equals half the pressure:

$$\mathbf{o\_r} = \frac{\mathbf{p}}{2}; \mathbf{s} \ll \mathbf{r\_m} \tag{1}$$

Here: σ<sup>r</sup> [MPa]: centrifugal tension,

p [MPa]: pressure applied,

da [mm]: outer diameter of the pipe,

s [mm]: pipe thickness,

rm <sup>¼</sup> da�<sup>s</sup> <sup>2</sup> [mm]: middle radius of the pipe.

**Figure 5** shows the circumferential tension σtan and other parameters occurring under the pressure applied on the pipe. Accordingly, the following equation is written in accordance with the principle of equality of forces in the horizontal direction:

$$\rightarrow \sigma\_{\tan} = \mathbf{p} \frac{\mathbf{r\_m}}{\mathbf{s}} \tag{2}$$

Here: σtan [MPa]: circumferential tension,

p [MPa]: pressure applied,

Dm [mm]: medium diameter of the pipe,

s [mm]: pipe thickness,

rm <sup>¼</sup> Dm <sup>2</sup> <sup>¼</sup> da�<sup>s</sup> <sup>2</sup> [mm]: medium radius of the pipe.

Thus, when the equivalent tension is geometrically collected, it is found as follows:

$$
\sigma\_{\rm E} = \sqrt{\sigma\_{\rm tan}^2 + \sigma\_{\rm r}^2 - \sigma\_{\rm tan} \sigma\_{\rm r}}
$$

$$
\sigma\_{\rm E} = \sqrt{\left(\mathbf{p}\frac{\mathbf{r}\_{\rm m}}{\mathbf{s}}\right)^2 + \left(\frac{\mathbf{p}}{2}\right)^2 - \mathbf{p}\frac{\mathbf{r}\_{\rm m}}{\mathbf{s}}\frac{\mathbf{p}}{2}}
$$

$$
\sigma\_{\rm E} = \mathbf{p}\sqrt{\left(\frac{\mathbf{r}\_{\rm m}}{\mathbf{s}}\right)^2 + \frac{\mathbf{1}}{4} - \frac{\mathbf{r}\_{\rm m}}{2\mathbf{s}}}\tag{3}
$$

**Figure 5.** *Peripheral (tangential) stress caused by the pressure pulse [13].*

Since the pipe is applied with a minimum degree of yield tension, the least required minimum pressure for this is found by the following formula:

$$\mathbf{p}\_{\rm min} = \frac{\mathbf{R}\_{\rm p0,2}}{\sqrt{\left(\frac{\mathbf{r}\_{\rm m}}{\rm s}\right)^2 + \frac{1}{4} - \frac{\mathbf{r}\_{\rm m}}{2\rm s}}} \tag{4}$$

$$\boldsymbol{\sigma}\_{\rm tan}\mathbf{A} = \mathbf{p}\mathbf{D}\_{\rm m}\mathbf{L} \to \boldsymbol{\sigma}\_{\rm tan}\mathbf{2sL} = \mathbf{p}\mathbf{D}\_{\rm m}\mathbf{L}$$

$$\rightarrow \boldsymbol{\sigma}\_{\rm tan} = \frac{\mathbf{p}\mathbf{D}\_{\rm m}\mathbf{L}}{2\rm sL}$$

Here: pmin [MPa]: required minimum pressure, Rp0,2 [MPa]: yield tension.

Example: Swaging circle fitting: R-29,4x2–6061-T6; Yield value: Rp0,2 ¼ 240 MPa.

! rm,a <sup>¼</sup> da�<sup>s</sup> <sup>2</sup> <sup>¼</sup> 29, 4�<sup>2</sup> <sup>2</sup> mm ¼ 13.7 mm; medium radius of the swaging circle fitting.

$$\rightarrow \mathbf{p}\_{\text{min},\mathbf{a}} = \frac{240 \text{ MPa}}{\sqrt{\left(\frac{13,7}{2}\right)^2 + \frac{1}{4} - \frac{13,7}{2 \cdot 2}}} = 33,7 \text{ MPa} = 337 \text{ bar}$$

Inner pipe: R-25,4x0,75–6061-T6;

Yield value: Rp0,2 ¼ 240 MPa*:* ! rm,i <sup>¼</sup> da,i�si <sup>2</sup> <sup>¼</sup> 25, 4�0, 75 <sup>2</sup> mm ¼ 12, 325 mm inner pipe medium radius.

$$p \to p\_{\min, i} = \frac{240 \text{ MPa}}{\sqrt{\left(\frac{12,325}{2}\right)^2 + \frac{1}{4} - \frac{12,325}{2 \star 0,75}}} = 14,8 \text{ MPa} = 148 \text{ bar}$$

Total required minimum pressure: pmin ¼ p min ,a þ p min ,i ¼ 33, 7 MPa þ 14, 8 MPa ¼ 48, 5 MPa.

In order to create not only elastic but also permanent plastic deflection in compressed pipes, it is recommended to apply a minimum pressure of more than 30% to the swaged flat fitting. Example: p ¼ 1, 3pmin ¼ 1, 3∙48, 5 MPa ¼ 63, 0 MPa.

Listed below are the reasons for the design features marked a to d in **Figure 6** that correspond to the requirements of the ring:

a. In principle, as shown in **Figure 4**, the deformation of the tube will be greater in the middle and decrease parabolic towards both sides of the ring.

*How Impact the Design of Aluminum Swaging Circle Fitting on the Sealing for Piping… DOI: http://dx.doi.org/10.5772/intechopen.99938*

**Figure 6.** *Swaging circle fitting design [13].*

Accordingly, the highest stresses in the inner tube will not be in the middle region, but in the region corresponding to the bottom of the fitting edge. Homogenizing the stress requires homogenizing the plastic deformation, requiring a more convenient beat circle connection. For this, a very small slope (<1.5°) is given on the outer surface of the ring.


The design features of the swaged circle fitting are illustrated in 3D in **Figure 6** and described above. The analyzes of these different design features are analyzed numerically using the finite element method (FEM).
