PSI - Issue 5

Hyung-Kyu Kim et al. / Procedia Structural Integrity 5 (2017) 63–68 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

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The result is provided in Fig. 2. It is found that the influence of the thickness tolerance on the uncertainty is much greater than that of the outer diameter. Further, the thickness uncertainty is the most critical factor among the parameters in Eq. (4) when the observation in section 3.1 is also included. The decrease of p cr by even more than 25% is shown in Fig. 2 although it is an example case of the above mentioned dimensions. 3.3. Uncertainties of the ovality It is readily evaluated from Eq. (6) that p y = σ ys / m and p cr when δ 1 = 0 and thus n = 0 (no ovality). More specifically, if we denote p y1 and p y2 for the larger and smaller values of p y , respectively, then p y 1 = p cr and p y 2 = σ ys / m if p cr ≥ σ ys / m ; to the contrary p y 1 = σ ys / m and p y 2 = p cr if p cr < σ ys / m . This implies that the smaller value of p y (= p y 2 ) always gives the critical buckling pressure of an oval tube regardless of t / r because it is certain that p cr ≤ σ ys / m within the elastic regime. This is shown in Fig. 3(a) as an example case of the Type 304 stainless steel tube of 0.02 ≤ t / r ≤ 0.2 . The mechanical properties are obtained from a web database (2017) such that E = 193 GPa, ν = 0.29, σ ys = 215 MPa. It is found p y = σ ys / m when t / r ≥ 0.066 and n = 0 . Except this, p y = p cr always for this example case. It is noted that p y = p cr if n ≠ 0 regardless of t / r . In turn, the influence of the ovality on the critical buckling pressure is examined. n = 0.01, 0.05 and 0.1 are considered for the ovality. Fig. 3(b) provides the result. It is surprising that the decrease of p y is tremendous as the ovality increases. This phenomenon becomes severer as the thickness reduces. In other words, the dropping of p y becomes smoother as the thickness increases. In this example, if 1% ovality is accommodated, the critical buckling pressure is reduced by 13-73% for 0.02 ≤ t / r ≤ 0.2 compared with its non-ovality case. If 10% ovality is allowed for t / r = 0.2, almost 95% of p cr of the non-ovality case is lost. Therefore, the ovality appears to be the most critical among the uncertainty parameters.

(a)

(b)

Fig. 3. Ovality effect on the critical buckling pressure: (a) comparison of p y ( r / t ) and σ ys , (b) reduction of the critical buckling pressure .

4. Application to thickness design Because a tube subject to an external pressure should not collapse during operation, the thickness needs to be determined to withstand it. To this end, the investigation result of section 3 can provide a guideline of the thickness design. On the uncertainties of the mechanical properties and dimension parameters, the present method used for Fig. 2 can give a safety factor of preventing the buckling failure. Then, the method for the ovality effect can be added to determine the final value of the thickness. In this circumstance, it is the designer ’ s responsibility to determine the necessary safety factor as well as the allowable ovality. If the safety factor and the ovality need to be increased, the thickness shoud be increased accordingly. For instance, Fig. 4 gives a reference of the tube thickness whose nominal outer diameter is 9.5 mm when the safety factor incorporating the uncertainties of the mechanical properties and dimension is set as 2.0, and the ovality of 1% is allowed. The range of E /(1 – ν 2 ) of the tube material is set as 100-400 GPa, and that of σ ys is 100-500 MPa in Fig. 4, which may cover most of the commercially available tube materials. If the nominal outer diameter, safety factor