PSI - Issue 5

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

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There is a well-known formula to evaluate the critical buckling pressure. It is composed of the mechanical properties of Young ’ s modulus and Poisson ’ s ratio, and the dimensions of the thickness and radius (or diameter). This implies that the critical buckling pressure calculated from the formula has a tolerance owing to the uncertainties of the above mentioned parameters. The uncertainties are associated with the measurement of the mechanical properties and the dimension tolerances. If the calculation result overestimates the critical buckling pressure owing to the uncertainties, the tube will be failed during service and thus a safety will be lost. Besides, another uncertainty of the critical buckling pressure is attributed to an irregular shape of the tube cross section. The tube cannot have a perfectly circular cross section owing to a shape tolerance in manufacturing. If the tube has an oval shape in its cross section, the critical buckling pressure will be reduced. This will also result in a premature buckling failure of a tube subject to an external pressure. Therefore, it is important to know the influence of the uncertainties on the critical buckling pressure to achieve a safe design of the tube, which is the purpose of the present work. To this end, the relevant formulae were revisited and possible variation of the parameters was investigated with consulting the previous works and standards of the tube fabrication. Then, the uncertainties of each parameter including the shape irregularity are analyzed. From this, a reference guideline of the tube thickness can be prepared for design purpose, an example of which is given here. Nomenclature A , B , C ,  constants composed of the tangential, secant and elastic moduli and Poisson ’ s ratio E, ν , σ ys elastic modulus, Poisson ratio and yield strength, in order t , r thickness and radius of a tube, respectively p cr , p y critical buckling pressure of a circular and an oval shaped cross sections, respectively β , δ deformation due to an external pressure in radial and circumferential directions, respectively δ 1 initial ovality Π potential energy stored in a deformed tube due to an external pressure

2. Revisit to formulae

The elastic buckling formulae have been developed by Griffin (1965), Morgan (1964) and Timoshenko and Gere (1961). The essentials of the derivation are as follows. It is assumed that the cross section of a tube, the thickness and mean radius of which are t and r , respectively, deforms like an ellipse, as illustrated in Fig. 1, when the tube of a perfectly circular cross section is buckled by an external pressure, p . In this case, the displacements in radial ( r ) and circumferential ( θ ) directions (denoted as u r and u θ , respectively) are expressed as

  cos2  r u ,

  sin 2  u



(1)

where δ and β are the maximum displacements in r - and θ -directions at θ = n π /2 and ( n π /2 + π /4), respectively.

Fig. 1. Centerlines of tube cross section before and after a deformation caused by an elastic buckling.