5.1 BUCKLING PRESSURE
The behavior of piping and tubing subject to external pressure has been well understood since the early 1900’s [Jasper, Roark, Saunders, Southwell, Timoshenko]. Consider a long, perfectly circular cylinder subject to uniform external pressure. By long cylinder, we mean a cylinder longer than a critical length given by
D=diameter, in t=wall thickness, in
If the external pressure is steadily increased, there will come a point where the cylinder will suddenly buckle. If the cylinder is long and thin, this buckling will occur while the cylinder wall is still elastic. The external pressure at which elastic buckling occurs is called the critical elastic pressure and is given by [Den Hartog]
PCE=critical elastic external pressure at buckling, psi v=Poisson ratio of material
E=modulus of elasticity, psi
I=cross section moment of inertia of cylinder wall per unit length (t3/12), in4 t=cylinder wall thickness, in
R=radius of cylinder, in n=integer equal to 2, 3,…
At the lowest critical pressure, the pressure corresponding to n=2, the cylinder buckles into an oval shape. If this shape is not physically possible because of external constraints, then the pressure will keep increasing till it reaches the value corresponding to n=3 and the cylinder will buckle in a profile with three lobes. For the first buckling mode (n=2) of a thin pipe, the critical pressure can also be written as [Den Hartog, Bednar]
The corresponding hoop strain is obtained by dividing the hoop stress by E
εCE=hoop strain at collapse pressure of a long section of pipe For steel, v=0.3, therefore
and the corresponding hoop strain is
For short sections of pipes (L<LC) with stiff ends, for example a short section between a flange and a valve acting as stiffeners, the stress in the pipe wall may reach the elastic limit before buckling occurs. The pipe will therefore undergo failure by plastic deformation rather than buckling. This occurs when the pressure reaches yield
PP=plastic limit pressure for short pipes, psi SY=material yield stress, psi.
For intermediate lengths the buckling pressure of steel cylinders can be approximated by [Harvey, Bednar]
PIC=buckling pressure of intermediate length pipe section, psi
The corresponding hoop strain is obtained by dividing the hoop stress by E
εIC=hoop strain at collapse pressure for an intermediate section of pipe
In reality, a pipe cross section is not perfectly round, it has an initial ovality, measured by
∆=ovality of cross section
D′=maximum measured diameter on oval cross section, in
D=nominal diameter, in
Unless the initial ovality ∆ is negligible, it is intuitive that the oval cross section will buckle more readily than a perfectly circular cross section. A 3% ovality limit is imposed in ASME B31.3 for piping subject to differential external pressure. The critical elastic buckling external pressure of an initially oval pipe is obtained by resolving the following equation
POB=buckling pressure of initially oval cross section, psi SY=material yield stress, psi
5.2 ASME CODE DESIGN
The design of piping subject to external pressure usually follows the rules for pressure vessels, as given in the ASME Boiler & Pressure Vessel Code. The ASME design is based on two sets of charts [ASME VIII, ASME II]. The first set of curves, provides the hoop strains εCE and εIC, called factor A, against the ratio L/D of the length L between stiffener divided by the pipe diameter D, plotted for different values of D/t. If L<LC (plastic collapse)
If L>LC (elastic collapse)
For example, for L/D=10 and D/t=20, we read from the ASME chart [ASME II]
A=0.003, which corresponds to A=1.1(1/20)2. The second set of ASME curves permits the calculation of the allowable external pressure Pa, which is defined as 1/3 the collapse pressure PCE or PIC (called PC) for D/t≥10 [Farr].
Pa=PC/3
Given factor A and the material’s Young modulus, we read from the second set of ASME curves a factor B that is half the hoop stress at the critical pressure PC
B=½[PCD/(2t)]
Therefore [ASME VIII]
Pa=(4/3) B/(D/t)
For example, given factor A=0.003 for carbon steel at ambient temperature, we read factor B=16,000 psi. Therefore, with B=16,000 psi and D/t=20, we calculate the allowable external pressure as Pa=1067 psi.
5.3 REFERENCES
ASME II, ASME Boiler and Pressure Vessel Code, Section II, Materials, Part D, Properties, Subpart 3, Charts and Tables for Determining Shell Thickness of Components Under External Pressure, American Society of Mechanical Engineers, New York.
ASME VIII, ASME Boiler and Pressure Vessel Code, Section VIII Division 1, Rules for Construction of Pressure Vessels, AG-28 Thickness of Shells and Tubes under External Pressure, American Society of Mechanical Engineers, New York.
Bednar, H.H., Pressure Vessel Design Handbook, Krieger Publishing Company, Florida.
Den Hartog, Advanced Strength of Materials, Dover Publications, New York.
Farr, J.R., Jawad, M.H., Guidebook for the Design of ASME Section VIII Pressure Vessels, ASME Press, New York.
Harvey, J.F., Theory and Design of Pressure Vessels, Van Nostrand Reinhold.
Jasper, T.M., Sullivan, J.W., The Collapsing Strength of Steel Tubes, Transaction of the American Society of Mechanical Engineers, vol. 53, 1931, American Society of Mechanical Engineers, New York.
Roark, R.J., Young, W.C., Formulas for Stress and Strain, McGraw Hill Book Company, New York.
Saunders, H.E., Windenberg, D.F., Strength of Thin Cylindrical Shells Under External Pressure, Transaction of the American Society of Mechanical Engineers, vol. 53, 1931, American Society of Mechanical Engineers, New York.
Southwell, R.V., On the Collapse of Tubes by External Pressure, Philos. Mag., vol. 29, p. 67, 1915.
Timoshenko, S., Theory of Plates and Shells, Engineering Societies Monograph, McGraw Hill, NewYork.