Elastic instability analysis of dished head

As one of the main components of the equipment, the structural form of the pressure vessel head has been very mature. If sorted according to the stress conditions, the best is the spherical head, followed by the elliptical head, and then the dished head. The flat cover head has the worst stress, and the calculated thickness of the head with good stress is relatively thin.

In our conventional design, the best choice is often between the stress and processing difficulty. Oval and dished heads are used most.
Elliptical head: the standard type with long short axis ratio of 2 is generally adopted.
Dished head: generally, the radius Ri of spherical part is taken as 0.9Di, and the inner radius r of head corner is taken as 0.1Di.
In addition to the above two common forms, many other forms of elliptical head and dished head are listed in the standard. For details, refer to pressure vessels (GB/T 150-2011) and pressure vessel heads (GB/T 25198-2010).
The focus of this paper comes. The requirements for minimum thickness of head are indicated in pressure vessels (GB/T 150.3-2011), taking dished head as an example “For dished heads with RI / R ≤ 5.5, the effective thickness shall not be less than 0.15% of the inner diameter of the head, and the effective thickness of other dished heads shall not be less than 0.3% of the inner diameter of the head. However, when determining the thickness of the head, the elastic instability under internal pressure has been considered, so it may not be limited.”

Theoretical basis of elastic instability under internal pressure

What is elastic instability under internal pressure?
A: there is high circumferential compressive stress in the corner area of dished head, especially the thinner head, which often loses stability and suffers damage within the elastic range.

Taking the dished head as an example, the dished head consists of a spherical shell with an inner radius of RI and a transition ring shell with an inner radius of R.

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Fig. 1 Composition of dished head

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Fig. 2 Stress and deformation of dished head

  • 1) The meridional stress is evenly distributed in the spherical shell of the head, which is the tensile film stress. It gradually decreases to the transition ring shell, and decreases to the bottom edge of the head, which is equal to the axial stress of the connected cylinder.
  • 2) The circumferential stress in the spherical shell is also the tensile film stress, and the value is equal to the meridional stress. On the transition ring shell, it is the circumferential compressive stress, and the compressive stress is the largest at the connection between the spherical shell and the ring shell, and the compressive stress gradually decreases along the meridional direction to the bottom edge, and is the minimum at the bottom edge.

In short, when the convex head is subjected to internal pressure, the shape tends to be spherical, the spherical shell part is axially stretched outward, and the transition section is circumferentially compressed inward. At the connection between the spherical shell and the transition section, due to the discontinuity of the structure, the transverse shear force and bending moment are generated. Due to the action of transverse shear force and bending moment, local membrane stress and bending stress are generated at the discontinuity, which is caused by superposition of internal pressure The total stress of the head is obtained from the membrane stress. The experiment shows that the membrane bending of the head is related to R / ri (extracted from the standard interpretation of GB/T 150 pressure vessels).

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Fig. 3 Variation curve of maximum total stress of dished head
The structure of the head determines the stress here. In addition to strength calculation, an eigenvalue buckling analysis should also be carried out here, because instability often occurs before strength failure.

FEA analysis of elastic instability under internal pressure

Case: take dished head as an example: design pressure 0.3MPa, dn2500, RI = 2250mm, r = 250mm, material Q345R (SM = 189MPa), temperature 20 ℃ (Young’s modulus of elasticity 2.01e5MPa, Poisson’s ratio 0.3)

The general calculation is as follows:

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Since: RI / r = 9 > 5.5, the effective thickness δ e=DiX0.3%=7.5mm
The nominal thickness shall be rounded and taken as at least 10mm (considering the negative deviation of steel plate and head stamping thinning)
As the calculated thickness is small, 6mmq345r steel plate is considered, and the minimum forming thickness of the head is 5.1mm. Try to calculate the elastic instability under internal pressure of the head.
The calculation conditions are divided into stress intensity and eigenvalue buckling analysis

The boundary conditions are as follows: the internal pressure is 0.3MPa, and the axial and circumferential displacement constraints are imposed on the end of the cylinder.

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Figure.4 Model

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Fig.5 Membrane stress

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Figure.6 Film plus bending stress

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Fig.7 Membrane stress of sub model

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Fig.8 Submodel film plus bending stress

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Figure.9 Eigenvalue buckling of sub model

Calculation

  • A) The stress intensity SII and SIV are qualified (according to JB4732).
  • B) The 1st to 5th steps are negative values, i.e. opposite to the direction of applying the design pressure. Since the equipment has no external pressure working condition, it is ignored temporarily.

The loadmultipler (linea) of the 6th order is 22.277, therefore: 22.277×0.3 = 6.6831MPa is the ultimate load value of equipment instability. According to the requirements of GB/T 150-2011, the safety factor for stability calculation is 3, so 6.6831 / 3 = 2.2277MPa > 0.3MPa, and the result is qualified.
That is, the thickness of the head of the equipment is 6mm, and the elastic instability under internal pressure is considered, which can not be limited by the effective thickness of conventional calculation.

Note: Reference: pressure vessels (GB/T 150-2011) and its interpretation.

Source: Network Arrangement – China Dished Head Manufacturer – Yaang Pipe Industry Co., Limited (www.metallicsteel.com)

(Yaang Pipe Industry is a leading manufacturer and supplier of nickel alloy and stainless steel products, including Super Duplex Stainless Steel Flanges, Stainless Steel Flanges, Stainless Steel Pipe Fittings, Stainless Steel Pipe. Yaang products are widely used in Shipbuilding, Nuclear power, Marine engineering, Petroleum, Chemical, Mining, Sewage treatment, Natural gas and Pressure vessels and other industries.)

If you want to have more information about the article or you want to share your opinion with us, contact us at sales@metallicsteel.com

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