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# Analysis and calculation of structural strength of square flanges with round holes

## 1. Overview

The bonnet in the valve and the valve body generally together to form a pressure-bearing shell, the common connection forms are flanged, threaded and self-fastening, etc.. The thickness of the bonnet flange can be calculated according to the open hole of the flat cover can also be calculated according to the overall flange, most of the introduction on the flange calculation is the calculation method of the round flange, this paper describes the calculation method for square valve covers with flange connections.

## 2. Analysis

Bonnet design is an essential part of the valve design, because it is subject to medium pressure, temperature and other technical parameters and the valve body is basically the same, so there are commonalities in the design of the two methods. Such as the minimum wall thickness of the bonnet neck, bonnet flange size are to be synchronized with the design of the valve body calculation, and the material, connection size is the same. Therefore, the minimum wall thickness of the valve cover neck can be calculated according to the relevant formula for the wall thickness of the valve body. The structural dimensions and thickness of the bonnet are calculated according to the overall flange, as the flange size of the bonnet connected to the valve body is the same as the size of the flange in the valve body, when the flange in the valve body strength test qualified, the bonnet can not be calibrated.
Common valve cover flange shapes are round, square, triangular and oval (Figure 1), and other special shapes. Although the structure of the flange is simple, but the force situation is more complex, and therefore used for the design and analysis of a variety of methods. But basically, they can be divided into three types of analysis methods based on material mechanics, analysis methods based on elasticity theory and analysis methods based on plastic limit load. Among them, the methods of stress calculation for each part of the flange according to the analysis method of elasticity theory are mainly the S. Timoshenko method and the E.O. Waters method. Although the Waters method is more complex than the Timoshenko method, the stress analysis is closer to the actual structure of the flange, so the ASME code, EN standards, JIS standards, China’s national standards GB150-1998 and GB/T 17186-1997 in the flange calculation A method are used in this method. Figure.1 Common flange shapes
(a) round; (b) square; (c) triangular; (d) oval
For the overall flange, the Waldorf method will be the overall flange hypothetically decomposed into a cylinder, conical neck, ring plate three parts (Figure 2). The maximum stress in each section is calculated by linking the boundary conditions with each other in accordance with the relationship of deformation coordination, and the level of the allowable stress is used to control the stress. Figure.2 Waldorf method force model
There are various assumptions when calculating flange stresses by the Waldorf method.

• (1) The materials are homogeneous and in an elastic state at the design temperature, without yielding, creep and plastic flow.
• (2) The ring plate is considered as a solid ring plate and the effect of the bolt holes in the ring plate is ignored.
• (3) The cylinder, cone neck and ring plate are symmetrical structure, ignoring the interaction between them.
• (4) The bolt load acting on the ring plate has been determined, and the bending deformation is produced under the action of the uniform bending moment.
• (5) The bolt load and the force arm are derived by assumption, and the product of the bolt load and the force arm is the external moment applied to the flange.
• (6) All the loads acting on the flange are equivalent to a force couple caused by a pair of forces of equal size acting on the inner and outer diameter of the ring plate.
• (7) Ignore the deformation of the middle surface of the ring plate and the radial displacement of the annular shape center caused by the torque acting evenly along the ring plate.
• (8) The deflection or corner of the ring plate is very small, so the two load systems (Figure 3, (a), (b)) and their elastic action are linearly related, and the whole problem can be used to superimpose the principle.
• (9) The conical neck and the cylinder are regarded as thin shells, the diameter of their faces can be replaced by the inner diameter approximation in the shell theory analysis, and the conical neck is assumed to have no radial displacement at the connection with the ring plate.
• (10) Whether it is a cylinder, cone neck or ring plate, only analyze the stresses generated under the action of the bending moment caused by the bolt force, omitting the internal or external pressure on each part directly caused by the film stress.

Under the premise of the assumptions and force models made, for the integral flange, both radial and circumferential bending stresses induced in the ring plate should meet the strength check conditions of the ring plate material. Since the tapered neck causes both axial and circumferential edge stresses, only the axial stresses can be checked from the design point of view, and the circumferential stresses can be disregarded. Similar to the tapered neck, the cylinder also has axial and circumferential edge stresses, and because the axial edge stresses are larger than the circumferential edge stresses, only the axial stresses need to be calibrated. The axial stress of the cylinder and the small end of the tapered neck is connected together and the same, so the axial stress of the cylinder can not be considered at the same time as the axial stress of the tapered neck check, only in the tapered neck axial stress check conditions to take into account the maximum stress may be in the small end of the tapered neck and the allowable stress of the cylinder material for check (Figure 4). Figure.3 Simplification of the load on the flange
(a) the assumed part I load; (b) the assumed part II load; (c) the actual load on the flange (slightly approximated, with (a) + (b) equal) Figure.4 Distribution of each stress on the flange
(a) stresses on the overall flange; (b) stresses on the ring plate
The flange is connected to the cylinder as a beam on the elastic base, and the bending moment MH0 and shear force QH0 are applied to the boundary with the conical neck of the flange.
The tapered neck part of the flange is assumed to have the same stress and deformation as the column and shell of variable section in the analysis, and the approximate relationship between stress and load is listed according to the static force and moment equilibrium conditions.
The ring plate part of the flange is a purely thin ring plate, and the equations of bending moment and shear force are listed according to the equations of the circular plate.
After establishing the relationship between the cylinder, tapered neck and ring plate, the three parts are then connected, and a set of stress equations can be derived by making the boundary internal force elements and displacements on the opposite sides equal according to the elastic analysis method. But in terms of flange strength, there is no need for a very complete stress analysis, so Walters method is only given to control the strength of the flange σH (axial stress of the tapered neck), σR (radial stress of the ring plate) and σT (annular stress of the ring plate) three main stress calculation formula (formula in the various symbols and meaning see GB150). For a flange with a given connection size, the flange moment can be found and the structural dimensions of its parts have been determined, so the stresses σH, σR and σT for the three controlling strengths of the flange can be found. The flange design can be completed by strength calibration according to the assumed flange thickness and strength conditions.

## 3. Method

Square flange is often used in the flange connection of small-diameter valves, small-diameter two-piece ball valves, such as the left and right valve body connection. But national standards or codes mainly provide for the design of round flange calculation method, the square flange with a round hole is rarely mentioned and its design method. Such as ASME Boiler and Pressure Vessel Code, Volume VIII, Book 1, mandatory Appendix 2, Section 2.10, AS 1210-1997, Section 3.21.8, BS EN 12516-2:2004, Section 10.4.1 and DIN 3840:1982, Section 8.1.4.1 all mention the design method of non-circular flanges with circular holes. Among them, AS Australian Pressure Vessel Standard and American ASME Code belong to a series, and BSEN British EU Standard and German DIN Standard belong to a series.
For non-circular flange with round hole, its outer diameter should be taken with the flange round hole concentric and can be completely connected in the flange outer boundary of the largest circle of diameter. Then use the bolt centerline drawn through the center of the outermost bolt hole to calculate the bolt load, moment and stress according to the round flange.
This is an equivalent calculation method for non-circular flanges, which is to quantify the non-circular flange as a circular flange for calculation. For square flanges, the same concept of quantifying non-round flanges to round flanges can be used to quantify round flanges. In the design of a square flange, the shape factor Z is 1 because the ratio of the long and short axes is 1. The design is based on the same principle as that of a round flange because only the strength conditions corresponding to the maximum stress are restricted. In the case of structural characteristic coefficient K, it is not necessary to multiply the shape coefficient Z determined by the ratio of long and short axis, that is, K is directly used as the structural characteristic coefficient in the calculation formula of the thickness of the square flat cover, and the length of the opposite side of the square instead of the calculated diameter of the circular flat cover Dc. Thus the calculated thickness of the flat cover δp is equivalent to the thickness of the circular flat plate with outer diameter D, which is √Z times of the calculated thickness, i.e. 1 times.
Therefore, for the square flange required thickness δf can be calculated in accordance with the length of the square on the side as the equivalent of the outer diameter of the flange round flange, equivalent to the width of the ring plate and the bolt center circle on the ring plate arrangement, that is, the bolt circle center line from the inner diameter side of the round flange, the outer diameter side of the distance and the original square flange bolt center line from the flange inside, outside the same distance. So when quantified after the round flange, on behalf of the original square flange, and safe and reliable.
This means that the calculation of the square flange in addition to the outside diameter of the flange Do with the square side of the maximum circle tangent that the length of the opposite side of the square as the outside diameter, the rest are in accordance with the calculation steps of the round flange can be (Figure 5). Figure 5: Structural dimensions of square valve cover

## 4. Conclusion

Understanding this equivalent circular flange of non-circular flange calculation method, you can use this method for a variety of other non-circular flange strength calculation. However, this method is only used in the case of the center of the flange open hole is round, the center of the flange is not a circular hole does not apply to this method.

Author: Li Long