Calculation of cold forming deformation rate of elliptical head
In accordance with Table 4 of the national standard Pressure Vessels Part 4: Manufacturing, Inspection and Acceptance (GB 150.4-2011), the control index of deformation rate of cold formed parts is used to determine whether the formed pressure parts need to be subjected to performance recovery heat treatment. The standard provides the calculation formula of deformation rate, but does not provide the method for determining the mid surface radius of oval head after cold forming. The radius of the circular arc transition zone of the elliptical head is obtained by geometric drawing and analytical methods, and then used as the mid surface radius of the formed elliptical head to calculate the deformation rate of the elliptical head.
1. Problem posing
The stress distribution on the elliptical head is continuous and uniform. Because of its good stress and easy processing, it is widely used in petrochemical equipment and is one of the important components of chemical equipment. During the forming process of elliptical head, the steel plate is subject to biaxial tension (see Figure 1). According to the control index of deformation rate of cold formed parts in Table 4 of the national standard Pressure Vessels Part 4: Manufacturing, Inspection and Acceptance (GB150.4-2011), it is necessary to calculate the deformation rate of cold formed elliptical head [1] to judge whether it needs to conduct performance recovery heat treatment after forming. Although the calculation formula of the deformation rate is given in the national standard, the value method of the key parameter of the cold forming deformation rate of the elliptical head, the mid radius after forming (R_{f}), is not given.
The cold forming deformation rate of elliptical head is calculated according to formula (1) [1]:
Deformation rate (%)=75 δ [1-(R_{f}/R_{o})]/R_{f}(1)
In the formula:
- δ—— Plate thickness, mm;
- R_{f} — radius of middle surface after forming, mm;
- R_{o} – radius of middle surface before forming (∞ for flat plate), mm.
By analyzing the deformation rate calculation formula (1), it can be seen that the middle radius R_{f} after forming is inversely proportional to the deformation rate. The smaller the value, the greater the deformation rate. Since the curvature of the elliptical head curve changes point by point [2], if the maximum deformation rate is obtained, the minimum radius of curvature is required.
Fig. 1 Schematic Diagram of Cold Forming Biaxial Tension of Elliptical Head
2. Stress analysis of elliptical head under internal pressure
As the geometric characteristics of elliptical heads cause smooth and continuous meridian curvature, the stress distribution in the heads is relatively uniform. As shown in Figure 2, the longitudinal stress σϕ It is always the tensile stress, and the vertex of the minor axis is the maximum value, and the endpoint of the major axis is the minimum value. For the circumferential stress in the latitude direction, there may be negative values. The maximum tensile stress is at the top of the minor axis, and the minimum tensile stress or maximum compressive stress is at the end of the major axis [2].
Fig. 2 Stress Direction Diagram of Elliptical Head [3]
Under the action of internal pressure, due to the geometric discontinuity at the connection between the head and the cylinder, transverse shear stress and bending moment are generated on the boundary of the head, and local membrane stress and bending stress are generated on the head near the connection between the head and the cylinder. The location, direction and size of the maximum stress on the head change with a/b as shown in the dotted line in Figure 3.
Fig. 3 The maximum stress on the head changes with a/b [2]
a. Long radius of elliptical head (mm) b. Short radius of elliptical head (mm) Di. Inner diameter of elliptical head (mm) hi. Depth of inner surface of elliptical head (mm) K. Shape coefficient of elliptical head
It can be seen from Figure 3 that:
- (1) When 1.0<a/b ≤ 1.2, the maximum stress of the head is the circumferential tensile stress, which is located at the bottom edge of the elliptical shell;
- (2) When 1.2<a/b ≤ 2.5, the maximum stress of the head is the longitudinal tensile stress, which is located in the inner wall of the elliptical transition zone;
- (3) When 2.5<a/b, the maximum stress of the head is the circumferential compressive stress, which is located on the outer wall of the elliptical transition zone.
It can be seen from the above analysis that the position of the maximum stress point of the elliptical head changes with the change of a/b. When a/b>1.2, the maximum stress is always at the position of the elliptical transition zone. The radius of curvature at the location of the elliptical transition zone is the smallest compared with the radius of curvature in the latitude and longitude directions and the radius of the sphere, as shown in Figure 4.
Fig. 4 Curvature Radius of Elliptic Head [3]
50. Spherical inner radius (mm) r. radius of transition zone (mm) h. Head depth (i.e. short radius of elliptical head, mm) R. long radius of elliptical head (mm) D. inner diameter of elliptical head (mm) t. wall thickness of elliptical head (mm) X. distance from any point in the transition zone of elliptical head to the short semi axis (mm) R_{L}. radius of curvature in latitude direction (mm) R_{m}. radius of curvature in longitude direction (mm)
It can be seen from the above analysis that the circular arc transition zone of the elliptical head is the place where the maximum stress often occurs. In actual production and manufacturing, this position is the place where the wall thickness is thinned most seriously, and the radius of the transition zone is the smallest among the curvature radii of the elliptical head. Therefore, it is reasonable to take the radius r of the transition zone as the middle surface radius R_{f} after forming.
3. Calculation method of middle plane radius R_{f}
3.1 Geometric drawing method
On the premise that the major axis and minor axis of the elliptical head are known, the four center method can be used to approximately draw the midplane curve of the elliptical head, so that the radius of the transition zone r can be measured as the midplane radius Rf after forming, and then the deformation rate can be calculated. The ellipsoidal head is drawn with the four center method as shown in Figure 5.
Fig. 5 Schematic Diagram of Drawing Elliptic Head with Four Center Method
- (1) First, draw the major axis (AB) and minor axis (CD) of the ellipse respectively;
- (2) Connect AC, make an arc with O as the center and OA as the radius to intersect the straight line of CD at point E, and make an arc with C as the center and CE as the radius to intersect AC at point F;
- (3) Make the vertical bisector of line segment AF (take A and F as the center respectively, draw an arc with the length greater than half of AF as the radius, and the intersection points are G and H respectively);
- (4) The straight line where G and H are located is the median line of segment AF, and the intersection points of the median line, the major axis and the minor axis are M and N respectively;
- (5) Make another “two centers”, namely P and Q, according to the above method. At this time, the four centers for drawing the elliptic curve have been found respectively, namely, points N, M, P and Q;
- (6) Take M as the center, MA as the radius, Q as the center, and QB as the radius to make arcs at both ends of the major axis;
- (7) Take N as the center, NC as the radius, P as the center, and PD as the radius to make arcs at both ends of the minor axis;
- (8) The above four arc lines are connected with each other and smoothly transited to obtain an elliptical curve.
The length of AM measured from Figure 5 is the radius r of the circular arc transition zone, which is substituted into Formula (1) as the mid surface radius Rf after forming, and then the deformation rate of the elliptical head can be calculated.
3.2 Analytical method
The radius r of the circular arc transition zone is calculated by the analytical method [4], as shown in Figure 6.
Fig. 6 Schematic Diagram of Elliptical Head Drawing by Analytical Method
The basic method for drawing elliptical heads based on geometric mapping is known: AO=a, OC=b, AC=, CF=a − b, AK=FK=(AC − CF)/2. obtain:
(1) In the right triangle AOC, α= arctanb/a, β= 90°− α.
(2) In the right triangle AKM, r=AM=AK/cosα, Substitute equation (2) into the above formula, namely:
It is easy to see from the above formula derivation that as long as the dimensions of the long radius a and short radius b of the elliptical head are known, the radius r of the circular arc transition zone can be obtained according to equation (3), and then it can be substituted into equation (1) as the mid surface radius Rf after forming, so that the deformation rate of the elliptical head can be calculated.
4. Conclusion
The radius r of the circular arc transition zone of the elliptical head is obtained through the geometric drawing method and the analytical method, and then it is used as the radius Rf of the middle surface after forming, so as to calculate the cold forming deformation rate of the elliptical head, which can be used as a reference for designers and engineering technicians. This method can also provide a reference for the calculation of the cold forming deformation rate of other forms of head.
Author: Liu Xu
Source: China Elliptical Head Manufacturer – Yaang Pipe Industry (www.steeljrv.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.)
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Reference
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- [4] Zhai Hongxu, Zhai Chunjiao, Zhai Chunkai. Practical Riveter Reader [M]. Beijing: Chemical Industry Press, 2002