Optimization of location and size of opening in a pressure vessel cylinder using ANSYS

AUTHORS

M. Javed Hyder: Department of Mechanical Engineering, Pakistan Institute of Engineering & Applied Sciences, Pakistan
M. Asif: Design and Development Centre, SES, Directorate, Plot No. 234, St. # 7, I-9/2, Industrial Area, Islamabad, Pakistan

Abstract

The main objective the research work is to optimize the location and size of opening (hole) in a pressure vessel cylinder using ANSYS. Analysis is performed for three thick-walled cylinders with internal diameters 20, 25 and 30 cm having 30 cm height and wall thickness of 20 mm. It is observed that as the internal diameter of cylinder increases the Von Misses stress increases. Optimization of hole size is carried out by making holes having diameter of 4, 8, 10, 12, 14, 16 and 20 mm located at center in each of the three cylinders, and it is observed that initially Von Misses stress decreases and then become constant with hole size. The optimum size of hole is found to be 8 mm for cylinder having internal diameter of 20 cm whereas a hole of size 10 mm for cylinder having internal diameter of 25 cm and 30 cm on the basis of lowest Von Mises stress value. Lastly, optimization of location of hole is carried out by making a 12 mm hole located at 1/16, 1/8, 2/8, 3/8 and 4/8 of cylinder height from top in all the three cylinders. The
Von Misses stress is maximum at the center i.e., 4/8 location and decreases in the direction away from center and then stress increases as the location is changed from 1/8 to 1/16 from
cylinder top due to the end effects. The optimum location of the hole is found to be at 1/8 of cylinder height.

Keywords: Damage; ANSYS; Pressure vessel

Article Outline

1. 

Introduction

2. 

Analysis of pressure vessel cylinder with out hole

2.1. Analytical analysis of stress distribution around a hole in the cylinder

2.2. ANSYS Analysis of Stress distribution around a hole in the cylinder

3. 

Conclusions

1. Introduction

The main purpose of this research is to perform a stress analysis on a thick-walled pressure vessel cylinder and optimize the location and size of opening using finite element analysis software namely, ANSYS. Finite element analysis offers a great deal of promise over other approaches mainly experimental, in the sense of low cost, high speed, complete information, and ability to simulate realistic and ideal conditions.

Pressure vessel cylinders find wide applications in thermal and nuclear power plants, process and chemical industries, in space and ocean depths, and fluid supply systems in industries. The failure of pressure vessel may result in loss of life, health hazards and damage of property. In addition to the pressure, the pressure vessels are also subjected to support loads that may be steady or variable (transient), piping reactions, and thermal shocks which require an overall knowledge of the stresses imposed by these conditions on various vessel shapes and appropriate design means to ensure safe and long life. Basic considerations in the design of pressure vessel include [1]:

1. Recognition of most likely modes of failure.

2. Stresses induced in vessel material due to pressure and temperature.

3. Selection of suitable material capable of withstanding the effects of pressure and thermal loads, and effects of environment.

4. Effect of concentration of stresses due to geometric discontinuities resulting from provision for supports, and openings for manhole, gauges etc.

In the present work, emphasis is on item four of the above list. It is shown that an appropriate location and size of the opening in a pressure vessel results is minimizing the stresses induced due to the stress concentration resulting from the end flanges.

The ever increasing use of vessels for storage, industrial processing, and power generation under unusual conditions of pressure, temperature, and environment has given special emphasis to analytical, experimental, and numerical methods for determining their operating stresses. Pressure vessels may be cylindrical, spherical, conical, ellipsoidal etc. The pressure vessels usually consist of a pressure resisting shell together with flange rings and fastening devices for assembly of the mating parts [2].

Strength is an inherent property of a mechanical element and is the characteristic of the material and is there even when no external load is applied on the mechanical element. To avoid the pressure vessel failure the design engineer must have positive assurance that stresses generated will never exceed the strength. Stress analysis of a pressure vessel is a very sophisticated area.

Holes in pressure vessels are frequent, in fact all riveted constructions make use of such means of fabrication, and all vessels must have openings. These geometric discontinuities alter the stress distribution in the neighborhood of discontinuity so that elementary stress equations no longer prevail. Such discontinuities are called “stress raisers” and the regions in which they occur are called the areas of stress concentrations. A theoretical, or geometric, stress concentration factor, K_t, is used to relate the actual maximum stress at the discontinuity to the nominal stress [3].

Stress concentration is a highly localized effect. The high stresses exist only in a very small region in the vicinity of the hole. In approaching the study of localized stresses it is well to note that their significance does not depend solely on their absolute value. It also depends upon [2]:

1. The general physical properties of the material.

2. The relative proportion of the member highly stressed to that under stressed which affects the reverse strength, it can develop in resisting excessive loads.

3. The kind of loading to which the pressure vessel is subjected.

The numerical approach adopted here is the analysis by using finite element analysis software, ANSYS which is a very powerful and versatile tool for structural, thermal, fluid, electric, magnetic, and electromagnetic analysis.

2. Analysis of pressure vessel cylinder with out hole

In the present study optimization of the location and size of opening in a pressure vessel cylinder has been done. The cylinder dimensions are such that the ratio of the thickness to radius of the cylinder is greater than 0.05 to satisfy the requirement of a thick-walled cylinder. The cylinder is subjected to internal pressure of P_i = 5 MPa, having a constant height (30 cm) and wall thickness (20 mm). Three different internal diameters, namely 20, 25 and 30 cm were used. Flanges (upper and lower) at the end of the cylinder were attached with 20 mm height and 40 mm thick in the radial direction. Initially, a comparison of the theoretical and ANSYS analysis is done with out any hole in the cylinders. Here the effects of the end flanges have been shown. For the purpose of comparison of theoretical and ANSYS analysis in a cylinder with out any hole following equations were used to calculate the theoretical maximum values. These maximum values were than compared with the ANSYS result. Designating the inside radius of the cylinder by a, the outside radius by b, the internal pressure by p_i, and the external pressure by p₀ the tangential and radial stresses are given by [3]

(1)And the maximum stresses occur at the inner surface, (r = a)

(2)It should be realized that longitudinal stress is given by

(3)All these three stresses basically represent the three principal stresses acting on a cylinder. Hence the Von Mises stress (also known as equivalent stress σ_eqv) with three principal stresses is given by

(4)The results of maximum tangential, longitudinal, radial and Von Mises stress are given in Table 1 for all three cylinders along with the results from ANSYS. The ANSYS result of tangential, longitudinal, radial and Von Mises stress distribution is shown in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5 and Fig. 6 for all the three cylinders. Graphically the maximum tangential, longitudinal, radial and Von Mises stresses for all the three cylinders is shown in Fig. 7. The analytical and ANSYS result shows the same trend i.e., both the analytical and ANSYS values of Von Mises stress are increasing by increasing the internal diameter of cylinder. The ANSYS results are slightly on the higher side due to the consideration of the constraints imposed by the end flanges which are kept fixed. Where as, in the analytical analysis end flanges effect could not be incorporated.

Table 1. Stresses in cylinder without hole

Stresses	Cylinder 1		Cylinder 2		Cylinder 3
	Analytical	ANSYS	Analytical	ANSYS	Analytical	ANSYS
σt (MPa)	27.73	29.2	33.94	36.1	40.16	42.5
σl (MPa)	11.36	13.1	14.47	16.1	17.58	19.3
σr (MPa)	−5	−6.74	−5	−9.35	−5	−10.2
σeqv (MPa)	28.34	30.7	33.72	37.1	39.11	43.7

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Fig. 1. Stresses for 20 cm cylinder without hole: (a) tangential and (b) longitudinal.

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Fig. 2. Stresses for 20 cm cylinder without hole: (a) radial and (b) Von Mises.

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Fig. 3. Stresses for 25 cm cylinder without hole: (a) tangential and (b) longitudinal.

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Fig. 4. Stresses for 25 cm cylinder without hole: (a) radial and (b) Von Mises.

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Fig. 5. Stresses for 30 cm cylinder without hole: (a) tangential and (b) longitudinal.

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Fig. 6. Stresses for 30 cm cylinder without hole: (a) radial and (b) Von Mises.

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Fig. 7. Comparison of theoretical and ANSYS result of hoop, longitudinal, radial, and Von Mises stresses in all cylinder without hole.

2.1. Analytical analysis of stress distribution around a hole in the cylinder

The cylinder is considered as a flat plate with hole in the center. The circumference of the cylinder is considered as the width and the cylinder height as the height of the plate. The stress distribution in the vicinity of a small circular hole of radius a, in a plate stretched elastically by a uniform tensile stress (σ), in the direction of the polar axis θ = 0, is given by [2]

(5)At the circumference of the hole, r = a and σ_r = 0, σ_t = σ (1–2 cos 2θ), σ_rt = 0. The tangential stress is a maximum at the points θ = 90 and θ = 270 located on the circumference of the hole, and on an axis perpendicular to the direction of the applied tension. At these points the stress σ_t = 3σ. For r = a, and θ = 0 or at θ = 180, σ_t = −σ. Thus it can be seen that a small hole in a plate subjected to tension in a given direction causes an increase in the stress in the vicinity of the hole to a maximum of three times of normal undisturbed portion of the plate. When ever a discontinuity occurs in the form of hole it results in maximum stress (σ_max) adjacent to the hole compared to the nominal stress (σ₀) away from the hole. This phenomena is represented by the stress concentration factor (K_t) as

(6)Stress concentration factor for a flat plate with transverse hole in tension is [4]

(7)

For d = 4, 8, 10, 12, 14, 16, 20 mm and w = 300 mm the maximum hoop stress has been calculated. The result is shown in Table 2 along with the ANSYS results. In this case Von Mises (equivalent) stress is the tangential (or the hoop) stress as other stresses are negligible as compared with hoop stress and hence have no effect on hole size.

Table 2. Von Mises stresses with different size hole located at the center of the height

Hole diameter (mm)	Equivalent stress in cylinder 1 (MPa)		Equivalent stress in cylinder 2 (MPa)		Equivalent stress in cylinder 3 (MPa)
	Analytical	ANSYS	Analytical	ANSYS	Analytical	ANSYS
4	81.97	86.7	100.33	108	118.71	129
8	80.64	86.6	98.70	107	116.78	126
10	80.08	87.4	98.02	106	115.98	126
12	79.47	87.8	97.27	107	115.10	127
14	78.86	88.6	96.52	108	114.22	127
16	78.36	89	95.91	108	113.49	128
20	77.23	91.2	94.52	110	111.85	129

2.2. ANSYS Analysis of Stress distribution around a hole in the cylinder

In the next ANSYS analysis holes of size 4, 8, 10, 12, 14, 16, 20 mm located at center of the height in all three cylinders for size optimization was carried. As a representative, the result of tangential, longitudinal, radial and Von Mises stress distribution in all the cylinders having a 12 mm hole at the center of the height is shown in Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12 and Fig. 13. Whereas the Fig. 14 shows a plot of Von Mises stress versus different hole sizes located at center under the same internal pressure. Both the analytical and ANSYS results are plotted for the three cylinders. The analytical values are decreasing by increasing the hole diameter. This is due to the fact that maximum stress at the edge of the hole is a function of nominal stress and the stress concentration factor which depends on the width of the assumed flat plate for the cylinder. As the diameter of the cylinder is increased the circumference of the diameter increases (i.e, width) so the stress concentration factor decreases. This decrease is reflected in the decrease in analytical values of the stress as the diameter is increased.

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Fig. 8. Stresses for 20 cm cylinder with hole: (a) tangential and (b) longitudinal.

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Fig. 9. Stresses for 20 cm cylinder with hole: (a) radial and (b) Von Mises.

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Fig. 10. Stresses for 25 cm cylinder with hole: (a) tangential and (b) longitudinal.

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Fig. 11. Stresses for 25 cm cylinder with hole: (a) radial and (b) Von Mises.

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Fig. 12. Stresses for 30 cm cylinder with hole: (a) tangential and (b) longitudinal.

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Fig. 13. Stresses for 30 cm cylinder with hole: (a) radial and (b) Von Mises.

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Fig. 14. Behaviour of Von Mises stress distribution for different hole sizes located at the center of the height of cylinders with three different diameters.

However, ANSYS results deviate a little from analytical values. Initially the stresses are decreasing and then become constant. But for cylinder 1 (20 cm internal diameter) stress is increasing from 10 mm hole diameter onwards. This is due to the fact that analytical results are calculated for finite plate with a circular hole under tension but in the ANSYS analysis a cylinder with a transverse hole is glued with a flange. So, the curvature of the cylinder and finiteness of the plate causes the difference in the results. The optimum hole diameter for cylinder 1 is 8 mm, for cylinder 2 is 10 mm and for cylinder 3 is 10 mm as these hole sizes have the lowest Von Mises stress values of 86.8, 106 and 126 MPa, respectively.

In Fig. 15, Fig. 16 and Fig. 17 the tangential, longitudinal and radial stress distribution in all the three cylinder is shown as the size of the hole is increased using ANSYS. Also the result is tabulated in Table 3. The magnitude of the longitudinal and radial stresses is relatively small as compared to the tangential stress. The tangential stress has the same behavior as the Von Mises stress distribution shown in Fig. 14, although the magnitude is slightly low.

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Fig. 15. Behaviour of tangential stress distribution for different hole sizes located at the center of the height of cylinder with three different diameters.

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Fig. 16. Behaviour of longitudinal stress distribution for different hole sizes located at the center of the height of cylinder with three different diameters.

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Fig. 17. Behaviour of radial stress distribution for different hole sizes located at the center of the height of cylinder with three different diameters.

Table 3. ANSYS result of stresses in three cylinders with different holes at center

Hole diameter (mm)	Tangential stress (MPa)			Longitudinal stress (MPa)			Radial stress (MPa)
	Cylinder 1	Cylinder 2	Cylinder 3	Cylinder 1	Cylinder 2	Cylinder 3	Cylinder 1	Cylinder 2	Cylinder 3
4	88.4	110	132	13	16.6	21	6.66	8	10.3
8	86.5	107	128	13.1	16.6	21.4	5.94	9.54	12.3
10	86.6	106	129	13.3	16.8	23.5	5.90	9.45	12.2
12	85.9	107	127	13.1	17.1	23.1	6.86	9.50	12.3
14	85.6	107	127	12.9	16.9	25.2	6.90	9.55	12.3
16	85.9	107	128	13	18.1	27.4	6.90	9.52	12.3
20	87.4	108	128	13.2	16.5	24.3	6.90	9.56	12.3

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Finally an analysis was carried out for a 12 mm hole located at 1/16, 1/8, 2/8, 3/8, and 4/8 mm from the top of the cylinder height in all three cylinders for location optimization. Fig. 18 shows a plot of Von Mises stress versus 12 mm hole at different locations of cylinder height. The result is also tabulated in Table 4. The Von Mises stress is maximum at the center (1/2 of cylinder height) and is decreasing away from center. Then from going 1/8 to 1/16 of cylinder height (from top of cylinder) the stress is increasing again. This is due to the discontinuity at the cylinder and upper flange interface, which acts as a stress raiser. The optimum hole location is 1/8 (0.125) of the cylinder height from top i.e., 0.0375 m for all the three cylinders.

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Fig. 18. Variation of Von Mises Stress (MPa) with 12 mm hole at different locations.

Table 4. ANSYS results for 12 mm diameter hole located at locations from top

Hole location (fraction of cylinder height)	Von Mises stress for cylinder1 (MPa)	Von Mises stress for cylinder 2 (MPa)	Von Mises stress for cylinder 3 (MPa)
1/16	56.1	68.7	75.8
1/8	53.5	57.9	60.3
2/8	82.4	94.8	105
3/8	87.9	106	123
4/8	87.8	107	127

3. Conclusions

It is concluded that the location and size of the hole depends on the size of the cylinder. For a specific application and size of the cylinder the location and size of the opening should be decided by caring out the finite element analysis (like ANSYS in this case) considering the end effects introduced by the flanges. The optimum location is where the Von Mises stress is minimum and also the hole size should be such that the Von Mises stresses are minimum around the vicinity of the hole.

References

[1] R.K. Jain, Machine design (3rd ed.), Khanna Publications (1983).

[2] F. Harvey John, Theory and design of modern pressure vessels (2nd ed.), Van Nostrand Reinhold Company (1974).

[3] Shigley and Mitchell, Mechanical engineering design (4th ed.), McGraw-Hill Book Company (1983).

[4] Norton RL. Machine design, an integrated approach, 2nd ed. Prentice-Hall International, Inc.

Corresponding author. Tel.: +92 512208025.

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