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discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/267611879

Vessel Under External Pressure

Conference Paper in American Society of Mechanical Engineers, Pressure Vessels and Piping Division (Publication)

PVP · January 2010

DOI: 10.1115/PVP2010-25173

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Proceedings of the ASME 2010 Pressure Vessels & Piping Division / K-PVP Conference

PVP2010

July 18-22, 2010, Bellevue, Washington, USA

PVP2010-25173

VESSEL UNDER EXTERNAL PRESSURE

Guru Nanak Dev Engineering College,

(Punjab Technical University, Jalandhar)

Gill Road, Ludhiana. 141006.

Punjab, India.

Tel: 91-9781991160

Email: brar.gurinder@gmail.com

Yogeshwar Hari

University of North Carolina at Charlotte

9201 University City Blvd.

Charlotte, NC 28223-0001 USA

Email: hari@uncc.edu

Dennis K. Williams

Sharoden Engineering Consultants, P.A

P.O. Box 1336

Matthews, NC 28106-1336 USA

Email: DennisKW@sharoden.com

external working pressure. Fifty simulated shells of geometry

Initial geometric imperfections have a significant effect on

similar to the example tower are generated by the Monte Carlo

the load carrying capacity of asymmetrical cylindrical pressure

method to calculate the nondeterministic buckling load. The

vessels. This research paper presents a comparison of a

representation of initial geometric imperfections in the

reliability technique that employs a Fourier series representation

cylindrical pressure vessel requires the determination of

of random asymmetric imperfections in a defined cylindrical

appropriate Fourier coefficients. The initial functional

pressure vessel subjected to external pressure. Evaluations as

description of the imperfections consists of an axisymmetric

prescribed by the ASME Boiler and Pressure Vessel Code,

portion and a deviant portion that appears in the form of a

Section VIII, Division 2 rules are also presented and discussed

double Fourier series. Multi-mode analyses are expanded to

in light of the proposed reliability technique presented herein.

evaluate a large number of potential buckling modes for both

The ultimate goal of the reliability type technique is to

predefined geometries and the associated asymmetric

statistically predict the buckling load associated with the

imperfections as a function of position within a given cylindrical

cylindrical pressure vessel within a defined confidence interval.

shell. The method and results described herein are in stark

The example cylindrical shell considered in this study is a

contrast to the dated “knockdown factor” approach currently

fractionating tower for which calculations have been performed

utilized in ASME B&PV Code.

in accordance with the ASME B&PV Code. The maximum

allowable external working pressure of this tower for the shell

thickness of 0.3125 in. is calculated to be 15.1 psi when utilizing

NOMENCLATURE

the prescribed ASME B&PV Code, Section VIII, Division 1

methods contained within example L-3.1. The Monte Carlo D Bending stiffness of cylindrical wall

method as developed by the current authors and published in the D0 Outside diameter of the shell

E Young’s Modulus Furthermore, buckling of cylindrical shells can occur when the

L Length of the shell structure is subjected to the individual or combined action of

Nxx Axial distributed in-plane force axial compression, external pressure, and torsion.

Nyy Circumferential in-plane force Buckling behavior (in particular, the critical buckling load)

P External design pressure is not accurately predicted by linear elastic equations. In

Pa Maximum allowable external working pressure contrast, classical buckling theories employing non-linear

Pcl Classical buckling load of a perfect shell equations have been utilized extensively in the past to predict

Pcr Critical buckling load of a shell with imperfections buckling behavior. However, classical theories include the

R Inside radius of the shell effect of pre-buckling deformations and post-buckling

Wn(ξ,θ) Initial imperfection function behaviors. In a shallow shell where the pre-buckling curvature

k Number of half waves in axial direction is small, the equilibrium conditions were shown by Donnell [1]

l Number of full waves in circumferential direction to be adequately described by Eq. (1) in a linearized form,

t Nominal shell wall thickness whereby the critical buckling load could be computed upon

w Radial displacement substitution of the applicable boundary conditions.

λ Non-dimensional buckling load

Poisson’s ratio 2 0 (1)

θ Non-dimensional circumferential coordinate

ξ Non-dimensional axial coordinate

It should be noted that represents the Laplacian operator

and that signifies the application of twice while four

INTRODUCTION TO BUCKLING OF SHELLS

Buckling is a failure mechanism that is associated with both times. Furthermore, represents the flexural

the application of a compressive load to a structural component rigidity of the shell.

and the instability of that component once any number of critical

loads are reached or exceeded. Shell buckling physically CLASSICAL & NONCLASSICAL BUCKLING LOADS

manifests itself by the appearance and growth (under continual

load) of bulges, ripples, and waves in both the circumferential Ultimately, the buckling behavior of thin cylindrical shells

and longitudinal direction of a cylindrical shell. Similar to is influenced in varying degrees by initial imperfections and

column buckling of bars and beams, shell buckling is variations in the geometry of the cylindrical shell. Variations in

encountered in long, shallow (i.e., relatively thin wall thickness) the shell wall can be manifested in the form of gradients in the

vessel and tank members when the members begin to exhibit particular loading, eccentricity from the ideal (i.e., perfect)

visibly large transverse displacements to an applied axial load or shape, variations in the material properties such as Young's

to an applied external pressure (or vacuum). A shell structure is modulus, imperfections in the shell wall thicknesses (i.e., local

considered to fail from buckling while subjected to a thin spots), and other miscellaneous parameters. Most large

compressive load; the structure undergoes a transition in diameter pressure vessels are manufactured by welding rolled

deformation from that of the direction of compressive load plates, creating both longitudinal and circumferential seams.

application to a deformation that is predominantly perpendicular Due to variations in manufacturing tolerances and techniques,

to the direction of load application. The load at which the initial fabricated cylindrical shells differ from perfect shape as

deformation transition and instability occurs is commonly evidenced by out-of-roundness and local thin wall conditions on

referred to as the critical buckling load. Often times this type of occasion. In the present work, a fractionating tower having

buckling failure is of a catastrophic nature, occurring without variations in the shell wall thickness (regarded as imperfections)

any visible precursor or form of warning to the user or operator. and subjected to external pressure is studied and the respective

Shell buckling can also produce a sudden collapse in a vessel or results are presented herein.

tank. Based upon the work by Saunders and Windenberg [2], an

Buckling failure is an important feature to be considered in approximation of the classical critical buckling load for a

many pressure vessel designs, especially when the vessel is cylindrical shell subjected to external pressure can be calculated

subjected to vacuum (i.e., external pressure) service. Typical as shown in Eq. (2):

failure theories based upon material strength, such as Tresca or

von Mises failure theories, have no method by which to address

0.807 (2)

buckling and the instability issue. Furthermore, the most

significant material properties affecting the resistance to

buckling failure are Young's modulus of elasticity and Poisson's When imperfections in shells exist and are considered in the

ratio. The most significant geometrical parameter is the aspect engineering design, the load carrying capacity of shells is

ratio comprised of the diameter to length (i.e., slenderness) ratio. reduced, as evidenced by tests. In an effort to relate the

The ultimate strength (or yield strength) does not play a predicted critical buckling load to the closed form classical

significant part in the prediction of the critical buckling load for form, a non-dimensional buckling load (λ) is defined as shown

any given shell geometry. Historically, buckling failures have in Eq. (3). The intent of utilizing λ is to allow the engineer to

occurred at calculated compressive stresses significantly less account for the effects of imperfections on the "actual" critical

than the ultimate compressive stress of the given shell material. buckling load of a given structure.

,0 (5a)

(3) ,0 2 (5b)

In general, two approaches can be used for determining the The length and inside radius of the cylindrical shell are

critical buckling load of a cylindrical shell: deterministic represented by L and R. The first half range cosine series

methods representing a host of closed form solutions, and summation term in Eq. (4) denotes the axisymmetric part of the

stochastic methods that employ any number of statistical imperfection and the second half range sine series summation

parameters. While the deterministic approach carries out term denotes the non-symmetric portion of the imperfection.

analysis on the basis of some physical laws, stochastic (or The axisymmetric imperfections as derived from Eq. (4) are

probabilistic) methods attempt to mimic several unknown given by Eq. (6):

factors (including the imperfection profile, for instance) that can

affect the critical buckling load or the given shell. Deterministic ∑ cos (6)

approaches do not include perturbations in the shell wall

thicknesses, which are admittedly known to exist in practice. In

the present analytical study, a stochastic approach is employed The initial asymmetric imperfections are represented by a

in an attempt to predict the probability of a given critical double Fourier sine series. To determine the non-dimensional

buckling load within a defined confidence interval. This critical buckling load, calculation of Fourier coefficients must

approach was previously presented by the authors [3, 4] for a first be completed. Fourier coefficients Ckl and Dkl, as

series of shells subjected to an external pressure. represented in Eq. (7), have to be determined in order to

The example cylindrical shell considered in this study is a represent the initial imperfections in a simulation of a number of

fractionating tower with a 14 ft. I.D., 21 ft. long bend line to shell geometries utilized in this study, herein after referred to as

bend line, fitted with fractionating trays, and designed for an the "GSB shells".

external design pressure of 15 psi at 700°F. The tower material

of construction is assumed to be SA-285, Gr. C carbon steel. , ∑ ∑ sin cos sin (7)

ASYMMETRIC VARIABLE WALL THICKNESS The particular cylindrical shell portion of the fractionating

Determination of the critical buckling load for a shallow tower has an internal diameter of 4.27 m, a height of 6.4 m, and

cylindrical shell containing small asymmetric thickness nominal wall thickness of 7.9375 mm (0.3125 in.). The

variations while subjected to an external pressure is performed significant geometric and material parameters for the

by a non-classical technique in the current study. This analysis fractionating tower are defined in Table 1 below:

was accomplished employing a reliability approach that

simulated of a number of shells using the Monte Carlo TABLE 1

GEOMETRIC AND MATERIAL PARAMETERS

technique, calculation of critical buckling loads using the multi-

mode method [5], and calculation of the non-dimensional

Nominal shell wall thickness, t 0.3125 in.

buckling load (λ) based on the reliability function. Similar to the

Stack length, L 39 in.

method described by Elishakoff et al. [6], any initial

Inside shell radius, R 84 in.

imperfection can be represented by series of trigonometric

Young’s Modulus, E 30 x 106 psi

functions, such as in a Fourier series. A review of the

previously defined work [6] revealed a multitude of errors and Poisson’s ratio, 0.31

omissions in the formulations and figures as published in the

open literature, thereby creating the necessity to revisit the bases Initial imperfections in the Fractionating tower of the

for the results as described and documented by the current current work are in the form of shell wall thickness variations.

authors [3]. With this in mind, as given by Elishakoff and In an effort to calculate the external working pressure, 20

Arbocz [7] and Arbocz and Williams [8], the initial imperfection cylindrical shells (identified as GSB1 through GSB20 shells)

function Wn(ξ,θ) can be represented as shown in Eq. (4) below: were simulated using a random number generator of a

commercially available symbolic math program. The software

sin cos utilizes the linear congruence method for generation of random

, ∑ cos ∑ ∑ (4) numbers. Consistent with the ASME Boiler and Pressure Vessel

sin sin

Code [9], “The reduction in thickness shall not exceed 1/32 in.

The chosen coordinate system for the cylindrical shell (1mm) or 10% of the nominal thickness of the adjoining surface,

utilizes axial (x) and circumferential (y) coordinates. In whichever is less,” the simulated shell wall thicknesses were

addition, ai, bkl, ckl are Fourier coefficients (multipliers) of the confined within a range that varied between 0.313 in. and 0.281

respective trigonometric terms. Equation (5) shows the in. These values were further consistent with the assumption

relationship for the non-dimensional coordinates ξ and θ in the that out-of-tolerance dimensions would be detected and

axial and circumferential directions, respectively. corrected as appropriate in a quality inspection and assurance

program of any reputable vessel manufacturer. A total of 144

"readings" were generated for each simulated shell; 12 readings BUCKLING LOAD MAPS & MODE COUPLING

axially and 12 circumferentially at each chosen elevation along

Buckling load maps consist of the predicted critical

the longitudinal direction. Table 3 (located at the end of this

buckling loads for different mode combinations of wave

paper) displays the simulated and generated shell wall thickness

numbers in the axial and circumferential directions. The

values for the GSB10 shell.

buckling loads were calculated for a perfect cylindrical shell

The shell wall thickness values thus generated represent

subjected to external pressure by first employing solutions to the

asymmetric imperfections (with respect to the circumferential

classical simply supported boundary conditions as previously

direction of the cylinder) and can be converted into

identified by Donnell [1] in Eq. (1). The imperfections (i.e.,

axisymmetric imperfections by taking arithmetic mean of all

wall thickness variations) are assumed to follow the double

values at a particular elevation. For example, Fig. 1 shows the

Fourier sine series as sown in Eq. (9). The load maps are then

asymmetric variations in the shell wall thicknesses for the

used to determine the dominant mode shape.

simulated GSB10 shell. These shell wall thickness variations

were transformed into axisymmetric form as shown below in

Fig. 2. sin cos sin sin (9)

between one axisymmetric mode with wave number (i,0) and

two asymmetric modes with wave numbers (k,l) and (m,n) will

occur, if the relationships i=׀k±l ׀and l=n are satisfied. For the

case of one axisymmetric (i,0) and one asymmetric (k,l) mode,

the coupling conditions reduce to the single relation i=2k. The

coupling between three asymmetric modes with wave numbers

(k,l), (m,n) and (p,q) will occur if the relations k+m+p= odd

integer and q=׀l±n ׀are satisfied. If all these coupling conditions

are satisfied, then the resulting critical buckling load of the shell

is generally lower than the buckling load were each mode to be

considered separately. The use of the former conditions results

in an 8-mode failure coupling that is ultimately employed in the

determination of the predicted critical buckling load. This mode

coupling is clearly depicted in Fig. 3 as shown below:

FIG. 1 RANDOM SHELL WALL THICKNESS (GSB10 SHELL)

Eq. (6) from which the respective Fourier coefficients were

calculated. The initial asymmetric imperfections are represented (2,0) (1,5) + (2,6) + (2,8)

by Eq. (7) and substitution of the specific range over which the + +

summation must be performed yields Eq. (8), from which the (1,6) (2,4)

respective Fourier coefficients were calculated. + +

(1,11) (1,10)

, ∑ ∑ sin cos sin (8)

FIG. 3 8-MODE COUPLING TREE FOR GB SHELLS

Donnell’s equilibrium based partial differential equation as

given by Eq. (1) was used for calculating and predicting the

critical buckling load of GSB shells subjected to external

pressure. The initial imperfection in the shell wall thickness, w,

is assumed to follow the 8 coupled buckling modes as

graphically defined in Fig. 3. It is this combination of

deformation modes that appear to create the minimum critical

buckling load for the simulated shells under the present

consideration for the external pressure load case.

A vertical bar graph depicting the number of buckled shells

for discrete ranges of the non-dimensional buckling load (λ) is

shown in Fig. 4. This histogram was employed in the

calculation of the reliability function from which the empirical

FIG. 2 AXISYMMETRIC WALL THICKNESS (GSB10 SHELL) value of the non-dimensional buckling load is determined for the

series of simulated shell geometries previously described herein.

maximum allowable external working pressure for the assumed

shell thickness of 0.3125 in. The steps in the Code calculations

are repeated below for the reader as follows:

STEP 1

For the assumed shell thickness (t) of 0.3125 in. and outside

diameter (D0) of 168.625 in., calculate ratio’s (L/D0) and (D0/t)

39

0.231

168.625

168.625

540

0.3125

STEP 2

Enter Fig. G at the value of L/D0 = 0.231; move horizontally to

the D0/t line of 540 and read the value of A of 0.0005.

STEP 3

FIG. 4 HISTOGRAM OF FOR 20 SIMULATED SHELLS Enter Fig. CS-2 at the value of A = 0.0005 and move vertically

to the material line for 700°F. Move horizontally and read B

Figure 5 illustrates the reliability function for the 20 value of 6100 on ordinate.

simulated shells. The value of the non-dimensional buckling

load (λ) can be calculated at any desired reliability from this STEP 4

curve, e.g., for a reliability of 0.95, λ is equal to 0.86. The maximum allowable external working pressure for the

At the 95% reliability level, the non-dimensional buckling assumed shell thickness of 0.3125 in. is

load value of the simulated shells utilizing the Monte Carlo 4 4 6100

technique, results in value of approximately 0.86, as shown in 15.1

3 3 540

Fig. 5. The classical critical buckling load for the predefined

fractionating tower subjected to external pressure as given by

Eq. (2) is 21.163 psi. Since, Pa is greater than the external design pressure P of 15 psi,

the assumed thickness is satisfactory.

The results presented herein clearly indicate that the effect

of shell wall thickness variation on buckling load deserves

special attention. Thus, in the absence of initial geometric

imperfections, this particular kind of thickness variation may

constitute the most important factor in the predicted buckling

load reduction. Under the current parameters of the work as

described in the preceding paragraphs, was determined to be

0.86 while employing the Monte Carlo technique. The results

clearly indicate that the mere presence of shell wall thickness

variations as a result of non-repeatability in any particular

manufacturing process (even within industry accepted tolerance

limits), that the load carrying capacity of the shell decreases by

approximately 14%. Furthermore, it can be concluded that

FIG. 5 RELIABILITY FUNCTION v. imperfections in shell wall thickness within the defined

tolerance limits have been sufficiently considered within the

ASME B&PV CODE SECTION VIII, DIVISION 1 ASME Code [10] as has been demonstrated in example L-3.1.

CALCULATION OF EXTERNAL PRESSURE The load carrying capacity must be reduced due to the

known presence of imperfections. The results obtained from the

The maximum allowable external working pressure evaluation of the non-dimensional buckling load when the

calculations are performed in ASME B&PV Code [10] as shown fractionating tower is subjected to external pressure while

in Appendix L, example L-3.1. The prescribed design employing the Monte Carlo technique and the ASME B&PV

information is related to a fractionating tower of 14 ft. inside Code [10] are shown in Table 2. For the simulated shell

diameter and designed for an external design pressure of 15 psi geometries considered in the present study, which are subjected

at 700F. The tower is assumed to be fabricated from SA-285, to external pressure, the non-dimensional buckling load becomes

Gr. C carbon steel. The design length is 39 in. Assuming a shell 0.86. The Monte Carlo technique considering asymmetric

wall thickness, t = 0.3125 in., the Code [10] calculates the imperfections results in a 14% decrease from the classically

computed values described in Eq. (2). In contrast, the load REFERENCES

carrying capacity of a shell under external pressure must

1. Donnell, L. H., 1934, “A New Theory for the Buckling of

decrease by 20% according to ASME Boiler and Pressure Vessel

Thin Cylinders Under Axial Compression and Bending,"

Code, Section VIII, Division 1 rules [10].

Transactions of the ASME, Aeronautical Engineering,

There is an obvious difference of approximately 3.2 psi in

AER-56-12, pp. 795-806, ASME, New York.

the working external pressure for the two methodologies. The

2. Saunders, H. E., and Windenberg, D. F., 1931, "Strength of

far more conservative results in the case of the ASME Boiler

Thin Cylindrical Shells Under External Pressure”,

and Pressure Vessel Code [10] approach may be due to the fact

Transactions of the ASME, 53(15), p. 207, ASME, New

that the Code has adopted a deterministic approach based upon

York.

empirical relations developed and published by various pressure

3. Brar, G. S., Hari, Y., and Williams, D. K., 2009, “Fourier

vessel design engineers based upon the results and experience

Series Analysis of a Cylindrical Pressure Vessel Subjected

gained by testing of cylindrical shells subjected to external

to External Pressure,” PVP2009-77854, ASME 2009

pressure. The Code does not employ specific measures for

Pressure Vessels and Piping Conference, ASME, New

addressing the effects of thickness perturbations throughout a

York.

given shell geometry subjected to external pressure. In contrast,

4. Brar, G. S., 2009, “Buckling Load Predictions in Pressure

the Monte Carlo simulations allow various statistical matching

Vessels Utilizing Monte Carlo Method”, Ph.D. Thesis,

procedures to specifically address any given range of geometric

University of North Carolina at Charlotte, USA.

parameters when seeking a solution to the equilibrium based

5. Arbocz, J. and Babcock, C. D., 1976, “Prediction of

differential equation long ago defined by Donnell [1].

Buckling Loads Based on Experimentally Measured Initial

In accordance with the rules of the ASME Boiler and Pressure

Imperfections”, Buckling of Structures, Budiansky B., ed.,

Vessel Code [10], the empirical relations developed account for

IUTAM Symposium, Cambridge, Mass., 1974, Springer

shape imperfections must be applied to the allowable stresses

Verlag, Berlin, pp. 291-311, New York.

utilized in the design calculations for external pressure. The

6. Elishakoff, I., Li, Y., and Starnes, J. H., Jr., 2001, Non-

calculation steps as reproduced in the previous section of this

Classical Problems in the Theory of Elastic Stability,

paper are used to calculate the external working pressure for un-

Cambridge University Press, Cambridge, UK.

stiffened cylindrical vessels in accordance with the ASME Code

7. Elishakoff, I. and Arbocz, J., 1985, "Reliability of Axially

[10] and are contained in Table 2 below.

Compressed Cylindrical Shells with General Non-

symmetric Imperfections," Journal of Applied Mechanics,

TABLE 2

52, pp. 122-128, ASME, New York.

WORKING PRESSURES FOR FRACTIONATING TOWER 8. Arbocz, J. and Williams, J. G., 1977, "Imperfection Surveys

on a 10 ft. Diameter Shell Structure," AIAA Journal, 15, N.

ASME B&PV Code 15 psi 7, pp. 949-956, Reston, VA.

Asymmetric (Monte Carlo) 18.2 psi 9. ASME, 2001, ASME Boiler & Pressure Vessel Code,

Section VIII, Division 2, American Society of Mechanical

Engineers, New York.

10. ASME, 2007, ASME Boiler & Pressure Vessel Code,

Section VIII, Division 1, American Society of Mechanical

Engineers, New York.

TABLE 3

SHELL WALL THICKNESS PROFILE OF GSB10 SHELL

θ\L(in.) 0.75 3.25 5.75 8.25 10.75 13.25 15.75 18.25 20.75 23.25 25.75 28.25

0 0.293 0.287 0.294 0.298 0.294 0.296 0.300 0.306 0.288 0.283 0.310 0.309

30 0.287 0.292 0.288 0.300 0.288 0.289 0.300 0.290 0.306 0.295 0.296 0.298

60 0.297 0.283 0.284 0.310 0.281 0.306 0.309 0.290 0.305 0.290 0.305 0.295

90 0.294 0.291 0.292 0.289 0.306 0.295 0.295 0.305 0.294 0.300 0.294 0.284

120 0.304 0.283 0.284 0.310 0.293 0.288 0.309 0.283 0.298 0.311 0.287 0.284

150 0.294 0.308 0.297 0.301 0.300 0.296 0.286 0.312 0.291 0.294 0.286 0.288

180 0.313 0.301 0.306 0.302 0.307 0.283 0.293 0.282 0.309 0.302 0.295 0.289

210 0.293 0.303 0.302 0.307 0.290 0.287 0.310 0.303 0.307 0.306 0.306 0.308

240 0.289 0.297 0.295 0.300 0.292 0.313 0.296 0.295 0.295 0.292 0.297 0.299

270 0.310 0.303 0.309 0.303 0.296 0.308 0.282 0.285 0.300 0.286 0.286 0.307

300 0.295 0.305 0.292 0.286 0.297 0.299 0.299 0.293 0.312 0.293 0.288 0.284

330 0.291 0.299 0.284 0.281 0.298 0.303 0.307 0.307 0.298 0.293 0.312 0.287