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*Corresponding author. Tel.: #416-978-5741; fax: #416-978-7753.E-mail address: [email protected] (S.A. Meguid).

International Journal of Mechanical Sciences 43 (2001) 1229}1242

Three-dimensional "nite element analysisof saddle supported pressure vessels

N. El-Abbasi, S.A. Meguid*, A. Czekanski

Engineering Mechanics and Design Laboratory, Department of Mechanical and Industrial Engineering,University of Toronto, 5 King's College Road, Toronto, ONT, Canada M5S 3G8

Received 16 November 1999; received in revised form 10 February 2000

Abstract

A three-dimensional "nite element analysis is made of a pressure vessel resting on #exible saddle supports.The analysis is carried out using a newly developed thick shell element and accounts for frictional contactbetween the support and the vessel. The seven-parameter shell element is capable of describing the variationof the stress and the strain "elds through its thickness. A variational inequalities based formulation is utilisedfor the accurate description of this class of frictional contact problems. Di!erent pressure vessel con"gura-tions are considered and the resulting contact stresses are examined. The e!ect of saddle radius, saddle width,plate extension, and support overhang on the resulting stress "eld in both vessel and support are evaluatedand discussed. ( 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Saddle supports; Pressure vessels; Contact; Thick shell elements

1. Introduction

Saddle supports are commonly used to hold pressure vessels (Fig. 1). The design of saddlesupports is inexpensive, and provides an e$cient method of carrying the vessel. The pressure vesselcan either be freely standing on the saddle supports or can be welded to them. In this work, westudy the case of a freely standing vessel resting on two #exible saddle supports.

The interaction between the saddle supports and the vessel body is one of the problem areas inpressure vessel design, since it involves highly localised contact stresses. The highest stresses areusually located at the uppermost position of the saddle, called the saddle horn [1}3]. In order to

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0020-7403/01/$ - see front matter ( 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 0 - 7 4 0 3 ( 0 0 ) 0 0 0 6 0 - 6

Nomenclature

f S body forcesg gap functionh element thickness¸P, ¸

Olength of pressure vessel and overhang

N unit normal vectorN two-dimensional isoparametric shape functionsq quadratic through-thickness displacement functionR

P, R

Sradius of pressure vessel and saddle support

Sij

second Piola}Kirchho! stress tensortS surface tractionu displacement vectorV@

3director vector connecting top and bottom shell surfaces

V3

unit vector in direction of V@3

V1,V

2unit vectors perpendicular to V

3wS

width of saddle supportx position vectora1,a

2rotational degrees of freedom

a3,a

4thickness and thickness gradient degrees of freedom

eij

Green}Lagrange strain tensorhS, h

Esaddle and saddle plate extension angles

m, g, f intrinsic variablesk coe$cient of friction

Subscripts and superscripts

B bottom-shell surfaceC contactM middle surfaceN normal componentt timeT top-shell surface or tangential component

alleviate this stress, several design modi"cations can be made. One of the commonly adoptedmodi"cations involves increasing the radius of the saddle. This introduces a gap between thesupport and the vessel, which permits the loaded vessel to deform radially without restraint.Consequently, the pinching e!ect of the support at the saddle horn is reduced. The saddle supportshould also be #exible in the longitudinal direction to avoid creating high localised stresses at itsedges. Accordingly, a wide saddle plate is usually welded to a thinner base, as depicted in Fig. 1. Analternative design modi"cation involves reducing the sti!ness of the support in the vicinity of thesaddle horn [3].

1230 N. El-Abbasi et al. / International Journal of Mechanical Sciences 43 (2001) 1229}1242

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Fig. 1. A schematic of pressure vessel and saddle supports.

The ASME and BS5500 pressure vessel codes [4,5] do not provide comprehensive details for thedesign of #exible saddle supports. However, a few references are listed which give some guidance.The most commonly used Refs. [1,6] propose a semi-empirical analysis technique based on beamtheory and on assuming that the vessel cross-section remains circular under load. However, moreaccurate analyses, based on cylindrical shell theory, are available [3,7,8]. The resulting system ofequations is commonly treated using double Fourier series expansion.

A limited number of attempts have been devoted to the "nite element analysis of saddlesupported pressure vessels. Most of them focused on the study of welded saddle supports, which aresimpler to treat due to the absence of contact nonlinearities. Perhaps the earliest "nite elementsimulation was due to Stoneking and Sheth [9]. They performed a linear two-dimensional analysisof a welded saddle in which both the vessel and support geometry were excessively simpli"ed. Thevessel head was modelled using beam elements and the supports were assumed rigid in both radialand longitudinal directions. Widera et al. [10] also performed a three-dimensional "nite elementanalysis of pressure vessels, but with welded supports.

Wilson and Tooth were the "rst to study yexible saddle supports using the "nite element method[2]. They used cylindrical shell theory to model the pressure vessel and a two-dimensional "niteelement program to simulate the #exible saddle. The solution of the shell problem was obtainedusing a double Fourier series expansion. Nash and Banks [11] used a standard penalty basedapproach to account for contact e!ects in sling-supported composite pressure vessels. Theirsolution was highly sensitive to the choice of the user-de"ned penalty parameter. A more accuratecontact formulation was presented by Bisbos et al. [12] for horizontal pipes loosely resting onsaddle supports. The unilateral contact conditions and Coulomb's friction law were used toformulate two linear complementarity problems in the normal and tangential directions. However,the solution of the complementarity problems was obtained using a Fourier series expansiontechnique, commonly applied to piping problems.

In the present work, we present a comprehensive three-dimensional "nite element analysis of#exible saddle-supported pressure vessels. A recently developed thick shell element [13], whichaccounts for the normal stresses through the shell thickness, and involves seven degrees of freedomper node, was used to model the pressure vessel. In order to accurately describe the frictional

N. El-Abbasi et al. / International Journal of Mechanical Sciences 43 (2001) 1229}1242 1231

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Fig. 2. Geometry and degrees-of-freedom of shell model.

contact problem, a variational inequalities based formulation was used [14}16]. A one-stepsolution to the variational inequality problem was adopted and the contact constraints wereaccurately imposed using Lagrange multipliers.

2. Theoretical developments

2.1. New thick shell element

Contact plays a fundamental role in the deformation behaviour of thin structures. However,commonly used shell elements involve basic assumptions which are not appropriate for theaccurate description of contact problems. Speci"cally, they do not account for the variation in thedisplacements and stresses in the transverse direction, and do not allow for double-sided contact.These restrictions severely hinder the accuracy of the solution, especially for moderately thickplates and shells [17]. Recently, El-Abbasi and Meguid [13}15] developed a thick shell elementwhich is capable of handling contact problems involving shell structures. In this section, however,we present the features of the new element that are relevant to the current pressure vessel supportproblem.

Consider the shell model shown in Fig. 2 in which a pair of points xT

and xB, that make up the

top and bottom faces of the element, are connected through a director vector V@3. The geometry of

the element can be expressed in terms of the mid-surface nodal coordinates, the director V@3, and

a quadratic function q, as follows:

tx" txM#

f2

tV@3#(1!f2) tq tV@

3(1)

The quadratic term tq, which is initially zero, is necessary to describe a complete linearly varyingstrain "eld through the shell thickness [18]. To obtain the displacement "eld, the followingincremental decomposition is used:

t`*txM" tx

M# tu

M,

t`*tV@3" t`*th t`*tV

3,


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t`*tV3" tR( ta

1, ta

2) tV

3,

t`*th" th# ta3,

t`*tq" tq# ta4, (2)

which results in seven degrees of freedom per node; viz

Mu1

u2

u3

a1

a2

a3

a4NT. (3)

The "rst three are the incremental translational components of the shell mid-surface, uM. The shell

director is decomposed into the product of a unit director V3

and a scalar thickness h. Thisdecoupling of rotational and extensional deformation avoids ill-conditioning in the thin shell limit[19]. The unit director V

3involves two rotations a

1and a

2about axes V

1and V

2(perpendicular to

V3) and is updated using orthogonal matrices for "nite rotations. The degree of freedom a

3repres-

ents the change in the shell thickness in the direction of V3

and a4, which is a quadratic transverse

displacement component, is also in the direction of V3.

By including a3

and a4

in the element formulation, a full linear variation of the strain throughthe thickness is allowed. This, in turn, enables the use of the standard 3D-continuum constitutiverelationship, without imposing the plane stress condition commonly applied to shells [18]. Thelarge rotation relationships of Eq. (2) are used to update the shell coordinates and the shell director.However, linearization with respect to the degrees of freedom, including linear and quadratic terms,is required for the development of a consistent tangent sti!ness matrix.

A four-noded element was developed based on the thick shell model. An assumed covarianttransverse shear strain interpolation was used in order to avoid shear locking [20]. In addition, theelement employed standard interpolation for the membrane and bending terms. In order to avoidthickness locking, an interpolation technique that preserves the magnitude of the director wasimplemented [13]

x" +%-%.

Ni(m, g)xi

M#

f2

1DV

3D+%-%.

Ni(m, g)hi Vi

3#(1!f2)

1DV

3D+%-%.

Ni(m, g)q thi Vi

3, (4)

where the modulus of the discretized director can be expressed as

DV3D"J(+N

k(m, g)<k

31)2#(+N

k(m, g)<k

32)2#(+N

k(m, g)<k

33)2. (5)

2.2. Variational inequality formulations

The contact formulations are governed by two inequality constraints: (i) the normal contactstress must be less than or equal to zero, and (ii) the displacements of the contacting surfaces mustnot result in inter-penetration. In addition, the tangential forces and displacements along thecontact surface are assumed to be governed by Coulomb's friction law.


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Fig. 3. Kinematic contact constraint for shell surfaces.

For every point on the shell surface !., a corresponding closest point on the target surfaces y can

be determined from the kinematics of the deformation (Fig. 3). This is de"ned for x3!., such that

DD tx!yH( tx) DD"miny|!S

DD tx! ty DD, s.t. N( tx) )N( ty)(0, (6a)

where N is a normal vector. The constraint prevents improper contact between the shell surfaces.The gap functions for the shell contact surfaces can then be de"ned as

t`*tgT@B(x)"[yHT@B

( t`*tx)! t`*txT@B

] ) t`*tNT@B*0, (6b)

where T and B refer to the top and bottom shell surfaces, respectively.Most existing contact formulations are based on the classical variational methods, see e.g. Ref.

[21]. However, frictional contact problems can be more accurately formulated in terms ofvariational inequalities (VI) [22]. The use of variational inequalities for modelling contact in thinstructures, however, has not been given its due attention. Ohtake et al. [23] were concerned withthe development of variational inequalities to treat contact in plate elements using von Karmanplate theory. El-Abbasi and Meguid [13}15] developed and implemented variational inequalitiesbased contact formulations for shell structures. In this section, the pertinent features of theseformulations are presented.

The general variational inequality formulation for frictional contact problems can be expressed,in a total Lagrangian framework, in terms of the contravariant second Piola}Kirchho! stress


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tensor S3 ij and the covariant Green}Lagrange strain tensor e8ij, as

a( tu, tv! tu)#j( tu, tv)!j( tu, tu)*f ( tv! tu) ∀ tv3K, (7)

where

a(u, v)"P0)

t`*t0S3 ij(u) t`*t

0e8ij(v) d 0),

f (v)"Pt`*t)

t`*tfB ) vd)#Pt`*t!t

t`*ttS ) v d!,

j(u, v)"Pt`*t!C

kDpN(u)D Dv

TD d!.

In the above expression, u represents the equilibrium con"guration and v represents any admissibledisplacement "eld. The a(u, v}u) term represents the virtual work of the elastic resistance ofdeformation. The f (v}u) term represents the virtual work done by the external forces, whilej(u, v)}j(u, u) is the contribution of the frictional forces. K is the space of all displacements for thepoints in the domain which satisfy the kinematic contact and boundary conditions.

Solution techniques for the variational inequality formulation of Eq. (7) are commonly obtainedby assuming that the normal stress within each time step is independent of the displacement"eld u [22]. In addition, the non-di!erentiable frictional term is regularized. This results ina di!erentiable symmetric tangent sti!ness matrix, which enables the use of standard "nite elementsolvers and signi"cantly decreases the computational requirements. The resulting formulations arefree from user-de"ned parameters that adversely a!ect the accuracy of solution, especially in shellstructures.

A consistent linearization of the general variational inequality formulation is essential formaintaining quadratic convergence. The linearized incremental total Lagrangian VI formulationcan be expressed as [15]

aw( t*u, tv)#SDjeq( t*u), tvT*Rw ( tv) ∀ tv3K, (8)

where q is the displacement relative to the con"guration corresponding to sticking friction, and w isthe total displacement. R(v), a(u, v) and Dj(u, v) include all the respective terms that contribute tothe loading vector, the element deformation, and the frictional forces [13]

aw(u, v)"P0)

t0C3 ijrs

0e8 L34(u)

0e8 Lij(v) d0)#P

0)t0S3 ij(w)uL

k,ivLk,j

d0)#P 0)t0S3 ij(w)SDe8 Q

ij(u),vTd0),

(9)

Rw (v)"Pt`*t)

t`*tfB ) vd)#Pt`*t!

t`*ttS ) vd!!P0)

t0S3 ij(w)

0e8 Lij(v) d0), (10)


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Table 1Details of geometry of pressure vessel and supports.

Vessel radius RP

2.0 m Vessel thickness 25.0 mmVessel length ¸

P40.0 m Vessel material Steel

Saddle overhang ¸O

4}12 m Saddle plate width wS

0.5}3.0 mSaddle angle h

S753 Saddle plate thickness 50.0 mm

Saddle radius RS

2.0}2.1 m Saddle material SteelCoe$cient of friction k 0.3 Plate extension h

E0}153

Fluid Water Fluid level Full

SDjeq(u), vT"G P t!C

ukDpNDA

MDq

TD!

qT?q

TDq

TD3 B v#

qT

DqTDkDp

NDvd t! for Dq

TD'e,

P t!C

uMe

kDpNDv#

qTe

kDpNDvd t! for Dq

TD)e,

(11)

where the M matrix isolates the tangential component of the displacement and e is a regularizationparameter. Superscripts L and Q refer to the strain component resulting from linear and quadraticdisplacement terms, respectively. Detailed expressions for the di!erent terms in Eqs. (9)}(11) areprovided in Ref. [13].

3. Results and discussion

In this section, we provide a detailed analysis of saddle-supported pressure vessels for variousvessel and saddle geometries. Table 1 gives the details of the geometry and loading conditionsconsidered, which are similar to those used in Refs. [3,7,9,10]. Due to symmetry, a quarter of thepressure vessel and saddle support were modelled (Fig. 4). The new seven-parameter shell elementwas used to model the pressure vessel, while the saddle, which is thicker and sti!er, was modelledusing solid elements. Contact and friction were accounted for using the variational inequalitiesformulation. Geometric nonlinearities resulting from large shell rotations were also taken intoaccount. Several trial runs were performed to determine an optimal mesh density. Attention wasthen devoted to studying the e!ect of the following parameters: (i) the saddle radius R

S, (ii) the

saddle plate extension hE, (iii) the saddle plate width w

S, and (iv) the overhang length ¸

O. In view of

their importance to the mechanical integrity of the vessel, the analysis focused on the hoop stressesnear the saddle support.

Fig. 5 shows the hoop stresses at the outer surface of the vessel for four saddle radii. A support ofthe same radius as the pressure vessel (R

S/R

P"1) results in high compressive hoop stresses at the

saddle horn and a smaller tensile region directly above that horn. Increasing the support radiusslightly (R

S/R

P"1.01}1.02) leads to a large reduction in the compressive stresses and an increase

in the tensile stresses. For both cases, the magnitude of the maximum stress is 50% less than thatcorresponding to R

S/R

P"1. A large saddle radius (R

S/R

P"1.05) results in a smaller support area,

leading to high tensile stresses over the saddle horn. A saddle ratio of 1.02 provides the least hoop


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Fig. 4. FE model of pressure vessel and saddle supports.

Fig. 5. E!ect of saddle to pressure vessel radius ratio RS/R

Pon the hoop stresses at the support.

stresses in the saddle region. These results are in agreement with the experimentally measured stressvalues of Ref. [2].

Fig. 6 shows the normal contact stress (interface pressure) acting on the vessel in the saddleregion for two saddle radii. The results show that for R

S/R

P"1.01 (and smaller values), the highest

contact stresses are concentrated near the saddle horn. For RS/R

P"1.02 (Fig. 6(c)), the maximum

stresses are closer to the saddle base. In both cases, the saddle horn is still in contact. Larger valuesof support radii result in higher interface pressures at the base, and total separation at and near thesaddle horn. Furthermore, it is evident that the stresses are not constant along the saddle width.Hence, a two-dimensional analysis cannot yield accurate results. Note that the saddle also


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Fig. 6. (a) Candidate contact surface between vessel and support, contact stress distribution for; (b) RS/R

P"1.01; and

(c) RS/R

P"1.02.

experiences tangential contact stresses due to friction. These frictional stresses were signi"cantlylower than the normal stresses.

Saddle plate extensions hE

of 0, 5, 10 and 153were examined, for a saddle-to-vessel radius ratio of1.0. The resulting hoop stresses at the outer surface of the vessel are shown in Fig. 7. The plateextension reduces the pinching e!ect at the saddle horn, which consequently leads to a reduction inthe maximum hoop stress. Saddle plate extensions of 5 and 103 result in approximately 25 and 40%reduction in maximum stress, respectively. However, a long unsupported plate extension su!ersfrom high localised stresses at its base. Since the saddle extension resembles a curved cantileverbeam, the stress at its root is proportional to the cube of its length. This localised bending stress inthe plate exceeds the hoop stress in the vessel for the case when h

E"153. Accordingly, a plate

extension of 5}103 is preferable for the selected geometry.The saddle plate width was also varied from 0.5 to 3.0 m for a pressure vessel with R

S/R

P"1.02.

The resulting hoop stresses at the outer surface of the vessel are shown in Fig. 8. Increasing theplate width from 0.5 to 2.0 m (w

S/R

P"0.25}1.0) results in a three-fold reduction in the stresses at

the supported region of the vessel (h"0}753). Increasing the plate width to 3.0 m and beyond doesnot a!ect the stresses in the vessel. However, the stresses in the saddle vary considerably with the


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Fig. 7. E!ect of saddle plate extension hE

on the hoop stresses at the support.

Fig. 8. E!ect of saddle plate width wS


plate width. For values greater than 2.3 m (wS/R

P"1.15), the saddle stresses exceed those in the

vessel.Finally, we examined the e!ect of the overhang ratio ¸

O/¸

Pon the stress state. According to

Ref. [24], this ratio should not exceed 0.25. Based on beam theory, an overhang of 0.195, whichminimises the longitudinal bending moments, was suggested in Ref. [10]. Fig. 9 shows thelongitudinal stresses acting on the outer surface of the vessel at h"03 for di!erent support


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Fig. 9. E!ect of saddle placement ¸O

on the longitudinal stresses at h"03.

Fig. 10. E!ect of saddle placement ¸O


locations for RS/R

P"1.02. The results indicate a preferable range of ¸

O"4}6 m which corres-

ponds to ¸O/¸

P"0.1}0.15. The resulting longitudinal stresses (and bending moments) are signi"-

cantly di!erent from the simpli"ed beam theory calculations. The e!ect of the saddle location onthe hoop stresses is shown in Fig. 10. Similar values for the maximum hoop stresses are obtainedfor ¸

O"4, 6 and 8 m, while ¸

O*10 m leads to higher stresses. These high stresses can be

attributed to the pinching e!ect at the saddle horn, caused by large vessel deformation. Since thevessel deformation is maximum at the centre, locating the saddles close to this region (¸

O*10 m)

subjects them to greater deformation leading to higher stresses.


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4. Concluding remarks

A detailed analysis of the contact stress state in saddle supported pressure vessels was conductedusing a new four-noded thick shell element, which accounts for the normal stress through the shellthickness. The element employs an assumed strain description to prevent shear locking and anenhanced director interpolation to avoid thickness locking. A variational inequalities basedformulation was also used to accurately model frictional contact in shells. Attention was devoted tostudy the e!ect of saddle radius, saddle width, plate extension and saddle overhang on the resultingstress "eld. The results, for the selected vessel and support geometries, reveal that a saddle radius1}2% larger than that of the vessel leads to a 50% reduction in stresses. A saddle plate extension of5}103 also leads to a reduction in the stress level in both vessel and support by 25}40%. Theoutcome of the study also indicates that for the selected dimensions, the saddle plate width shouldbe within the range 1.0}2.0 m, in order to reduce stresses in both saddle and support. Finally,the optimal horizontal locations of the saddles was found to correspond to an overhang ratio¸O/¸

P"0.1}0.15.

Acknowledgements

The "nancial support provided by ALCOA, ALCOA Foundation of the USA and NaturalSciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged.

References

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