Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 1, pp. 59-68, Warsaw 2015
DOI: 10.15632/jtam-pl.53.1.59

THE EFFECT OF MANHOLE SHAPE AND WALL THICKNESS ON

STRESS STATE IN A CYLINDRICAL PRESSURE VESSEL

Jerzy Lewiński

Poznan University of Technology, Institute of Applied Mechanics, Poland

e-mail: jerzy.lewinski@put.poznan.pl

The paper describes attempts to find advantageous shape and configuration of a manhole
located in the upper part of a cylindrical pressure vessel.Manholes of circular and elliptical
cross sectionshavebeen considered.Themanhole is placedvertically and symmetricallywith
respect to the cylindrical vessel. Three variants of the manhole walls have been taken into
account. In thefirst variant, themanholewall reachesonly the cylinder surface.Themanhole
walls in two other variants are elongated inside the cylindrical vessel to various lengths. In
the case of ellipticalmanholes two other variantswere been considered, i.e. with theirmajor
axes situated in parallel or perpendicularly to the axis of symmetry of the cylindrical vessel.
Results of the numerical calculation are presented in figures and tables.

Keywords: pressure vessel, manhole, manhole-vessel strength

1. Introduction

The problems related to pressure vessels, as being very important for safety and reliability of
these structures, are extensively considered and investigated by many authors. Many works
are devoted to optimal shape of the vessel. Special attention is paid to the junction between
the vessel and its head since this region is usually subject to considerable stress concentration
due to the edge effect arising there. This aspect was considered by Ventsel and Krauthammer
(2001) who delivered amonograph presenting the strength and stability problems of plates and
shells with the edge effect of cylindrical shells. In the paper by Błachut andMagnucki (2008), a
review of strength, static stability and structural optimization of horizontal pressure vessels is
presented.

Magnucki andLewiński (2000) described the stress state arising in an untypical torispherical
head composed of circular and polynomial parts.Magnucki and Lewiński (2003) presented opti-
mal design of the ellipsoidal headwith consideration of various thicknesses of the shell.Magnucki
et al. (2004) presented optimization of ground-based cylindrical tanks with consideration their
strength and stability. Lewiński and Magnucki (2010) considered a dished head of a pressure
vessel subject to internal uniform pressure. They used a trigonometric series in order to find
proper shape of the profile. A similar approach to the shape of a dished head of the pressure
vessel was presented by the same authors (Lewiński andMagnucki, 2012). In order to design the
required profile of the head they used Bézier curves. Further development of the method was
demonstrated by Krużelecki and Proszowski (2012). They considered a larger range of shapes
of a vessel head approximated by convex Bézier polynomials or by functions with free parame-
ters. The optimal solutions were obtained with the use of the simulated annealing algorithm.
Magnucki et al. (2002) solved the problem of stress minimization of a vessel with an ellipsoidal
head.

Other regions of stress concentration occur in the vicinity of holes that must be placed in
the vessel shell. Such holes are designed for placing nozzles ormanholes in the vessel. According



60 J. Lewiński

to specific purposes, the holes may be located in the cylindrical part of the vessel or in its
head. These problems were considered by Carbonari et al. (2011) who presented integrated
optimization of the shape of an axisymmetric vessel provided with a nozzle. They minimized
the Huber-Mises mechanical stress in the whole vessel and, particularly, in the nozzle/vessel
junction. Giglio (2003) compared two different methods of designing vessel nozzles, included
in the ASME and VCR 1995 standards. He focused on the fatigue analysis of various pressure
vessel nozzles. Liu et al. (2001) sought the optimal shapes of intersecting pressure vessels. They
considered optimal profile of variable thickness nozzle connecting a spherical shell pressurevessel
to a cylindrical nozzle. They have shown that the design with a protruding nozzle gives a better
stress distribution than a flush nozzle.

The problems of proper quality of the manhole/vessel junction was considered by Frum
(1995a,b). Hedescribedattempts aimedat avoiding circumferential cracking of the reactor vessel
wall initiated in the vicinity of themanhole neck. Firstly, hemeasured temperature distribution
at the reactor jacket. Afterwards, he developed a program including four basic steps: reduction
of thermal stresses, elimination of discontinuities, improvement of material properties and im-
provement of the structure itself. Such an approach appeared to be very effective. Lianghai et
al. (1996) investigated the effect of thermal overload on strength of welded structures. Their
research pertained mainly to pressure vessels and the fittings that were welded to the vessels,
inclusive of manholes. They found that the thermal overload is an effective method for reducing
residual stress and the risk of vessel failure. Rao et al. (1997) discussed the reasons of failures
of stainless steel pressure vessels and components welded to them. They focused chiefly on im-
proper care during fabrication, welding, testing, and storage of the vessels, andmentioned such
factors as residual stress, sensitization, iron contamination and saline atmosphere that worsened
their strength. Bolton et al. (2002) considered the strength of Magnox reactor pressure vessel
and the pipingwelded to it under condition of irradiation. They investigatedmainly thewelding
technology used during the manufacturing process of the vessel. Borla (2002) presented the re-
sults of investigation of the effects of thermal fatigue, vibration and hydrogen attack onmaterial
and welds degradation in reactors. He considered possible repairs of the vessel and focused on
appropriate production, service and repair procedures.

The present paper considers a cylindrical pressure vessel equipped with a manhole. The
manhole is located vertically and symmetrically with respect to the vessel. In search of themost
advantageous shape and configuration of themanhole, many variants divided into separate sets
have been analyzed. It is assumed that the material of the manhole/vessel junction is of the
same quality as the one of thewhole vessel. Itmeans that thewelds should be perfectly laid and
smoothed.

2. Formulation of the problem

Vessels are usually provided with holes that may be located in various positions. The present
paper considers the strength of a pressure vessel subject to internal uniform pressure and equ-
ippedwith amanhole vertically penetrating the upper part of the cylindrical shell of the vessel.
The manhole axis intersects the axis of symmetry of the vessel. Only a dimensionless profile of
the vessel is considered with radius of the cylindrical part of the vessel equal to r =1. Dimen-
sionless thickness of the shell in the examples calculated here amounts to t=0.02. In the case of
a cylindrical vessel without any holes or nozzles, the equivalent dimensionless stress generated
in the shell by internal pressure is equal to σ̃eq =

√
3/2 = 0.866, according to Lewiński and

Magnucki (2010) . It is obvious that a manhole significantly disturbs the stress pattern in the
vessel shell. Hence, the present paper is aimed at finding such a shape and configuration of the
manhole as to minimize the stress concentration arising in the shell.



The effect of manhole shape and wall thickness on stress state... 61

Let us assume that the manhole is of circular cross section with its dimensionless diameter
equal to D = 2rm = 1. Its cover is placed at dimensionless height above the upper generatrix
of the vessel being equal to h= 0.24. (Fig. 1a). It is also assumed that the manhole is located
far enough from the vessel heads and, in consequence, the stress concentration arising in the
head/cylinder junction does not collide with the one resulting from the manhole. This allows
one to consider the case as a cylinder of infinite length with the manhole located anywhere in
it. Such a system is distinguished by the vertical plane of symmetry that allows taking into
account only its one-fourth part with a certain length of the cylindrical shell. Such an approach
requires imposing proper boundary conditions on themodel edges. The upper and bottom edges
of the vessel, as laying at the vertical lengthwise plane of symmetry, must not displace in the
lateral direction, i.e. ux = 0. On the other hand, the cross-section of the vessel, which divides
themanhole in half, should remain planar. Itmeans that uz =0 at this edge. The upper edge of
the manhole remains circular and, therefore, ux =uz =0. Additionally, the bottom vessel edge
should be restrained in the vertical direction, i.e. uy = 0. The boundary conditions are shown
in Fig. 1a.

Fig. 1. Model of one-fourth part of the vessel with a manhole; (a) geometry, dimensions and boundary
conditions marked at the edges; (b) the load: internal pressure and forces acting on the model

boundaries, stress concentration regions R1◦ and R2◦

The model is loaded with internal pressure. Its value is so adjusted as to ensure easy com-
parison of the system to its dimensionless version, i.e. to ensure dimensionless membrane stress
equal to σ̃eq =0.866 for the vesselwithout themanhole.Theupper edge of themanhole is loaded
with a vertical upward force equivalent to the pressure acting on the manhole cover. Similarly,
the lateral edge of the vessel, located in the left hand side of the model, is loaded with a force
caused by the vessel head, which is also subject to the pressure. The scheme of the load is shown
in Fig. 1b. The regions R1◦ andR2◦ denoted in Fig. 1b indicatemaximum stress concentrations
arising in various variants of the manhole considered in the present paper.

All the manhole variants have been analyzed with the help of the finite element method
by means of the educational version of the SolidWorks software. The meshes have been built
of solid tetrahedral elements (with four Gauss points). An example of the mesh is shown in
Fig. 2.Dimensions of the finite elements are significantly reduced in the region ofmanhole/vessel
junction to improve the accuracy of computation. In all variants considered for the purposes of
the present paper the meshes are similar, having about 650000 nodes and 450000 elements.



62 J. Lewiński

Fig. 2. Example of a part of the finite element mesh locally refined in the critical region

3. Numerical results

In the first basic variant, the side walls of the manhole reach exactly the vessel shell without
penetrating it, as shown in Figs. 1a and 1b. The manhole wall thickness is equal to the one
of the vessel. In this case, the maximum dimensionless equivalent stress amounts to 6.503 and
occurs in the region R1◦ (Fig. 3).

Fig. 3. Distribution of dimensionless equivalent stress in the region of manhole/vessel junction. The
manhole reaches the vessel shell, manhole wall thickness equal to the one of the vessel

In order to improve the situation, the manhole wall has been elongated to a dimensionless
length equal to L = 0.5 (Fig. 4) so as to protrude from the inner surface inside the vessel for
a moderate length. As a result, the manhole obtained form of a cylinder of length L intersec-
ting with the vessel. In this variant, the maximum dimensionless equivalent stress is equal to
σ̃eq =6.218, approximately in the same place as before. Hence, it gives only insignificant impro-
vement.
A better result has been obtained by rounding the sharp edges of the manhole/cylinder

junction. Dimensionless radius of the edge rounding is equal to ρ=0.03, and finally themanho-
le/vessel region assumed form shown in Fig. 4. Such an approach allows reducing themaximum
dimensionless stress down to σ̃eq =4.43, which occurs in the same location as before.
Further attempts to reduce the stress concentration consist in changing the manhole wall

thickness, which is no longer equal to the one of the vessel. Dimensionless thickness of the
manhole wall vary from tm =0.012 to tm =0.060 and the results obtained in this series of the
variants are shown inTable 1. In the case of low thickness of thewall, changing from tm =0.012



The effect of manhole shape and wall thickness on stress state... 63

Fig. 4. Distribution the dimensionless equivalent stress in the region of manhole/vessel junction. The
manhole embedded into the vessel, its wall thickness equal to that of the vessel

Table 1.Maximumdimensionless equivalent stress and location of stress concentration obtained
for various values of manhole wall thickness

Manhole wall Dimensionless Location of stress
thickness ×10−3 equivalent stress concentration

12 6.254
16 5.084
20 4.430
22 4.148
24 3.903 R1◦

25 3.779
25.4 3.733
25.6 3.722
25.8 3.705

25.9 3.798
26 4.013
28 5.292 R2◦

40 5.729
60 9.304

to tm = 0.0258, the maximum stress appears in the location marked as R1
◦. The lowest value

of the maximum stress corresponds just to dimensionless thickness equal to tm = 0.0258 and
amounts to σ̃eq =3.705. Further growth of the thickness induces highermaximum stress but, at
the same time, the location of stress concentration changes to the region R2◦ (Fig. 5).

The manhole having dimensionless length equal to L = 0.5 penetrates the vessel, with
ρ=0.03 rounding of sharp edges of the manhole/vessel junction.

Such a change of themaximum stress location is presumably caused by excessive stiffness of
the junction between themanhole and vessel. Once the thickness exceeds themost advantageous
value equal to tm =0.0258, the stiffness grows together with increasing thickness of themanhole
wall causing further growth of the stress.

Aquestionariseswhether themanholemustbeof circular cross section.Hence, analternative
elliptical shape has also been checked with the unchanged surface area of the manhole cross
section. Such amanhole shape ensures better comfort of theworker who enters inside the vessel.
An ellipse having the semi-axes ratio equal to 5/4 is assumed.As a result, dimensionless lengths
of its semi-axes are equal to a = 0.559 and b = 0.447. Such a semi-axes ratio corresponds to



64 J. Lewiński

Fig. 5. Distribution of the dimensionless equivalent stress in the region of manhole/vessel junction. The
manhole penetrates the vessel, its dimensionless wall thickness is equal to 0.028

an ellipse which does not much deviate from circular shape and, at the same time, the passage
through it is quite comfortable.

For the elliptical manhole cross section, two basic orientations of its shape are possible:
with the major semi-axis situated length-wise or cross-wise with regard to the vessel axis of
symmetry. In the preliminary case, when the side walls of the elliptic manhole have thickness
equal to thickness of the vessel and reach the vessel shell without penetrating it, the maximum
dimensionless equivalent stress amounts to σ̃eq = 8.583 for the length-wise and σ̃eq = 5.321
for the cross-wise orientation of the ellipse major semi-axis. In both cases, it occurs in the
location R1◦.

Comparison to the stress level equal to σ̃eq = 6.503 which has been obtained in the similar
case of the circularmanhole shape allows one to presume that the cross-wise orientation ismore
advantageous.

Similarly as before, the manhole wall has been elongated to dimensionless length equal
to L = 0.5. As a result, the manhole obtained form of an elliptical cylinder penetrating the
vessel. Without rounding the sharp edges arising in the manhole/vessel junction, the maxi-
mum dimensionless equivalent stress amounts this time to σ̃eq = 7.833 for the length-wise and
σ̃eq =5.431 for the cross-wise orientation of the ellipse major semi-axis. In both cases, it occurs
again in the location R1◦. These two cases are presented in Table 2.

Table 2.Maximum dimensionless equivalent stress and location of maximum stress concentra-
tion obtained for elliptical cross-section of the manhole without rounding sharp edges of the
manhole/vessel junction. Dimensionless manhole thickness is equal to tm =0.02

Length-wise Cross-wise Location
orientation orientation of stress

of major semi-axis of major semi-axis concentration

Manhole wall reaches
8.583 5.321

R1◦
vessel surface
Manhole wall elongated to

7.833 5.431
dimensionless length L=0.5



The effect of manhole shape and wall thickness on stress state... 65

The cross-wise orientation of the elliptic shape gives promising results, encouraging one to
round the edges of the manhole/vessel junction as before, and to check what is the effect of
variation of themanholewall thickness. A series of such numerical experiments has been carried
out. The results are shown in Table 3.

Table 3.Maximum dimensionless equivalent stress and location of maximum stress concentra-
tion obtained for elliptical cross-section of the manhole, with ρ=0.03 rounding of sharp edges
of the manhole/vessel junction

Manhole wall Length-wise orientation Cross-wise orientation Location of stress
thickness×10−3 of major semi-axis of major semi-axis concentration

12 8.740 6.717
14 7.959 7.417 R1◦, R2◦

16 7.218 8.046

20 6.367 9.056
22 6.972 9.585 R2◦, R2◦

24 7.608 10.134

As opposed to the previous case of circular cross section of the manhole, the rounding
worsens the situation. It is presumably caused by the fact that for the cross-wise orientation of
the ellipse, themanhole reaches farther from the symmetry plane of the vessel and the rounded
edges stiffen the junctionmore efficiently than in the case of the circular profile of themanhole.
The region R1◦ is subject to the maximum stress only for the length-wise orientation of the
major semi-axis and dimensionless thickness not exceeding tm = 0.016. In all other cases, the
stress concentration moves to R2◦, and the stiffening effect becomesmore evident with growing
thickness of the manhole wall.

Fig. 6. Distribution of the dimensionless equivalent stress in the region of junction between the elliptical
manhole and the vessel with cross-wise orientation of the ellipse major semi-axis. Themanhole
enbedded into the vessel and flushed with the inner wall of the vessel in the region R2◦, its

dimensionless wall thickness equal to t
m
=0.036

Excessive stiffness of the regionR2◦ of themanhole/vessel junction suggests another solution
of the manhole configuration. It seems that themanhole should be so elongated as to avoid too
deep penetration into the vessel. The elliptical cylinder of the manhole should be flushed with
the inner wall of the vessel in the region R2◦. Such a solution is shown in Fig. 6. The stress



66 J. Lewiński

concentration in it has been examined for a series of variants with dimensionless thickness of the
manhole wall varying from tm =0.02 to tm = 0.05. The maximum stresses and their locations
are shown in Table 4.

Table 4.Maximumdimensionless equivalent stress and location of stress concentration obtained
for various values of the manhole wall thickness. The elliptic manhole with semi-axes equal to
a=0.559 and b=0.447 is elongated so as to be flushed with the inner wall of the vessel in the
region R2◦. Sharp edges of the manhole/vessel junction are rounded with radius ρ=0.03

Manhole wall Dimensionless Location of stress
thickness ×10−3 equivalent stress concentration

18 3.765

R1◦

20 3.479
22 3.220
24 2.992
30 2.450
31 2.375

32 2.366
34 2.335
36 2.305 R2◦

40 2.251
50 2.114

Among all the above considered manhole variants, this one is the most advantageous. With
the growing wall thickness, the stress drops monotonically, only the region of maximum stress
concentration moves from R1◦ to R2◦ for the thickness exceeding tm = 0.031. Such a result
allows one to assume that the elliptical manhole profile with the cross-wise orientation of the
ellipse and the elliptical cylinder of the manhole flushed with the inner wall of the vessel in the
region R2◦ should be recommended as the best solution.

Onemight suppose that themanholewall could be shorter also near the regionR1◦. Figure 6
shows that the dimensionless equivalent stress in the bottom part of the manhole wall slightly
drops below 1 in this location, approximating the level σ̃eq = 0.866 existing in the vessel wall.
Therefore, such length of the manhole may be considered as appropriate.

4. Conclusions

The proper shaping and location of a manhole situated in a pressure vessel appears to be very
important from the point of view of the strength and reliability of the structure. In the case of
a manhole fixed vertically and symmetrically in the upper wall of the pressure vessel there are
some possible variants of its configuration.

• Manholes usually have circular cross section. In such a case, the manhole cylinder should
be extended to a certain depth into the vessel and, at the same time, sharp edges of the
manhole/vessel junction should be rounded. Depending on thickness of the manhole wall,
the maximum stress concentration arises either in the region R1◦ or R2◦ (Fig. 1b). For
themanhole thickness tm below the limit value equal about to 1.30 of the vessel thickness
(tm < 1.30tv), the stress concentration is located in R1

◦. The lowest stress occurs for
tm =1.30tv and it is themost advantageous solution of such amanhole. Once themanhole
thickness exceeds this limit and increases, the stress concentration translocates toR2◦ and
the stress level grows (Table 1).



The effect of manhole shape and wall thickness on stress state... 67

• A manhole of elliptical shape has been assumed as an alternative solution. In order to
enable its comparison to the circular one, themanhole cross-section area has beenmainta-
ined equal to the previous circular case. Finally, an ellipse having the semi-axes ratio equal
to 5/4 has been assumed. Such amanhole shape ensures better comfort of theworker who
enters inside the vessel. The elliptical manhole may be pointed with its major semi-axis
situated length-wise or cross-wise with regard to the vessel axis of symmetry. It appears
that the cross-wise orientation is a better solution. Moreover, the elliptical cylinder of the
manhole should be extended into the vessel so as to be flushed with the inner wall of the
vessel in the regionR2◦ (Fig. 6). In sucha case, themaximumstress decreaseswith growing
thickness of themanhole. Nevertheless, its excessive growth results only in an insignificant
stress reduction. An examplary solution shown in Fig. 6, in which the manhole thickness
is nearly twice as large as the one of the vessel (tm =1.80tv) seems to be sufficient.

It should be noticed that in the case of such a solution of the manhole/vessel junction, the
maximum stress remains relatively high as compared to the reference level of dimensionless
equivalent stress σ̃eq = 0.866 existing in the vessel shell. Further improvement of the situation
might be presumably achieved by more sophisticated shaping of the manhole/vessel junction,
with special consideration of the region R2◦.

References

1. BłachutJ.,MagnuckiK., 2008,Strength, stability, andoptimizationofpressurevessels:Review
of selected problems,Applied Mechanics Reviews, 61, 6, 1-33

2. Bolton C.J., Bischler P.J.E., Wootton M.R., Moskovic R., Morri J.R., Pegg H.C.,
Haines A.B., Smith R.F., Woodman R., 2002, Fracture toughness of weld metal samples re-
moved from a decommissioned Magnox reactor pressure vessel, International Journal of Pressure
Vessels and Piping, 79, 685-692

3. Borla J.Z., 2002, Parentmaterial and weldments degradation on SASOL reduction reactors due
to combined effect of thermal fatigue, vibration and hydrogen attack, International Journal of
Pressure Vessels and Piping, 79, 577-584

4. Carbonari R.C., Munoz-Rojas P.A., Andrade E.Q., Paulino G.H., Nishimoto K., So-
lva E.C.N., 2011, Design of pressure vessels using shape optimization: An integrated approach,
International Journal of Pressure Vessels and Piping, 88, 198-212

5. Frum Y., 1995a, Elimination of Circumferential Cracking in Reactor’s Internal Jacket, Interna-
tional Journal of Pressure Vessels and Piping, 61, 383-409

6. Frum Y., 1995b, Time Related Temperature Field over a Reactor Internal Surface, International
Journal of Pressure Vessels and Piping, 61, 367-381

7. GiglioM., 2003,Fatigue analysis of different types of pressure vessel nozzle, International Journal
of Pressure Vessels and Piping, 80, 1-8

8. Krużelecki J., Proszowski R., 2012, Shape optimization of thin-walled pressure vessel end
closures, Structural and Multidisciplinary Optimization, 46, 739-754

9. Lewiński J., Magnucki K., 2010, Shaping of a middle surface of a dished head of a circular
cylindrical pressure vessel, Journal of Theoretical and Applied Mechanics, 48, 2, 297-307

10. Lewiński J.,Magnucki K., 2012,Optimal shaping ofmiddle surface of a dished head of circular
cylindrical vessel with the help of Bézier curve, EngOpt 2012 – 3rd International Conference on
Engineering Optimization, Rio de Janeiro, Brazil

11. Lianghai Z., XuemongS.,Hengyun J., 1996, Influence owarmoverloading ion stress-corrosion
cracking of welded structures, International Journal of Pressure Vessels and Piping, 69, 91-96



68 J. Lewiński

12. Liu J.S, Parks G.T., Clarkson P.J., 2001, Shape optimization of axisymmetric cylindrical
nozzles in spherical pressure vessels subject to stress constraints, International Journal of Pressure
Vessels and Piping, 78, 1, 1-9

13. MagnuckiK.,Lewiński J., 2000,Fully stressedheadofapressurevessel,Thin-Walled Structures,
38, 167-178

14. Magnucki K., Lewiński J., 2003,Optimal design of an ellipsoidal head of a pressure cylindrical
vessel,Proceedings in Applied Mathematics and Mechanics, 3, 517-518

15. Magnucki K., Lewiński J., Stasiewicz P., 2004, Optimal sizes of a ground-based horizontal
cylindrical tank under strength and stability constraints, International Journal of Pressure Vessels
and Piping, 81, 227-233

16. Magnucki K., Szyc W., Lewiński J., 2002, Minimization of stress concentration factor in
cylindrical pressure vessels with ellipsoidal heads, International Journal of Pressure Vessels and
Piping, 79, 841-846

17. Rao B.S.C., Madeswaran R., Chandromohan R., 1997, In-fabrication and pre-service care
on stainless steel pressure vessels, International Journal of Pressure Vessels and Piping, 73, 53-57

18. VentselE.,KrauthammerT., 2001,ThinPlates andShells. Theory, Analysis andApplications,
Basel: Marcel Dekker, Inc, NewYork

Manuscript received May 30, 2013; accepted for print July 17, 2014