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Beijing, China, August 7-12, 2005 SMiRT18-F07-4

APPLICATION OF FEM ANALYSIS METHODS TO A

CYLINDER-CYLINDER

INTERSECTION STRUCTURE

Liping XUE

Center for Joining and Manufacturing

Assembly

Marquette University

Milwaukee, WI 53233, USA

Phone: 01-414-933-2903

E-mail: liping.xue@mu.edu

G.E.O. WIDERA

Center for Joining and Manufacturing

Assembly

Marquette University

Milwaukee, WI 53233, USA

Phone:01-414-288-3543,Fax:

414-288-1647

E-mail:geo.widera@mu.edu

Zhifu SANG

Department of Mechanical and Power Engineering

Nanjing University of Technology

Nanjing, Jiangsu 210009, China

Phone: 01186-25-83240932

E-mail: zfsang@njut.edu.cn

ABSTRACT

The objective of this paper is to study a particular cylindrical shell intersection (d/D=0.526) by use of both linear elastic and elastic-plastic stress analyses via the finite element method using the FEA software ANSYS. The former mainly focuses on the calculation of the stress concentration and flexibility factors in the intersection area before the structure experiences plastic behavior. When an elastic-plastic analysis method is employed, the limit load and burst pressure need to be determined. In this study, two different methods, the “double elastic-slope method” and the “tangent intersection method” are both employed to determine the limit pressure. To predict the burst pressure and failure location, the “arc-length method” in ANSYS is used to solve the nonlinear problem.Finally, the FEA results are compared to experimental data and the agreement is shown to be good.

Keywords: Stress concentration factor, flexibility factor, limit pressure, burst pressure

1. INTRODUCTION

Cylindrical shell intersections are structural configurations commonly used in many industries. The action of various loads leads to high local stresses in the intersection region resulting in stress concentrations there. In addition, due to difficulties that may arise during welding and the resulting defects, the intersection region then becomes the weakest point and failure source of the whole structure. Thus, it is very important to be able to accurately obtain the magnitude and location of stresses and strains so as to allow one to examine the failure modes of these structures.

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factors are two important factors which need to be determined. When elastic-plastic analysis is considered, limit load and burst pressure analyses are two of the phenomena that can be investigated.

Extensive investigations of stress intensification and flexibility factors for cylindrical shell intersections and similar configurations have been performed by Fujimoto and Soh (1988), Willians and Clark (1996), Wais et al. (1999) and so on. The Pressure Vessel Research Council (PVRC) has sponsored several projects to determine limit loads so as to be able to refine the design criteria for pressure vessels and pipe connections. The particular studies which were conducted to obtain the limit loads for internal pressure and external loads on the nozzle by use of nonlinear finite element analyses have included Break (1990) and Junker (1982). No similar body of knowledge is available for burst pressure prediction of cylindrical shell intersections by use of FEA.

2. EXPERIMENTAL STUDY 2.1 Description of Test Vessel

2.1.1 Structure and Dimension

A test vessel was designed and fabricated for the experimental study. It consists of a main vessel, a nozzle, two elliptical heads, filling and venting connections, etc. It should be noted that thedimensions of the test vessel were determined by a Pressure Vessel Research Council (PVRC) oversight committee. Table 1 illustrates the configuration and geometric dimensions of the test vessel.

Table 1 Geometric Dimensions of Test Vessel

Scheme of Test Vessel

Di mm

T mm

do mm

t mm

Lv mm

Ln

mm d/D t/T D/T

Dimension

600 6 325 6 1200 600 0.526 1.0 101

Di: inside diameter of vessel T: thickness of vessel D: mean diameter of vessel t: thickness of nozzle do: outside diameter of nozzle Lv: length of vessel

d: mean diameter of nozzle Ln: length of nozzle

2.1.2 Material Properties

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Fig.1 Typical Engineering Stress-Strain Diagram for Q235-A

2.1.3 Fabrication of Test Vessel

Single V-groove butt joints, Fig.2 (a), were used for both the longitudinal and circumferential welds used in the manufacture of the test vessel. A single bevel groove fillet weld, Fig.2 (b), was employed for the vessel-nozzle corner joint. The details of the welds are shown in Fig.2. Tungsten-inert gas arc welding (root of welds) and manual electric-arc welding (deposit of weld rod material) were used for all welds.

(a)

(b)

Fig.2 Welds for Test Vessel

2.1.4 Experimental Procedure

The test was performed using electrical resistance gages to obtain the load versus strain plots for both the elastic and plastic regions. The strain gages were installed in the longitudinal and circumferential sections on the outside surface of the vessel and nozzle, as dictated by the PVRC oversight committee. The strain gage rosettes attached to the vessel and nozzle need to be located as close as possible to the weld in the intersection area. The location and distribution of the strain gages throughout the vessel are shown in Fig.3. The test vessel was pressurized with a hydraulic test pump. The test was conducted by increasing the pressure gradually in increments of about 0.25 MPa and strain gage readings were recorded during each load step.

2.2 Experimental Results 2.2.1 Elastic Analysis

(4)

5

.

50

6

2

606

0

.

1

2

0

=

×

×

=

=

T

PD

σ

(1)

80

.

4

5

.

50

3

.

242

0

max

=

=

=

σ

σ

SCF

(2)

20 16 14 14 14 16 20 80 80 110

14 14 16 20 50 50

1 2 3 4 5 6

78 9 10 11 12 50 50 20 16 14 1920 21 22 23 1314 15 16 17 18 24 80 70

2.2.2 Limit Pressure

The experimental limit pressures,

P

LTφ and T Lt

P

, are obtained by use of two methods from the pressure-hoop strain curve of the critical gages located near the junction of the vessel and nozzle (see Fig.4). Here,

P

LTφrepresents the double elastic-slope method (intersection of the pressure-strain curve with a line through

the origin having double the slope of the elastic portion of the curve). Further,

P

LtT is the limit pressure using the

tangent intersection method (intersection of the tangents drawn to the curve in the elastic and plastic regions).

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(a) Strain Gage No.1

(b) Strain Gage No.7

Fig.4 Pressure-Strain Curves and Limit Pressures

2.2.3 Burst Pressure

The test vessel was pressurized in small increments to burst. The burst pressure is 7.4 MPa. The initiation of the fracture occurred off the longitudinal plane at the junction between the vessel and nozzle. Figure 5 shows the rupture of the test vessel.

0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 4. 5 5 5. 5 6 6. 5

0 0. 02 0. 04 0. 06 0. 08 0. 1

Hoop St r ai n

P (MPa)

MPa P

MPa P

T Lt T L

00 . 3

85 . 2 = =

φ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Hoop Strain

P (MPa)

MPa P

MPa P

T Lt T L

15 . 3

14 . 3 =

(6)

Fig.5 Rupture of Test Vessel

3. FINITE ELEMENT ANALYSIS 3.1 Finite Element Model

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Fig.6 Finite Element Model for Internal Pressure Load Case

Fig.7 Finite Element Model for External Load Case

3.2 Boundary and Loading Conditions

For the internal pressure load case, symmetry boundary conditions are employed on the two symmetry planes. To simulate the contained pressure, equivalent axial stresses are imposed as boundary conditions at the ends of the vessel and nozzle.

An in-plane moment on the nozzle represents symmetric loading about the X-Z plane (see Fig.7), while the out-of-plane moment represents an antisymmetric loading about the X-Z plane. Therefore, symmetry boundary conditions are imposed on all nodes at the X-Z plane for in-plane moment, while antisymmetry boundary conditions are imposed on all nodes at the X-Z plane for out-of-plane moment load case. Further, the left end of the vessel is fixed while the right end of the vessel and the end of the nozzle are free.

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Fig.8 Multilinear Material Mode Curve

3.3 Results

3.3.1 Elastic Analysis

3.3.1.1 Stress Concentration Factor

The maximum elastic stress from FEA occurs at the inner crotch corner. The stress concentration factor for this particular structure, which equals to 5.11, can be obtained by divided the maximum stress by the nominal hoop stress in the vessel

σ

0 (see Eq. 1).

Decock (1975) devised an empirical SCF for pressure loading based on his collection of 156 experimental results from steel and araldite models of branch junctions. His proposed equation for predicting the SCF at the inner crotch corner was as follows:

(

) ( )

(

)(

)

( ) (

32

)

12

2 1 2

1 2 3

1

25

.

1

2

2

D

d

T

t

T

D

D

d

T

t

D

d

SCF

+

+

+

=

(3)

Substituting D/T, d/D and t/T of the test vessel into the above equation gives: SCF=5.43.

Table 2 Comparisons of SCF’s

Experiment FEA Eq. 3

SCF 4.80 5.11 5.43

Table 2 shows the comparisons of SCF’s determined from experiment, FEA and Eq.3, respectively. Although Eq.3 yields a more conservative SCF, (considering the location where the experimental SCF was determined), one can see that the SCF’s agrees quite well with each other.

3.3.1.2 Flexibility Factors

The flexibility factors

k

of cylindrical shell intersections are defined by the virtual spring of rotation:

(

)

o n beam n

Md

EI

k

=

θ

θ

(4)

where

0 100 200 300 400 500 600

0 0.05 0.1 0.15

True strain

(9)

n

θ

= rotation at the top of nozzle by FEA

beam

θ

= nominal rotation at the top of a nozzle by beam theory

E

= Young’s Modulus

n

I

= moment of inertia of nozzle

v

I

= moment of inertia of vessel

M

= in-plane or out-of-plane moment at the top of nozzle

o

d

= outside diameter of nozzle

To determine the in-plane and out-of-plane flexibility factors

k

i,

k

ofrom the numerical results,

θ

nand

beam

θ

will be substitute into Eq.4. For in-plane moment loading

M

i

Y n

θ

θ

=

(5)





+

=

v v n n i beam

I

L

I

L

E

M

1

.

0

θ

(6)

For out-of-plane moment loading

M

o

X n

θ

θ

=

(7)





+

=

v v n n o beam

I

L

I

L

E

M

1

.

3

θ

(8)

Where

X

θ

,

θ

Y= rotations at the top of nozzle (from FEA)

Fujimoto (1988) developed empirical formulas for the flexibility factors by fixing both ends of vessel. The lengths of the vessel and nozzle are Lv=2.89D and Ln=3.05D, respectively. Wais (1999) also developed correlation equations based on average flexibility factors of one vessel end fixed and both vessel ends fixed. The lengths of the vessel and nozzle are Lv=2D and Ln=1.5D, respectively.

Table 3 shows the present in-plane and out-of-plane flexibility factors for shorter, medium, longer length models as well as flexibility factors from empirical formulas from Fujimoto and Wais. From Table 3, it is seen that the agreement for

k

i is quite good. In addition, the results show that the lengths of the vessel and nozzle

do affect the flexibility factors, especially for

k

o. It is worthwhile to note that there is no “conservative” value

in evaluating flexibility factors in piping systems. Therefore, it is desirable that the vessel and nozzle be long enough to eliminate their effect on flexibility factors prior to performing a parametric finite element analysis.

Table 3 Comparison of Flexibility Factors

Present Shorter Model Present medium Model Present Longer Model Fujimoto (1988) Wais (1999) Lv=2D, Ln=D Lv=4D, Ln=2.5D Lv=7D, Ln=7D Lv=2.89D, Ln=3.05D Lv=2D,Ln=1.5D

i

k

13.01 13.09 14.48 10.38 12.71

o

k

82.08 104.92 122.83 78.49 66.63

3.3.2 Limit Pressure

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and tangent intersection methods, the limit pressure at the locations mentioned above for the test vessel are also illustrated in Fig.9.

(a) Strain Gage No.1

(b) Strain Gage No.7

Fig.9 Limit Pressure by FEM

Table 4 shows the comparison of limit pressure from FEA with those from experiment. From Table 4, it is seen that the numerically determined limit pressures agree reasonably well with the experimental results. In addition, the double elastic-slope method yields more conservative limit pressure. Further, the limit pressures at stain gage No.1 are lower that those at strain gage No.7. It can thus be concluded that the failure of the structure is more likely to occur on the vessel rather than no

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

0 0.02 0.04 0.06 0.08 0.1

Hoop Strain

P (MPa)

MPa P

MPa P

FEM Lt

FEM L

88 . 2

75 . 2

= =

φ

0 1 2 3 4 5 6

0 0.02 0.04 0.06 0.08

Hoop Strain

P (MPa)

MPa P

MPa P

FEM Lt

FEM L

05 . 3

80 . 2

= =

(11)

Table 4 Comparison of Limit Pressure

Strain Gage No.1 Strain Gage No.7

T L

P

φ 2.85 3.14

Experiment

T Lt

P

3.00 3.15

FEM L

P

φ 2.75 2.80

FEA

FEM Lt

P

2.88 3.05

3.3.3 Burst Pressure

To predict the burst pressure, a failure criterion is necessary. In reference (1999), failure is defined as the pressure for which the structure approached dimensional instability, i.e., unbounded displacement for a small increment in pressure. This condition is symptomatic of an ill-conditioned boundary problem caused by the combined changes in geometry and material stiffness leading to a physical instability. The failure location is determined based on the maximum average equivalent plastic strain across the thickness of the specimen.

Figure 10 shows the pressure versus equivalent plastic strain curve at the critical node where the maximum equivalent plastic strain occurs. From Fig.10, it is seen that when the pressure reaches 8.6 MPa, the equivalent plastic strain increases significantly with little change of pressure, i.e., the slope of the pressure-equivalent plastic strain curve becomes vanishingly small. Based on the criteria mentioned above, the total pressure (8.6 MPa) that causes this condition is the incipient failure pressure. Therefore, the pressure (near zero load-strain slope) can be deemed as the burst pressure. Compared with the experimental burst pressure 7.4 MPa, the predicted burst pressure is 16% higher. For this cylindrical shell intersection, the maximum average equivalent plastic strain across the thickness occurs about 26 degrees off the longitudinal plane at the juncture on the vessel. The failure location coincides with the experimental result (See Fig.5).

Fig.10 Pressure versus Equivalent Plastic Strain curve

0 1 2 3 4 5 6 7 8 9 10

0% 10% 20% 30% 40% 50% 60%

Equivalent plastic strain

Pressu

(12)

4. CONCLUSIONS

The elastic and plastic behaviors of a thin-walled cylindrical shell intersection with a large diameter (

D

/

T

100

) have been discussed by use of finite element analysis. From a comparison of the FEA results with the experimental data, the following conclusions can be reached:

1. From the point of view of stress concentration, limit pressure and burst pressure, the weakest location occurs at the intersection area of vessel.

2. The lengths of the vessel and nozzle do affect the flexibility factors. In order to eliminate the effect of this factor on the flexibility factors, the lengths of the final FE models should be ascertained prior to performing any finite element analysis.

3. The limit design method circumvents the complicated stress categories, and is a simple and advanced design technique. The key point of this method is to determine the limit load of the vessel. The double elastic-slope method for determining the limit load is conservative.

4. The burst pressure and failure location of a cylindrical shell intersection can be predicted by employing a deformation based failure criterion.

5. Finite element simulations can be employed with sufficient accuracy to study the various failure modes associated with cylindrical shell intersection structures.

5. REFERENCES

Swanson Analysis System Inc., (2003), “Analysis Engineering Analysis Systems User’s Mannual”. Break, E. G. and Gerdeen, J.C., (1990), Journal of Pressure Vessel Technology, Vol.112, pp.138.

Decock, J., (1975),Rep. No. MT104, Centre de Recherches Scientifiques et Techniques de L’ Industrie des Fabrications Metalliques.

Fujimoto,T. and Soh, T., (1988), Journal of Pressure Vessel Technology, Vol.110, pp.374.

Jones, D. P., Holliday, J. E. and Larson, L. D., (1999), Journal of Pressure Vessel Technology, Vol.121, pp.149. Junker, A.T., (1982), American Society of Mechanical Engineers Paper, 82-PVP-10.

Wais, E. A., Rodabaugh, E C. and Carter, R., (1999), American Society of Mechanical Engineers, Pressure Vessels & Piping Division (Publication), Vol.383, pp.159.

Williams, D. K. and Clark, J. R., (1996), Pressure Vessels and Piping Design, Analysis, and Severe Accidents, ASME, Vol.331, pp.55.

Xue, L. and Widera, G. E. O., (2002), Parametric Finite Element Analysis of Large Diameter Ratio Shell Intersections Subjected to External Loadings: Flexibility Factors for External Moments on Nozzle, PVRC Report.

Figure

Table 1 Geometric Dimensions of Test Vessel
Fig.8  Multilinear Material Mode Curve
Fig.9   Limit Pressure by FEM  Table 4 shows the comparison of limit pressure from FEA with those from experiment
Table 4  Comparison of Limit Pressure

References

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