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COMPUTATIONAL MECHANICSNew Trends and Applications

S. Idelshon, E. Oñate and E. Dvorkin (Eds.)©CIMNE, Barcelona, Spain 1998

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ANALYSIS OF THE COLLAPSE OF STEEL TUBES UNDEREXTERNAL PRESSURE

Andrea P. Assanelli*, Rita G. Toscano** and Eduardo N. Dvorkin***

Center for Industrial Research, FUDETECAv. Córdoba 320

1054 Buenos Aires, Argentina*e-mail: [email protected]

**e-mail: [email protected]***e-mail: [email protected]écnica de.unl.edu.ar/gtm-

Key words: Finite elements, Collapse, Tubes, Elasto-plasticity.

Abstract. The collapse of steel tubes under external pressure is described using finite elementmodels. The models are validated comparing their predictions against the results of collapsetests performed in our laboratory.

Andrea P. Assanelli, Rita G. Toscano, and Eduardo N. Dvorkin.

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1 INTRODUCTION

The external collapse pressure of very thin steel tubes is governed by classical elastic bucklingformulas1,2; however, for thicker tubes more involved elasto-plastic considerations have to be takeninto account. There are many factors that have some degree of influence on the external pressure thatproduce the collapse of a steel tube, among them3-9:

1. The relation (outside diameter/thickness) (D/t ratio).2. The yield stress of the tube steel (σy).3. The work - hardening of the tube steel.4. The shape of the tube sections (outside diameter shape and thickness distribution).5. The residual stresses locked in the tube steel.6. The localized imperfections introduced either in the tubes production, in the tubes handling or

due to localized wear.In this paper we discuss the experimental techniques and the hierarchy of finite element models

that we have developed to study the collapse behavior of seamless steel tubes.In Section 2 we comment on our facilities for performing external pressure collapse tests and we

also present an imperfections measuring system that we have implemented at our laboratory basedon the device described by Yeh et al10 and Arbocz et al11,12.

In Section 3 we describe some simple 2D finite element models that we have developed as a firstapproach for the simulation of the external pressure collapse test. We compare the results that these2D models provide with the experimental results obtained at our laboratory; the comparisondemonstrates that a 2D geometrical characterization of the tubes does not contain enoughinformation to assess on their collapse strength.

The tube shape imperfections and the wall thickness normally change along the tube length; alsolocalized imperfections can be found in the tubes. In Section 4 we present the 3D finite elementmodels that we have developed to overcome the limitations of the simpler 2D models. The 3D modelpredictions have been qualified via experimental testing at our laboratory. The geometricalinformation on the tested samples has been acquired using our imperfections measuring system andhas been used as input data for our 3D finite element models.

2 EXPERIMENTAL PROCEDURES

In the present Section we describe our laboratory facilities for performing external pressurecollapse tests and we also present the imperfections measuring system that we have developedbased on the previous work of Yeh et al10 and Arbocz et al11,12.

2.1 External pressure collapse tests

A schematic representation of our laboratory assembly for performing external pressure collapsetests is presented in Fig. 1.


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Chamber length > 10 D

D

Packer

Free to slide

Figure 1. Collapse chamber

It is important to notice that:1. The experimental set-up does not impose any axial restrain on the tubes.2. The length of the sample being tested fulfills in all cases the relation: L/D > 10.3. The collapse of the samples is detected by a drop in the pressure of the water used to

pressurize them.

2.2 Imperfections measuring system

In order to acquire information on the geometry of the tube samples that are going to be tested inthe collapse chamber, the shape of the external surface is mapped and the wall thickness is measuredat a number of points.

2.2.1. Mapping of the samples external surface

To map the external surface of a sample we rotate it in a lathe and an LVDT (Linear VariableDisplacement Transducer) placed on the lathe cross-slide (see Fig. 2) measures, at regular intervalsof axial and angular position, the distance ri shown in Fig. 3.


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We have developed the following algorithm to transform the data acquired with the LVDT intothe Fourier series description of the samples external surface.

Figure 2. Mapping of the samples external surface

Figure 3. Algorithm to process the data acquired with the LVDT


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Data

ri: radial distance from the rotation axis to the external surface (see Fig. 3).qi: total turns corresponding to the i-th section, measured from an arbitrary defined zero.

Algorithm

1. We can define:

( )( )Θ i i iq INT q= ⋅ ⋅ −2 π

Note: the data is acquired along a spiral path path (with a pitch less than half of the wallthickness); however, in subsequent analyses we will consider that the points correspondingto a turn are located on a planar section, at an axial distance x K from an arbitrary origin.

( )x x INT qK i= ⋅∆

2. For the k-th section we can define a "best-fit circle", with radius Ro and with its center located at(xo yo) in a Cartesian system, contained in the section plane and with its origin at the sectionrotation center10. For determining Ro, xo and yo we solve the following minimization problem:

( ) ( )[ ]( )[ ]

( ) [ ] ( )

R x y minE R x y

E r g R x y

g R x y x y sin R x sin y

o o o o o o

i i o o oi

i o o o o i o i o o i o i

, , arg , ,

, , ,

, , , cos cos

=

= −

= + + − −

∑

2

22

2 2

Θ

Θ Θ Θ Θ Θ

To solve the above nonlinear minimization problem we apply the Levemberg-Marquard method13,using as first trial a simplified solution in which the term in (xo yo) inside the square root in theexpression for g, is neglected14.

3. Once the center of the "best-fit circle" is determined we reduce the acquired data to it,

( ) ( )

$ cos

$

$ $ $

$ $

$

x r x

y r sin y

r x y

tany

x

i i i o

i i i o

i i i

ii

i

= −

= −

= +

=

−

Θ

Θ

Θ

2 2

1


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We expand ( )$r Θ using a discrete Fourier transform

( ) ( ) ( )[ ]$ $ cos $ $ $r R a j b sin jo j jj

NΘ Θ Θ= + +

=∑

1

where,

( )[ ]

( )[ ]

$ $ cos $ $

$ $ $ $

a r j

b r sin j

j k k kk

M

j k k kk

M

=

=

=

=

∑

∑

1

1

1

1

π

π

Θ ∆Θ

Θ ∆Θ

In Figs. 4 we present the Fourier decomposition obtained using the above described algorithm inthe case of a tube with a nominal outside diameter of 9 5/8" and a metric weight of 47 lb/ft.

The length of each sample is approx. 2.5 mts, the pitch of the cross-slide is 3mm/turn and theangular distance between the acquired points is one degree.

Figure 4. Fourier decomposition of the outside radius

2.2.2. Wall thickness measurement

We measure the wall thickness at a number of points evenly distributed on the sample externalsurface, using manual ultrasonic gages; the obtained thickness map is represented in Fig. 5.


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Thickness[mm]

A...I::axial locations

1,...8 :angular locations

Figure 5. Thickness map

3. TWO DIMENSIONAL FINITE ELEMENT MODELS

In this Section we discuss the 2D finite element models that we have implemented to simulate thebehavior of ideal "long specimens" in the external pressure collapse test. We also compareexperimental results with the predictions of these 2D models: the comparison demonstrates that a 2Dgeometrical characterization of the tubes does not contain enough information to assess on theircollapse strength. However, the 2D models can provide useful information on the trends of theexternal collapse pressure value when some parameters -such as residual stresses, shapeimperfections, etc.- are changed.

3.1. Formulation of the 2D models

We develop the 2D finite element models using a total Lagrangian formulation15 thatincorporates:

1. Geometrical nonlinearities due to large displacements/rotations (infinitesimal strainsassumption);

2. Material nonlinearity: an elasto-plastic constitutive relation is used for modeling the steelmechanical behavior (Von Mises associated plasticity16).


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We carry out the finite element analyses using a special version of the general purpose finiteelement code ADINA that incorporates the quadrilateral QMITC element17-19.

For modeling the external hydrostatic pressure we use follower loads2, and we introduce in ourmodels the residual stresses via temperature gradients through the thickness.

In order to validate our procedure for simulating the residual stresses, in Fig. 6 we present theresult of the 2D finite element simulation of a slit-ring test; which is the test that we use at ourlaboratory to measure the residual stresses in the tube samples. The model results, that we present inFig. 6, agree with the experimental observations; hence,we can consider that our residual stressessimulation is realistic enough.

Figure 6. Two dimensional simulation of the slit-ring test

It is important to recognize that the actual collapse test is not modeled exactly neither by planestrain nor by plane stress models:

1. The absence of longitudinal restrains imposes a plane stress situation at the sample edges.2. The length of the samples (L/D > 10) approximates a plane strain situation at its center.

In this Section, in order to explore the limitations of the 2D models, we analyze the collapse testusing both, plane stress and plane strain finite element models.

3.2. Two dimensional finite element results vs. experimental results

The 2D finite element models have been developed considering an elastic-perfectly plastic


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material constitutive relation (we will show in Section 3.3 that disregarding the steel work-hardeningintroduces only a negligible error in the calculated collapse pressures).

For standard samples in which the tube geometry is not analyzed using the above describedimperfections measuring system, the laboratory keeps records of the following geometricalparameters:

1. Average, maximum and minimum outside diameter at three sections (the central section andthe two end sections)

2. Thicknesses at the two end sections (eight points per section).We construct the geometry of the 2D finite element models using:

1. The ovality [Ov = (Dmax - Dmin)/Daverage] and the average outside diameter of the centralsection.

2. The eccentricity ([ε = [(tmax -tmin)/taverage] and t is the thickness]) obtained by averaging theeccentricities of the two end sections and the average thickness obtained by averaging thesixteen thickness determinations.

The residual stresses for each sample are measured using a slit-ring test and its values areimposed on the model as discussed above. The actual transversal yield stress of the sample materialis measured and its value is used for the elastic-perfectly plastic material constitutive relation.

To determine the collapse loads of the sample models we calculate the nonlinearload/displacement path and seek for its horizontal tangent.

To perform the numerical analyses we use, for half the section, a 2D mesh with 720 QMITCelements and 1572 d.o.f.(see Fig. 7.a). To assess on the quality of this mesh we analyze the planestrain collapse of an infinite tube and we compare our numerical results against the analytical resultsobtained using the formulas in Timoshenko et al1.

Average OD 245.42 mmAverage thickness 12.61 mm

D/t 19.47Ov 0.18%σy 9079 kg/cm2

pcr as per [1] / pcr FEA 0.992

Table 1. Qualification of the 2D finite element model

From the results in Table 1 we conclude that the proposed 2D mesh of QMITC elements isaccurate enough to represent the collapse of very long specimens.

From the analyses of different cases we have identified two basic types of load/displacementpaths:

1. In Fig. 7.b we present the load/displacement path of an eccentric tube with (D/t =22.13).2. In Fig. 7.c we present the load/displacement path of another eccentric tube but with (D/t =

17.67). This second case presents an "inverse collapse behavior" that was previously


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described for the case of collapse under external pressure and bending20,21.

a) Finite element mesh(720 QMITC elements - 1572 d.o.f.)

Displacement [cm]

b) Eccentric tube: direct collapse behavior


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Displacement [cm]

c) Eccentric tube: inverse collapse behavior

Figure 7. 2D models of the external pressure test

A total of 32 collapse tests, for tubes 9 5/8"x47lb/ft and 7"x26lb/ft both Grade 95, have beenanalyzed using plane stress and plane strain models. The comparisons between the numerical andexperimental results are plotted in Figs. 8.a (plane stress models) and Fig. 8.b (plane strain models).

It is important to observe that the collapse pressures determined using the 2D models present asignificant error. Also, for most cases the numerical determined collapse pressure values are lowerthan the experimental ones. Some reasons for this behavior are:

1. The middle section ovality is not fully representative of the sample geometry. Thenumerical/experimental technique to be presented in Section 4, that combines ourimperfections measuring system and 3D finite element models, will remove this source ofdisagreement between numerical and experimental results.

2. In developing the 2D models the measured ovality is entirely assigned to the first elasticbuckling mode and this conservative approach partially accounts for the fact that thenumerical values are lower than the actual ones. Again, the 3D finite element models willremove this source of disagreement between numerical and experimental results because theywill incorporate the actual tube geometry.

3. The experimental set-up (see Fig. 1) imposes on the samples unilateral radial restraints atboth ends, these restraints are not described by the 2D models; the difference between thenumerical and actual boundary conditions also partially accounts for the fact that thenumerical values are in general lower than the actual ones. The 3D models will also removethis disagreement incorporating those end restraints to the external pressure collapse testsimulation.


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FEA collapse pressure/experimental collapse pressure

a) 9 5/8" 47lbs/ft and 7" 26lbs/ft Grade 95 - Plane stress

FEA collapse pressure/experimental collapse pressure

b) 9 5/8" 47lbs/ft and 7" 26lbs/ft Grade 95 - Plane strain

Figure 8. 2D models vs. experimental results

It is obvious, from the 2D results in Fig. 8, that the geometrical characterization of only onesection of a tube is not enough for assessing on its collapse performance. However, the 2D modelsare very useful for performing parametric studies on the relative weight of different factors that affectthe external collapse pressure of tubes.


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3.3. Strain hardening effect

We have previously assessed that the strain hardening of the tube material does not play animportant role in the determination of its external collapse pressure. In this Section we examine theabove assessment using 2D finite element models.

We analyze two 9 5/8" tubes (a thin and a thick one) using a bilinear material model; in order toexplore different hardenings we consider three values for the constant tangential modulus:

ET = 0.0 (perfect plasticity); 0.057E and 0.10 E.

where E is the Young modulus. We also consider two ovality (Ov) values but we do not include inthis analysis neither the tube eccentricities nor its residual stresses. We summarize the numericalresults of our analyses in Table 2:

D/t Ov ET = 0.0 ET = 0.057E ET = 0.10E17.66 0.75% 611 614 61617.66 0.35% 684 686 68824.37 0.75% 272 272 27224.37 0.35% 296 297 297

Table 2. Analysis of the work-hardening effect on the external collapse pressure

As it can be seen from the above table, for all the analyzed cases, the strain hardening has anegligible effect on the external collapse pressure value.

It is important to recognize that we have considered σy to be independent of ET. This is not thecase if as "yield stress" we adopt the one corresponding to a relatively large permanent offset.

3.4. Effect of ovality, eccentricity and residual stresses

Using 2D finite element models we develop in this Section a parametric study aimed at theanalysis of the effect, on the tube collapse pressure, of the following parameters:

1. Ovality (Ov);2. Eccentricity (ε)3. Residual stresses (σR)

In the present analyses the ovality is considered to be concentrated in the shape corresponding tothe first elastic buckling mode and the eccentricity is modeled considering non-coincident OD and IDcenters.

In Fig. 9 we plot the results of our parametric study, normalized with the collapse pressurecalculated according to API standards. It is obvious from these results that the main influence on theexternal collapse pressure comes from the ovality and from the residual stresses; however, the effectof the residual stresses diminishes when the ratio (D/t) evolves from the plastic collapse range to theelastic collapse range. The eccentricity effect, in the case of the external collapse test with neither


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axial nor bending loads, is minor.

Figure 9. Parametric study of the effect of out-of-roundness, eccentricity andresidual stresses on the casing collapse pressure. Performed using 2D finite element models

4. THREE DIMENSIONAL FINITE ELEMENT MODELS

In this Section we discuss the 3D finite element models that we have implemented to simulate theexternal pressure collapse test. We use the 3D models with the geometrical data acquired with ourimperfections measuring system, described in Section 2, and we compare the numerical collapsepressure predictions with the experimental determinations carried out at our laboratory.


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4.1. Formulation of the 3D models

The (radius/thickness) relation of the tubes to be analyzed ranges from 7.61 to 12.08, hence ashell model that incorporates shear deformations is a suitable one. Following what we did for the 2Dmodels we use a total Lagrangian formulation15 that incorporates the geometrical nonlinearitiescoming from the large displacements/rotations and the material nonlinearity coming from the elasto-plastic constitutive relation. We carry out the finite element analyses using the code ADINA and theMITC4 shell element22-24.

As in the previous Section, follower loads are used to model the hydrostatic external pressure andthe residual stresses are modeled via thermal gradients through the thickness.

To check the capability of our 3D finite element models to simulate different residual stresspatterns we consider a 9 5/8" x 47 lb/ft tube with σy = 77.3 kg/mm2 and σR = 0.2 σy.

We calculate the value of the samples opening in a slit-ring-test:1. Using the analytical relation between residual stresses and sample openings that we presente

in Table 3.

Sample σR

L=3D (approx. plane strain)

( )a t E

R v

⋅ ⋅

⋅ ⋅ ⋅ −4 12 2π

L=1" (approx. plane stress) a t E

R

⋅ ⋅⋅ ⋅4 2π

Table 3. Post-processing formulas for the slit-ring test

2. Using the finite element model shown in Fig. 10, where the residual stresses are simulated viathermal gradients through the thickness and the slit of the cylindrical sample is simulated byremoving a row of elements.

The results are:

Sample length FEA predicted opening

Analytical predicted opening

1" 1.023D 0.99

Table 4. Validation of the residual stresses simulation using 3D finite element models

Hence, our 3D finite element procedure for representing the residual stresses can be consideredas realistic enough.

4.2. Three dimensional simulation of the external pressure collapse test

In order to validate our 3D finite element models of the external pressure collapse test, we


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analyze three tubular samples. The meshes are generated using only the first 12 modes (see Fig. 11)in the external surface Fourier decomposition.

a) Lateral view

b) Front view

Figure 10. Three dimensional finite element simulation of the slit-ring test


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a) Lateral view

Imperfections amplified 50X

b) Front view

Figure 11. Finite element mesh developed considering the first 12 modes (sample #1)

The actual transversal yield stress of the samples and the residual stresses measured using "long"slit-ring samples are shown in Table 5:

Sample σy [Kg/mm2] σR/σy

1 86.15 0.17


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2 82.24 0.383 84.7 0.27

Table 5. Mechanical properties of the three analyzed samples

The comparison between the FEA determined collapse pressures and the experimentallydetermined ones is presented in Table 6:

Sample pcr FEA / pcr exp.1 1.0052 1.0253 0.984

Table 6. Comparison between 3D FEA and experimentally determined collapse pressures

From the above results it is evident that the developed 3D finite element models simulate veryaccurately the external pressure collapse test.

4.3. Effect of low and high shape modes

The numerically calculated values of external collapse pressure that we report in Table 6 havebeen calculated considering 12 shape modes in the external surface Fourier decomposition (see Fig.4). In order to investigate the effect of the different shape modes we re-calculate the collapsepressures but considering only the first 4 modes. The obtained numerical results indicate that for the3 analyzed cases:

p pcres

cres4 12mod mod=

Hence, we can assess that the external pressure collapse load is determined only by the firstmodes of the external surface Fourier decomposition.

5. CONCLUSIONS

Experimental/numerical procedures for analyzing the behavior of tube samples in the externalpressure collapse test were discussed.

It was shown that 2D finite element models do not contain enough information on the samplesgeometry to provide accurate information on the tubes collapse pressure. However, the 2D modelsare very useful for developing parametric studies: using 2D finite element models it was shown thatthe tube ovality and residual stresses have a strong influence on the value of the external collapsepressure.

Three dimensional finite element models, developed using shell elements, provide a very accurateprediction of the value of the external collapse pressure, when the input data contains a detaileddescription of the tube geometry and residual stresses.


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An imperfection measuring system was developed to acquire detailed information on the tubularsamples geometry.

Using 3D finite element models it was shown that the high frequency modes of the externalsurface Fourier decomposition do not have a significant effect on the value of the external collapsepressure.

Acknowledgement

We acknowledge the experimental data provided by D.H. Johnson from the Center for IndustrialResearch laboratory.

We acknowledge the financial support for this research of SIDERCA (Campana, Argentina) andTAMSA (Veracruz, Mexico).

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[9] H. Mimura, T. Tamano and T. Mimaki, "Finite element analysis of collapse strengh of casing",Nippon Steel Technical Report, 34, 62-69 (1987).

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[11] J. Arbocz and C.D. Babcock, "The effect of general imperfections on the buckling of cylindricalshell", ASME, J. Appl. Mechs., 36, 28-38 (1969).

[12] J. Arbocz and J.G. Williams, "Imperfection surveys of a 10-ft diameter shell structure", AIAAJournal, 15, 949-956 (1977).


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[13] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Veherling, Numerical Recipes,Cambridge University Press, Cambridge (1986).

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[15] K.J. Bathe, Finite Element Procedure, Prentice Hall, Upper Saddle River, New Jersey(1996).

[16] R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, New York (1971).[17] E.N. Dvorkin and S.I. Vassolo, "A quadrilateral 2-D finite element based on mixed interpolation

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[20] E. Corona and S. Kyriakides, "An unusual mode of collapse of tubes under combined bendingand pressure", ASME, J. Pressure Vessel Technol., 109, 302-304 (1987).

[21] J.R. Fowler, B. Hormberg and A. Katsounas, "Large scale collapse testing", SES Report,prepared for the Offshore Supervisory Committee, American Gas Association (June 1990).

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[24] K.J. Bathe and E.N. Dvorkin, "A formulation of general shell elements - The use of mixedinterpolation of tensorial components", Int. J. Numerical Methods in Engng., 22, 697-722(1986).

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