The collapse resistance of casing is influenced by many factors, among which the level of residual stresses is an essential one. This paper presents the results of research activities aimed to determine such stress level in seamless tube specimens made of grade N80 steel.
These activities included: the selection – based on experimental and theoretical studies – of the most adequate method to determine the residual circumferential stresses in tubes for well casing; a series of experimental tests to define the level of such residual stresses and its variation across the tube thickness; finally, an evaluation of the influence of such stresses upon the value of the collapse pressure of casing.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/1156

Dumitrescu, A.

Zisopol, D.G.

SocioEconomic Challenges Journal (Sumy State University)

 «Вісник СумДУ», №1(73), 2005 25 
ANALYSIS OF THE RESIDUAL STRESS LEVEL IN CASING AND 
ITS INFLUENCE ON THE COLLAPSE STRENGTH 
 
Andrei Dumitrescu, Ph. D. Eng.,  Dragoş Gabriel Zisopol, Ph. D. Eng. 
“Petroleum-Gas” University of Ploieşti, Romania 
 
1 INTRODUCTION 
One of the phenomenons which can decisively affect the reliability of oil 
well casing is collapse under the effect of external hydrostatic pressure 
(leading to ovalisation followed by flattening of tubulars). The collapse 
mechanism differs essentially with the value of the ratio between the outside 
diameter, D, and the wall thickness, t, of casing: 
- for greater values of this ratio (D/t > 35), collapse occurs by means of 
an elastic flattening, before the tube material reaches its yield strength; in 
case of elastic collapse of a perfect circular tube, the critical value of the 
external pressure (the elastic collapse pressure, pE) can be calculated using 
the following equation [7]: 
 
2 3
2 1
1 /
E
E
p
D t
,  (1.1) 
where E is Young’s elastic modulus, and  Poisson’s coefficient of the 
casing material; 
- for smaller values of the D/t ratio (under 15...20), collapse will take 
place in the plastic field; in such case, if considering the thin-wall tubes 
theory, the critical value of the external pressure (the plastic collapse 
pressure, pF) can be calculated as follows [3]: 
  pF = 2 c·t/D  (1.2) 
where c is the minimum specified yield strength (SMYS) of the casing 
material; 
- for D/t = 20...35, the failure mechanism is much more complex (elastic-
plastic collapse); a gradual passage from elastic failure to plastic failure will 
take place and different calculation methods, detailed in [2], have been 
proposed in such case by various researchers. 
The collapse resistance capacity of casing is importantly affected by 
some factors, among which the level of residual stress, the pipe geometrical 
imperfections (mainly its initial ovality), and the pipe material anisotropy. 
The effect of these factors can be accounted for when assessing the critical 
value of the external pressure (the collapse pressure, pc), whatever the 
failure mechanism, by using the following equation [3]: 
  pc = k·pF = kr·kδ kξ pF  (1.3) 
where k ≤ 1 is a reduction factor, considering the effect of residual stresses 
(by means of the coefficient kr), of initial tube ovality (kδ), and of initial tube 
eccentricity (kξ). 
 «Вісник СумДУ», №1(73), 2005 26 
Residual stress is an important factor affecting the collapse strength of 
casing and therefore its effect has been studied in various papers [3, 5, 6, 8, 
9]. Such stresses are a direct consequence of the operations performed 
during the manufacturing process of tubulars, mainly rolling, thermal 
treatment (quenching, normalizing, etc.) and straightening. 
The effect of the residual stress level on the collapse pressure value can 
be assessed using the following equation, developed in [8], to calculate kr 
coefficient (from eq. 1.3): 
 «Вісник СумДУ», №1(73), 2005 27 
   
1 + 0,8 r/ c if  r < 0 
 
kr =  (1.4) 
  
2
2 1 0,8 /
1 1 2 /
r c
t D
 if  r > 0 
where r is the level of residual stresses at the inside wall of the tube. The 
first equation above applies in case yielding develops at the inside wall of 
the tube (compression at the inside), while the second one applies in case of 
yielding at the outside wall (traction at the inside). 
The effect of a compressive stress ( r < 0) is to cause early yielding on 
the inside and thus a reduction of wall thickness, while a moderate level of 
traction stress ( r > 0) on the inside has a positive effect by reducing the 
value of collapse pressure, pc. 
In such context, the research activities described in this paper aimed to 
investigate the level of the residual circumferential stresses in seamless 
tubes for well casing and its variation across the tube thickness, and to 
evaluate – based on eq. 1.4 above – the influence of such stresses on the 
critical collapse pressure of casing. 
 
2 DEVELOPMENT OF AN EXPERIMENTAL METHOD FOR THE 
EVALUATION 
OF THE RESIDUAL STRESSES IN TUBES FOR CASING 
In order to properly determine the actual level of residual circumferential 
stresses in seamless tubes, an extended research activity has been performed 
to define the most adequate experimental method, simple, easy to apply, and 
also accurate. 
After analysing various methods, we reached the conclusion that the most 
suited one for measuring residual stresses which do not vary 
circumferentially is the slit ring method, described in [4]. In this procedure, 
a ring with the height equal to h is cut from the tube (with the outside 
diameter De, the inside diameter Di, and the wall thickness t) and then slit 
axially (fig. 1). The movement across the slit gives a measure of the residual 
stress; an opening of the ring indicates compressive stress on the inside wall 
( r < 0). Various values have been proposed for the ring height, h, ranging 
between 51 mm (2 in) [9] and 10 mm [4]. 
In order to define the most adequate calculation method to assess the 
residual stress level, given the measured opening of the ring, Δa, or the 
variation of the outside diameter of the ring, several selected methods have 
been investigated. Their results were compared with the results of 
experimental tests performed to determine the actual stress level in the ring 
 «Вісник СумДУ», №1(73), 2005 28 
– both at the inside and outside walls – by means of strain gauges, placed as 
per figure 1. 
The first calculation method considered was proposed by Lari [4], based 
on the straight beams theory. This method, neglecting the effect of the axial 
force, N, and bending moment, M, developed in the slit zone (see fig. 2), 
recommends the following equation: 
 
2
2,77r
a
E t
D
.  (2.1) 
 
 
Figure 1 - Ring Specimen Geometry: 
1 – marks on the ring; 2 – slit zone; 3 – strain gauges 
 
The second method considered has been developed by the authors of the 
present paper using the slender beams theory. If considering the relative 
displacement, , and relative rotation, , between the two sides of the slit 
(fig. 2), the following equations can be written: 
 3  r
2
N + 2  r
2
M = E I”  , 
 2  r
2
N + 2  r M = E I” , (2.2) 
where r is the average radius of the ring, and I” is the modified inertia 
moment, given by: 
 «Вісник СумДУ», №1(73), 2005 29 
 I” = 
2
1
1 1
'Ih t r
, 
 I’ = 
2 43 3 3
1
12 20 112
h t t t
r r
. (2.3) 
The values of N and M will be determined by solving equations 2.2 
above, given  and  – calculated based on accurate measures of distances a 
and b from figure 2. Then, the stress values at the gauges (see fig. 1), 
respectively at the outside and at the inside of the ring specimen, can be 
calculated as follows: 
    ,
' 2 0.5
r ext
rN M M t
h t h t r I r t
, 
 int
' 2 0.5
r ,
rN M M t
h t h t r I r t
. (2.4) 
 
 
Figure 2 - Sectional Efforts in the Ring Specimen after Splitting 
The method above has been applied firstly considering the calculated 
value for the relative rotation, , and then assuming  ≡ 0, due to the fact 
that such rotation is mostly the effect of a radial displacement of the two 
sides of the slit and not the effect of residual stress. 
 «Вісник СумДУ», №1(73), 2005 30 
The third method considered has been developed by NKK Corporation 
[6] and is based on the following equation, considering the variation of the 
outside diameter of the ring: 
 
2
1
1 1
1
r
o
E
t
D D
, (2.5) 
where Do is the initial outside diameter, and D1 is the outside diameter after 
sectioning, measured in a direction perpendicular to the one of the slit. If D1 
> Do, the circumferential residual stress at the inside of the tube is 
compressive ( r < 0). 
Equation 2.5 is a particular case of the equation developed by Sachs, 
based on the elasticity theory, in order to assess the residual stress inside the 
tube wall at a given distance, x, from the outside surface of the tube: 
 1
2
1
( ) ( 2 )
1
o
r
o
D DE
x t x
D D
. (2.6) 
Finally, the calculation method proposed by Verner [9] which uses the 
equation below has also been considered: 
 
2
0,889 1r
a D
E
t t
. (2.7) 
The characteristics of the five ring specimens used to determine the 
residual stress by means of strain gauges are summarised in Table 1, 
together with the test results. All specimens were cut from seamless tubes 
made of N80 steel (SMYS value – 552 MPa). Table 2 compares the residual 
stress values at the outside of the specimens, obtained as test results, with 
the ones calculated by using the four methods presented above. 
 
Table 1 - Main Characteristics of the Ring Specimens and Measured Values 
 
Ring 
No. 
Effective 
Tube 
Outside 
Diameter 
Effective 
Tube 
Wall 
Thickness 
Measured Geometrical 
Characteristics (before and 
after slitting the ring) 
Strain 
at 
Gauge 
D t Do D1    
mm mm mm mm mm rad m/m 
1 140.0 9.5 139.69 140.25 2.569 0.016 240 
2 140.0 9.5 139.93 140.51 1.710 0.061 210 
3 136.8 7.9 136.62 138.12 4.127 0.038 650 
4 140.0 8.0 139.75 142.12 6.734 0.061 1050 
5 140.0 8.7 139.87 140.14 1.030 0.014 140 
 
 
 «Вісник СумДУ», №1(73), 2005 31 
Table 2 - Calculated and Experimental Values for the Circumferential 
Residual Stress at the Outside Wall of the Ring Specimens (in MPa) 
 
Ring 
No. 
Calculate
d 
Calculated with 
eqs. 2.2-2.4 
Calculated Calculated Test 
with eq. 
2.1 
  0   0 with eq. 2.7 with eq. 2.5 Result 
1 230.5 189.6 187.4 261.0 64.0 50.4 
2 205.8 - 700.8 167.3 233.0 64.7 44.1 
3 346.4 - 185.3 276.5 384.0 144.9 136.5 
4 545.9 - 289.1 435.4 604.4 224.7 220.5 
5 93.7 - 111.3 75.4 104.8 27.6 29.4 
It can be concluded that the method proposed by NKK and based on eq. 
2.5 is the most accurate as it gives values very close to the test results for all 
specimens investigated (see Table 2). This method has been used during all 
tests described in the next section. 
Experimental studies have also been performed by using strain gauges to 
determine the influence of the height, h, of the ring specimens on the results 
of the residual stress assessment. After investigating ring specimens with 
various values of the height (h = 10, 20, 50 mm), cut from the same tube, no 
influence of h on the results has been observed. 
 
3 TEST RESULTS FOR RESIDUAL STRESSES IN TUBES FOR 
CASING 
The level of circumferential residual stress has been investigated using 
specimens cut from seamless tubes for casing with the nominal outside 
diameter D = 139,7 mm, made of N80 steel (SMYS = 552 MPa). Ten 
specimens have been tested using the method defined in the previous 
section: five specimens have been taken from tubes before being subject to 
the thermal treatment (normalizing), and five specimens have been taken 
from tubes after being normalized and straightened. 
The results are reported in Table 3 for specimens cut before thermal 
treatment and in Table 4 for specimens normalized and straightened. In all 
cases, the residual stress values refer to the outside wall and were found to 
be positive; therefore, the residual stresses at the inside wall are compressive 
which has negative effects on the collapse resistance of casing. 
Based on the test results reported, the medium (σmed), maximum (σmax), 
and minimum (σmin) values of the residual circumferential stress have been 
calculated and are respectively: 
- before thermal treatment: σmed = 30.06 MPa; σmax = 40.5 MPa; σmin = 
22.85 MPa; 
- after normalizing and straightening: σmed = 131.68 MPa; σmax = 204.7 
MPa; σmin = 47.5 MPa. 
 
 «Вісник СумДУ», №1(73), 2005 32 
Table 3 - Test Results for Tube Specimens before Thermal Treatment 
 
Spec. 
No. 
Ring 
No. 
Measured Geometrical 
Characteristics (before and after 
slitting the ring) 
Circumferential Residual 
Stress Values 
Effective 
W.T.,  t 
Do D1 for each 
ring 
Medium for 
specimen 
mm mm mm MPa MPa 
I 
1 
8.5 
140.73 141.02 29.34 
33.16 2 140.58 140.90 32.43 
3 140.65 141.05 40.48 
4 140.93 141.24 30.40 
II 
5 
8.7 
140.71 141.05 34.39 
29.73 6 141.13 141.40 27.16 
7 139.87 140.14 27.65 
III 
8 
8.7 
141.21 141.56 35.15 
29.18 9 140.91 141.20 29.26 
10 141.15 141.38 23.14 
IV 11 8.8 141.10 141.38 28.43 28.43 
V 12 9.0 141.47 141.69 22.85 22.85 
 
Table 4 - Test Results for Tube Specimens after Normalizing and 
Straightening 
Spec. 
No. 
Ring 
No. 
Measured Geometrical 
Characteristics (before and after 
slitting the ring) 
Circumferential Residual 
Stress Values 
Effective 
W.T.,  t 
Do D1 
for each 
ring 
Medium for 
specimen 
mm mm mm MPa MPa 
VI 
13 
8.5 
140.67 142.27 156.82 
181.73 14 140.48 142.57 204.69 
15 140.75 142.63 183.69 
VII 
16 
8.4 
140.53 141.01 47.51 
96.64 17 140.75 142.46 167.28 
18 140.48 141.66 116.31 
19 140.82 141.39 55.44 
VIII 
20 
8.5 
140.89 142.92 197.90 
160.18 21 140.80 142.81 196.08 
22 140.78 141.66 86.55 
IX 23 8.2 140.94 141.78 79.48 79.48 
X 24 8.5 140.58 141.75 88.46 88.46 
 
 «Вісник СумДУ», №1(73), 2005 33 
It can be seen that the residual stress level is considerably greater (about 
4.4 times on average) after normalizing and straightening. As a 
consequence, the residual stress in casing is mainly the effect of thermal 
treatment and straightening operations 
In addition, for six specimens (three not subject to thermal treatment and 
three normalized and straightened), the variation of the residual 
circumferential stress across the tube thickness has been investigated. To 
that purpose, eight rings have been cut from each of these specimens and 
their thickness has been reduced with various values (between 1 and 4 mm) 
by machining four of them at the inside and the other four at the outside, as 
shown in figure 3. The value of the residual stress was obtained in each case 
by using eq. 2.5. 
 
Figure 3 - Ring Specimen Used to Investigate Residual Stress Variation: 
1, 2 – material stratum machined at the outside (1) or at the inside (2) of the specimen 
 
 
The results of this investigation are summarised in figure 4 for the 
specimens not subject to thermal treatment and in figure 5 for specimens 
normalized and straightened.  
 
 «Вісник СумДУ», №1(73), 2005 34 
 
Figure 4 - Residual Stress Variation before Normalizing and Straightening 
 
Figure 5 - Residual Stress Variation after Normalizing and Straightening 
These results show that the residual stress values are not uniformly 
distributed across the tubes thickness, the maximum value being reached at 
about 2-3 mm from the inside wall. Such value corresponds to a 
compressive stress and is about 45 MPa for the specimens not subjected to 
thermal treatment and about 220 MPa for the specimens normalized and 
-60
-50
-40
-30
-20
-10
0
10
20
30
40
0 1 2 3 4 5 6 7 8 9
R
es
id
u
a
l 
S
tr
es
s 
(M
P
a
) 
Distance from Inside, s (mm)  
Spec. I
Spec. IV
Spec. V
-250
-200
-150
-100
-50
0
50
100
150
0 1 2 3 4 5 6 7 8 9
R
es
id
u
a
l 
S
tr
es
s 
(M
P
a
) 
Distance from Inside, s (mm)  
Spec. VII
Spec. IX
Spec. X
 «Вісник СумДУ», №1(73), 2005 35 
straightened. This last value is important, corresponding to about 40% of 
SMYS of the specimens’ material. 
Finally, an evaluation of the effect of the residual stress upon the value of 
the collapse pressure of casing was performed based on eq. 1.4 and the 
results of the test performed. 
If considering the average value of the residual circumferential stress of 
casing (corresponding to specimens subjected to thermal treatment and 
straightening operations), i.e. 132 MPa (compression stress), eq. 1.4 shows a 
reduction with 17.3% of the collapse pressure value with respect to a tube 
without residual stresses. However, if the maximum value obtained during 
the tests performed is taken into account (250 MPa for specimen X – fig. 5) 
such reduction becomes important (about 33%). 
 
4 CONCLUSIONS  
 The experimental results described in this paper aimed at 
investigating the level of residual circumferential stresses in seamless tubes 
for well casing, its variation across the tube thickness, and its influence on 
the collapse resistance. 
 The most adequate (simple, easy to apply, and accurate) method to 
determine the actual value of residual circumferential stresses in tubes for 
casing was found to be the slit ring method combined with NKK calculation 
method (eq. 2.5) for assessing the stress level at the inside wall. 
 The test results showed for all specimens that the residual stress is 
positive (traction) at the outside wall of casing and negative (compression) 
at the inside, corresponding to the most unfavourable situation from the 
point of view of the collapse resistance. 
 The residual stress values are about 4.4 times (on average) greater in 
tubes normalized and straightened with respect to the tubes not yet subjected 
to thermal treatment; therefore, the main source of these residual stresses in 
casing are the thermal treatment and the straightening operation. 
 The residual stresses are not uniformly distributed across the tube 
thickness, the maximum value being a compressive stress reached at about 
2-3 mm from the inside wall; such value is about 220 MPa (about 40% of 
SMYS of the specimens’ material) for tubes normalized and straightened. 
 The average value of the circumferential residual stress in casing 
after normalizing and straightening is about 132 MPa – corresponding to a 
reduction of about 17.3% of the collapse resistance – while the maximum 
value is about 250 MPa – corresponding to an important reduction of about 
33% of such resistance. 
 
ABSTRACT 
 
The collapse resistance of casing is influenced by many factors, among which the level of residual stresses is 
an essential one. This paper presents the results of research activities aimed to determine such stress level in 
seamless tube specimens made of grade N80 steel. 
 «Вісник СумДУ», №1(73), 2005 36 
These activities included: the selection – based on experimental and theoretical studies – of the most adequate 
method to determine the residual circumferential stresses in tubes for well casing; a series of experimental tests to 
define the level of such residual stresses and its variation across the tube thickness; finally, an evaluation of the 
influence of such stresses upon the value of the collapse pressure of casing. 
 
REFERENCES 
 
1. API, “Bulletin 5C3 on Formulas and Calculations for Casing, Tubing, Drill Pipe and Line Pipe Properties”, 
3rd edition, American Petroleum Institute, Dallas, Texas, 1980. 
2. Dumitrescu, A., “Studies on Improving Resistance and Stability Characteristics of Subsea Pipelines”, Ph. D. 
Thesis, “Petroleum-Gas” University of Ploieşti, 1998. 
3. Langner, C. G., “Introduction – History and Review of Collapse”, Proceedings, Seminar on Collapse of 
Offshore Pipelines, American Gas Association, 1990. 
4. Lari, I., “Turbo-Electrical Drilling Method and Studies Concerning Residual Stresses in the Drill Pipe 
Coupling”, Ph. D. Thesis, I. P. G. G. Bucureşti, 1967. 
5. Nara, Y. a. o., “Theoretical Study on Casing Collapse”, Sumitomo Metal Industries Ltd., Osaka, Japan, Jan. 
1981. 
6. NKK, “Evaluation of Factors which Affect on Collapse Strength”, N. K. K. Corporation, March 1990. 
7. Timoshenko, S. P., Gere, J. M., Theory of Elastic Stability, McGraw-Hill Book Company, New York, 1961. 
8. Tokimasa, K., Tanaka, K., “FEM Analysis of the Collapse Strength of a Tube”, ASME Paper No. 84-PVP-
104, 1984. 
9. Verner, E. A., Langner, C. G., Reifel, M. D., “Collapse of Thick Wall Pipe in Ultra-Deep Water”, Journal of 
Transportation Engineering, Vol. 111, No. 5, p. 485-509, Sept. 1985. 
 
Надійшла до редакції 19 лютого 2005р. 
  
 «Вісник СумДУ», №1(73), 2005 37 
 


English

Analysis of the residual stress level in casing and its influence on the collapse strength

Electronic Sumy State University Institutional Repository

 «Вісник СумДУ», №1(73), 2005 25 ANALYSIS OF THE RESIDUAL STRESS LEVEL IN CASING AND ITS INFLUENCE ON THE COLLAPSE STRENGTH  Andrei Dumitrescu, Ph. D. Eng.,  Dragoş Gabriel Zisopol, Ph. D. Eng. “Petroleum-Gas” University of Ploieşti, Romania  1 INTRODUCTION One of the phenomenons which can decisively affect the reliability of oil well casing is collapse under the effect of external hydrostatic pressure (leading to ovalisation followed by flattening of tubulars). The collapse mechanism differs essentially with the value of the ratio between the outside diameter, D, and the wall thickness, t, of casing: - for greater values of this ratio (D/t > 35), collapse occurs by means of an elastic flattening, before the tube material reaches its yield strength; in case of elastic collapse of a perfect circular tube, the critical value of the external pressure (the elastic collapse pressure, pE) can be calculated using the following equation [7]:  2 32 11 /EEpD t,  (1.1) where E is Young’s elastic modulus, and  Poisson’s coefficient of the casing material; - for smaller values of the D/t ratio (under 15...20), collapse will take place in the plastic field; in such case, if considering the thin-wall tubes theory, the critical value of the external pressure (the plastic collapse pressure, pF) can be calculated as follows [3]:   pF = 2 c·t/D  (1.2) where c is the minimum specified yield strength (SMYS) of the casing material; - for D/t = 20...35, the failure mechanism is much more complex (elastic-plastic collapse); a gradual passage from elastic failure to plastic failure will take place and different calculation methods, detailed in [2], have been proposed in such case by various researchers. The collapse resistance capacity of casing is importantly affected by some factors, among which the level of residual stress, the pipe geometrical imperfections (mainly its initial ovality), and the pipe material anisotropy. The effect of these factors can be accounted for when assessing the critical value of the external pressure (the collapse pressure, pc), whatever the failure mechanism, by using the following equation [3]:   pc = k·pF = kr·kδ kξ pF  (1.3) where k ≤ 1 is a reduction factor, considering the effect of residual stresses (by means of the coefficient kr), of initial tube ovality (kδ), and of initial tube eccentricity (kξ).  «Вісник СумДУ», №1(73), 2005 26 Residual stress is an important factor affecting the collapse strength of casing and therefore its effect has been studied in various papers [3, 5, 6, 8, 9]. Such stresses are a direct consequence of the operations performed during the manufacturing process of tubulars, mainly rolling, thermal treatment (quenching, normalizing, etc.) and straightening. The effect of the residual stress level on the collapse pressure value can be assessed using the following equation, developed in [8], to calculate kr coefficient (from eq. 1.3):  «Вісник СумДУ», №1(73), 2005 27    1 + 0,8 r/ c if  r < 0  kr =  (1.4)   22 1 0,8 /1 1 2 /r ct D if  r > 0 where r is the level of residual stresses at the inside wall of the tube. The first equation above applies in case yielding develops at the inside wall of the tube (compression at the inside), while the second one applies in case of yielding at the outside wall (traction at the inside). The effect of a compressive stress ( r < 0) is to cause early yielding on the inside and thus a reduction of wall thickness, while a moderate level of traction stress ( r > 0) on the inside has a positive effect by reducing the value of collapse pressure, pc. In such context, the research activities described in this paper aimed to investigate the level of the residual circumferential stresses in seamless tubes for well casing and its variation across the tube thickness, and to evaluate – based on eq. 1.4 above – the influence of such stresses on the critical collapse pressure of casing.  2 DEVELOPMENT OF AN EXPERIMENTAL METHOD FOR THE EVALUATION OF THE RESIDUAL STRESSES IN TUBES FOR CASING In order to properly determine the actual level of residual circumferential stresses in seamless tubes, an extended research activity has been performed to define the most adequate experimental method, simple, easy to apply, and also accurate. After analysing various methods, we reached the conclusion that the most suited one for measuring residual stresses which do not vary circumferentially is the slit ring method, described in [4]. In this procedure, a ring with the height equal to h is cut from the tube (with the outside diameter De, the inside diameter Di, and the wall thickness t) and then slit axially (fig. 1). The movement across the slit gives a measure of the residual stress; an opening of the ring indicates compressive stress on the inside wall ( r < 0). Various values have been proposed for the ring height, h, ranging between 51 mm (2 in) [9] and 10 mm [4]. In order to define the most adequate calculation method to assess the residual stress level, given the measured opening of the ring, Δa, or the variation of the outside diameter of the ring, several selected methods have been investigated. Their results were compared with the results of experimental tests performed to determine the actual stress level in the ring  «Вісник СумДУ», №1(73), 2005 28 – both at the inside and outside walls – by means of strain gauges, placed as per figure 1. The first calculation method considered was proposed by Lari [4], based on the straight beams theory. This method, neglecting the effect of the axial force, N, and bending moment, M, developed in the slit zone (see fig. 2), recommends the following equation:  22,77raE tD.  (2.1)   Figure 1 - Ring Specimen Geometry: 1 – marks on the ring; 2 – slit zone; 3 – strain gauges  The second method considered has been developed by the authors of the present paper using the slender beams theory. If considering the relative displacement, , and relative rotation, , between the two sides of the slit (fig. 2), the following equations can be written:  3  r2N + 2  r2M = E I”  ,  2  r2N + 2  r M = E I” , (2.2) where r is the average radius of the ring, and I” is the modified inertia moment, given by:  «Вісник СумДУ», №1(73), 2005 29  I” = 211 1'Ih t r,  I’ = 2 43 3 3112 20 112h t t tr r. (2.3) The values of N and M will be determined by solving equations 2.2 above, given  and  – calculated based on accurate measures of distances a and b from figure 2. Then, the stress values at the gauges (see fig. 1), respectively at the outside and at the inside of the ring specimen, can be calculated as follows:     ,' 2 0.5r extrN M M th t h t r I r t,  int' 2 0.5r ,rN M M th t h t r I r t. (2.4)   Figure 2 - Sectional Efforts in the Ring Specimen after Splitting The method above has been applied firstly considering the calculated value for the relative rotation, , and then assuming  ≡ 0, due to the fact that such rotation is mostly the effect of a radial displacement of the two sides of the slit and not the effect of residual stress.  «Вісник СумДУ», №1(73), 2005 30 The third method considered has been developed by NKK Corporation [6] and is based on the following equation, considering the variation of the outside diameter of the ring:  211 11roEtD D, (2.5) where Do is the initial outside diameter, and D1 is the outside diameter after sectioning, measured in a direction perpendicular to the one of the slit. If D1 > Do, the circumferential residual stress at the inside of the tube is compressive ( r < 0). Equation 2.5 is a particular case of the equation developed by Sachs, based on the elasticity theory, in order to assess the residual stress inside the tube wall at a given distance, x, from the outside surface of the tube:  121( ) ( 2 )1oroD DEx t xD D. (2.6) Finally, the calculation method proposed by Verner [9] which uses the equation below has also been considered:  20,889 1ra DEt t. (2.7) The characteristics of the five ring specimens used to determine the residual stress by means of strain gauges are summarised in Table 1, together with the test results. All specimens were cut from seamless tubes made of N80 steel (SMYS value – 552 MPa). Table 2 compares the residual stress values at the outside of the specimens, obtained as test results, with the ones calculated by using the four methods presented above.  Table 1 - Main Characteristics of the Ring Specimens and Measured Values  Ring No. Effective Tube Outside Diameter Effective Tube Wall Thickness Measured Geometrical Characteristics (before and after slitting the ring) Strain at Gauge D t Do D1    mm mm mm mm mm rad m/m 1 140.0 9.5 139.69 140.25 2.569 0.016 240 2 140.0 9.5 139.93 140.51 1.710 0.061 210 3 136.8 7.9 136.62 138.12 4.127 0.038 650 4 140.0 8.0 139.75 142.12 6.734 0.061 1050 5 140.0 8.7 139.87 140.14 1.030 0.014 140    «Вісник СумДУ», №1(73), 2005 31 Table 2 - Calculated and Experimental Values for the Circumferential Residual Stress at the Outside Wall of the Ring Specimens (in MPa)  Ring No. Calculated Calculated with eqs. 2.2-2.4 Calculated Calculated Test with eq. 2.1   0   0 with eq. 2.7 with eq. 2.5 Result 1 230.5 189.6 187.4 261.0 64.0 50.4 2 205.8 - 700.8 167.3 233.0 64.7 44.1 3 346.4 - 185.3 276.5 384.0 144.9 136.5 4 545.9 - 289.1 435.4 604.4 224.7 220.5 5 93.7 - 111.3 75.4 104.8 27.6 29.4 It can be concluded that the method proposed by NKK and based on eq. 2.5 is the most accurate as it gives values very close to the test results for all specimens investigated (see Table 2). This method has been used during all tests described in the next section. Experimental studies have also been performed by using strain gauges to determine the influence of the height, h, of the ring specimens on the results of the residual stress assessment. After investigating ring specimens with various values of the height (h = 10, 20, 50 mm), cut from the same tube, no influence of h on the results has been observed.  3 TEST RESULTS FOR RESIDUAL STRESSES IN TUBES FOR CASING The level of circumferential residual stress has been investigated using specimens cut from seamless tubes for casing with the nominal outside diameter D = 139,7 mm, made of N80 steel (SMYS = 552 MPa). Ten specimens have been tested using the method defined in the previous section: five specimens have been taken from tubes before being subject to the thermal treatment (normalizing), and five specimens have been taken from tubes after being normalized and straightened. The results are reported in Table 3 for specimens cut before thermal treatment and in Table 4 for specimens normalized and straightened. In all cases, the residual stress values refer to the outside wall and were found to be positive; therefore, the residual stresses at the inside wall are compressive which has negative effects on the collapse resistance of casing. Based on the test results reported, the medium (σmed), maximum (σmax), and minimum (σmin) values of the residual circumferential stress have been calculated and are respectively: - before thermal treatment: σmed = 30.06 MPa; σmax = 40.5 MPa; σmin = 22.85 MPa; - after normalizing and straightening: σmed = 131.68 MPa; σmax = 204.7 MPa; σmin = 47.5 MPa.   «Вісник СумДУ», №1(73), 2005 32 Table 3 - Test Results for Tube Specimens before Thermal Treatment  Spec. No. Ring No. Measured Geometrical Characteristics (before and after slitting the ring) Circumferential Residual Stress Values Effective W.T.,  t Do D1 for each ring Medium for specimen mm mm mm MPa MPa I 1 8.5 140.73 141.02 29.34 33.16 2 140.58 140.90 32.43 3 140.65 141.05 40.48 4 140.93 141.24 30.40 II 5 8.7 140.71 141.05 34.39 29.73 6 141.13 141.40 27.16 7 139.87 140.14 27.65 III 8 8.7 141.21 141.56 35.15 29.18 9 140.91 141.20 29.26 10 141.15 141.38 23.14 IV 11 8.8 141.10 141.38 28.43 28.43 V 12 9.0 141.47 141.69 22.85 22.85  Table 4 - Test Results for Tube Specimens after Normalizing and Straightening Spec. No. Ring No. Measured Geometrical Characteristics (before and after slitting the ring) Circumferential Residual Stress Values Effective W.T.,  t Do D1 for each ring Medium for specimen mm mm mm MPa MPa VI 13 8.5 140.67 142.27 156.82 181.73 14 140.48 142.57 204.69 15 140.75 142.63 183.69 VII 16 8.4 140.53 141.01 47.51 96.64 17 140.75 142.46 167.28 18 140.48 141.66 116.31 19 140.82 141.39 55.44 VIII 20 8.5 140.89 142.92 197.90 160.18 21 140.80 142.81 196.08 22 140.78 141.66 86.55 IX 23 8.2 140.94 141.78 79.48 79.48 X 24 8.5 140.58 141.75 88.46 88.46   «Вісник СумДУ», №1(73), 2005 33 It can be seen that the residual stress level is considerably greater (about 4.4 times on average) after normalizing and straightening. As a consequence, the residual stress in casing is mainly the effect of thermal treatment and straightening operations In addition, for six specimens (three not subject to thermal treatment and three normalized and straightened), the variation of the residual circumferential stress across the tube thickness has been investigated. To that purpose, eight rings have been cut from each of these specimens and their thickness has been reduced with various values (between 1 and 4 mm) by machining four of them at the inside and the other four at the outside, as shown in figure 3. The value of the residual stress was obtained in each case by using eq. 2.5.  Figure 3 - Ring Specimen Used to Investigate Residual Stress Variation: 1, 2 – material stratum machined at the outside (1) or at the inside (2) of the specimen   The results of this investigation are summarised in figure 4 for the specimens not subject to thermal treatment and in figure 5 for specimens normalized and straightened.    «Вісник СумДУ», №1(73), 2005 34  Figure 4 - Residual Stress Variation before Normalizing and Straightening  Figure 5 - Residual Stress Variation after Normalizing and Straightening These results show that the residual stress values are not uniformly distributed across the tubes thickness, the maximum value being reached at about 2-3 mm from the inside wall. Such value corresponds to a compressive stress and is about 45 MPa for the specimens not subjected to thermal treatment and about 220 MPa for the specimens normalized and -60-50-40-30-20-100102030400 1 2 3 4 5 6 7 8 9Residual Stress (MPa) Distance from Inside, s (mm)  Spec. ISpec. IVSpec. V-250-200-150-100-500501001500 1 2 3 4 5 6 7 8 9Residual Stress (MPa) Distance from Inside, s (mm)  Spec. VIISpec. IXSpec. X «Вісник СумДУ», №1(73), 2005 35 straightened. This last value is important, corresponding to about 40% of SMYS of the specimens’ material. Finally, an evaluation of the effect of the residual stress upon the value of the collapse pressure of casing was performed based on eq. 1.4 and the results of the test performed. If considering the average value of the residual circumferential stress of casing (corresponding to specimens subjected to thermal treatment and straightening operations), i.e. 132 MPa (compression stress), eq. 1.4 shows a reduction with 17.3% of the collapse pressure value with respect to a tube without residual stresses. However, if the maximum value obtained during the tests performed is taken into account (250 MPa for specimen X – fig. 5) such reduction becomes important (about 33%).  4 CONCLUSIONS   The experimental results described in this paper aimed at investigating the level of residual circumferential stresses in seamless tubes for well casing, its variation across the tube thickness, and its influence on the collapse resistance.  The most adequate (simple, easy to apply, and accurate) method to determine the actual value of residual circumferential stresses in tubes for casing was found to be the slit ring method combined with NKK calculation method (eq. 2.5) for assessing the stress level at the inside wall.  The test results showed for all specimens that the residual stress is positive (traction) at the outside wall of casing and negative (compression) at the inside, corresponding to the most unfavourable situation from the point of view of the collapse resistance.  The residual stress values are about 4.4 times (on average) greater in tubes normalized and straightened with respect to the tubes not yet subjected to thermal treatment; therefore, the main source of these residual stresses in casing are the thermal treatment and the straightening operation.  The residual stresses are not uniformly distributed across the tube thickness, the maximum value being a compressive stress reached at about 2-3 mm from the inside wall; such value is about 220 MPa (about 40% of SMYS of the specimens’ material) for tubes normalized and straightened.  The average value of the circumferential residual stress in casing after normalizing and straightening is about 132 MPa – corresponding to a reduction of about 17.3% of the collapse resistance – while the maximum value is about 250 MPa – corresponding to an important reduction of about 33% of such resistance.  ABSTRACT  The collapse resistance of casing is influenced by many factors, among which the level of residual stresses is an essential one. This paper presents the results of research activities aimed to determine such stress level in seamless tube specimens made of grade N80 steel.  «Вісник СумДУ», №1(73), 2005 36 These activities included: the selection – based on experimental and theoretical studies – of the most adequate method to determine the residual circumferential stresses in tubes for well casing; a series of experimental tests to define the level of such residual stresses and its variation across the tube thickness; finally, an evaluation of the influence of such stresses upon the value of the collapse pressure of casing.  REFERENCES  1. API, “Bulletin 5C3 on Formulas and Calculations for Casing, Tubing, Drill Pipe and Line Pipe Properties”, 3rd edition, American Petroleum Institute, Dallas, Texas, 1980. 2. Dumitrescu, A., “Studies on Improving Resistance and Stability Characteristics of Subsea Pipelines”, Ph. D. Thesis, “Petroleum-Gas” University of Ploieşti, 1998. 3. Langner, C. G., “Introduction – History and Review of Collapse”, Proceedings, Seminar on Collapse of Offshore Pipelines, American Gas Association, 1990. 4. Lari, I., “Turbo-Electrical Drilling Method and Studies Concerning Residual Stresses in the Drill Pipe Coupling”, Ph. D. Thesis, I. P. G. G. Bucureşti, 1967. 5. Nara, Y. a. o., “Theoretical Study on Casing Collapse”, Sumitomo Metal Industries Ltd., Osaka, Japan, Jan. 1981. 6. NKK, “Evaluation of Factors which Affect on Collapse Strength”, N. K. K. Corporation, March 1990. 7. Timoshenko, S. P., Gere, J. M., Theory of Elastic Stability, McGraw-Hill Book Company, New York, 1961. 8. Tokimasa, K., Tanaka, K., “FEM Analysis of the Collapse Strength of a Tube”, ASME Paper No. 84-PVP-104, 1984. 9. Verner, E. A., Langner, C. G., Reifel, M. D., “Collapse of Thick Wall Pipe in Ultra-Deep Water”, Journal of Transportation Engineering, Vol. 111, No. 5, p. 485-509, Sept. 1985.  Надійшла до редакції 19 лютого 2005р.    «Вісник СумДУ», №1(73), 2005 37  

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Analysis of the residual stress level in casing and its influence on the collapse strength

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