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Chelyabinsk Physical and Mathematical Journal. 2021. Vol. 6, iss. 2. P. 208-216.
DOI: 10.47475/2500-0101-2021-16206
CRITICAL STATES OF LONGITUDINAL SEAM PIPES OF LARGE DIAMETER UNDER AXIAL COMPRESSION AND INTERNAL PRESSURE
V.L. Dilman", T.V. Karpetab
South Ural State University (National Research University), Chelyabinsk, Russia " [email protected], [email protected]
We consider the conditions of loading of a thin-walled cylindrical shell containing a layer of less strong material along the generatrix, including a longitudinal seam pipe of a large diameter, under its axial compression and internal pressure. The base shell material and the layer material are elastoplastic and hardenable. The ultimate strengths and yield points of the layer are lower than that of the base material. We investigate critical states of the layer and shell arising from general or local plastic strains. The purpose of the article is to establish dependence of critical strains, stresses and pressures on the shell on its mechanical and geometric parameters and loading conditions. The method of the study is based on the application of the Swift — Marciniak criterion to lose the stability of the process of plastic strain of the layer material. We obtain explicit analytical expressions for the required quantities. The results allow to determine critical pressures under the given conditions of loading and wall thickness at a given operating pressure.
Keywords: thin-walled cylindrical shell, longitudinal seam pipe, plastic stability, Swift — Marciniak criterion, critical strains and stresses, localization of plastic strain.
Introduction
In their near-weld zone (heat-affected zone or fusion zone), longitudinal seam welded pipes, including thermally hardened ones, can contain areas of weakening, i.e. layers of metal, the yield point and temporary resistance of which are lower than that of the base pipe metal. Mechanical heterogeneity of welded joints, the thickness and location of less strength layers have a significant influence on the strength of the pipe. Despite widespread use of longitudinal seam pipes of large diameter, their work as part of pipelines under difficult conditions was not studied enough. Carrier capacity of main pipelines is significantly dependent on conditions of their work. The pipeline axial load, leading to biaxial loading of the pipe wall, can significantly effect on the critical pressure in the pipeline [1; 2]. Axial load caused by temperature fluctuations and special operating conditions of the pipeline (under water, in an area with mobile soils, in mountainous, in areas with high seismic activity [3; 4]), can lead to axial stresses a2 in the shell wall opposite in sign to the annular stresses o\ generated by internal or external pressure.
Influence of the coefficient of pipe biaxiality loading m = a\/a2 on carrying capacity of pipelines were studied in [5-12] and others. The works [7-12] use the Swift — Marciniak (SM) criterion to lose stability of plastic strain of thin-walled cylindrical shell (TWCS) [13-15] in order to determine critical states of TWCS made of hardened material, including large diameter pipes. In the works [7-12], it was assumed that m > 0.
The work considers the stress-strain state (SSS) of a longitudinal layer made of hardened material as part of TWCS subject to monotonic static loading by both internal pressure and axial compressive force at the critical moment of loading. The purpose of the work is to find SSS of a less strong layer of TWCS at the critical moment of loading having the specified type based on the use of the SM criterion; then, on this basis, to find analytical expressions for critical stresses and strains in the TWCS layer (pipes of large diameter), critical pressures and axial loads on TWCS depending on the mechanical and geometric parameters of the shells and layers under axial compression and internal pressure required both to analyse the strength of longitudinal "soft" layers of such shells and to evaluate the carrier capacity of longitudinal seam pipes. It is assumed that strain diagrams of both base material and layer material are approximated by dependence of the form (4) (see below).
1. Notation and Assumptions
Let o\, 72, 73 be axial, annular and radial normal stresses in the TWCS wall, respectively, Si, e2, s3 be axial, annular and radial logarithmic strains of the TWCS wall, respectively, p be an internal or external pressure, N be an axial tensile or compressive force, m = <7\/<72 be a coefficient of biaxiality loading of the TWCS wall. In order to simplify the formulas, we use the following notation [15]: s = Vm2 — m + 1. Within the framework of the paper, the subscript cr indicates the critical value, and signs + and — given as superscripts denote the ratio of the value to the base material (BM) of the TWCS and the layer material, respectively.
The shell and layer materials are assumed to be elastoplastic, homogeneous and isotropic, moreover, the "hypothesis of a single curve" is assumed to be fulfilled, that is, the diagram of strain of the hardened TWCS material has the form:
7i = f (Si), (1)
where 7i and si are intensities of stresses and strains in the shell wall or layer. In the work, we consider the main parameters of the TWCS and the longitudinal layer described below.
As regards geometrical parameters of the TWCS, we note the inner radius R and the wall thickness t. At the initial moment of loading, we have R = R0 and t = t0. The relationship of R and t with logarithmic strains is as follows:
R = Ro exp s2, t = to exp s3. (2)
The relative layer thickness is x = h/t , where h is a layer thickness.
The mechanical properties of the TWCS material and the layer material determine the following parameters: 1) the ultimate strengths 7+ and 7- of the shell BM and the layer, respectively, the coefficient of mechanical heterogeneity of the connection K = 7+/7- ; 2) the yield points 7+ and 7- of the shell BM and the layer, respectively; 3) the Young moduli + and - of the shell BM and the layer, respectively; 4) the strength indicators n+ and n- of the shell BM and the layer, respectively, under their strains in the plastic zone.
If a statement refers to the material of both the shell and the layer, the subscripts + and — are not used.
In the zone of plastic strain, for many materials, the power-law approximation of the strain diagram (1)
7i = Asn, A = enn-n7B
has satisfactory experimental confirmation [16]. Among disadvantages of this approximation, we note its noticeable discrepancy with real dependence for small strains and the fact that the approximation does not distinguish materials with the same ultimate strength, but different yield points. The work uses a more general approximation of dependence (1), which, in addition to the strength indicator n, has one more parameter B when describing the strain process of the the strength material in the plastic section. This approximation has the form
Vi = / (£i)
E£i, 0 < £ < £t , (3)
A(£ + B)n, A = en-B(n - B)-nVB, £ > £t. (3)
Here e is the Euler constant, £T is the strain rate corresponding to the onset of plastic material flow: / (£T) = vt. In next formula (4), the coefficient A is calculated using the SM criterion [10, p. 27-30]. Whereas n and vb are material characteristics, the parameter B needs to be determined. Continuity of dependence (3) for £i = £T implies that E£T = A(£T + B)n and £T = vt/E , therefore, we arrive at the following transcendental equation for the unknown B:
_n VT _ , ,-n-B( vt
e-n^ = (n - B) e ^E + B) . (4)
2. Critical State of Layer Material
In the wall of the TWCS, the main stresses are calculated [17, p. 220] by the formulas
RP + N Rp 0
Vl = + V2 = T; = 0. (5)
The third equality of (5) is satisfied approximately. Since t ^ R in the TWCS, the second equality of (5) implies that p ^ v2, therefore a3 ^ v2. Hence a3 ~ 0. In a weaker layer, a3 may deviate significantly from zero due to the occurrence of a complex stress state leading to hardening of the layer material. At the critical moment of loading the layer, we assume that BM is at the stage of elastic strain or stable plastic hardening and has restraining effect on the layer, which, in the axial direction, undergoes a strain determined by a relatively small strain of BM. Therefore, in the axial direction, we assume that the strain of the layer is £- ~ 0. According to the theory of small strains,
3£i £l £2 Vi + V2 + V3
=-=-, Vo =-o-, (6)
Vi — Vo V2 — Vo 3
therefore v- ~ v-, i.e. 2v- ~ v- + v- (in particular, m- ~ 1/2 due to v3 ~ 0). It follows from (6) that
3£- 2£-
2 - - - (7) 2 Vi V2 — V3
The median plane between contact surfaces is the section of the layer with the smallest area (net section), along which the critical load should be calculated. In the net section of the layer, due to symmetry, shear stresses are equal to zero (the normal stresses 3 are nonzero). In this case, the plasticity condition has the form
V- — V- = 2k- = ^V3. (8)
Let g, 1 < g < K, be the coefficient of contact hardening layer material [10, pp. 105, 232]. In wide layers, for x > 1, there is no contact hardening, i.e. g = 1;
for 0.5 < x < 1, contact hardening is small, i.e. g ~ 1, and can be ignored [18; 19]. Methods to determine g are given in [18-20] and others. At the critical moment of layer loading, due to 73 > 0 (in thin layers), the average value over the width of the layer 72 > 2k is as follows:
7— = 2gk- = -2| 7-. (9)
From (7) and (8) it follows that, on the net section,
_ V3 _
S- = T"S-.
(10)
In order to calculate the stress intensity on the net section at the critical moment of loading, we consider a rectangular fragment of a flat sheet sample containing a layer of less strong material whose contact boundaries are parallel to the end faces of the sheet and are orthogonal to the stress direction a2 (we neglect the curvature of the TWCS due to the thinness). Let h = h0 expe- be the layer thickness, t and L be the thickness and width of the fragment, V = tLh be its volume, which is not changes during plastic strain, N2 be the external force acting in the direction of a2 at the critical moment of loading. At this moment, the force stops increase, i. e. is constant. Then (2) implies that
(To
N2 tL
N2h N2h
2h0
V
V
exp 1^2
Therefore, from (9) and (10) it follows that
<7„-
_ N2^0
-a- =--
2g 2 2g V
exp I e2
V3 N2 ho
2g V
exp
V3 _
T £2
(11)
In accordance with the SM criterion, the derivatives of dependencies (3) and (11) on the variable s- should be equated. Taking into account that V and N2 are constant, we have
2n-
73
- B2
As it follows from (10) and (12), in the middle section of the layer,
- - ^R-
S2 cr = n — B .
(12)
(13)
Substitute the right-hand side of (12) into (3) to obtain the formula for calculating the critical stress intensity:
<7„-
Indeed,
a
B
n
3B
2e
a
B
V3 (1 - B) e
B
2
2
2e
a- « (1 + 1.144n)
R2 R3
BB (1 - B) eB « 1----« 1
a
B.
(14)
since usually B < 0.03. From (14) it is obvious that the intensity of critical stress increases by about 1.12 times when the indicator of strain hardening n changes from 0.1
F
n
n
n
n
to 0.2. On the other hand, the influence of vt on critical stresses is insignificant. Indeed, relation (14) does not include the parameter B.
Calculate the value of the critical pressure, which is limited by the strength of the layer. Since £1 = 0, then £3 = —£2. Then t = t0 exp (£-) = t0 exp (—£-) . From (9) and (14) it follows that
- _ 21+n gen _
V 2cr =
73
1+n 7 B•
Hence, the second formula of (5) and (13) imply that
Per
7 2er t
R
G <
2 er
iQ exp (-^2er)
21+r
g exp
'V3
B2
R0
V31+n RQ
to
-a
b •
(15)
From (15), it is obvious that the parameter B makes more precise the formula for
/3 2
calculating the critical pressure to ^ B .
3. Critical Pressure in TWCS. Computational Experiments
In a seamless TWCS [21, 22], dependence of critical pressures on the coefficient of biaxial loading m can be made more precise if use strain diagram (3). In this case (here n = n+, B = B+)
p+ =
er
2
m
n '
2-m exp B | —--1
2s
?1-n RQaB'
m1
3 exM W ^-1
t0
(16)
, - <7+, -1 < m < 0.5.
1nB
S1-n Rq
2
t
0
Fig. 1. Dependence of the critical pressure pcr on m under internal pressure and axial compression at g = 1, aT/ = 0.75, a+ = 600 MPa, a- = 400 MPa, to = 17 mm, R0 = 610 mm, n = 0.10; 0.15; 0.20 from bottom to top, respectively
Fig. 2. Dependence of the critical pressure pcr on m under internal pressure and axial compression at g = 1.1, aT/aB = 0.75, a + = 600 MPa, a- = 400 MPa, t0 = 17 mm, R0 = 610 mm, n = 0.10; 0.15; 0.20 from bottom to top, respectively
Obviously, the critical pressure of TWCS containing a longitudinal seam of less strong material is defined as
Per = min (P-riP+r) ? (17)
where p— and p+ are found by (15) and (16). Computational experiments were carried out on the basis of these formulas. Dependence of critical pressures (17) on m G [-2; 0.5] for K = /a- = 1.5 and n+ = n- are shown in Figs. 1-4.
Fig. 3. Dependence of the critical pressure pcr on m under internal pressure and axial compression at g = 1.2, aT/aB = 0.75, a+ = 600 MPa, a- = 400 MPa, t0 = 17 mm, R0 = 610 mm, n = 0.10; 0.15; 0.20 from bottom to top, respectively
Fig. 4. Dependence of the critical pressure pcr on m under internal pressure and axial compression at g = 1.3, aT/aB = 0.75, a + = 600 MPa, a- = 400 MPa, t0 = 17 mm, R0 = 610 mm, n = 0.10; 0.15; 0.20 from bottom to top, respectively
These computational experiments show that the coefficient of biaxial loading m significantly affects the value of critical pressure. With increasing axial compressive loads, the critical pressure decreases rapidly. Moreover, despite of the large mechanical inhomogeneity K = 1.5 used in the experiments, the less strong layer does not reduce the critical pressure even in the absence of contact hardening of the layer and even at m < -0.5 (Fig. 1). If contact hardening of the layer takes place (Figs. 2-4), the layer may not decrease the strength at m < 0 (Fig. 4). In some neighborhood of the point m = 0, a less strong layer is responsible for loss of stability of the process of plastic strain of the shell. At large modulus values of m, the effect of a less strong layer stops. The influence of the hardening indicator n is less significant, but is noticeable (Figs. 1-4).
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Article received 16.03.2021 Corrections received 30.04.2021
216
У.Ь. БПтап, Т.У. Кагре1а
Челябинский физико-математический журнал. 2021. Т. 6, вып. 2. С. 208-216.
УДК 539.374 Б01: 10.47475/2500-0101-2021-16206
КРИТИЧЕСКИЕ СОСТОЯНИЯ ПРЯМОШОВНЫХ ТРУБ БОЛЬШОГО ДИАМЕТРА ПРИ ОСЕВОМ СЖАТИИ И ВНУТРЕННЕМ ДАВЛЕНИИ
В. Л. Дильман", Т. В. Карпета6
Южно-Уральский государственный университет
(национальный исследовательский университет), Челябинск, Россия
" [email protected], [email protected]
Рассматриваются условия нагружения тонкостенной цилиндрической оболочки, содержащей слой из менее прочного материала вдоль образующей, в том числе прямо-шовной трубы большого диаметра, при её осевом сжатии и внутреннем давлении. Основной материал оболочки и материал слоя упруго-пластические упрочняемые. Пределы прочности и текучести у слоя ниже, чем эти величины у основного материала. Исследуются критические состояния слоя и оболочки, возникающие вследствие общих или локальных пластических деформаций. Цель статьи — установить зависимости критических деформаций, напряжений и давлений на оболочку от её механических и геометрических параметров и условий нагружения. Метод исследования основан на применении критерия Свифта — Марциньяка потери стабильности процесса пластического деформирования материала слоя. Получены явные аналитические выражения для искомых величин. Результаты позволяют определять критические давления при данных условиях нагружения и толщины стенок при заданном рабочем давлении.
Ключевые слова: тонкостенная цилиндрическая оболочка, прямошовная труба, пластическая стабильность, критерий Свифта — Марциньяка, критические деформации и напряжения, локализация пластической деформации.
Поступила в редакцию 16.03.2021 После переработки 30.04.2021
Сведения об авторах
Дильман Валерий Лейзерович, доктор физико-математических наук, доцент, заведующий кафедрой математического анализа и методики преподавания математики, ЮжноУральский государственный университет, Челябинск, Россия; [email protected]. Карпета Татьяна Васильевна, кандидат физико-математических наук, доцент кафедры прикладной математики и программирования, Южно-Уральский государственный университет, Челябинск, Россия; [email protected].
