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An axisymmetric method of creep analysis for primary and secondary creep
International Journal of Pressure Vessels and Piping 80 (2003) 597–606 www.elsevier.com/locate/ijpvp
An axisymmetric method of creep analysis for primary and secondary creep Hamid Jahed*, Jalal Bidabadi Department of Mechanical Engineering, Iran University of Science and Technology, Tehran 16844, Iran Received 30 December 2002; revised 14 June 2003; accepted 18 June 2003
Abstract A general axisymmetric method for elastic – plastic analysis was previously proposed by Jahed and Dubey [ASME J Pressure Vessels Technol 119 (1997) 264]. In the present work the method is extended to the time domain. General rate type governing equations are derived and solved in terms of rate of change of displacement as a function of rate of change in loading. Different types of loading, such as internal and external pressure, centrifugal loading and temperature gradient, are considered. To derive specific equations and employ the proposed formulation, the problem of an inhomogeneous non-uniform rotating disc is worked out. Primary and secondary creep behaviour is predicted using the proposed method and results are compared to FEM results. The problem of creep in pressurized vessels is also solved. Several numerical examples show the effectiveness and robustness of the proposed method. q 2003 Elsevier Ltd. All rights reserved. Keywords: Robust solution; Primary creep; Secondary creep; Time hardening; Strain hardening; Plane stress
1. Introduction Predictions of creep life in many axisymmetric problems such as gas turbine discs and pressure vessels are a very important and complex task. Even the most elaborate finite element procedure provides results that are very time consuming and are not always satisfactory. Robustness is the general ability to provide acceptable results on the basis of a less-than-ideal model together with conceptual insight and economy of computational effort. Therefore, robust solutions are very important especially in the early stages of design where design parameters are subjected to changes. Many papers in creep mechanics have been published during the last 50 years. These were mostly influenced by various approaches in establishing suitable constitutive equations or by the practical use of the proposed constitutive equations in structural analysis (thin-walled structures like tubes,discs,plates, shells,etc.).Thestateofartwasreportedby numerous authors in different papers, monographs or proceedings (a collection of such works is reported in Ref. [2]). A great deal of work has been dedicated to the area of steady-state creep behaviour. Efforts are usually conducted * Corresponding author. Tel.: þ 98-217491228; fax: þ 98-217454050. E-mail address: [email protected] (H. Jahed). 0308-0161/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0308-0161(03)00136-4
to study the creep mechanisms associated with this stage [3 –7] or they involve the development and use of a life predicting time-temperature parameter such as the SherbyDorn [8], Larson-Miller [9], or Manson-Haferd [10] parameters [11,12]. This makes sense, since the majority of a component’s creep life is spent in this secondary stage. Axisymmetric creep analysis has also been of interest. Wall [13] was first to analyze axisymmetric secondary creep problems. He considered different loading of a rotating disc. Millenson and Manson [14] solved creep of a rotating disc under a temperature gradient. They used the finite difference approach. Mendelson [15] considered primary and secondary creep for a disc with restricted types of boundary condition. This method was based on direct integration and finite difference. Penny and Marriott [2] considered creep of a thin rotating disc. They proposed a method for multiaxial creep calculation of a rotating disc. Here, the fundamental solution for creep of a uniform disc given by Ref. [2] is used and extended to nonuniform discs and cylinders under different loading conditions. The variable material properties (VMP) method was proposed by Jahed and Dubey [1] for elastic – plastic analysis of axisymmetric problems. The application of this method to pressure vessels [16 – 18] and rotating discs [19] has been developed extensively. The method takes a linear elastic
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E Young’s modulus n Poisson’s ratio a coefficient of thermal expansion U radial displacement se Mises effective stress 1e Mises effective strain aij i; j ¼ 1; 2 coefficient matrix bi i ¼ 1; 2 coefficient matrix F r ; F u rate of radial and tangential force per unit length F ri ; F ro rates of radial force per unit length at inner and outer radius C1 ; C2 ; C3 ; C4 integration constants Dt Time interval
Nomenclature h r sr su g g v 1 K; n; q 1r ; 1u 1r;c ; 1u;c
thickness radius radial stress tangential stress density gravitational acceleration angular velocity strain material constants for Norton’s law radial and tangential strain radial and tangential creep strain
solution of an axisymmetric problem to generate its inelastic solution. In this paper, the VMP method is extended to the time domain to predict creep behaviour of an axisymmetric problem. First, the governing equations of axisymmetric problems (e.g. rotating discs and pressure vessels) are reviewed. Rate type governing equations based on the rate of displacement are then derived and a general solution for uniform members is presented. A method for consideration of inhomogeneouity due to temperature gradients and nonuniformity of members is proposed. Different numerical examples of a rotating disc are solved and compared to available results. Creep of the pressure vessel is also studied.
2. Governing equations
Elastic and thermal strain may be obtained from Hooke’s law and Fourier’s law of thermal conductivity, respectively. Creep strain components are assumed to follow Norton’s law [20]. 1creep ¼ K sne tq
ð3Þ
q; n; K; are material constants, s and t are the stress and time parameter, respectively. In the case of plane stress, the compatibility equations take the following form: dU 1 ¼ ½ðsr 2 nsu Þ þ aT þ 1r;c dr E U 1 ¼ ½ðsu 2 nsr Þ þ aT þ 1u;c 1u ¼ r E
1r ¼
ð4Þ
These equations for the plane strain case are: If the thickness of a disc is small compared to its radius, the axial stress is negligible and hence the disc is in a state of plane stress. On the other hand if the length of the cylinder is large compared to its cross sectional dimensions the axial strain is negligible and the cylinder is then in a state of plane strain. The material is assumed to be isotropic but inhomogeneous along the radial direction. The loading of an axisymmetric member may contain internal and external pressures, centrifugal loading and temperature gradients. The essential boundary conditions are arbitrary. Now, let a differential element of an axisymmetric member be in equilibrium. Then from equilibrium considerations of this element, the following equation is derived. 1 dh ds r s 2 su g þ r þ v2 r ¼ 0 s þ h dr r dr r g
ð1Þ
where sr ; su are the radial and tangential stresses, respectively, g is the gravitational acceleration, v is the angular velocity of the disc, g ¼ gðrÞ is the density of the disc material and h ¼ hðrÞ is the thickness of the disc at distance r: Total strains are assumed to be composed of three parts: elastic, thermal and creep strain. 1total ¼ 1elastic þ 1thermal þ 1creep
ð2Þ
1r ¼
dU 1 2 n2 nð1 þ nÞ ¼ sr 2 su þ aT þ 1r;c dr E E
1u ¼
U 1 2 n2 nð1 þ nÞ ¼ su 2 sr þ aT þ 1u;c r E E
ð5Þ
in which E ¼ EðrÞ; n ¼ nðrÞ; a ¼ aðrÞ; T ¼ TðrÞ; 1r;c and 1u;c are Young’s modulus, Poisson’s ratio, the coefficient of thermal expansion, the temperature field (which is taken to be steady and axisymmetric), radial and tangential creep strain, respectively. The radial and tangential stresses sr and su for the state of plane stress are given by: E dU U þ 2ð1 sr ¼ n þ n 1 Þ2ð1þ n Þ a T ð6Þ r;c u;c r 12 n2 dr E dU U þ 2ðn1r;c þ1u;c Þ2ð1þ nÞaT su ¼ n r 12 n2 dr And for the state of plane strain are given by: E dU U ð12 nÞþ n sr ¼ ð122nÞð1þ nÞ dr r 2ðð12 nÞ1r;c þ n1u;c Þ2 aT
ð7Þ
H. Jahed, J. Bidabadi / International Journal of Pressure Vessels and Piping 80 (2003) 597–606
E dU U þð12 nÞ su ¼ n ð122nÞð1þ nÞ dr r 2ðn1r;c þð12 nÞ1u;c Þ2 aT
Hence, the governing rate equations become: d2 U dU F 32 þf1 þf2 U¼ 2 dr dr
Substituting Eq. (6) for plane stress and Eq. (7) for plane strain into Eq. (1) the governing equation for an inhomogeneous rotating disc with arbitrary varying thickness and also an inhomogeneous cylinder loaded by a temperature gradient are obtained. d2 U dU þf2 U ¼ f31 þf32 þf1 dr dr 2 In which f1 ;f2 ;f31 ;f32 for plane stress are: 1 d hE ln f1 ¼ þ r dr 12 n2 1 n d hE 1 dn ln f2 ¼ 2 2 þ þ r dr r dr r 12 n2 f31 ¼
ð8Þ
ð9Þ
d 12 n2 2 d hE ðð1þ nÞaTÞ2 ln gv r þð1þ nÞaT dr dr Eg 12 n2 12 n d hE ln þ dr r 12 n2 d1 d1 n 21 d hE dn þn ln þ1u;c þ þ r;c þ n u;c r dr dr dr dr 12 n2
f32 ¼ 1r;c
And for plane strain are: 1 d Ehð12 nÞ ln f1 ¼ þ r dr ð1þ nÞð122nÞ 1 ð1þ nÞð122nÞ d Ehn f2 ¼ 2 2 þ ð12 nÞrhE dr ð1þ nÞð122nÞ r
599
ð10Þ
ð12Þ
In general, there are no exact solutions to these equations. However, there have been numerous numerical methods for the treatment of such problems. Millenson and Manson [14] determined the elastic –plastic stresses in a gas turbine disc using the finite difference method. Mendelson [15] used an iterative scheme to obtain the thermoplastic solution for rotating discs based on Lame´’s solution. Genta [21] used the finite element method for the analysis. The boundary element method was also used by Abdul-Mihsen [22] for treating the same problem. Kia-Yuan and Han [23] proposed a systematic formulation to predict the elastic stress in an inhomogeneous rotating disc with arbitrary thickness under thermal loading. Jahed and Sherkati [19] applied the VMP method [1] to an inhomogeneous disc with variable thickness. Now the method in Ref. [1] is extended to the time domain for creep analysis of an inhomogeneous axisymmetric body. For a disc with arbitrary varying thickness and inhomogeneous properties an approximate method is employed. This involves discretization of the disc into a finite numbers of rings with constant material properties and thicknesses. These rings are subjected to a stress rate at the inner and outer radius, as illustrated in Ref. Fig. 1. For each annular disc, following Eq. (12), the rate governing equation becomes: U d2 U 1 dU dg g 2 2 ¼ þ 2 þ 2 r dr r dr dr r
t.0
ð13Þ
where g1 ; g2 for plane stress are: g 1 ¼ ð1 2 nÞð1r;c 2 1u;c Þ
d ð1þ nÞð122nÞ 2 ð1þ nÞð122nÞ ðaT Þ2 gv r þ dr Eð12 nÞ Ehð12 nÞ d Eh aT dr ð1þvÞð122nÞ
f31 ¼ ð12 nÞ
ð14Þ
g 2 ¼ ð1r;c þ n1u;c Þ and for plane strain are: g 1 ¼
ð1 2 2nÞ ð1r;c 2 1u;c Þ 12n
ð15Þ
122n d Ehð12nÞ þ ð12nÞr dr ð1þnÞð122nÞ 2n21 ð1þnÞð122nÞ d Ehn dn þ þ1u;c þ rðn21Þ Ehð12nÞ dr ð1þnÞð122nÞ dr
f32 ¼1r;c
þ
d1r;c n d1u;c þ dr 12n dr
If axisymmetric bodies are free from initial creep strains, the thermo-elastic behaviour is governed by: d2 U dU þf1 þf2 U¼f31 dr dr 2
ð11Þ
Fig. 1. A rotating disc under internal pressure and a ring taken from radius r:
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g 2 ¼ 1r;c þ
n 1 1 2 n u;c
Radial and tangential strain rate can be obtained from the Prandtl Reuss flow rule [3]. For plane stress: 1r;c ¼ 1u;c
1e ð2sr 2 su Þ 2s e
ð16Þ
where C3 ;C4 are integration constants. The integration constants are calculated by applying the following boundary conditions: r ¼ ri sðrÞ ¼ si r ¼ ro sðrÞ ¼ so which yield the following equations for plane stress:
1 ¼ e ð2su 2 sr Þ 2se
C3 ¼
so ro2 2 si ri2 ðIO Þro2 n Þ2 ð12 Eðro2 2ri2 Þ ð1þ nÞðro2 2ri2 Þ
C4 ¼
ðso 2 si Þri2 ro2 ðIO Þro2 ri2 n Þ2 ð1þ Eðro2 2ri2 Þ ð12 nÞðro2 2ri2 Þ
and for plane strain: 3 1e 1r;c ¼ ðs 2 s u Þ 4 se r 1u;c ¼
ð17Þ
where se ; 1e are Mises effective stress and strain rate, respectively. The general solution of Eq. (13) is:
so ro2 2 si ri2 ðIO Þro2 n Þð122 n Þ2 ð1þ Eðro2 2ri2 Þ ðro2 2ri2 Þ
C4 ¼
D þ Ir U ¼ Cr þ r
ðso 2 si Þro2 ri2 ðIO Þro2 ri2 ð1þ nÞ2 2 2 Eðro 2ri Þ ð12 nÞðro2 2ri2 Þ
where IO is defined as:
ð18Þ
IO ¼ ðI2 þI3 Þr¼ro
where ðr g 1 ðr 1 dr g 1 2 2g2 r dr þ 2 2r ri ri 2r
with the help of Eq. (4), one may obtain: E ð1 2 nÞ sr ¼ C ð1 þ n Þ 2 C þ I þ I 3 4 2 3 1 2 n2 r2 E ð1 2 nÞ su ¼ C3 ð1 þ nÞ þ C4 1 2 n2 r2 2 I2 þ I3 2 ð1 2 n2 Þ1u where I2 ; I3 are: 1 2 n ðr I2 ¼ ðg1 2 2g2 Þr dr 2r 2 ri 1 þ n ðr g 1 I3 ¼ dr 2 ri r
ð25Þ
and for plane strain the forms are:
3 1e ðs 2 s r Þ 4 se u
C3 ¼
I¼2
ð24Þ
ð19Þ
ð20Þ
ð26Þ
ð27Þ
Rates of displacement at the inner and outer radius of each ring may be related in the following matrix form: #( ) " # ( ) " b1 u i a11 a12 sri ¼ þ ð28Þ a21 a22 sro b2 u o where for plane stress
ð21Þ
a11 ¼
ri3 ðn 21Þ2ri ro2 ð1þ nÞ Eðro2 2ri2 Þ
a12 ¼
2ri ro2 Eðro2 2ri2 Þ
a21 ¼
22ro ri2 Eðro2 2ri2 Þ
a22 ¼
ro3 ð12 nÞþri2 ro ð1þ nÞ Eðro2 2ri2 Þ
ð29Þ
Similarly, in the case of plane strain, the equation will take the following form. E C4 C 2 ð1 2 2nÞ 2 þ I2 þ I3 sr ¼ ð22Þ ð1 þ nÞð1 2 2nÞ 3 r E C 2n 21 C3 þð122nÞ 24 2I2 þI3 þ su ¼ 1u;c ð1þ nÞð122nÞ 12 n r
and
where I2 ;I3 are: 122n ðr ðg1 22g2 Þr dr I2 ¼ 2r 2 ri 1 ðr g 1 I3 ¼ dr 2 ri r
For plane strain these coefficients take the following form. ð23Þ
b1 ¼ 2ro2 ri
IO 2 ðro2 2ri2 Þ 12 n2
ð30Þ
! IO ro3 ro ri2 þro ðIÞr¼ro b2 ¼ 2 2 2 þ ðro 2ri Þ 1þ n 12 n
a11 ¼
2ri3 ðn þ1Þð122nÞ2ri ro2 ð1þ nÞ Eðro2 2ri2 Þ
a12 ¼
2ri ro2 ð12 n2 Þ Eðro2 2ri2 Þ
ð31Þ
H. Jahed, J. Bidabadi / International Journal of Pressure Vessels and Piping 80 (2003) 597–606
a21 ¼
22ro ro2 ð12 n2 Þ Eðro2 2ri2 Þ
a22 ¼
ro3 ð1þ nÞð122nÞþri2 ro ð1þ nÞ Eðro2 2ri2 Þ
for plane strain. In these equations n; Kand q are material constant. However, the strain hardening theory assumes the creep strain rate is a function of stress, temperature and accumulated creep strain and is given in the following form for plane stress:
and
IO 2ð12 nÞ 2 b1 ¼ 2ro ri 2 2 ðro 2ri Þ 122n ! IO ro ri2 3 þðIÞr¼ro b2 ¼ 2 2 2 r o þ 122n ðro 2ri Þ
ð32Þ
F F ð33Þ sr ¼ r su ¼ u h h where F r ; F u are the rate of radial and tangential force per unit length and h is the thickness of each ring. Therefore, Eq. (29) can be written as follows: " #21 ( ) 8 9 " # " # < F ri = a11 a12 21 b1 u i 1 a11 a12 þ ¼ ð34Þ : F ; h a a a a b u 22
o
ro
21
22
ð35Þ
Using Eq. (34) the rate of This may be solved for {U}: force per unit length can be calculated at the inner radius and outer radius of each ring. Then the distribution of radial and tangential forces is obtained. Now to obtain creep strains, two common models, i.e. time hardening and strain hardening are employed [3]. The time hardening hypothesis, takes the creep strain rate as a function of stress, time and temperature in the following form for plane stress: 1r;c ¼
K sen21 ð2sr 2 su Þtq21 2q
1u;c ¼
K sn21 e ð2su 2 sr Þtq21 2q
ð38Þ
q21 q K sen21 1c ð2su 2 sr Þ ¼ n 2q K se
and q21 q 3 ksn21 1c e ð sr 2 su Þ 1r;c ¼ n 4 q K se 1u;c ¼
ð39Þ
q21 q 3 ksn21 1c e ðsu 2 sr Þ n 4 q K se
for plane strain.
2
where F ri ; F ro are the rates of radial force per unit length at the inner radius and outer radius, respectively. After obtaining Eq. (34) for each ring, the continuity equations at the common boundary of any two annular rings are satisfied through the assembling process. This will yield a system of linear equations of the following form: ¼ {F} þ ½K½B ½K{U}
q21 q K sn21 1c e 1r;c ¼ ð2sr 2 su Þ 2q K sne 1u;c
To consider the thickness variation, rates of radial and tangential stresses are taken as:
21
601
ð36Þ
3. Method of creep analysis The following outlines the procedure for creep analysis. 1. The distribution of radial and tangential stresses is calculated by the method described in Ref. [19]. This method uses the basic Lame´’s solution for the homogeneous uniform rotating disc and generates the solution for a disc with variable thickness and properties. This is done by discretizing the disc to number of rings with constant properties. 2. The distributions of radial and tangential strain rates are obtained by using both time hardening and strain hardening theories. 3. The distributions of the rates of radial and tangential stresses are calculated by using the method in Section 2. 4. By selecting a suitable time interval Dt; the radial and tangential stresses at the next time step ðsr Þiþ1 are obtained by assuming. ðsr Þiþ1 ¼ ðsr Þi £ Dt þ ðsr Þi
ð40Þ
ðsu Þiþ1 ¼ ðsu Þi £ Dt þ ðsu Þi ð1r Þiþ1 ¼ ð1r Þi £ Dt þ ð1r Þi
ð41Þ
and 3 K sn21 e ðsr 2 su Þtq21 1r;c ¼ 4 q 1u;c
3 K sen21 ðsu 2 sr Þtq21 ¼ 4 q
ð1u Þiþ1 ¼ ð1u Þi £ Dt þ ð1u Þi ð37Þ 5. Steps 2 –4 are continued for each time step until the radial and tangential stress distributions approach a steady state condition.
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4. Numerical examples Numerical examples are presented here to illustrate the merits of the above method of solution. These results are then compared with those obtained from exact solution and FEM solutions using the commercial FEM code ANSYS. In all cases good agreement is observed. 4.1. Example 1 The properties of the disc in this example are chosen in such a way that an exact solution may be obtained. This will ease the evaluation of the method described above. Consider a disc with inner radius of 5 cm and outer radius of 30 cm, which is rotated at 837.76 rad/s. The disc temperature is T ¼ 900 8C and elasticity modulus, isc thickness, density and creep strain are assumed to be E ¼ 1:5 £ 108 r 0:1 N=cm2 ; h ¼ 100 r 20:1 cm (r is measured in cm) r ¼ 0.0681/981 kg/cm3 and 1creep ¼ 5:2 e226 s6e t; respectively. The values are selected in a way, where an analytical solution may be obtained. By substituting the disc properties in Eq. (11), this equation simplifies and has a solution of the following form: C 1 ðr r ðr uðrÞ ¼ C1 r þ 2 2 f ðzÞz2 dz þ f ðzÞdz ð42Þ 2r ri 2 ri r C1 ; C2 are integration constants and are obtained by applying the following boundary conditions: uðri Þ ¼ 0;
Fr ðro Þ ¼ 0
ð43Þ
For this example, C1 ; C2 are: C1 ¼ 83:317 £ 10
26
C2 ¼ 22056:38 £ 10
26
ð44Þ
The radial displacement, radial and tangential stress distributions will then take the following form: 2056:38 3 2 0:037r £ 1026 cm ð45Þ uðrÞ ¼ 83:317r 2 r 1 sr ¼ 150r0:1 119:02 þ 1581:83 2 2 0:1398r2 N=cm2 r 1 su ¼ 150r 0:1 119:02 2 1581:83 2 2 0:07715r2 N=cm2 r The governing equation is in the form of Eq. (13) and its solution is similar to Eq. (18). Applying the following rate boundary conditions u ðri Þ ¼ 0;
F r ðro Þ ¼ 0:
ð46Þ
The integration constant C3 and C4 may be obtained as: C3 ¼ 21:97 £ 1025
C4 ¼ 49:24 £ 1025
ð47Þ
Radial displacement rate and rate of radial and tangential stresses at time zero are then calculated: 49:24 u ðrÞ ¼ 21:97r þ 2 Ir £ 1025 cm=h ð48Þ r ! 34:468 £ 1025 0:1 25 sr ¼ 1:684r 22:561 £ 10 2 þ I2 þ I3 r2 £ 108
N cm2 h
sr ¼1:684r0:1 22:561 £ 1025 þ ! 2 I2 þ I3 þ 0:911u;c £ 108
Fig. 2. Elastic and steady state stress distributions of the disc in example 1.
34:468 £ 1025 r2
N : cm2 h
H. Jahed, J. Bidabadi / International Journal of Pressure Vessels and Piping 80 (2003) 597–606
603
Fig. 3. Radial and tangential stress rate distributions after first time interval.
By selecting a suitable time interval Dt; radial and tangential stresses and strains at the next time step may be calculated using Eqs. (40 and 41). The progress in time is continued until the radial and tangential stress distribution approach a steady state condition. Fig. 2 shows the radial and tangential stress in the elastic and steady state case. The radial and tangential stress rates at time zero are shown in Fig. (3). The same problem is solved using the method proposed in this paper and results are compared to the exact solution and are shown in Figs. (2 and 3). The agreement is very good.
4.2. Example 2 In this example the first and second stages of creep in an inhomogeneous rotating disc are discussed. Consider a disc with inner radius of 31.8 mm and outer radius of 152.4 mm which is rotated at 15000 rpm, inside and outside temperatures are Ti ¼ 600 8C, To ¼ 900 8C. The disc is bladed at its outer rim and the blade effect is considered as a tensile stress at the outer radius of the disc with a value of 20 MPa. The modulus of elasticity, density, disc thickness and creep strain in the first stage
Fig. 4. Elastic and steady state stress distributions in the disc (example 2: sðri Þ ¼ 0; sðro Þ ¼ 20 MPa).
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Fig. 5. Variation of tangential stress at inner and outer radius of disc (example 2: sðri Þ ¼ 0; sðro Þ ¼ 20 MPa).
and second stage are assumed to be, respectively EðrÞ¼ð21:5184r 3 þ0:5764r 2 20:0827rþ0:0186Þ£104 GPa
r ¼7750kg=m3 hðrÞ¼256:3707r 3 þ19:5898r 2 22:5488rþ0:1630m 0:5 254 6:2 1c ¼9:9£10256 s6:2 se t e t ;1c ¼6:83£10
This example is solved for two different types of boundary condition. The first is a boundary condition
on stress only sðri Þ¼0; sðro Þ¼20MPa; while the second is on displacement and stress Uðri Þ¼0; sðro Þ¼20MPa: These two problems are solved by using the proposed method. Also, the ANSYS finite element software was used to obtain FEM solutions to the same problems. The results are compared as follows. Fig. 4 shows the radial and tangential stresses in the elastic and steady state cases for the first boundary conditions. Tangential stress rates at the inner and outer radius versus time are shown in Fig. 5. Fig. 6 shows the radial and tangential stresses in the elastic and steady state cases for
Fig. 6. Elastic and steady state stress distributions in the disk (example 2: Uðri Þ ¼ 0; sðro Þ ¼ 20 MPa).
H. Jahed, J. Bidabadi / International Journal of Pressure Vessels and Piping 80 (2003) 597–606
605
Fig. 7. Variation of tangential stress at inner and outer radii of disc (example 2: Uðri Þ ¼ 0; sðro Þ ¼ 20 MPa).
the second boundary conditions. Tangential stress rates at the inner and outer radius versus time are shown in Fig. 7. Again very good agreement between the present method and the FEM results is observed in these figures. The computational time in this example using the present method is about 1/100 of that of standard FEM analysis. This shows the robustness of the present method. 4.3. Example 3 In this example the first and second stages of creep in an inhomogeneous pressure vessel are obtained.
Consider a cylinder with inner radius of 30 cm and outer radius of 40 cm, inside and outside temperatures are Ti ¼ 600, To ¼ 800 8C. The pressure at the inner radius is assumed to be 40 MPa. The modulus of elasticity and creep strain for primary and secondary creep are assumed to be, EðrÞ ¼ ð2490:8344r 2 2 135:3803r þ 184:91Þ GPa, 1c ¼ 0:5 9:9 £ 10256 s6:2 1c ¼ 6:83 £ 10254 s6:2 e t e t; respectively. Fig. 8 shows the radial and tangential stresses in the elastic and steady states. Tangential stresses at the inner and outer radius versus time are shown in Fig. 9.
Fig. 8. Elastic and steady state stress distributions of the cylinder in example 3.
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Fig. 9. Variation of tangential stress at inner and outer radii of cylinder in example 3.
5. Conclusions A general axisymmetric method for creep analysis for primary and secondary creep has been presented. The method uses the basic solution of a uniform homogeneous axisymmetric member and generates the solution for a nonuniform, inhomogeneous one. The method is capable of predicting primary and secondary creep. There is no modeling involved and the convergence to the final solution is very fast. Also, it has been shown that the results agree very well with known solutions. However, the time consumed by other methods for solving the same problem is much higher. The robustness of the present method makes it suitable for use in the early stages of design.
References [1] Jahed H, Dubey RN. An axisymmetric method of elastic plastic analysis capable of predicting residual stress field. ASME J Pressure Vessels Technol 1997;119:264– 73. [2] Penny RK, Marriott DL. Design for Creep, 2nd ed. London: Chapman and Hall; 1995. [3] Nabarro FRN. Report of a conference on the strength of solids. London: Physical Society; 1948. [4] Herring C. Diffusional viscosity of a polycrystalline solid. J Appl Phys 1950;21:437 –45. [5] Coble RL. A model for boundary diffusion controlled creep in polycrystalline materials. J Appl Phys 1963;34:1679–82. [6] Weertman J. Steady-state creep through dislocation climb. J Appl Phys 1957;28:362– 4. [7] Raj R, Ashby MF. On grain boundary sliding and diffusional creep. Metall Trans 1971;2:1113–27. [8] Sherby OD, Dorn JE. Creep correlations in alpha solid solutions of aluminum. Trans AIME, J Metals 1952;959–64.
[9] Larson FR, Miller J. A time–temperature relationship for rupture and creep stresses. Am Soc Mech Engng 1952;51-A-36:765–75. [10] Manson SS., Haferd AA. A linear time–temperature relation for extrapolation of creep and stress rupture data, National Advisory Committee for Aeronautics. Technical Note. 1953; 2890. [11] Conway JB. Stress Rupture Parameters: Origin, Calculation and Use. New York: Gordon and Breach; 1969. [12] Goldhoff RM. A comparison of parameter methods for extrapolating high-temperature data. J Basic Engng 1959;629–43. [13] Wall AM, Sankey GA, Manjoine MJ, Shoemaker E. Creep tests of rotating disc at elevated temperature and comparison with theory. ASME J ApplMech 1954;21:222–35. [14] Millenson, M. and Manson, SS. Determination of stresses in gas turbine disks subjected to plastic flow and creep, NACA, 1948; TR-906. [15] Mendelson A. Plasticity: theory and application. New York: Macmillan Company; 1968. [16] Jahed H, Sethuraman R, Dubey RN. A variable material property approach for solving elastic–plastic problems. Int J Pressure Vessels Piping 1997;71(3):285–91. [17] Parker AP. Autofrettage of open end tubes—pressure, stresses, strains and code comparisons. ASME J Pressure Vessels Technol 2001;123:1–11. [18] Jahed H, Ghanbari G. Actual unloading behavior and its significance on material removal. Proc GT Conf, Oxford, UK 2002;2002. [19] Jahed H, Shirazi R. Loading and unloading behavior of a thermoplastic disk. Int J Pressure Vessel Piping 2001;78(9):637 –45. [20] Bailey RW. The utilization of creep test data in engineering design. Proc IMechE 1935;131. [21] Genta G, Gola M, Gugliotta A. Axisymmetric computation of the stress distribution in orthotropic rotating disks. Int J Mech Sci 1982; 24:21–6. [22] Abdoul-Mihsein MJ, Bakr AA, Packer AP. Stresses in axisymmetric rotating bodies determined by the boundary equation method. J Strain Anal 1985;20:70–86. [23] Kia-Yuan Y, Han RPS. Analysis of high-speed rotating disks with variable thickness and inhomogeneity. J Appl Mech 1994;61: 186 –91.
An axisymmetric method of creep analysis for primary and secondary creep Hamid Jahed*, Jalal Bidabadi Department of Mechanical Engineering, Iran University of Science and Technology, Tehran 16844, Iran Received 30 December 2002; revised 14 June 2003; accepted 18 June 2003
Abstract A general axisymmetric method for elastic – plastic analysis was previously proposed by Jahed and Dubey [ASME J Pressure Vessels Technol 119 (1997) 264]. In the present work the method is extended to the time domain. General rate type governing equations are derived and solved in terms of rate of change of displacement as a function of rate of change in loading. Different types of loading, such as internal and external pressure, centrifugal loading and temperature gradient, are considered. To derive specific equations and employ the proposed formulation, the problem of an inhomogeneous non-uniform rotating disc is worked out. Primary and secondary creep behaviour is predicted using the proposed method and results are compared to FEM results. The problem of creep in pressurized vessels is also solved. Several numerical examples show the effectiveness and robustness of the proposed method. q 2003 Elsevier Ltd. All rights reserved. Keywords: Robust solution; Primary creep; Secondary creep; Time hardening; Strain hardening; Plane stress
1. Introduction Predictions of creep life in many axisymmetric problems such as gas turbine discs and pressure vessels are a very important and complex task. Even the most elaborate finite element procedure provides results that are very time consuming and are not always satisfactory. Robustness is the general ability to provide acceptable results on the basis of a less-than-ideal model together with conceptual insight and economy of computational effort. Therefore, robust solutions are very important especially in the early stages of design where design parameters are subjected to changes. Many papers in creep mechanics have been published during the last 50 years. These were mostly influenced by various approaches in establishing suitable constitutive equations or by the practical use of the proposed constitutive equations in structural analysis (thin-walled structures like tubes,discs,plates, shells,etc.).Thestateofartwasreportedby numerous authors in different papers, monographs or proceedings (a collection of such works is reported in Ref. [2]). A great deal of work has been dedicated to the area of steady-state creep behaviour. Efforts are usually conducted * Corresponding author. Tel.: þ 98-217491228; fax: þ 98-217454050. E-mail address: [email protected] (H. Jahed). 0308-0161/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0308-0161(03)00136-4
to study the creep mechanisms associated with this stage [3 –7] or they involve the development and use of a life predicting time-temperature parameter such as the SherbyDorn [8], Larson-Miller [9], or Manson-Haferd [10] parameters [11,12]. This makes sense, since the majority of a component’s creep life is spent in this secondary stage. Axisymmetric creep analysis has also been of interest. Wall [13] was first to analyze axisymmetric secondary creep problems. He considered different loading of a rotating disc. Millenson and Manson [14] solved creep of a rotating disc under a temperature gradient. They used the finite difference approach. Mendelson [15] considered primary and secondary creep for a disc with restricted types of boundary condition. This method was based on direct integration and finite difference. Penny and Marriott [2] considered creep of a thin rotating disc. They proposed a method for multiaxial creep calculation of a rotating disc. Here, the fundamental solution for creep of a uniform disc given by Ref. [2] is used and extended to nonuniform discs and cylinders under different loading conditions. The variable material properties (VMP) method was proposed by Jahed and Dubey [1] for elastic – plastic analysis of axisymmetric problems. The application of this method to pressure vessels [16 – 18] and rotating discs [19] has been developed extensively. The method takes a linear elastic
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E Young’s modulus n Poisson’s ratio a coefficient of thermal expansion U radial displacement se Mises effective stress 1e Mises effective strain aij i; j ¼ 1; 2 coefficient matrix bi i ¼ 1; 2 coefficient matrix F r ; F u rate of radial and tangential force per unit length F ri ; F ro rates of radial force per unit length at inner and outer radius C1 ; C2 ; C3 ; C4 integration constants Dt Time interval
Nomenclature h r sr su g g v 1 K; n; q 1r ; 1u 1r;c ; 1u;c
thickness radius radial stress tangential stress density gravitational acceleration angular velocity strain material constants for Norton’s law radial and tangential strain radial and tangential creep strain
solution of an axisymmetric problem to generate its inelastic solution. In this paper, the VMP method is extended to the time domain to predict creep behaviour of an axisymmetric problem. First, the governing equations of axisymmetric problems (e.g. rotating discs and pressure vessels) are reviewed. Rate type governing equations based on the rate of displacement are then derived and a general solution for uniform members is presented. A method for consideration of inhomogeneouity due to temperature gradients and nonuniformity of members is proposed. Different numerical examples of a rotating disc are solved and compared to available results. Creep of the pressure vessel is also studied.
2. Governing equations
Elastic and thermal strain may be obtained from Hooke’s law and Fourier’s law of thermal conductivity, respectively. Creep strain components are assumed to follow Norton’s law [20]. 1creep ¼ K sne tq
ð3Þ
q; n; K; are material constants, s and t are the stress and time parameter, respectively. In the case of plane stress, the compatibility equations take the following form: dU 1 ¼ ½ðsr 2 nsu Þ þ aT þ 1r;c dr E U 1 ¼ ½ðsu 2 nsr Þ þ aT þ 1u;c 1u ¼ r E
1r ¼
ð4Þ
These equations for the plane strain case are: If the thickness of a disc is small compared to its radius, the axial stress is negligible and hence the disc is in a state of plane stress. On the other hand if the length of the cylinder is large compared to its cross sectional dimensions the axial strain is negligible and the cylinder is then in a state of plane strain. The material is assumed to be isotropic but inhomogeneous along the radial direction. The loading of an axisymmetric member may contain internal and external pressures, centrifugal loading and temperature gradients. The essential boundary conditions are arbitrary. Now, let a differential element of an axisymmetric member be in equilibrium. Then from equilibrium considerations of this element, the following equation is derived. 1 dh ds r s 2 su g þ r þ v2 r ¼ 0 s þ h dr r dr r g
ð1Þ
where sr ; su are the radial and tangential stresses, respectively, g is the gravitational acceleration, v is the angular velocity of the disc, g ¼ gðrÞ is the density of the disc material and h ¼ hðrÞ is the thickness of the disc at distance r: Total strains are assumed to be composed of three parts: elastic, thermal and creep strain. 1total ¼ 1elastic þ 1thermal þ 1creep
ð2Þ
1r ¼
dU 1 2 n2 nð1 þ nÞ ¼ sr 2 su þ aT þ 1r;c dr E E
1u ¼
U 1 2 n2 nð1 þ nÞ ¼ su 2 sr þ aT þ 1u;c r E E
ð5Þ
in which E ¼ EðrÞ; n ¼ nðrÞ; a ¼ aðrÞ; T ¼ TðrÞ; 1r;c and 1u;c are Young’s modulus, Poisson’s ratio, the coefficient of thermal expansion, the temperature field (which is taken to be steady and axisymmetric), radial and tangential creep strain, respectively. The radial and tangential stresses sr and su for the state of plane stress are given by: E dU U þ 2ð1 sr ¼ n þ n 1 Þ2ð1þ n Þ a T ð6Þ r;c u;c r 12 n2 dr E dU U þ 2ðn1r;c þ1u;c Þ2ð1þ nÞaT su ¼ n r 12 n2 dr And for the state of plane strain are given by: E dU U ð12 nÞþ n sr ¼ ð122nÞð1þ nÞ dr r 2ðð12 nÞ1r;c þ n1u;c Þ2 aT
ð7Þ
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E dU U þð12 nÞ su ¼ n ð122nÞð1þ nÞ dr r 2ðn1r;c þð12 nÞ1u;c Þ2 aT
Hence, the governing rate equations become: d2 U dU F 32 þf1 þf2 U¼ 2 dr dr
Substituting Eq. (6) for plane stress and Eq. (7) for plane strain into Eq. (1) the governing equation for an inhomogeneous rotating disc with arbitrary varying thickness and also an inhomogeneous cylinder loaded by a temperature gradient are obtained. d2 U dU þf2 U ¼ f31 þf32 þf1 dr dr 2 In which f1 ;f2 ;f31 ;f32 for plane stress are: 1 d hE ln f1 ¼ þ r dr 12 n2 1 n d hE 1 dn ln f2 ¼ 2 2 þ þ r dr r dr r 12 n2 f31 ¼
ð8Þ
ð9Þ
d 12 n2 2 d hE ðð1þ nÞaTÞ2 ln gv r þð1þ nÞaT dr dr Eg 12 n2 12 n d hE ln þ dr r 12 n2 d1 d1 n 21 d hE dn þn ln þ1u;c þ þ r;c þ n u;c r dr dr dr dr 12 n2
f32 ¼ 1r;c
And for plane strain are: 1 d Ehð12 nÞ ln f1 ¼ þ r dr ð1þ nÞð122nÞ 1 ð1þ nÞð122nÞ d Ehn f2 ¼ 2 2 þ ð12 nÞrhE dr ð1þ nÞð122nÞ r
599
ð10Þ
ð12Þ
In general, there are no exact solutions to these equations. However, there have been numerous numerical methods for the treatment of such problems. Millenson and Manson [14] determined the elastic –plastic stresses in a gas turbine disc using the finite difference method. Mendelson [15] used an iterative scheme to obtain the thermoplastic solution for rotating discs based on Lame´’s solution. Genta [21] used the finite element method for the analysis. The boundary element method was also used by Abdul-Mihsen [22] for treating the same problem. Kia-Yuan and Han [23] proposed a systematic formulation to predict the elastic stress in an inhomogeneous rotating disc with arbitrary thickness under thermal loading. Jahed and Sherkati [19] applied the VMP method [1] to an inhomogeneous disc with variable thickness. Now the method in Ref. [1] is extended to the time domain for creep analysis of an inhomogeneous axisymmetric body. For a disc with arbitrary varying thickness and inhomogeneous properties an approximate method is employed. This involves discretization of the disc into a finite numbers of rings with constant material properties and thicknesses. These rings are subjected to a stress rate at the inner and outer radius, as illustrated in Ref. Fig. 1. For each annular disc, following Eq. (12), the rate governing equation becomes: U d2 U 1 dU dg g 2 2 ¼ þ 2 þ 2 r dr r dr dr r
t.0
ð13Þ
where g1 ; g2 for plane stress are: g 1 ¼ ð1 2 nÞð1r;c 2 1u;c Þ
d ð1þ nÞð122nÞ 2 ð1þ nÞð122nÞ ðaT Þ2 gv r þ dr Eð12 nÞ Ehð12 nÞ d Eh aT dr ð1þvÞð122nÞ
f31 ¼ ð12 nÞ
ð14Þ
g 2 ¼ ð1r;c þ n1u;c Þ and for plane strain are: g 1 ¼
ð1 2 2nÞ ð1r;c 2 1u;c Þ 12n
ð15Þ
122n d Ehð12nÞ þ ð12nÞr dr ð1þnÞð122nÞ 2n21 ð1þnÞð122nÞ d Ehn dn þ þ1u;c þ rðn21Þ Ehð12nÞ dr ð1þnÞð122nÞ dr
f32 ¼1r;c
þ
d1r;c n d1u;c þ dr 12n dr
If axisymmetric bodies are free from initial creep strains, the thermo-elastic behaviour is governed by: d2 U dU þf1 þf2 U¼f31 dr dr 2
ð11Þ
Fig. 1. A rotating disc under internal pressure and a ring taken from radius r:
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g 2 ¼ 1r;c þ
n 1 1 2 n u;c
Radial and tangential strain rate can be obtained from the Prandtl Reuss flow rule [3]. For plane stress: 1r;c ¼ 1u;c
1e ð2sr 2 su Þ 2s e
ð16Þ
where C3 ;C4 are integration constants. The integration constants are calculated by applying the following boundary conditions: r ¼ ri sðrÞ ¼ si r ¼ ro sðrÞ ¼ so which yield the following equations for plane stress:
1 ¼ e ð2su 2 sr Þ 2se
C3 ¼
so ro2 2 si ri2 ðIO Þro2 n Þ2 ð12 Eðro2 2ri2 Þ ð1þ nÞðro2 2ri2 Þ
C4 ¼
ðso 2 si Þri2 ro2 ðIO Þro2 ri2 n Þ2 ð1þ Eðro2 2ri2 Þ ð12 nÞðro2 2ri2 Þ
and for plane strain: 3 1e 1r;c ¼ ðs 2 s u Þ 4 se r 1u;c ¼
ð17Þ
where se ; 1e are Mises effective stress and strain rate, respectively. The general solution of Eq. (13) is:
so ro2 2 si ri2 ðIO Þro2 n Þð122 n Þ2 ð1þ Eðro2 2ri2 Þ ðro2 2ri2 Þ
C4 ¼
D þ Ir U ¼ Cr þ r
ðso 2 si Þro2 ri2 ðIO Þro2 ri2 ð1þ nÞ2 2 2 Eðro 2ri Þ ð12 nÞðro2 2ri2 Þ
where IO is defined as:
ð18Þ
IO ¼ ðI2 þI3 Þr¼ro
where ðr g 1 ðr 1 dr g 1 2 2g2 r dr þ 2 2r ri ri 2r
with the help of Eq. (4), one may obtain: E ð1 2 nÞ sr ¼ C ð1 þ n Þ 2 C þ I þ I 3 4 2 3 1 2 n2 r2 E ð1 2 nÞ su ¼ C3 ð1 þ nÞ þ C4 1 2 n2 r2 2 I2 þ I3 2 ð1 2 n2 Þ1u where I2 ; I3 are: 1 2 n ðr I2 ¼ ðg1 2 2g2 Þr dr 2r 2 ri 1 þ n ðr g 1 I3 ¼ dr 2 ri r
ð25Þ
and for plane strain the forms are:
3 1e ðs 2 s r Þ 4 se u
C3 ¼
I¼2
ð24Þ
ð19Þ
ð20Þ
ð26Þ
ð27Þ
Rates of displacement at the inner and outer radius of each ring may be related in the following matrix form: #( ) " # ( ) " b1 u i a11 a12 sri ¼ þ ð28Þ a21 a22 sro b2 u o where for plane stress
ð21Þ
a11 ¼
ri3 ðn 21Þ2ri ro2 ð1þ nÞ Eðro2 2ri2 Þ
a12 ¼
2ri ro2 Eðro2 2ri2 Þ
a21 ¼
22ro ri2 Eðro2 2ri2 Þ
a22 ¼
ro3 ð12 nÞþri2 ro ð1þ nÞ Eðro2 2ri2 Þ
ð29Þ
Similarly, in the case of plane strain, the equation will take the following form. E C4 C 2 ð1 2 2nÞ 2 þ I2 þ I3 sr ¼ ð22Þ ð1 þ nÞð1 2 2nÞ 3 r E C 2n 21 C3 þð122nÞ 24 2I2 þI3 þ su ¼ 1u;c ð1þ nÞð122nÞ 12 n r
and
where I2 ;I3 are: 122n ðr ðg1 22g2 Þr dr I2 ¼ 2r 2 ri 1 ðr g 1 I3 ¼ dr 2 ri r
For plane strain these coefficients take the following form. ð23Þ
b1 ¼ 2ro2 ri
IO 2 ðro2 2ri2 Þ 12 n2
ð30Þ
! IO ro3 ro ri2 þro ðIÞr¼ro b2 ¼ 2 2 2 þ ðro 2ri Þ 1þ n 12 n
a11 ¼
2ri3 ðn þ1Þð122nÞ2ri ro2 ð1þ nÞ Eðro2 2ri2 Þ
a12 ¼
2ri ro2 ð12 n2 Þ Eðro2 2ri2 Þ
ð31Þ
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a21 ¼
22ro ro2 ð12 n2 Þ Eðro2 2ri2 Þ
a22 ¼
ro3 ð1þ nÞð122nÞþri2 ro ð1þ nÞ Eðro2 2ri2 Þ
for plane strain. In these equations n; Kand q are material constant. However, the strain hardening theory assumes the creep strain rate is a function of stress, temperature and accumulated creep strain and is given in the following form for plane stress:
and
IO 2ð12 nÞ 2 b1 ¼ 2ro ri 2 2 ðro 2ri Þ 122n ! IO ro ri2 3 þðIÞr¼ro b2 ¼ 2 2 2 r o þ 122n ðro 2ri Þ
ð32Þ
F F ð33Þ sr ¼ r su ¼ u h h where F r ; F u are the rate of radial and tangential force per unit length and h is the thickness of each ring. Therefore, Eq. (29) can be written as follows: " #21 ( ) 8 9 " # " # < F ri = a11 a12 21 b1 u i 1 a11 a12 þ ¼ ð34Þ : F ; h a a a a b u 22
o
ro
21
22
ð35Þ
Using Eq. (34) the rate of This may be solved for {U}: force per unit length can be calculated at the inner radius and outer radius of each ring. Then the distribution of radial and tangential forces is obtained. Now to obtain creep strains, two common models, i.e. time hardening and strain hardening are employed [3]. The time hardening hypothesis, takes the creep strain rate as a function of stress, time and temperature in the following form for plane stress: 1r;c ¼
K sen21 ð2sr 2 su Þtq21 2q
1u;c ¼
K sn21 e ð2su 2 sr Þtq21 2q
ð38Þ
q21 q K sen21 1c ð2su 2 sr Þ ¼ n 2q K se
and q21 q 3 ksn21 1c e ð sr 2 su Þ 1r;c ¼ n 4 q K se 1u;c ¼
ð39Þ
q21 q 3 ksn21 1c e ðsu 2 sr Þ n 4 q K se
for plane strain.
2
where F ri ; F ro are the rates of radial force per unit length at the inner radius and outer radius, respectively. After obtaining Eq. (34) for each ring, the continuity equations at the common boundary of any two annular rings are satisfied through the assembling process. This will yield a system of linear equations of the following form: ¼ {F} þ ½K½B ½K{U}
q21 q K sn21 1c e 1r;c ¼ ð2sr 2 su Þ 2q K sne 1u;c
To consider the thickness variation, rates of radial and tangential stresses are taken as:
21
601
ð36Þ
3. Method of creep analysis The following outlines the procedure for creep analysis. 1. The distribution of radial and tangential stresses is calculated by the method described in Ref. [19]. This method uses the basic Lame´’s solution for the homogeneous uniform rotating disc and generates the solution for a disc with variable thickness and properties. This is done by discretizing the disc to number of rings with constant properties. 2. The distributions of radial and tangential strain rates are obtained by using both time hardening and strain hardening theories. 3. The distributions of the rates of radial and tangential stresses are calculated by using the method in Section 2. 4. By selecting a suitable time interval Dt; the radial and tangential stresses at the next time step ðsr Þiþ1 are obtained by assuming. ðsr Þiþ1 ¼ ðsr Þi £ Dt þ ðsr Þi
ð40Þ
ðsu Þiþ1 ¼ ðsu Þi £ Dt þ ðsu Þi ð1r Þiþ1 ¼ ð1r Þi £ Dt þ ð1r Þi
ð41Þ
and 3 K sn21 e ðsr 2 su Þtq21 1r;c ¼ 4 q 1u;c
3 K sen21 ðsu 2 sr Þtq21 ¼ 4 q
ð1u Þiþ1 ¼ ð1u Þi £ Dt þ ð1u Þi ð37Þ 5. Steps 2 –4 are continued for each time step until the radial and tangential stress distributions approach a steady state condition.
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4. Numerical examples Numerical examples are presented here to illustrate the merits of the above method of solution. These results are then compared with those obtained from exact solution and FEM solutions using the commercial FEM code ANSYS. In all cases good agreement is observed. 4.1. Example 1 The properties of the disc in this example are chosen in such a way that an exact solution may be obtained. This will ease the evaluation of the method described above. Consider a disc with inner radius of 5 cm and outer radius of 30 cm, which is rotated at 837.76 rad/s. The disc temperature is T ¼ 900 8C and elasticity modulus, isc thickness, density and creep strain are assumed to be E ¼ 1:5 £ 108 r 0:1 N=cm2 ; h ¼ 100 r 20:1 cm (r is measured in cm) r ¼ 0.0681/981 kg/cm3 and 1creep ¼ 5:2 e226 s6e t; respectively. The values are selected in a way, where an analytical solution may be obtained. By substituting the disc properties in Eq. (11), this equation simplifies and has a solution of the following form: C 1 ðr r ðr uðrÞ ¼ C1 r þ 2 2 f ðzÞz2 dz þ f ðzÞdz ð42Þ 2r ri 2 ri r C1 ; C2 are integration constants and are obtained by applying the following boundary conditions: uðri Þ ¼ 0;
Fr ðro Þ ¼ 0
ð43Þ
For this example, C1 ; C2 are: C1 ¼ 83:317 £ 10
26
C2 ¼ 22056:38 £ 10
26
ð44Þ
The radial displacement, radial and tangential stress distributions will then take the following form: 2056:38 3 2 0:037r £ 1026 cm ð45Þ uðrÞ ¼ 83:317r 2 r 1 sr ¼ 150r0:1 119:02 þ 1581:83 2 2 0:1398r2 N=cm2 r 1 su ¼ 150r 0:1 119:02 2 1581:83 2 2 0:07715r2 N=cm2 r The governing equation is in the form of Eq. (13) and its solution is similar to Eq. (18). Applying the following rate boundary conditions u ðri Þ ¼ 0;
F r ðro Þ ¼ 0:
ð46Þ
The integration constant C3 and C4 may be obtained as: C3 ¼ 21:97 £ 1025
C4 ¼ 49:24 £ 1025
ð47Þ
Radial displacement rate and rate of radial and tangential stresses at time zero are then calculated: 49:24 u ðrÞ ¼ 21:97r þ 2 Ir £ 1025 cm=h ð48Þ r ! 34:468 £ 1025 0:1 25 sr ¼ 1:684r 22:561 £ 10 2 þ I2 þ I3 r2 £ 108
N cm2 h
sr ¼1:684r0:1 22:561 £ 1025 þ ! 2 I2 þ I3 þ 0:911u;c £ 108
Fig. 2. Elastic and steady state stress distributions of the disc in example 1.
34:468 £ 1025 r2
N : cm2 h
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603
Fig. 3. Radial and tangential stress rate distributions after first time interval.
By selecting a suitable time interval Dt; radial and tangential stresses and strains at the next time step may be calculated using Eqs. (40 and 41). The progress in time is continued until the radial and tangential stress distribution approach a steady state condition. Fig. 2 shows the radial and tangential stress in the elastic and steady state case. The radial and tangential stress rates at time zero are shown in Fig. (3). The same problem is solved using the method proposed in this paper and results are compared to the exact solution and are shown in Figs. (2 and 3). The agreement is very good.
4.2. Example 2 In this example the first and second stages of creep in an inhomogeneous rotating disc are discussed. Consider a disc with inner radius of 31.8 mm and outer radius of 152.4 mm which is rotated at 15000 rpm, inside and outside temperatures are Ti ¼ 600 8C, To ¼ 900 8C. The disc is bladed at its outer rim and the blade effect is considered as a tensile stress at the outer radius of the disc with a value of 20 MPa. The modulus of elasticity, density, disc thickness and creep strain in the first stage
Fig. 4. Elastic and steady state stress distributions in the disc (example 2: sðri Þ ¼ 0; sðro Þ ¼ 20 MPa).
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Fig. 5. Variation of tangential stress at inner and outer radius of disc (example 2: sðri Þ ¼ 0; sðro Þ ¼ 20 MPa).
and second stage are assumed to be, respectively EðrÞ¼ð21:5184r 3 þ0:5764r 2 20:0827rþ0:0186Þ£104 GPa
r ¼7750kg=m3 hðrÞ¼256:3707r 3 þ19:5898r 2 22:5488rþ0:1630m 0:5 254 6:2 1c ¼9:9£10256 s6:2 se t e t ;1c ¼6:83£10
This example is solved for two different types of boundary condition. The first is a boundary condition
on stress only sðri Þ¼0; sðro Þ¼20MPa; while the second is on displacement and stress Uðri Þ¼0; sðro Þ¼20MPa: These two problems are solved by using the proposed method. Also, the ANSYS finite element software was used to obtain FEM solutions to the same problems. The results are compared as follows. Fig. 4 shows the radial and tangential stresses in the elastic and steady state cases for the first boundary conditions. Tangential stress rates at the inner and outer radius versus time are shown in Fig. 5. Fig. 6 shows the radial and tangential stresses in the elastic and steady state cases for
Fig. 6. Elastic and steady state stress distributions in the disk (example 2: Uðri Þ ¼ 0; sðro Þ ¼ 20 MPa).
H. Jahed, J. Bidabadi / International Journal of Pressure Vessels and Piping 80 (2003) 597–606
605
Fig. 7. Variation of tangential stress at inner and outer radii of disc (example 2: Uðri Þ ¼ 0; sðro Þ ¼ 20 MPa).
the second boundary conditions. Tangential stress rates at the inner and outer radius versus time are shown in Fig. 7. Again very good agreement between the present method and the FEM results is observed in these figures. The computational time in this example using the present method is about 1/100 of that of standard FEM analysis. This shows the robustness of the present method. 4.3. Example 3 In this example the first and second stages of creep in an inhomogeneous pressure vessel are obtained.
Consider a cylinder with inner radius of 30 cm and outer radius of 40 cm, inside and outside temperatures are Ti ¼ 600, To ¼ 800 8C. The pressure at the inner radius is assumed to be 40 MPa. The modulus of elasticity and creep strain for primary and secondary creep are assumed to be, EðrÞ ¼ ð2490:8344r 2 2 135:3803r þ 184:91Þ GPa, 1c ¼ 0:5 9:9 £ 10256 s6:2 1c ¼ 6:83 £ 10254 s6:2 e t e t; respectively. Fig. 8 shows the radial and tangential stresses in the elastic and steady states. Tangential stresses at the inner and outer radius versus time are shown in Fig. 9.
Fig. 8. Elastic and steady state stress distributions of the cylinder in example 3.
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Fig. 9. Variation of tangential stress at inner and outer radii of cylinder in example 3.
5. Conclusions A general axisymmetric method for creep analysis for primary and secondary creep has been presented. The method uses the basic solution of a uniform homogeneous axisymmetric member and generates the solution for a nonuniform, inhomogeneous one. The method is capable of predicting primary and secondary creep. There is no modeling involved and the convergence to the final solution is very fast. Also, it has been shown that the results agree very well with known solutions. However, the time consumed by other methods for solving the same problem is much higher. The robustness of the present method makes it suitable for use in the early stages of design.
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