Scholarly article on topic 'Stress Analysis and Parameter Optimization of an FGM Pressure Vessel Subjected to Thermo-Mechanical Loadings'

Stress Analysis and Parameter Optimization of an FGM Pressure Vessel Subjected to Thermo-Mechanical Loadings Academic research paper on "Mechanical engineering"

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Abstract of research paper on Mechanical engineering, author of scientific article — Z.W. Wang, Q. Zhang, L.Z. Xia, J.T. Wu, P.Q. Liu

Abstract The thermo-mechanical behavior and material design of functionally graded materials (FGMs) has been gaining increasing attention in the past two decades. However, little work can obtain the exact solution of FGM vessel consisting of a finite length hollow cylinder and two closed ends due to the difficult in calculating the axial strain. By constructing an exponentially function determining the material properties and based on the steady state thermo-elastic theory and Euler-Cauchy formula, the closed-form analytical solutions of the thermo-mechanical stresses for the FGM vessel subjected to an internal pressure and a thermal load were firstly established taking the effect of closed ends into consideration. And then, a numerical model of the FGM pressure vessel was developed using the finite element (FE) method for validating the analytical solution. Lastly, a double-variable optimization model was proposed for determining the optimal material distribution of the FGM pressure vessel. The results demonstrated the accuracy and efficiency of the proposed stress formulations and optimization model in this paper, which would be helpful for better understanding and scientific design of FGM vessel or related structures.

Academic research paper on topic "Stress Analysis and Parameter Optimization of an FGM Pressure Vessel Subjected to Thermo-Mechanical Loadings"

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Procedía Engineering 130 (2015) 374 - 389

Procedía Engineering

www.elsevier.com/locate/procedia

14th International Conference on Pressure Vessel Technology

Stress Analysis and Parameter Optimization of an FGM Pressure Vessel Subjected to Thermo-Mechanical Loadings

Z.W. Wanga, Q. Zhang3, L.Z. Xia3, J.T. Wua, P.Q. Liu3 *

aSchool of Chemical Machinery, Dalian University of Technology, Dalian, 116024, China

Abstract

The thermo-mechanical behavior and material design of functionally graded materials (FGMs) has been gaining increasing attention in the past two decades. However, little work can obtain the exact solution of FGM vessel consisting of a finite length hollow cylinder and two closed ends due to the difficult in calculating the axial strain. By constructing an exponentially function determining the material properties and based on the steady state thermo-elastic theory and Euler-Cauchy formula, the closed-form analytical solutions of the thermo-mechanical stresses for the FGM vessel subjected to an internal pressure and a thermal load were firstly established taking the effect of closed ends into consideration. And then, a numerical model of the FGM pressure vessel was developed using the finite element (FE) method for validating the analytical solution. Lastly, a doublevariable optimization model was proposed for determining the optimal material distribution of the FGM pressure vessel. The results demonstrated the accuracy and efficiency of the proposed stress formulations and optimization model in this paper, which would be helpful for better understanding and scientific design ofFGM vessel or related structures. © 2015 The Authors.PublishedbyElsevierLtd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICPVT-14

Keywords: Thermal stress; Pressure vessel; Parameter optimization; Boundary effect

1. Introduction

FGM is a kind of composite material that the physical and mechanical properties of the material vary spatially along specific directions over the entire domain to achieve specific thermal and/or mechanical properties for some specific applications, such as the heat-shielding materials in space planes, nuclear fusion reactors, energy conversion

* Corresponding author. Tel.: +86-15904963546; fax: +86-411-84986275. E-mail address: wangzewu@dlut.edu.cn

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ICPVT-14

doi: 10.1016/j.proeng.2015.12.230

and thermo generators [1, 2]. Various problems of FGM have attracted considerable attention in recent years. In particular, the thermo-mechanical behavior of the FGM structures, such as the hollow cylinders [3-6], rotating disks [7] cylindrical vessel [8-10], and spherical vessel [11, 12], has also been widely investigated by many researchers.

In general, the plane strain or plane stress assumptions were used for the FGM cylinders [13-16] with infinite length, thus the axial strain was often negligible compared with the hoop and radial strain. However, the assumption for the infinite long cylinder is difficult to be justified in the stress analysis of cylindrical pressure vessel with closed ends. Moreover, the results obtained according to the plane strain assumption are debatable for the FGM cylindrical vessels [8, 10] due to the axial strain generated from its closed ends [17, 18]. Therefore, it is necessary to reestablish the closed-form analytical solution taking the influence of closed ends into consideration.

It is important, but difficult, to validate the theoretical stress formulations for complex structures, and only few works have been reported for FGM structures in open literatures. For example, earlier work on the hoop stress formulation [8] can be improved as it underestimates the axial stress due to ignoring the closed ends, which was also mentioned in the appendix C in the literature [14, 18]. More efforts should therefore be devoted to find another means to verify related theoretical solutions, such as the numerical method using FE theory [19].

Meanwhile, FGM makes sagaciously tailoring of material composition possible so as to obtain the optimal benefits. Na and Kim [20] investigated the volume fraction optimization of FGMs considering stress and critical temperature. Vel and Pelletier [21] presented a method for the multi-objective optimization of material distribution of FGM cylindrical shells for the steady thermo-mechanical process. Ootao et al. [22] ever applied a neural method to solve an optimization problem of material compositions for a functionally graded hollow circular cylinder. Therefore, optimization of the material distribution is a critical step in the design of FGM structure. However, limited works have been carried out on how to obtain the optimal material distribution of an FGM pressure vessel.

The main objectives in this work are to investigate the closed-form analytical solutions of the thermal-mechanical stresses for the FGM pressure vessel considering its influence of closed ends, and to construct a numerical model using FE method to verify the analytical solutions, and to develop an optimization model for determining the optimal material distribution of FGM pressure vessel. The whole work will be helpful for better understanding and scientific design ofFGM pressure vessel or related structures in the future.

2. Derivation of closed-form analytical solutions

2.1. Construction of an exponentially function determining the material properties

The variation of material properties in an FGM is usually obtained by adjusting the volume fractions of two or more compatible constituents, and variations in the material properties such as Young's modulus and Poisson's ratio may be arbitrary functions of the radial coordinate. Three kinds of material models were often used: (a) simple power law with constant Poisson's ratio [8, 23], (b) exponentially-varying properties [3], and (c) volume fractions of the constituents [4, 15]. In this work, Poisson's ratio is assumed to be a constant, while the Young's modulus, the coefficient of thermal expansion and the thermal conductivity of the FGM were defined to obey a power law [10, 24] as follows

E(r) = E„

, a(r) = ao

f \<p r

V Ro y

where r, E, a and X are the radial radius, Young's modulus, coefficient of thermal expansion and thermal conductivity, respectively. Ra, Ea, a0 and À0 are the radial radius, Young's modulus, coefficient of thermal expansion and thermal conductivity of the outside wall, respectively. And p, <p, and y are the graded factors of FGM pressure vessel.

2.2. Steady-state heat conduction equation

Generally, the temperature variation through the thickness of FGM pressure vessel is slight [10, 25]. Hence, the heat conduction model can be expressed by the steady-state equation as below

IK > ^ } =»

where T{r) is the radial distribution function of the temperature along the wall thickness direction. The boundary conditions of the inside and outside wall surfaces are given as below

T|r=R = T, T\r__Ro = To (3)

where Ri is the radius of the inside wall, T and T0 are the temperatures of the inside and outside walls, respectively. Using Eqs. (l)-(3), the temperature distribution function can be written as

T (r} = r-"T - T )- ToR7+ TA' r;' - r/

2.3. Thermal and mechanical stresses

(1) Basic equations

According to the infinitesimal strain theory, the equilibrium equation and the strain-displacement relations for an axisymmetric structure can be written, respectively, as

da <r -a0

du u dw

£r = , £q = y £z =

dr r dz

For the uncoupled thermo-elastic steady-state conduction problem, the constitutive equations of the pressure vessel subjected to the mechanical and thermal loads is given as

E v i £r H--( £ r + £„ _ r 1 - 2v r e aET (r)

1 + v 1 - 2v

E v i £„ +-\£r + £„ _ s 1 -2v r e + e. ) aET (r)

1 + v 1 - 2v

E v t z 1 -2v r e + ) aET (r)

1 + v 1 - 2v

where <Jr, and <JZ are the radial stress, hoop stress, and axial stress, respectively. er, £e and ez are the radial strain, hoop strain and axial strain, respectively, u and w are the radial and axial displacement, respectively. And v denotes Poisson's ratio. In Eq. (5), the body force is ignoring.

For cylinders with infinite length, the plain strain assumption is usually applied, thus, sz - 0, and the value of axial stress was ignored or decreased [8, 10, 13, 16]. But for the vessel consisting of two closed ends and a finite

length cylinder, sz 0, and the closed ends of the vessel can generate a significant component of the axial stress. In this work, the axial strain is considered and assumed to be constant taking the large length-radius ratio into consideration.

(2) Solving process

Under the linear-elastic and uncoupled thermo- mechanical conduction condition, the combined stresses can be divided as two parts

-ar -OV

crT crT

where uf , uf and uf are the mechanical stresses in the radial, hoop, and axial directions, respectively. r , ° &T

and z are the thermal stresses in the radial, hoop, and axial directions, respectively. Moreover, the coupling effect between the mechanical stress and thermal stress was ignored. In order to reduce the difficulty of solving the complex equations, the mechanical stresses and the thermal stresses will be deduced separately. ® For the mechanical stresses

If only under the internal pressure load, thus, Ti To, then, - 0.

By combining Eqs. (1) (5)~(7), the governing equation in the cylindrical part of pressure vessel can be derived as

r V + (P + l)ru' + (v'p - l)u = -rfiv'sf (9)

where u is the radial displacement and v' = W(1 — v). Eq. (9) is similar to the Euler-Cauchy equation [26]. The characteristic equation of Eq. (9), is m2 + ¡5m + (v*P -1) = 0 , and its roots are

^ = - Apv" + 4)

12 = -AWA2 -+ 4)

And the solution of Eq. (9) is given as

u = AMrm + BMrm -vsfr (10)

where Am and Bm are unknown constants of integration that need be determined by the boundary conditions. For the axisymmetric vessel only subjected to internal pressure, the boundary conditions are listed as follows.

aM = 0 when r = Ro

aM =- p when r = R (11)

¡R°aM 2nrdr = pnR2

where p is the internal pressure.

By combining Eqs. (1), (6), (7), (10) and (11), the constants of integration AM, BM and the axial strain ezm are determined as follows. Combing Eqs. (7), (10) and (11), the analytical solutions of the mechanical stresses in the radial, hoop, and axial directions can be obtained.

A„ =--

(1 + v)(l - 2v)

E; ((A)m' - (A)m )(v + (1 - v)mj )R„m'

K R„

p \ ^ - ^ Ro '

(1 + r)(l - 2v)

E, <( A)"- < A >">)(v + (1 - v )m,)Rm R R

0.5 pR} Rßo +

(mx +1)( Rm+ß+1 - Rm+ß+1) (m2 +1)( R? +ß+1 - Rm+ß+1)

(ml +ß + l)(v + (1 - v )m0 Rm (m2 +ß +1) (v + (1 - v )m2) R,

«M = - p

( f \ml+£-1 r

m2 +fi-1 N \m, +£-1

s m2 +£-1 N

z (/ \ m+ß-i / \m2+ß-i \

(m, +1) rm+ß-1__(m2 +1) rm +ß-1

{v + m, - mv)Rm(v + m2 - m2v)Rm

p (mxv +1 -v) rm+ß-1 (m2v +1 -v) rm+ß-1

if R mi +ß~ (R m2+ß-l \ (v + mx - my)Rm (v + m2 - m2v )Rm (12b)

{R / ^ V o J lK J 7

© For the thermal stresses

By combining Eqs. (1) and (4)~(7), the governing equation in the cylindrical part of the FGM pressure vessel can be derived as

rV+(^+l) ru'+[ vß-i) u=pT~r*'p +QQ*'P —ßve^r where

q, (v +1 ){<P + ß~r)(T, -T)

(R?- R:%\-v)r: a, (v + \)(jp + ß){TR;r - TR)

For Eq. (13), the characteristic equation is m2 + ¡3m + (v'P -1) = 0 , and its roots are

»,=1 ) ;m2=± [-P + 4P1 - Av'p + 4 )

m , , , ,, , ,

2V " r z 2'

And the solution is given as

u = ATrm1 + BTrm + C/-r+<p + C2 r^1 - veTzr (14)

where At and Bt are unknown constants of integration that need to be determined according to the boundary conditions. And

C =_«. jv+l ){p+p-y) (T0 - T)_

1 ((1 -y+vf +p(\-Y+q>)+vp-\W7- RTr)(l-v)/RT

«o (v + l){<p + fi){TR-J-ToR7r) ' ((1 + <p)2 + p{ 1 + <p) + v'p-1)(R7 -R~r)(l- v)Rf

For the axisymmetric pressure vessel only subjected to thermal loads, the boundary conditions are listed as follows.

<JTr = 0 when r - Rt

<jTr - 0 when r - Ro ^^

a Ifirdr = 0

By combining Eqs.(l), (6) , (7), (14) and (15), the constants of integration AT, BT and the axial strain e/ are determined as

Rm"C4 -Rm"C, Rm-C. -Rm"C,

A __._4_o_3_ D ___._4_o_3_

((l -v)ml +v){Rm~lRm-Rm) ' T " ((1 -v)m2 "R^"1 -Rm~1R"2)

A = Cv ao (1 + v) To-T . = C v «o (1 + v) TR7~ ToR7 1 1 R:(2-r + q>) R-J-2 R:(2 + <p) R~/-

2(A (Rfr+2 -R?~r+2) + A (Rf2 -Rf+2)+ ATv(Rp*1 -Rm+1 ) + BTv[Rm+1 -Rm+1 )) = (1 + v)(l - 2v)(R2-R2 )

C3 = CR" ((1 - v)(1 -r + v) + v) + C2R ((1 -v)(<p + l) + v)~ (1 + vKr-t/r: C4 = C, Rf ((1 - v) (1 -y + V) + v) + C2 R: ((1 -v){(p +1) + v) - (1 + v)a0T0

Combing Eqs. (7), (14) and (15), the analytical solutions of the thermal stresses in the radial, hoop, and axial directions can be obtained as follows

~(Rm-1C4 -R"-1C3)r"-1 ~(R"-1C4 -R"-1C3)r i R" ~lR"1 — R" ~lRm \

I o i oil

" (1 + v)(l - 2v)

(R?->C4 -~'C3) r"-1 (m1v +1 -v) R "C -R"-JC3) r"-1 (m2v +1 -v) - V )m +v)(ro™-1Rm1 - Rm-1Ri™-1) ((1 -v)m2 +v){r^ -1 - R^ _1RI"2)

(1 + v)(l - 2v)

(R,"2-JC4 -R^"1C3) v^™1 (m +1) (Rm-'C4 -Rm-1C3) vrm(m2 +1)

((1 -v)m1 +v)(R" -1 - R" -1) ((1 -v)m2 + v)(R" ^R" - RO"1 "'R,™2 )

(1 + v)(l - 2v)

L = Clrp~r ((l-v)(l+ + + C2rp((l+ l) + v)-(l + v)aT{r) G = Cxr9~r ((-y + p)v +1) + C2{<pv +1) - a (1 + v) T (r) H = C1rr^"r( 2+ + + 2 )-a(l + T (r ) + (l -v-2r2 )ezT

More details on how to derive the mechanical and thermal stresses for the FGM spherical pressure vessel were given in Appendix A.

3. FE modeling and Results Analysis

3.1. FE modeling

For verifying above-mentioned analytical solutions, a thermo-mechanical coupling FE model of an FGM vessel,

consisting of a hollow cylindrical body and two hemispherical closed ends, was constructed as shown in Fig. 1.

Figure 1 (a) shows the heterogeneous cross-section of the FGM hollow cylinder (R,=800 mm, RO=840 mm, height=1600 mm), in which the Young's modulus changes from the inside surface to the outside surface. Since there

is no authorized element in all existing FE codes to simulate the mechanical behavior of FGM, stepwise composition-based FE model has been generally used by some scholars [15, 27]. Similarly, in this work, a composition-based FE model was also developed and implemented into the main program. By changing the continuous material variation into stepwise variation, the heterogeneous cross-section in Fig. 1 (a) was discretized into ten individual homogeneous layers as shown in Fig. 2 (b), including a spherical head (R,=800 mm, RO=820 mm). Thus, each layer has the same material properties and is isotropic. But the material properties is not the same among each layer, and was changed in gradient along the wall thickness according to the Eq. (1).

And then the solid model in Fig. 1(b) was converted into an FE model as shown in Fig. 1(c). Considering the axis-symmetry of the geometrical structure and loads as well as boundary conditions, a quarter of the FGM structure was established with the axisymmetrical 8-node thermal element Plane77 and the 8-node structural element Planel83 using the FE code ANSYS. The symmetry boundary condition was applied at the bottom of the model. And ten areas were meshed into 7200 elements and 23051 nodes as shown in Fig. 1(c). The internal pressure is lOMPa. The temperature of the inner surface was 120 °C, and the temperature of the outer surface was 20 °C. Two

primary materials are AI2O3 for ceramic and stainless steel for metal, respectively, and Eo=210GPa, v=0.3, and its grade factors were randomly given as yff=-6.4024, ^=7.2554 and 7=35.5406 only for the purpose of theoretical analysis, respectively.

(a) (b) (c)

Fig. 1. Material, Geometric and FE model of an FGM pressure vessel.

An indirect method was used to calculate the thermo-mechanical coupling stresses of the FGM pressure vessel composed often-layer homogeneous cylindrical body and hemispherical closed end. The solution procedure was as follows: Firstly, the thermal model was created and the temperature boundary condition inputted, and the temperature distribution calculated. Secondly, the element type was converted from the thermal element PLANE77 to the structural element PLANE183, and hence the thermal stress was obtained. Finally, the internal pressure was imposed on the inner layer of the FE model ofthe pressure vessel, and the three stresses (o-r, ce, Oz) were determined.

3.2. Results and discussion

The radial stresses, hoop stresses and axial stresses determined by the analytical solution and the FE method are shown in Fig. 2, Fig. 3 and Fig. 4, respectively. In these figures, the square symbol denotes the stresses of the cylindrical body, and the circular symbol denotes the stresses ofthe hemispherical closed end.

Fig. 2. Curves ofthe radial stress vs radius.

Fig. 3. Curves ofthe hoop stress vs radius.

Fig. 4. Curves ofthe axial stress vs radius.

Fig. 5. Curves ofthe stresses along DB path.

It can be seen that the analytical solutions agree well with the FE results in Fig. 2 and 3. The discrepancy of the axial stressed of the cylindrical vessel in Fig. 4 is less than 6%. Thus it illustrates that the analytical solutions are good agreement with the FE solutions, which indicates that the proposed model is credible and feasible in this work.

Figure 5 shows the change trend of the hoop and axial stresses along the DB path as shown in Fig. 1(C), it can be seen that the two stresses both appeared sharp fluctuation in the joint zone between the cylinder and hemispherical end, and the sharp change stress is generally named as the edge stress, which was caused by the discontinuous geometrical structure in the joint zone. Compared to the analytical solution, the FE analysis can determine the edge stress more accurately.

According to the empirical equations [28], the attenuation range of the edge stress for a cylinder is given as x = 2.5 (17)

For a hemispherical end, Eq.(17) is modified as

e = (18)

where x is the distance from point O to point B, and 0 is the angle of rotation from point O to point D (in radians) as shown in Fig. 5.

According to Eqs. (17) and (18), the results are x=458.3 mm and 0=31.3°. On the other hand, the FE solutions are x=494.4 mm and 0=36.7°, which indicates that the effective zone of the edge stress obtained by the FE method keeps good agreement with the empirical equations. Besides, it also indicates that the influence of structure discontinuity in a FGM pressure vessel can also be neglected, because the scope of the edge stress is limited with obvious attenuating trend. Therefore, the principle stress of the area far away from the connection is a major consideration for the FGM pressure vessel design.

4. Optimization modeling for seeking the optimal graded factor

4.1. Construction of optimization objective function

The maximal stress is generally used as the design condition according to the existing pressure vessel design code, such as ASME, EN13445, GB150 (Chinese code) and so on. However, it can be seen from the Fig.3 that the distribution of hoop stress is not uniform along the cross-section of the wall thickness for the FGM pressure vessel, thus its bear-loading capacity is not fully utilized. It is suggested that the stress should be designed as uniformly as possible by adjusting the graded factors of FGMs for fully utilizing its bear-loading capacity. However, it is difficult to seek the optimal graded factors. In this work, an optimization method was proposed by coupling the FE model and optimization method. Firstly, the optimization objective function is established as

min(^ = — (a-ae (0))) \n i=1

S.t. s = max(cre )< [cr] (19)

fie (0,r) (0,f)

where n denotes the number of layers in the AB path in Fig.l (c), a is the average stress in the AB path, cr (i) is the von Mises stress in the nth layer, and 77 is the average variance, [a] is the allowable stress of the FGM pressure vessel, max(ae) is the maximal von Mises, r is a constant given according to the maximal value of p, and e is another constant given according to the maximal value of q>. Notes that the thermal grade factor y was not optimized

for reducing the calculation scale. The computation is reiterated until the magnitude of r/ of the objective function is minimized.

4.2. Optimization results and discussion

The optimization model was performed based on above-mentioned FE model, and double design variables (fi and (p) optimization procedure was developed in this work by compiling a sub-program using the ANSYS parametric design language (APDL). The solution is achieved when the average variance {rj) of the objective function becomes the smallest. As there are two unknown parameters (fi and (p) need to be optimized, the zero-order optimization method has been adopted in order to obtain the accurate solution. It should be noted that the optimization algorithm is feasible at finding the local optimization solution within a specified tolerance. For this reason, multi-step optimization was carried out by adjusting the optimization range. In this work, n=10, [c] =200MPa. The design variables (t and e) were both given as 18 at the initial optimization and multi-adjusted into a reasonable local subrange.

Table 1 shows the optimization computational results, the optimal graded factor is y§=-11.355 and ^=17.946 through the repeated iterative, in which the average variance of the von Mises stress arrives at the smallest.

Table 1. Iterative process ofthe graded factor.

n 1 2 3 4 5 *6

Ißl 15.61 17.38 17.31 10.83 11.26 11.35

M 16.60 17.65 13.87 17.68 17.90 17.95

s 188.8 191.0 193.8 184.7 184.6 184.7

10.26 11.57 14.77 8.842 8.030 7.961

Figure 6 shows the curves of the von Mises stresses along the AB path (the wall thickness of FGM cylinder) as shown in Fig. 1(c), it can be seen that the maximal stress was decreased from 229.1 MPa to 175.8 MPa, and the von Mises stress becomes more uniform along the AB path, which illustrates that the bear-loading capacity would be improved about 23% if using the optimized graded factors. The results will be helpful in better and reasonable design of the FGM pressure vessel.

Fig. 6. Curves ofthe Mises stress along AB path.

5. Conclusions

In this study, the analytical solutions and FE solutions for determining the mechanical behavior of the FGM pressure vessel subjected to thermo-mechanical loading, and the innovation is to implement the axial strain into the stress equations. The stress calculation formulations obtained are further improved compared with the existing ones. Moreover, the analytical solutions were in good agreement with the FE solutions, which proved that the analytical solutions of the stresses of the FGM pressure vessel were credible. Another innovation is to propose an optimization model for seeking the optimal graded factors of the FGM pressure vessel, and the results illustrated that the optimization model developed was feasible. The findings in this work will be helpful for accurate stress calculation and rapid parameter design of the FGM pressure vessel in the future.

Acknowledgement

The authors would like to thank the National Natural Science Foundation of China (51005030) and the Fundamental Research Funds for the Central Universities (DUT11LK25) for their support of this work.

Appendix A. Derivation of the thermal stresses for spherical FGM pressure vessel

For the spherical FGM pressure vessel, the equilibrium equations for axisymmetric hollow spherical pressure vessel are modified as,

dar + 2 (g;-ae) dr r

And the strain-displacement relations are

For the spherical FGM pressure vessel, the constitutive equation are modified as

E [2v£e+(l-v)er ] aET (r ) (1 + v)(l - 2v) (1 - 2v) E{£g+v£r) aET (r ) (1 + v)(l - 2v) (1 - 2v)

A.1. For the mechanical stresses

For the spherical pressure vessel, the control equation can be transformed as

r2u" + (2 + J3)ru' + 2(y'p - l)u = 0

For Eq. (A.4), the characteristic equation is z2 + {fi +1)z + 2(v*ft -1) = 0, and its roots are

z {-fi-1+ + 2 P-8 v'P + 9 )

The solution is given as

u = CMrz + DMrz

For the spherical vessel only subjected to internal pressure p, the boundary conditions are

a'M = - p when r - R aM = 0 when r = R

The constants of integration CM and DM are determined by Eq. (A.5) and (A.6) as

C„ =--

p( R -v)(1 - 2v ) R

E. ((R)z - (R)Z2 )(2v + (1 -v)z1 R-R R

D,. =-

p( r -v)(1 - 2v) R

Eo((Rr ~(R}Z )(2v + (1 " v}^ R

Then the analytical solutions of the mechanical stresses for the spherical pressure vessel are obtained as

+ 0-1 f V2 + 0-1 ^ If,

/fi ^ I

z, +f3-\ r V2 +f3-\ ^

(A.7a)

if R Ï z, +0-1 f R ^ z2+0-1N

H R, IR ; y

ZjV + 1

R j 2v + (1 -v)z

z2v +1

2v + (1 - v) z2

A.2. For the thermal stresses

For the spherical pressure vessel, the control equation can be transformed as

rLu" + (p + 2)ru' + 2 (v'p- l) u = Prl~r+9 + Qr1*9 The characteristic equation for Eq. (A.8) is z2 + (fi +1)z + 2(v P -1) = 0, and its roots are

(~P~1-4P1 + 2 P~ + 9 )

(A. 7b)

^[-P-1+4P2+2P-9 )

The solution is given as

u = CTrz1 + DTrZ2 + V + S2 r^+1

(1 -y + çf +{/3 + 0 (1 ~y + ç) + 2 [n'P-1)

S2 - ,, \2

(1 + +(/? +1)(1 + ç) + 2 1) For the spherical pressure vessel only subjected to thermal loads, the boundary conditions are

crTr - 0 when r - Rl cl - 0 when r - R

The constants of integration CT and DT are determined by Eq.(A.9) and (A. 10) as

Rz*-1Si - Rz2-'S,

i_4_o_3_

((l -v)z1 + 2v) [R?"R -1 - Rz "R-1 )

DT =-■

Rz"S - Rz-'S,

i_4_o_3_

((l -v)z2 + 2v) -R-1 - RZ -R-1 )

(A. 10)

53 = S,Rt((l -v)(1 -y + ç) + 2v) + S2Rf((l-v){<p +1) + 2v)-(l + v)a0R<TllR0

54 = SR7 ((1 - v)(1 -y + v) + 2v) + SR ((1 -v){(p +1) + 2v)-(l + v)a^

Then the analytical solutions of the thermal stresses for the spherical pressure vessel are obtained as

(.R22-lSA -R0-1S3)rz-1 ~(RZ-1S4 -Rz)rZï-l

ÎR,2-1 RZ] -1 _ R^-R -i \ \ o l oil

(A. 11a)

(1 + v)(l - 2v)

(Riz-'S4 - RZ -S,) rz-1 (1 + zy) (RZ -'S4 - RZ -'S,) r"1 -1 (1 + z2v)

((l-v)z1 + 2v)(/£ -R -1 - R^R2 ((l-v)z> + 2v)(R? "R^1 - RrR -1)

(A.llb)

(l + v)(l - 2v)

W = Slr9~r ((1 -y + y)( 1 -v) + 2v) + S2r^((l-v)(v +1) + 2v)-a(\ + v)T(r) K = Slrv~r{\ + (1 -y + y)v) + S2rv{\ + + \)v)-a(\ + v) T (r)

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