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Evaluation of the optimum pre-stressing pressure and wall thickness determination of thick-walled spherical vessels under internal pressure
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Journal of the Franklin Institute 344 (2007) 439–451 www.elsevier.com/locate/jfranklin
Evaluation of the optimum pre-stressing pressure and wall thickness determination of thick-walled spherical vessels under internal pressure M.H. Kargarnovin, H. Darijani, R. Naghdabadi Department of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, Tehran 11365, I.R. Iran Received 7 February 2006; accepted 8 February 2006
Abstract In the present study, in the first part for a spherical vessel with known dimensions and working pressure, two methods of hoop and equivalent stress optimization across the wall thickness are employed to determine the best autofrettage pressure. In the next part for a predefined working pressure the minimum wall thickness of the vessel is calculated using two other design criteria i.e. (A) optimizing the hoop stress, and (B) assuming a suitable percent for the penetration of yielding within the wall thickness. Finally, the optimum thickness and the necessary strengthening pressure are extracted and different plots are introduced for different types of structural materials under different internal pressures. r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Thick-walled vessels; Elasto-plasticity; Linear work hardening; Pre-stressing; Isotropic hardening; Hoop and equivalent stress optimization
1. Introduction The ever increasing demands for axisymmetrical pressure vessels have high applications in industries such as chemical, nuclear, fluid transmitting plant, power plant and on the other hand in the military equipments, have turned the attention of many designers to this particular area of engineering. The progressive, hardly appropriate scarcity of some materials and also the high cost of their production have caused researchers not to confine their design to the customary elastic regime and have attracted their attention to the Corresponding author. Tel.: +98 21 6616 5510; fax: +98 21 6600 0021.
E-mail address: [email protected] (M.H. Kargarnovin). 0016-0032/$30.00 r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2006.02.013
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elastic–plastic approach which offers a more efficient use of such materials. In brief, increasing the strength to weight ratio and extending the fatigue life are some of the main objectives of optimal design for the thick-walled sphere. The optimization can be carried out primarily by generating a residual stress field in the sphere wall, known as the autofrettage process. The study of this technique has been the subject of numerous researches for many years, such as those given by Majzoobi [1], Manning [2], Chen [3,4], Franklin [5], and has been discussed by Kargarnovin [6] and Ruilin Zhu [7] for elastoperfectly plastic materials. For the linear work hardening material, however, little progress has been made, as the analysis is a more complicated compared to the above-mentioned behavior. Most of the earlier solutions for residual stresses are based on the assumption of elastic unloading and only a few researchers have considered a criterion known as the reverse yielding point [8]. Hence, it has been noticed that the inclusion of plastic behavior on the unloading phase has been an untouched area of work to be considered in our study in this paper. To do this, a closed form solution of residual stresses in autofrettaged spheres is obtained. In the next step, a simple, accurate and reasonable analytical equation for determining the best autofrettage pressure with known dimensions of the sphere and the working pressure is derived. Moreover, for the cases in which the working pressure and inner radius are known, the minimum acceptable thickness is determined. In the present research for the thick-wall sphere the following basic assumptions are made: the material is assumed to be elastic-linear work hardening which yields according to Tresca/von-Mises criterions and hardens isotropically, and the Bauschinger effect is ignored. 2. Elasto-plastic loading Consider a thick-walled sphere with inner radius a, and outer radius b, which is subjected to the inner pressure Pi (see Fig. 1). The following non-dimensional parameters are used: b r c c0 Pi Pw ; r ¼ ; rc ¼ ; r0c ¼ ; P ¼ ; pw ¼ , a a a a s0 s0 sr sy se Eer Eey Sr ¼ ; Sy ¼ ; S ¼ ¼ S y S r ; 2r ¼ ; 2y ¼ , s0 s0 s0 s0 s0 b¼
Fig. 1. Different parameters using a thick-walled sphere under internal pressure.
ð1Þ
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where Pw, Pi, s0 , se , sr , sy and E are the working pressure, the autofrettage pressure, the initial yielding stress, the equivalent stress, the radial stress, the hoop stress and Young’s modulus, respectively. Moreover, c and c0 are the radii for the elasto-plastic boundaries on loading and unloading, respectively. The equilibrium and compatibility relations in dimensionless form for this specific problem have the form [9] dS r Sr Sy þ2 ¼ 0, dr r
(2)
d2y 2y 2r þ ¼ 0, dr r
(3)
while the stress–strain relations are [9] 2r ¼ S r 2vS y þ 2pr , 2y ¼ ð1 vÞS y vS r þ 2py ,
ð4Þ
where 2r ; 2y are the total strains, and 2pr ; 2py represent the corresponding plastic strains. The boundary conditions are Sr ð1Þ ¼ P;
S r ðbÞ ¼ 0.
(5)
During loading, the elastic–plastic solution induces stresses in the plastic zone ð1ororc Þ as [9] 1 1 1m 1m lnðrc Þ ðS r;c þ PÞ , Sr ¼ P þ 1 3 C 1 þ r 1v m 2m 1 1 1m 1m lnðrc Þ ðS r;c þ PÞ , ð6Þ Sy ¼ P þ 1 3 C 1 þ 2r 1v m 2m where the work-hardening parameter m is defined as the ratio of the slope of the workhardening part of the stress–strain curve (tangent modulus) to the elastic modulus (m ¼ E t =E) Fig. 2. Moreover, the stresses in elastic zone of the same sphere ðrc orobÞ can be derived as Sr ¼
2 r3 b3 1 , 3 r3 b3c
Sy ¼
2 2r3 þ b3 1 . 3 2r3 b3c
ð7Þ
The other important parameter which we need, is the value of the pressure for an arbitrary initial plastic zone radius, which is given in [9] as P¼
ð4=3Þð1 vÞmðb3 1=b3 Þr3c þ 2ð1 mÞ ln ðrc Þ þ ð2ð1 mÞ=3b3c Þðb3c 1Þ . 2mð1 vÞ þ ð1 mÞ
(8)
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σ′e
σe
Et
Ep σe
E σe σ0 Ep
σe
εp
ε′p
Fig. 2. Stress–strain curve of the loading and unloading.
3. Elastic–plastic unloading When the pressure is removed from the sphere under consideration, if it causes plastic flow over part of the sphere, some residual stresses result in. In order to calculate this induced residual stresses, it is necessary to superpose the obtained stresses in loading phase due to the internal pressure P and stresses caused during unloading phase i.e. P. It is clear that the equilibrium and compatibility relations during unloading are similar to those given in Eqs. (2), (3) in which the dimensionless parameters are superscripted by prime. It should be noted that these primed parameters correspond to the system of stresses in the sphere caused due to internal pressure P. However, the boundary conditions during unloading are different which are S 0r ð1Þ ¼ þP;
S 0r ðbÞ ¼ 0.
(9)
The method of analysis during unloading is the same as the one described in the phase of loading represented in [9]. To do this one should substitute the constitutive relations into the compatibility equation and then combine the obtained result with the equilibrium equations. After integrating of the these relations, one would get the following equations: R r 20 Pr 1 1 S0r ¼ 1v dr þ 1 þ C 01 þ P; 3 1 r r R r 20 Pr 1 1 S0y ¼ P þ 1 þ 2r1 3 C 1 þ 2ð1vÞ 20 Pr þ 1v (10) 1 r dr; 20 P
r S 0 ¼ 2r3 3 C 01 þ 2ð1vÞ :
In order to calculate the exact form of stress distribution, the answer for the integral in a form of relations given in Eq. (10) should be known in advance. To do this during unloading phase it is necessary to know: (a) The final value of yield stress at the end of loading phase for each material point within the plastic zone.
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(b) The plastic part of stress–strain curve during unloading follows the same trend as in loading phase. Moreover, it is assumed that yielding occurs in the region of 1oror0c with r0c orc during unloading and the material follows an isotropic hardening rule. In this case, reverse yielding is controlled by ( 2S; rprc ; 0 S0 ¼ (11) ðS þ 1Þ; rXrc ; where S 00 is the dimensionless initial yielding stress in the reverse yielding, and also S is given by [9] " 3 # 2mð1 vÞ 1m rc þ S¼ . (12) 2mð1 vÞ þ 1 m 2mð1 vÞ r As it was mentioned before, the stress at the initial step of unloading phase is the same as the stress at the end of loading phase for each material point. As it is seen from Fig. 2, the unloading strain–stress curve is then constructed from the end point of loading phase with an elastic range with its limit at a stress difference of S00 and a hardening part with the same slope as the loading phase. By implementing the yield criterion for linear work hardening material, it follows from Fig. 2 that 1m 0 ðS þ S 00 Þ. (13) m Now, supposing that the plastic zone extends to r ¼ r0c then, from Eq. (15) and making use of the equilibrium relation on unloading, it follows that 8 3 rc > 4ð1mÞ Ln r 2mð1vÞ 0 1m 4 3 > P S þ r ; rpr0c ; > Z r 0P c r 1mþ2mð1vÞ 1mþ2mð1vÞ 3 r < 2m 2r dr ¼ 4 rc 3 > r 4ð1mÞ Ln r0c 2mð1vÞ 0 1m 3 1 > > P S þ r ; rXr0c ; 0 : 2m c r;c 1mþ2mð1vÞ 3 1mþ2mð1vÞ r0 0 20p r ¼ 2p ¼
c
(14) S 0r;c0
S0r
where is the value of at r ¼ substituting into last of Eqs. (10), gives
r0c .
0
And, since S ¼ 2S when r ¼
C 01 ¼ 43r03c S ;r0c , where S ;r0c is the value of S at r ¼ Eqs. (10) gives
r0c ,
then by (15)
r0c .
Substituting the Eqs. (14) and (15) into last of
" # 3 4ð1 mÞ2 4 ð1 mÞðmÞð1 vÞ r c P¼ ln r0c r3c 3 1 m þ 2mð1 vÞ2 r0c ½1 m þ 2mð1 vÞ2 4 3 1 b3 1m 1 1 3 þ r0c S;r0c 3 1 m þ 2mð1 vÞ r0c b3
ð16Þ
which relates the pressure P to the secondary plastic zone radius r0c during unloading. The minimum value of the applied pressure (P) which causes the vessel to yield during unloading phase is defined by P*. To calculate P*, it is enough to substitute
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r0c ¼ 1 and S0r;c0 ¼ P in Eq. (16). Therefore, by performing the appropriate some mathematical manipulations we obtain 4 b3 1 2mð1 vÞ 1m 3 þ rc . P ¼ (17) 3 b3 1 m þ 2mð1 vÞ 2mð1 vÞ By considering the above relation, S 0r;c0 , which is the critical pressure during unloading acting on a sphere with inner radius c0 and outer radius 00 b00 , can be obtained. This can be done by replacing 00 a00 with c0 in Eq. (17) " # 4 b3c0 1 2mð1 vÞ rc 3 1m 0 S r;c0 ¼ þ . (18) 3 b3c0 1 m þ 2mð1 vÞ rc0 2mð1 vÞ For the zone ðr0c orobÞ we calculate the elastic stress distribution of a sphere with inner radius r0c and outer radius b, subjected to an internal pressure S 0r;c0 . Therefore, the stress distributions in this elastic zone ðr0c orobÞ are S0r ¼
r3 b3 S0 0 , r3 ðb03c 1Þ r;c
S 0y ¼
2r3 þ b3 S0 0 . 2r3 ðb03c 1Þ r;c
ð19Þ
4. Determination of the residual stresses Depending on the geometrical factor b and the amount of removed pressure from the vessel, different deformation modes are produced on the sphere. If P4Pn then during unloading, yielding will occur otherwise, the unloading is entirely elastic. In order to analyze the effect of these changes, at first the value of the two aforementioned pressures, i.e. P and P*, have to be equated. In order to get to this state, the value of P from Eq. (8) is set equal to the value of P*, obtained from Eq. (17). The result is m¼
3b2 lnðrc Þ þ b3 þ r3c 2 . r3c b3 ð2n 2Þ þ r3c ð3 2nÞ þ b3 ð1 3 lnðrc ÞÞ 2
(20)
It can be proved that the maximum value for m equals to 0.088 if n ¼ 0:3. Furthermore, for bo1:7 the value of m becomes less then zero. Therefore, it can be said that for ðm40:088 or bo1:7Þ, the deformation mode during unloading is entirely elastic. Moreover, for ðmo0:088 or b41:7Þ, no specific judgment can be made on the mode of deformation, unless the values of P and P* are known. In general, if PoP*, unloading is entirely elastic and the stresses in this phase are given as S0r ¼
r3 b3 P, r3 ðb3 1Þ
S 0y ¼
2r3 þ b3 P. 2r3 ðb3 1Þ
ð21Þ
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On the other hand, if P4Pn , then during unloading the plastic zone extends to r ¼ r0c and the stress components in the reverse yielding zone ð1oror0c Þ are determined from Eq. (10) and stress distributions in elastic zone ðr0c orobÞ are determined from Eq. (19). Finally, the residual stresses are found by superposing the stress components obtained during loading and unloading phases, i.e. 0 Sre r ¼ Sr þ Sr , 0 Sre y ¼ Sy þ Sy .
ð22Þ
The superscript 00 re00 refers to the residual stress component.
5. Equivalent stress and hoop stress optimization Now, suppose that the dimensions of the sphere and also the working pressure are known. The optimum autofrettage pressure is an internal pressure which is applied to the sphere before it is being put into operation. The main task of this pressure is to initiate yielding in the inner surface of the sphere and beside; it optimizes the distribution of the hoop stress, or equivalent stress, throughout the wall thickness.
5.1. Method of hoop stress optimization The analysis begins by assuming a radius for the elasto-plastic boundary for which the residual stress and moreover, the total hoop stress (summation of residual hoop stress and working hoop stress) are calculated. This operation is repeated several times for different radii. The optimal case is the one in which the maximum hoop stress in that case is the lowest amongst other maximums. Fig. 3 illustrates the variation of the hoop stress distribution for different initial elasto-plastic radius rc for m ¼ 0:02; pw ¼ 1:3; b ¼ 2:2. In this figure, the points with negative maxima represent the position of r0c during unloading and points with positive maxima represent the position of rc during loading. By close examination of these curves, it can be noted that an enveloped curve can be passed through all points having maximum value in each curve. Furthermore, it can be seen that this envelope curve has a minimum at a certain point. Therefore, in order to obtain the position of this optimum condition, it is enough to impose the minimization rule on this curve, i.e., to take the derivative of the curve functions with respect to rc and set the result to zero. The mathematical form of hoop stress at this point is equal to the combination of hoop stress expressions due to (i) primary loading, (ii) unloading, (iii) working condition pressure, i.e.: Sy ¼ S re y;rc þ
2r3c þ b3 pw 2r3c ðb3 1Þ
(23)
in which the Sre y;rc is the summation of hoop stresses due to primary loading (case i) and unloading (case ii) at r ¼ rc . First, we assume that the unloading becomes entirely elastic (m40.088 or bo1.7), then an explicit form of Eq. (23) is obtained. In this case by setting the derivative of Eq. (23)
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Fig. 3. The hoop stress distribution for different initial elastic–plastic boundaries m ¼ 0:02, pw ¼ 1:3, b ¼ 2:2.
with respect to rc equal to zero, one would get 27b3 ¼
ð36m þ 36 36r3c;opt þ 36mr3c;opt Þð2m lnðrc;opt Þ pw þ 2 lnðrc;opt Þ þ 2pw mn mpw Þ2 ðpw þ 2pw mn þ 2 lnðrc;opt Þ 2m lnðrc;opt Þ mpw Þ3
r3c;opt .
ð24Þ Then, rc;opt is obtained from Eq. (24) and after it is substituted in Eq. (8), the best autofrettage pressure is determined. When the condition of (mo0.088 or b41.7) is not prevailing, we are not sure that at the point rc;opt the vessel behaves elastically. In this case we again assume an elastic behavior at this point, then we obtain the value of Popt at rc;opt from Eq. (8) and check the condition of reverse yielding. If PoP*opt then our assumption is correct otherwise, we should determining Popt, utilizing the mechanism of pre-stressing pressure which was explained before i.e. solving Eq. (23) numerically. From the data shown in Fig. 3, the optimal values of rc;opt ¼ 1:47; Popt ¼ 1:25 and Sopt ¼ 0:62 can be determined. 5.2. Optimizing the distribution of equivalent stress Here in order to determine an optimized distribution for the equivalent stress we follow the same approach as in the previous section. By carrying out the corresponding calculation, the equivalent stress distribution for different initial elasto-plastic radius, rc , is obtained and the results for m ¼ 0:1, Pw ¼ 0:7 and b ¼ 2 are shown in Fig. 4. Again, by close examination of these curves, it can be noted that an envelope curve can be passed
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through all the points having a maximum value on each curve. Furthermore, it can be seen that this envelope curve has a minimum at a certain point. It can be determined from Fig. 3 that in this case the optimal values are ropt ¼ 1:5; Popt ¼ 1:3 and S opt ¼ 0:7. The position of this minimum point of the envelope curve rc;opt is found by applying the minimization rule with respect to rc on the following equation: 3 3pw b 0 Se;total ¼ S e þ S e þ 3 . (25) 3 2r b 1 c For the case of entirely elastic unloading, the position of rc;opt can be obtained as rc;opt ¼ eðpw ðmð2n1Þ1=2ðm1ÞÞÞ ,
(26)
which relates the optimum elasto-plastic radius rc;opt to pw . It is observed from Eq. (26) that there exists a minimum optimum point on the envelope curve if pw oð2ð1 mÞ=mð2n 1Þ 1Þ ln b, otherwise, i.e. for pw 4ð2ð1 mÞ=mð2n 1Þ 1Þ ln b this curve follows a descending trend, hence, the optimum point approaches to the end of curve or at the outer radius for which the P** is the best autofrettage pressure that causes the sphere to yield completely on loading. 6. Thickness optimization In Figs. 3 and 4 it can be seen that the value of the equivalent stress in radius rc;opt is less than the value of the initial yield stress. This means that in this case the vessel is capable to
Fig. 4. The equivalent stress distribution for different initial boundaries elasto–plastic m ¼ 0:1; pw ¼ 0:7; b ¼ 2.
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withstand higher level of working pressure pw. On the other hand, if the value of pw is prescribed, then the optimum thickness and initial strengthening pressure also can be obtained simultaneously. The design criterion is to consume less material in the production of the vessel. Where, by introducing two new design criteria i.e. (A) optimizing the hoop stress, and (B) assume a suitable percent for the yield zone in the wall thickness, the minimum thickness of the vessel can be calculated. Based on criterion A we have q ðS y;max Þ ¼ 0. qrc
(27)
Note that Eq. (27) is a function of b and rc;opt and hence it cannot be solved directly. Thus, in order to come up with an optimum case, another design condition is needed. When the sphere in the first step undergoes of an autofrettage process and then in the next step is loaded by a predefined working pressure, again another yielding process initiates. It is clear that this secondary yielding would begin again from the inner radius. Therefore, according to our yielding criteria, we have 3 3 S re e;1 þ 3pw b =2ðb 1Þ ¼ S Y;1
(28)
in which, S Y;1 is the value of the new yielding stress at inner radius. If the unloading is elastic, then, S Y;1 ¼ S e;1 , otherwise, SY;1 ¼ S e;1 S0e;1 . It is known that in the autofrettage process, the inner layer of the sphere is in the most critical yielding condition and furthermore, we cannot increase the working pressure indefinitely. Therefore, to come up with a practical value for its value, still we need to introduce another constraint for the failure mechanism in the inner layer of the sphere. We choose this additional constraint to be S Y;1 p1:7 during autofrettage process. Now we are ready to obtain the values of rc;opt and b by using following equations: 9 8 q > > > > ðS Þ ¼ 0 condition 1; > > = < qrc y;max (29) 3 3 re S þ 3pw b =2ðb 1Þ ¼ S Y;1 condition 2; > > > > > > e;1 ; : SY;1 p1:7 condition 3: It is notable to verify that in the case of elastic–perfectly plastic behavior (m ¼ 0), from Eq. (28) one can conclude that 3pw b3 =2ðb3 1Þp2.
(30)
The inequality in Eq. (30) is valid only for pw p1:33. Therefore, for pw p1:33 the system of Eq. (29) for m ¼ 0 has a unique solution. Fig. 5 illustrates the variations of the dimensionless radius b and the dimensionless optimum autofrettage pressure, Popt, versus the working pressure pw. Based on above-mentioned criterion (B), we begin by choosing an appropriate value for the percent of yielded zone in the autofrettage process. Again, it has to be noted that it is very important to select a justified value for this elasto-plastic radius. To elaborate further on this point it is easy to see that if the percent value for this yielded zone is taken to be 100%, consequently, the elasto-plastic radius will shift towards the external radius. This means that the entire wall thickness undergoes plastic deformation in the first cycle of loading and therefore, this would not be a suitable design principle. On the other hand, if
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Fig. 5. Variation of dimensionless design parameters vs. pw .
Fig. 6. Variation of dimensionless design parameters vs. pw (assume, 75% thickness yields).
the percent value for this yielded zone is taken to be 0%, the elasto-plastic radius will shift towards the internal radius; again this would not be suitable design criteria, because in this case the wall thickness becomes very large. In this study, we choose the percent value for the yielded zone to be 75%. Therefore, it is clear that rc ¼ 1 þ 0:75ðb 1Þ.
(31)
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To solve Eq. (31), we have to come up with another criterion; hence, we recall Eq. (28) and rephrase it as following: 3pw b3 =2ðb3 1Þ ¼ S Y;1 S re e;1 .
(32)
Eqs. (31), (32) can be solved simultaneously for different working pressures. Fig. 6 illustrates the variations of the dimensionless radius b and the dimensionless autofrettage pressure Popt versus different working pressure for different values of ‘‘m’’. 7. Conclusion In this paper based on the theory of elasto-plasticity for optimal design of thick-walled spherical vessels, the governing equations were derived and solved. Based on the obtained results, it is concluded that (1) Autofrettage considerably reduces stress level in the wall of a sphere. (2) The method presented in this paper is very simple, applicable, accurate and reasonable compared to other existing methods. (3) The autofrettage pressure must be always greater than the working pressure. (4) Refered to Figs. 5 and 6 it can be said that for dimensionless working pressure, (pw) up to 0.6, the effect of variation of m ðE t =EÞ in the wall thickness optimization is immaterial. (5) By considering elastic–perfectly plastic behavior, the design problem has a solution, only for pw p1:33. It is apparent that the wall thickness calculated from the current method is less than the thickness usually obtained by employing the criterion of initiation of yielding in the inner radius. Moreover, by reviewing the results depicted in Figs. 5 and 6, it can be seen that the difference between the design curves are negligible for a low level of working pressure. In the case of elastic–perfectly plastic (m ¼ 0) material, when pw ! 1:33, the geometric dimensions of the vessel approaches infinity. Furthermore, from the same figures, it is noted that, by increasing the value of m, the allowed design pressure and the thickness will decrease, and, moreover, for a fixed value of geometric dimension when m is increased, the allowed working pressure is increased. Finally, under optimum condition when unloading is elastic, Popt ¼ pw irrespective of the value of m, and this condition is valid up to the point where the Popt start nonlinear behavior (slope is 1). Reference [1] G.H. Majzoobi, G.H. Farrahi, A.H. Mahmoudi, A finite element simulation and an experimental study of autofrettage for strain hardened thick-walled cylinder, Mater. Sci. Eng. (2003) 326–331. [2] W.R.D. Manning, Trans. ASME, J. Pressure Vessel Technol. 100 (1978) 374–381. [3] P.C.T. Chen, Stress and deformation analysis of autofrettaged high pressure vessels, ASME Special Publication, vol. 110, PVP. ASME United Engineering Center, New York, 1986, pp. 61–67. [4] P.C.T. Chen, The Bauschinger and hardening effect on residual stresses in an autofrettaged thick-walled cylinder, J. Pressure Vessel Technol. 108 (1986) 108–112. [5] G.J. Franklin, J.L.M. Morrison, Autofrettage of cylinder: prediction of pressure/expansion curves and calculation of residual stresses, Proc. Inst. Mech. Eng. 174 (1960) 947–974.
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[6] M.H. Kargarnovin, A. Rezai Zarei, Thickness optimization of thick-walled cylinderical vessels using prestressing pressure, The Third World Conference on Integrated Design and Process Technology, Berlin, Germany, July 6–9, 1998, pp 139–146. [7] J. Zhu, R. Yang, Autofrettage of thick cylinder, Int. J. Vessel Piping 75 (1998) 443–446. [8] La. Paolo, Li. Paolo, Different solutions for stress and strain fields in autofrettaged thick-walled cylinders, Int. J. Vessel Piping 71 (1997) 231–238. [9] A. Mendelson, Plasticity: Theory and Application, Macmillan, New York, 1968.
Journal of the Franklin Institute 344 (2007) 439–451 www.elsevier.com/locate/jfranklin
Evaluation of the optimum pre-stressing pressure and wall thickness determination of thick-walled spherical vessels under internal pressure M.H. Kargarnovin, H. Darijani, R. Naghdabadi Department of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, Tehran 11365, I.R. Iran Received 7 February 2006; accepted 8 February 2006
Abstract In the present study, in the first part for a spherical vessel with known dimensions and working pressure, two methods of hoop and equivalent stress optimization across the wall thickness are employed to determine the best autofrettage pressure. In the next part for a predefined working pressure the minimum wall thickness of the vessel is calculated using two other design criteria i.e. (A) optimizing the hoop stress, and (B) assuming a suitable percent for the penetration of yielding within the wall thickness. Finally, the optimum thickness and the necessary strengthening pressure are extracted and different plots are introduced for different types of structural materials under different internal pressures. r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Thick-walled vessels; Elasto-plasticity; Linear work hardening; Pre-stressing; Isotropic hardening; Hoop and equivalent stress optimization
1. Introduction The ever increasing demands for axisymmetrical pressure vessels have high applications in industries such as chemical, nuclear, fluid transmitting plant, power plant and on the other hand in the military equipments, have turned the attention of many designers to this particular area of engineering. The progressive, hardly appropriate scarcity of some materials and also the high cost of their production have caused researchers not to confine their design to the customary elastic regime and have attracted their attention to the Corresponding author. Tel.: +98 21 6616 5510; fax: +98 21 6600 0021.
E-mail address: [email protected] (M.H. Kargarnovin). 0016-0032/$30.00 r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2006.02.013
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elastic–plastic approach which offers a more efficient use of such materials. In brief, increasing the strength to weight ratio and extending the fatigue life are some of the main objectives of optimal design for the thick-walled sphere. The optimization can be carried out primarily by generating a residual stress field in the sphere wall, known as the autofrettage process. The study of this technique has been the subject of numerous researches for many years, such as those given by Majzoobi [1], Manning [2], Chen [3,4], Franklin [5], and has been discussed by Kargarnovin [6] and Ruilin Zhu [7] for elastoperfectly plastic materials. For the linear work hardening material, however, little progress has been made, as the analysis is a more complicated compared to the above-mentioned behavior. Most of the earlier solutions for residual stresses are based on the assumption of elastic unloading and only a few researchers have considered a criterion known as the reverse yielding point [8]. Hence, it has been noticed that the inclusion of plastic behavior on the unloading phase has been an untouched area of work to be considered in our study in this paper. To do this, a closed form solution of residual stresses in autofrettaged spheres is obtained. In the next step, a simple, accurate and reasonable analytical equation for determining the best autofrettage pressure with known dimensions of the sphere and the working pressure is derived. Moreover, for the cases in which the working pressure and inner radius are known, the minimum acceptable thickness is determined. In the present research for the thick-wall sphere the following basic assumptions are made: the material is assumed to be elastic-linear work hardening which yields according to Tresca/von-Mises criterions and hardens isotropically, and the Bauschinger effect is ignored. 2. Elasto-plastic loading Consider a thick-walled sphere with inner radius a, and outer radius b, which is subjected to the inner pressure Pi (see Fig. 1). The following non-dimensional parameters are used: b r c c0 Pi Pw ; r ¼ ; rc ¼ ; r0c ¼ ; P ¼ ; pw ¼ , a a a a s0 s0 sr sy se Eer Eey Sr ¼ ; Sy ¼ ; S ¼ ¼ S y S r ; 2r ¼ ; 2y ¼ , s0 s0 s0 s0 s0 b¼
Fig. 1. Different parameters using a thick-walled sphere under internal pressure.
ð1Þ
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where Pw, Pi, s0 , se , sr , sy and E are the working pressure, the autofrettage pressure, the initial yielding stress, the equivalent stress, the radial stress, the hoop stress and Young’s modulus, respectively. Moreover, c and c0 are the radii for the elasto-plastic boundaries on loading and unloading, respectively. The equilibrium and compatibility relations in dimensionless form for this specific problem have the form [9] dS r Sr Sy þ2 ¼ 0, dr r
(2)
d2y 2y 2r þ ¼ 0, dr r
(3)
while the stress–strain relations are [9] 2r ¼ S r 2vS y þ 2pr , 2y ¼ ð1 vÞS y vS r þ 2py ,
ð4Þ
where 2r ; 2y are the total strains, and 2pr ; 2py represent the corresponding plastic strains. The boundary conditions are Sr ð1Þ ¼ P;
S r ðbÞ ¼ 0.
(5)
During loading, the elastic–plastic solution induces stresses in the plastic zone ð1ororc Þ as [9] 1 1 1m 1m lnðrc Þ ðS r;c þ PÞ , Sr ¼ P þ 1 3 C 1 þ r 1v m 2m 1 1 1m 1m lnðrc Þ ðS r;c þ PÞ , ð6Þ Sy ¼ P þ 1 3 C 1 þ 2r 1v m 2m where the work-hardening parameter m is defined as the ratio of the slope of the workhardening part of the stress–strain curve (tangent modulus) to the elastic modulus (m ¼ E t =E) Fig. 2. Moreover, the stresses in elastic zone of the same sphere ðrc orobÞ can be derived as Sr ¼
2 r3 b3 1 , 3 r3 b3c
Sy ¼
2 2r3 þ b3 1 . 3 2r3 b3c
ð7Þ
The other important parameter which we need, is the value of the pressure for an arbitrary initial plastic zone radius, which is given in [9] as P¼
ð4=3Þð1 vÞmðb3 1=b3 Þr3c þ 2ð1 mÞ ln ðrc Þ þ ð2ð1 mÞ=3b3c Þðb3c 1Þ . 2mð1 vÞ þ ð1 mÞ
(8)
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σ′e
σe
Et
Ep σe
E σe σ0 Ep
σe
εp
ε′p
Fig. 2. Stress–strain curve of the loading and unloading.
3. Elastic–plastic unloading When the pressure is removed from the sphere under consideration, if it causes plastic flow over part of the sphere, some residual stresses result in. In order to calculate this induced residual stresses, it is necessary to superpose the obtained stresses in loading phase due to the internal pressure P and stresses caused during unloading phase i.e. P. It is clear that the equilibrium and compatibility relations during unloading are similar to those given in Eqs. (2), (3) in which the dimensionless parameters are superscripted by prime. It should be noted that these primed parameters correspond to the system of stresses in the sphere caused due to internal pressure P. However, the boundary conditions during unloading are different which are S 0r ð1Þ ¼ þP;
S 0r ðbÞ ¼ 0.
(9)
The method of analysis during unloading is the same as the one described in the phase of loading represented in [9]. To do this one should substitute the constitutive relations into the compatibility equation and then combine the obtained result with the equilibrium equations. After integrating of the these relations, one would get the following equations: R r 20 Pr 1 1 S0r ¼ 1v dr þ 1 þ C 01 þ P; 3 1 r r R r 20 Pr 1 1 S0y ¼ P þ 1 þ 2r1 3 C 1 þ 2ð1vÞ 20 Pr þ 1v (10) 1 r dr; 20 P
r S 0 ¼ 2r3 3 C 01 þ 2ð1vÞ :
In order to calculate the exact form of stress distribution, the answer for the integral in a form of relations given in Eq. (10) should be known in advance. To do this during unloading phase it is necessary to know: (a) The final value of yield stress at the end of loading phase for each material point within the plastic zone.
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(b) The plastic part of stress–strain curve during unloading follows the same trend as in loading phase. Moreover, it is assumed that yielding occurs in the region of 1oror0c with r0c orc during unloading and the material follows an isotropic hardening rule. In this case, reverse yielding is controlled by ( 2S; rprc ; 0 S0 ¼ (11) ðS þ 1Þ; rXrc ; where S 00 is the dimensionless initial yielding stress in the reverse yielding, and also S is given by [9] " 3 # 2mð1 vÞ 1m rc þ S¼ . (12) 2mð1 vÞ þ 1 m 2mð1 vÞ r As it was mentioned before, the stress at the initial step of unloading phase is the same as the stress at the end of loading phase for each material point. As it is seen from Fig. 2, the unloading strain–stress curve is then constructed from the end point of loading phase with an elastic range with its limit at a stress difference of S00 and a hardening part with the same slope as the loading phase. By implementing the yield criterion for linear work hardening material, it follows from Fig. 2 that 1m 0 ðS þ S 00 Þ. (13) m Now, supposing that the plastic zone extends to r ¼ r0c then, from Eq. (15) and making use of the equilibrium relation on unloading, it follows that 8 3 rc > 4ð1mÞ Ln r 2mð1vÞ 0 1m 4 3 > P S þ r ; rpr0c ; > Z r 0P c r 1mþ2mð1vÞ 1mþ2mð1vÞ 3 r < 2m 2r dr ¼ 4 rc 3 > r 4ð1mÞ Ln r0c 2mð1vÞ 0 1m 3 1 > > P S þ r ; rXr0c ; 0 : 2m c r;c 1mþ2mð1vÞ 3 1mþ2mð1vÞ r0 0 20p r ¼ 2p ¼
c
(14) S 0r;c0
S0r
where is the value of at r ¼ substituting into last of Eqs. (10), gives
r0c .
0
And, since S ¼ 2S when r ¼
C 01 ¼ 43r03c S ;r0c , where S ;r0c is the value of S at r ¼ Eqs. (10) gives
r0c ,
then by (15)
r0c .
Substituting the Eqs. (14) and (15) into last of
" # 3 4ð1 mÞ2 4 ð1 mÞðmÞð1 vÞ r c P¼ ln r0c r3c 3 1 m þ 2mð1 vÞ2 r0c ½1 m þ 2mð1 vÞ2 4 3 1 b3 1m 1 1 3 þ r0c S;r0c 3 1 m þ 2mð1 vÞ r0c b3
ð16Þ
which relates the pressure P to the secondary plastic zone radius r0c during unloading. The minimum value of the applied pressure (P) which causes the vessel to yield during unloading phase is defined by P*. To calculate P*, it is enough to substitute
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r0c ¼ 1 and S0r;c0 ¼ P in Eq. (16). Therefore, by performing the appropriate some mathematical manipulations we obtain 4 b3 1 2mð1 vÞ 1m 3 þ rc . P ¼ (17) 3 b3 1 m þ 2mð1 vÞ 2mð1 vÞ By considering the above relation, S 0r;c0 , which is the critical pressure during unloading acting on a sphere with inner radius c0 and outer radius 00 b00 , can be obtained. This can be done by replacing 00 a00 with c0 in Eq. (17) " # 4 b3c0 1 2mð1 vÞ rc 3 1m 0 S r;c0 ¼ þ . (18) 3 b3c0 1 m þ 2mð1 vÞ rc0 2mð1 vÞ For the zone ðr0c orobÞ we calculate the elastic stress distribution of a sphere with inner radius r0c and outer radius b, subjected to an internal pressure S 0r;c0 . Therefore, the stress distributions in this elastic zone ðr0c orobÞ are S0r ¼
r3 b3 S0 0 , r3 ðb03c 1Þ r;c
S 0y ¼
2r3 þ b3 S0 0 . 2r3 ðb03c 1Þ r;c
ð19Þ
4. Determination of the residual stresses Depending on the geometrical factor b and the amount of removed pressure from the vessel, different deformation modes are produced on the sphere. If P4Pn then during unloading, yielding will occur otherwise, the unloading is entirely elastic. In order to analyze the effect of these changes, at first the value of the two aforementioned pressures, i.e. P and P*, have to be equated. In order to get to this state, the value of P from Eq. (8) is set equal to the value of P*, obtained from Eq. (17). The result is m¼
3b2 lnðrc Þ þ b3 þ r3c 2 . r3c b3 ð2n 2Þ þ r3c ð3 2nÞ þ b3 ð1 3 lnðrc ÞÞ 2
(20)
It can be proved that the maximum value for m equals to 0.088 if n ¼ 0:3. Furthermore, for bo1:7 the value of m becomes less then zero. Therefore, it can be said that for ðm40:088 or bo1:7Þ, the deformation mode during unloading is entirely elastic. Moreover, for ðmo0:088 or b41:7Þ, no specific judgment can be made on the mode of deformation, unless the values of P and P* are known. In general, if PoP*, unloading is entirely elastic and the stresses in this phase are given as S0r ¼
r3 b3 P, r3 ðb3 1Þ
S 0y ¼
2r3 þ b3 P. 2r3 ðb3 1Þ
ð21Þ
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On the other hand, if P4Pn , then during unloading the plastic zone extends to r ¼ r0c and the stress components in the reverse yielding zone ð1oror0c Þ are determined from Eq. (10) and stress distributions in elastic zone ðr0c orobÞ are determined from Eq. (19). Finally, the residual stresses are found by superposing the stress components obtained during loading and unloading phases, i.e. 0 Sre r ¼ Sr þ Sr , 0 Sre y ¼ Sy þ Sy .
ð22Þ
The superscript 00 re00 refers to the residual stress component.
5. Equivalent stress and hoop stress optimization Now, suppose that the dimensions of the sphere and also the working pressure are known. The optimum autofrettage pressure is an internal pressure which is applied to the sphere before it is being put into operation. The main task of this pressure is to initiate yielding in the inner surface of the sphere and beside; it optimizes the distribution of the hoop stress, or equivalent stress, throughout the wall thickness.
5.1. Method of hoop stress optimization The analysis begins by assuming a radius for the elasto-plastic boundary for which the residual stress and moreover, the total hoop stress (summation of residual hoop stress and working hoop stress) are calculated. This operation is repeated several times for different radii. The optimal case is the one in which the maximum hoop stress in that case is the lowest amongst other maximums. Fig. 3 illustrates the variation of the hoop stress distribution for different initial elasto-plastic radius rc for m ¼ 0:02; pw ¼ 1:3; b ¼ 2:2. In this figure, the points with negative maxima represent the position of r0c during unloading and points with positive maxima represent the position of rc during loading. By close examination of these curves, it can be noted that an enveloped curve can be passed through all points having maximum value in each curve. Furthermore, it can be seen that this envelope curve has a minimum at a certain point. Therefore, in order to obtain the position of this optimum condition, it is enough to impose the minimization rule on this curve, i.e., to take the derivative of the curve functions with respect to rc and set the result to zero. The mathematical form of hoop stress at this point is equal to the combination of hoop stress expressions due to (i) primary loading, (ii) unloading, (iii) working condition pressure, i.e.: Sy ¼ S re y;rc þ
2r3c þ b3 pw 2r3c ðb3 1Þ
(23)
in which the Sre y;rc is the summation of hoop stresses due to primary loading (case i) and unloading (case ii) at r ¼ rc . First, we assume that the unloading becomes entirely elastic (m40.088 or bo1.7), then an explicit form of Eq. (23) is obtained. In this case by setting the derivative of Eq. (23)
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Fig. 3. The hoop stress distribution for different initial elastic–plastic boundaries m ¼ 0:02, pw ¼ 1:3, b ¼ 2:2.
with respect to rc equal to zero, one would get 27b3 ¼
ð36m þ 36 36r3c;opt þ 36mr3c;opt Þð2m lnðrc;opt Þ pw þ 2 lnðrc;opt Þ þ 2pw mn mpw Þ2 ðpw þ 2pw mn þ 2 lnðrc;opt Þ 2m lnðrc;opt Þ mpw Þ3
r3c;opt .
ð24Þ Then, rc;opt is obtained from Eq. (24) and after it is substituted in Eq. (8), the best autofrettage pressure is determined. When the condition of (mo0.088 or b41.7) is not prevailing, we are not sure that at the point rc;opt the vessel behaves elastically. In this case we again assume an elastic behavior at this point, then we obtain the value of Popt at rc;opt from Eq. (8) and check the condition of reverse yielding. If PoP*opt then our assumption is correct otherwise, we should determining Popt, utilizing the mechanism of pre-stressing pressure which was explained before i.e. solving Eq. (23) numerically. From the data shown in Fig. 3, the optimal values of rc;opt ¼ 1:47; Popt ¼ 1:25 and Sopt ¼ 0:62 can be determined. 5.2. Optimizing the distribution of equivalent stress Here in order to determine an optimized distribution for the equivalent stress we follow the same approach as in the previous section. By carrying out the corresponding calculation, the equivalent stress distribution for different initial elasto-plastic radius, rc , is obtained and the results for m ¼ 0:1, Pw ¼ 0:7 and b ¼ 2 are shown in Fig. 4. Again, by close examination of these curves, it can be noted that an envelope curve can be passed
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through all the points having a maximum value on each curve. Furthermore, it can be seen that this envelope curve has a minimum at a certain point. It can be determined from Fig. 3 that in this case the optimal values are ropt ¼ 1:5; Popt ¼ 1:3 and S opt ¼ 0:7. The position of this minimum point of the envelope curve rc;opt is found by applying the minimization rule with respect to rc on the following equation: 3 3pw b 0 Se;total ¼ S e þ S e þ 3 . (25) 3 2r b 1 c For the case of entirely elastic unloading, the position of rc;opt can be obtained as rc;opt ¼ eðpw ðmð2n1Þ1=2ðm1ÞÞÞ ,
(26)
which relates the optimum elasto-plastic radius rc;opt to pw . It is observed from Eq. (26) that there exists a minimum optimum point on the envelope curve if pw oð2ð1 mÞ=mð2n 1Þ 1Þ ln b, otherwise, i.e. for pw 4ð2ð1 mÞ=mð2n 1Þ 1Þ ln b this curve follows a descending trend, hence, the optimum point approaches to the end of curve or at the outer radius for which the P** is the best autofrettage pressure that causes the sphere to yield completely on loading. 6. Thickness optimization In Figs. 3 and 4 it can be seen that the value of the equivalent stress in radius rc;opt is less than the value of the initial yield stress. This means that in this case the vessel is capable to
Fig. 4. The equivalent stress distribution for different initial boundaries elasto–plastic m ¼ 0:1; pw ¼ 0:7; b ¼ 2.
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withstand higher level of working pressure pw. On the other hand, if the value of pw is prescribed, then the optimum thickness and initial strengthening pressure also can be obtained simultaneously. The design criterion is to consume less material in the production of the vessel. Where, by introducing two new design criteria i.e. (A) optimizing the hoop stress, and (B) assume a suitable percent for the yield zone in the wall thickness, the minimum thickness of the vessel can be calculated. Based on criterion A we have q ðS y;max Þ ¼ 0. qrc
(27)
Note that Eq. (27) is a function of b and rc;opt and hence it cannot be solved directly. Thus, in order to come up with an optimum case, another design condition is needed. When the sphere in the first step undergoes of an autofrettage process and then in the next step is loaded by a predefined working pressure, again another yielding process initiates. It is clear that this secondary yielding would begin again from the inner radius. Therefore, according to our yielding criteria, we have 3 3 S re e;1 þ 3pw b =2ðb 1Þ ¼ S Y;1
(28)
in which, S Y;1 is the value of the new yielding stress at inner radius. If the unloading is elastic, then, S Y;1 ¼ S e;1 , otherwise, SY;1 ¼ S e;1 S0e;1 . It is known that in the autofrettage process, the inner layer of the sphere is in the most critical yielding condition and furthermore, we cannot increase the working pressure indefinitely. Therefore, to come up with a practical value for its value, still we need to introduce another constraint for the failure mechanism in the inner layer of the sphere. We choose this additional constraint to be S Y;1 p1:7 during autofrettage process. Now we are ready to obtain the values of rc;opt and b by using following equations: 9 8 q > > > > ðS Þ ¼ 0 condition 1; > > = < qrc y;max (29) 3 3 re S þ 3pw b =2ðb 1Þ ¼ S Y;1 condition 2; > > > > > > e;1 ; : SY;1 p1:7 condition 3: It is notable to verify that in the case of elastic–perfectly plastic behavior (m ¼ 0), from Eq. (28) one can conclude that 3pw b3 =2ðb3 1Þp2.
(30)
The inequality in Eq. (30) is valid only for pw p1:33. Therefore, for pw p1:33 the system of Eq. (29) for m ¼ 0 has a unique solution. Fig. 5 illustrates the variations of the dimensionless radius b and the dimensionless optimum autofrettage pressure, Popt, versus the working pressure pw. Based on above-mentioned criterion (B), we begin by choosing an appropriate value for the percent of yielded zone in the autofrettage process. Again, it has to be noted that it is very important to select a justified value for this elasto-plastic radius. To elaborate further on this point it is easy to see that if the percent value for this yielded zone is taken to be 100%, consequently, the elasto-plastic radius will shift towards the external radius. This means that the entire wall thickness undergoes plastic deformation in the first cycle of loading and therefore, this would not be a suitable design principle. On the other hand, if
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Fig. 5. Variation of dimensionless design parameters vs. pw .
Fig. 6. Variation of dimensionless design parameters vs. pw (assume, 75% thickness yields).
the percent value for this yielded zone is taken to be 0%, the elasto-plastic radius will shift towards the internal radius; again this would not be suitable design criteria, because in this case the wall thickness becomes very large. In this study, we choose the percent value for the yielded zone to be 75%. Therefore, it is clear that rc ¼ 1 þ 0:75ðb 1Þ.
(31)
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To solve Eq. (31), we have to come up with another criterion; hence, we recall Eq. (28) and rephrase it as following: 3pw b3 =2ðb3 1Þ ¼ S Y;1 S re e;1 .
(32)
Eqs. (31), (32) can be solved simultaneously for different working pressures. Fig. 6 illustrates the variations of the dimensionless radius b and the dimensionless autofrettage pressure Popt versus different working pressure for different values of ‘‘m’’. 7. Conclusion In this paper based on the theory of elasto-plasticity for optimal design of thick-walled spherical vessels, the governing equations were derived and solved. Based on the obtained results, it is concluded that (1) Autofrettage considerably reduces stress level in the wall of a sphere. (2) The method presented in this paper is very simple, applicable, accurate and reasonable compared to other existing methods. (3) The autofrettage pressure must be always greater than the working pressure. (4) Refered to Figs. 5 and 6 it can be said that for dimensionless working pressure, (pw) up to 0.6, the effect of variation of m ðE t =EÞ in the wall thickness optimization is immaterial. (5) By considering elastic–perfectly plastic behavior, the design problem has a solution, only for pw p1:33. It is apparent that the wall thickness calculated from the current method is less than the thickness usually obtained by employing the criterion of initiation of yielding in the inner radius. Moreover, by reviewing the results depicted in Figs. 5 and 6, it can be seen that the difference between the design curves are negligible for a low level of working pressure. In the case of elastic–perfectly plastic (m ¼ 0) material, when pw ! 1:33, the geometric dimensions of the vessel approaches infinity. Furthermore, from the same figures, it is noted that, by increasing the value of m, the allowed design pressure and the thickness will decrease, and, moreover, for a fixed value of geometric dimension when m is increased, the allowed working pressure is increased. Finally, under optimum condition when unloading is elastic, Popt ¼ pw irrespective of the value of m, and this condition is valid up to the point where the Popt start nonlinear behavior (slope is 1). Reference [1] G.H. Majzoobi, G.H. Farrahi, A.H. Mahmoudi, A finite element simulation and an experimental study of autofrettage for strain hardened thick-walled cylinder, Mater. Sci. Eng. (2003) 326–331. [2] W.R.D. Manning, Trans. ASME, J. Pressure Vessel Technol. 100 (1978) 374–381. [3] P.C.T. Chen, Stress and deformation analysis of autofrettaged high pressure vessels, ASME Special Publication, vol. 110, PVP. ASME United Engineering Center, New York, 1986, pp. 61–67. [4] P.C.T. Chen, The Bauschinger and hardening effect on residual stresses in an autofrettaged thick-walled cylinder, J. Pressure Vessel Technol. 108 (1986) 108–112. [5] G.J. Franklin, J.L.M. Morrison, Autofrettage of cylinder: prediction of pressure/expansion curves and calculation of residual stresses, Proc. Inst. Mech. Eng. 174 (1960) 947–974.
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[6] M.H. Kargarnovin, A. Rezai Zarei, Thickness optimization of thick-walled cylinderical vessels using prestressing pressure, The Third World Conference on Integrated Design and Process Technology, Berlin, Germany, July 6–9, 1998, pp 139–146. [7] J. Zhu, R. Yang, Autofrettage of thick cylinder, Int. J. Vessel Piping 75 (1998) 443–446. [8] La. Paolo, Li. Paolo, Different solutions for stress and strain fields in autofrettaged thick-walled cylinders, Int. J. Vessel Piping 71 (1997) 231–238. [9] A. Mendelson, Plasticity: Theory and Application, Macmillan, New York, 1968.