Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 1, pp. 129-137, Warsaw 2014 OPTIMAL DESIGN OF COMPRESSED COLUMNS WITH CORROSION TAKEN INTO ACCOUNT Mark M. Fridman Metallurgical Institute, Krivoy Rog, Ukraina e-mail: mark17@i.ua This study is devoted to the optimal design of compressed columns with a circular cross section under axial compressive forces and exposed to a corrosive environment. The initial volume of the structure is taken as an optimality parameter. The main constraint is the buckling of a loaded column at the final time of its operation.Gutman-Zaynullin’s exponen- tial stress corrosionmodel is adopted for the analysis. Analytical and numerical results are derived for optimal variation of the cross-sectional area of the bar along its axis. Key words: corrosion, optimization, stability 1. Introduction Elements of many engineering structures are exposed not only to loads and temperatures, but also to various corrosive environments. These factors often appear in highly unfavorable combi- nations, reducing the load carrying capacity and service life of the structure.Neglecting corrosive environments in the analysis may lead to premature and often emergent halting of the system operation, causing great damage to both the environment and economy. Based on experimental results, we conclude that the corrosive process of a structure in an aggressive environment is determined by the temperature, stress-strain state, nature of the aggressive environment and time span during with the structure resides in the corrosive envi- ronment. In certain circumstances, the governing parameters may also include fluid pressure, speed of the fluid or of aggressive gas, characteristic location of elements in the structure and other factors. Different researchers offered various models to describe the same process. The influence of stress on corrosion speed, known as the corrosion rate, was apparently first considered by Dolinsky (1967), who dealt with the strength of thin-walled pipes subjected to a continuous corrosion rate as a linear function of stress. An exponential dependence of corrosion in the stress of the structure was proposed by Gutman and Zaynullin (1984). Papers by Potchman and Fridman (1995-1997), Fridman and Życzkowski (2001), Fridman (2002) utilized Dolinsky’smodel for the optimization study under corrosion. This study extends the above papers to the exponential stress corrosion model by Gutman and Zaynullin to study the stability optimization in the corrosive environment. 2. Problem statement Weconsider theproblemofoptimaldesign employing thecriterionofminimumweight of columns with a circular cross section loaded with an axial compressive F force (Fig. 1) and subjected to uniform corrosion. We adopt the corrosion model introduced by Gutman and Zaynullin, which reads dr dt =−αexp(ηt)exp(γσ) (2.1) 130 MarkM. Fridman Fig. 1. Bar under axial compressive load P where α, η, γ are constant coefficients, σ(t)=F/π[r(t)]2 is the stress, r0 and r(t) is the initial and current radius of the circular column, respectively. It is assumed that the rack is subject to uniform corrosion on the perimeter of its cross-section. Separation of variables in Eq. (2.1) leads, with the notation a= γF/π, to ∫ exp ( − a r2 ) dr=−α ∫ exp(ηt) dt (2.2) Taking into account the familiar series expansion ex = ∞ ∑ 0 xn n! (2.3) we get from Eq. (2.2) ∫ ( 1− a r2 + ∞ ∑ 2 (−a)n n! r−2n ) dr= r+ a r + ∞ ∑ n=2 (−a)n n! r1−2n 1−2n =− α η exp(ηt)+C (2.4) Invoking the initial conditions at t=0, we obtain C = r0+ a r2 + ∞ ∑ n=2 (−a)n n! r1−2n0 1−2n + α η (2.5) When t = T , with T denoting the durability, while rT – radius of the section at the time instant T , we have η α ( r0−rT +a ( 1 r0 − 1 rT ) + ∞ ∑ n=2 (−a)n n! r1−2n0 −r 1−2n T 1−2n ) =exp(ηT)−1 (2.6) 3. Lagrange function and optimality conditions Theminimumweight of the column is achieved by the optimal distribution of the initial radius of the cross section r0 along the columns length. In terms of the volume V =2 l/2 ∫ 0 Adx=2π l/2 ∫ 0 r20(x) dx (3.1) The conditions of optimality are expressed as the Euler-Lagrange equations δyf = fy− d dx fy′ + d2 dx2 fy′′ =0 δr0f = df dr0 =0 (3.2) Optimal design of compressed columns with corrosion taken into account 131 where f is the so-called Lagrange function with an additional condition in form of equation (2.6) as follows f =πr20+λ(x) [ − η α ( r0−rT+a ( 1 r0 − 1 rT ) + ∞ ∑ n=2 (−a)n n! r1−2n0 −r 1−2n T 1−2n ) +exp(ηT)−1 ] (3.3) The governing equation for buckling of the column reads EIy′′+Fy=0 (3.4) Assuming that the loss of stability occurs at time t > T (and considering that I = πr4/4) from the equation of buckling, rT is defined as rT = ( − 4Fy πEy′′ ) 1 4 (3.5) In these circumstances, Eq. (3.3) becomes f =πr20 +λ(x) { − η α [ r0− ( − 4Fy πEy′′ ) 1 4 + a r2 −a (−4Fy πEy′′ )− 1 4 + ∞ ∑ n=2 (−a)n n! 1 1−2n [ r1−2n0 − (−4Fy πEy′′ ) 1−2n 4 ] ] +exp(ηT)−1 } (3.6) Evaluation of the derivatives yields fy =λ(x) [ − η α ( (−4Fy πEy′′ )− 3 4 F πEy′′ −a (−4Fy πEy′′ )− 5 4 F πEy′′ + ∞ ∑ n=2 (−a)n n! ( − 4Fy πEy′′ )− 3 4 − n 2 F πEy′′ )] fy′ =0 fy′′ =λ(x) [ − η α ( (−4Fy πEy′′ )− 3 4 Fy πE(y′′)2 −a (−4Fy πEy′′ )− 5 4 Fy πE(y′′)2 + ∞ ∑ n=2 (−a)n n! ( − 4Fy πEy′′ )− 3 4 − n 2 Fy πE(y′′)2 )] (3.7) Substituting these expressions into Eq. (3.2)1 results in δyf =λ(x) [( (−4Fy πEy′′ )− 3 4 F πEy′′ −a (−4Fy πEy′′ )− 5 4 F πEy′′ + ∞ ∑ n=2 (−a)n n! ( − 4Fy πEy′′ )− 3 4 − n 2 F πEy′′ )] + { λ(x) [ y y′′ ( (−4Fy πEy′′ )− 3 4 F πEy′′ −a (−4Fy πEy′′ )− 5 4 F πEy′′ + ∞ ∑ n=2 (−a)n n! ( − 4Fy πEy′′ )− 3 4 − n 2 F πEy′′ )]} =0 (3.8) 4. Chentsov’s method Hereinafter, we utilize themethod proposed byChentsov (1936), Rzhanitsyn (1955) to solveEq. (3.8). We introduce the following notation k= λy y′′ [ (−4Fy πEy′′ )− 3 4 F πEy′′ −a (−4Fy πEy′′ )− 5 4 F πEy′′ + ∞ ∑ n=2 (−a)n n! (−4Fy πEy′′ )− 3 4 − n 2 F πEy′′ ] (4.1) 132 MarkM. Fridman Multiplying Eq. (3.8) by y, we represent it in the form ky′′−yk′′ =0 (4.2) One can check by direct differentiation that the integral of the latter equation is ky′−yk′ =C (4.3) where C is an arbitrary constant. For further integration, one should take into account that C = 0 for the adopted boundary conditions. Indeed, assuming the buckling mode as being symmetric with respect to themiddle cross-section of the column, we establish that y, y′′, y/y′′ and k are even functions of x. Hence, the derivatives of y as well as k would be odd functions of x and thus vanish at x=0. Therefore, letting in Eq. (4.3) x=0, we get C =0. Thus, Eq. (4.3) becomes ky′−yk′ =0 (4.4) Its integral is evaluated by separation of variables k=C1y (4.5) where C1 is a new arbitrary constant. Returning to the original notation in Eq.(4.1), we find λy y′′ [ (−4Fy πEy′′ )− 3 4 F πEy′′ −a (−4Fy πEy′′ )− 5 4 F πEy′′ + ∞ ∑ n=2 (−a)n n! (−4Fy πEy′′ )− 3 4 − n 2 F πEy′′ ] =C1y (4.6) Additionally, δr0f =2πr0+λ [ − η α ( 1− a r0 2 + ∞ ∑ n=2 (−a)n n! r−2n0 )] =0 (4.7) fromwhich we determine the Lagrange multiplier λ= 2πr0α η ( 1− a r0 + ∞ ∑ n=2 (−a)n n! r−2n0 ) (4.8) Substituting λ into (4.6), we obtain 2r0αF ηE [ (−4Fy πEy′′ )− 3 4 −a (−4Fy πEy′′ )− 5 4 + ∞ ∑ n=2 (−a)n n! (−4Fy πEy′′ )− 3 4 − n 2 ] = ( 1− a r20 + ∞ ∑ 2 (−a)n n! r−2n0 ) C1(y ′′)2 (4.9) Or, given that rT (Eq. (3.5)), we finally have 2r0αF ηE ( 1 r3T − a r5T + ∞ ∑ n=2 (−a)n n! 1 r3+2nT ) = ( 1− a r20 + ∞ ∑ 2 (−a)n n! r−2n0 ) C1(y ′′)2 (4.10) Optimal design of compressed columns with corrosion taken into account 133 5. Particular cases First let us consider case 1 when in the expansion of exponent in Eq. (2.3) the first two terms are retained. In this case, we get a model of corrosion similar to the model by Dolinsky. The equations (2.6) and (4.10) become, respectively η α [ r0−rT +a ( 1 rT − 1 r0 )] =exp(ηT)−1 2r0αF ηE ( 1 r3T − a r5T ) = ( 1− a r20 ) C1(y ′′)2 (5.1) We introduce the following non-dimensional quantities A=exp(ηT)−1 B= γF π η2 α2 R0 = r0η α RT = rTη α = η α ( − 4Fy πEy′′ ) 1 4 = η α ( − 4F πE l2 4 V V ′′ ) 1 4 =C ( − V V ′′ ) 1 4 (5.2) In view of y y′′ = l2 4 V V ′′ (5.3) we get C = η α (Fl2 πE ) 1 4 V = √ αEC1 ηF 4 l2 y V ′′ξξ = √ αEC1 ηF y′′ ξ= 2x l (5.4) In this case, Eqs. (5.1) become, respectively R0−RT +B ( 1 R0 − 1 RT ) =A 2R30 R5T R2T −B R20−B =V ′′ 2 (5.5) From Eq. (5.5)1, we find R0 = D+ √ D2−4R2TB 2RT (5.6) where D=R2T +ART +B. Substituting the expression into Eq. (5.6) with (5.2), we arrive at f1(A,B,C,V,V ′′)= 0 (5.7) where f1(A,B,C,V,V ′′) depend on A,B,C, V , V ′′. Now we turn to numerical implementation of the procedure. Taking into account symmetry of the buckling mode with respect to the y axis, we divide half-length of the column into elementary parts of the length ∆ξ as shown in Fig. 2. To determine the shape of the column in this case, in addition to Eq. (5.7), we use a re- lationship between the buckling mode and its second derivative in the central finite difference setting Vi−1 =∆ξ 2V ′′i +2Vi−Vi+1 (5.8) Starting from the arbitrary negative value of V11 and V10 = V ′′10 = 0 (i.e. V9 = −V11), we determine the value V ′′9 from Eq. (5.7) by using a random search algorithm (Gurvich et al., 134 MarkM. Fridman Fig. 2. Discretization of the half-column 1979). Then, from Eq. (5.8) we find V8. The process is repeated until values V0 and V ′′0 are determined. The solution is validated by evaluating V ′0, since the latter must vanish due to V1 ≈V0. After the values of Vi and V ′′i are found, values of RTi and R0i can be directly evaluated. Let us consider particular case 2 when n= 2. In this case, equations (2.6) and (4.10) lead to the following η 2α [ r0−rT +a ( 1 r0 − 1 rT ) − a2 6 ( 1 r30 − 1 r3T )] =exp(ηT)−1 2r0αF ηE ( 1 r3T − 1 r5T + a2 2r7T ) = ( 1− a r20 + a2 2r40 ) C1(y ′′)2 (5.9) or the non-dimensional quantities R0−RT +B ( 1 R0 − 1 RT ) − B2 6 ( 1 R30 − 1 R3T ) =A 2R50 R7T R4T −BR 2 T + B2 2 R40−BR 2 T + B2 2 =V ′′ 2 (5.10) We find out R0 in Eq. (5.10)1. We obtain the following equation R40+ ( B2 6R3T − B RT −RT −A ) R30+BR 2 0− B2 6 =0 (5.11) To solve, it we apply the method by Ferrari. Then R0 = y− b (5.12) where b= 1 4 ( B2 6R3T − B RT −RT −A ) y= √ t 2 − √ −p− t 2 − q √ 2t p=3c−3b2 q=4b3−6cb c= B 6 t= z1− r1 3 z1= 3 √ − q1 2 + √ D∗+ 3 √ − q1 2 − √ D∗ D∗ = (p1 3 )3 + (q1 2 )3 p1 = 3S1−r21 3 q1 = 2r31 27 − r1s1 3 + t1 s1 = p2−r r=6cb2+e−3b4 e=− B2 6 t1 =− q2 2 Optimal design of compressed columns with corrosion taken into account 135 Defining R0 and substituting into Eq. (5.10)2 with (5.2), we arrive at f2(A,B,C,V,V ′′)= 0 (5.13) where f2(A,B,C,V,V ′′) depend on on A,B,C, V , V ′′. A more accurate analytical solution for n=3 is not possible because of obvious reasons. 6. Numerical results The optimum shape of the initial radius of the cross section bar R0(ξ) and its form at t≈ T : RT(ξ) was derived for alloy D16T with the following rates of corrosion models (1): α = 4.8 · 10−4m/year, gamma=0.588 ·10−5m2/kN, η=0.0911/year, E =7 ·1010Pa. Theparameters were fixed at h = 10−2m, F = 10kN, T = 10years, l = 1m, C1 = 2m3. In this case, the non-dimensional quantities are A=1.484, B =0.6727, C =2.768. In all cases, the optimized shapes turn out to be almost identical (Fig. 3). Fig. 3. Optimum initial shape and its final form for B=0.6727 Similar results were obtained with the F =14.86kN, T =7.62years, l=1m. In this case, the associated non-dimensional quantities are A = 1, B = 1, C = 3.06. The optimum shape of the initial radius of the cross section bar R0(ξ) and its form at t≈ T : RT(ξ) are shown in Fig. 4a. Fig. 4. Optimum initial and final shapes for (a) B=1 and (b) B=5 Figure 4b shows the numerical results obtained for the following data: F = 14.86kN, T = 7.62years, l = 1m, A = 1, C = 3.057, B = 5. The new value of B with remaining 136 MarkM. Fridman parameters kept constant is associated with a dramatic increase in the corrosion rate, i.e. in the parameter γ = (0.588 · 10−5) · 5 = 2.94 · 10−5m2/kN. The optimum shape of the initial radius of the cross section bar R0(ξ) and its form when t ≈ T , namely RT(ξ), are shown in Fig. 4b. 7. Conclusion In this paper, the general and particular optimization solutions are obtained for the Gutman- Zaynullin corrosion model. The results of numerical evaluation shown in Fig. 3 and Fig. 4a show that the increase in the number of terms in the expansion in Eq. (2.3) does not produce significant changes for the chosen sets of parameters. In all cases, the optimal shape of the initial radius of the cross section bar R0(ξ) and of its counterpart for t≈T , namely RT(ξ), are almost identical. This closeness in the results may have the following explanation. By substituting the values obtained for R0,RT in Eq. (5.10)1, one observes that the terms in front of B2 tend to zero. the increasing of the value of factor B (with other values fixed at α = 4.8 · 10−4m/year, γ = 0.588 · 10−5m2/kN, η = 0.0911/year, E = 7 · 1010Pa) should lead also to an increase in the force F, which – in turn – leads to a sharp increase in R0, RT . Since these terms appears in the denominators in (5.10)1 then, respectively, the contribution of terms containing B2 is decreased. An increased slight difference (between 3.7-21%) in all cases occurs when the corrosion rate depends strongly on B, as shown in Fig. 4b. Summing up the above, the use of the corrosion model by Dolinsky allows one to get a sufficiently accurate optimization problem for bars under axial compression in a corrosive envi- ronment, when the corrosion rate does not depend strongly on B. However, when the corrosion rate strongly depends on B, one is recommended to utilize themodel byGutman andZaynullin. The results of his study include, as a particular case, the analysis associated with Dolinsky’s corrosion model. References 1. Chentsov N.G., 1936, Columns of minimal weight (in Russian), Trudy Centr. Aerogidrodinam. Inst., 265, 20-26 2. Dolinsky V.M., 1967, Analysis of loaded tubes, subjected to corrosion,Khim. i Neft. Mashino- stroyenye, 2, 21-30 3. Fridman M.M., 2002, Conceptual approaches for optimal design of structures operating under extreme conditions (inUkrainian), Scientific and Technical. Sbor. “Soprotivlenie Materialov i Teo- riya Soorugeniy”, Kiev, KNUBA, 70, 158-175 4. Fridman M.M., Życzkowski M., 2001, Structural optimization of elastic columns under stress corrosion conditions, Structural and Multidisciplinary Optimization, 21, 3, 218-228 5. Gurvich I.B., Zaharchenko B.G., Pochtman Y.M., 1979, Randominized algorithm for solu- tion of problems of nonlinear programming (in Russian), Izv. Ac. Sci. USSR, Engineering Cyber- netics, 5, 15-17 6. Gutman E.M., Zaynullin R.S., 1984, Kinetics of mechanochemical collapse and durability of structural elements under tension during elastic-plastic deformation (in Russian), Fiz.-Khim. Me- khanika Materialov, 2, 14-17 7. Pochtman Y.M., Fridman M.M., 1995,Optimal design of pressure vessels including the effects of environment,Computer Assisted Mechanics and Engineering Sciences, 2, 1, 19-23 Optimal design of compressed columns with corrosion taken into account 137 8. Pochtman Y.M., Fridman M.M., 1996, Optimization of cylindrical shells subjected to pitting corrosion,Computer Assisted Mechanics and Engineering Sciences, 4, 3, 1-5 9. Pochtman Y.M., Fridman M.M., 1997,Methods of Reliability Analysis and Optimal Design of StructuresunderExtremalConditions (inRussian),Nauka iObrazovanye,Dnepropetrovsk,Ukraine 10. RzhanitsynA.R., 1955,StabilityofEquilibriumofElasticSystems (inRussian),Gos. Izd.Tekhn.- Teor. Literature Manuscript received May 6, 2013; accepted for print July 12, 2013