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On the Method of Calculation of Buckling and Post-Buckling Behavior of Laminated Shells with Small Arbitrary Imperfections

On the Method of Calculation of Buckling and Post-Buckling Behavior of Laminated Shells with... Acta Sci. Pol. Architectura 20 (2) 2021, 17–25 content.sciendo.com/aspa ISSN 1644-0633 eISSN 2544-1760 DOI: 10.22630/ASPA.2021.20.2.11 ORIGINAL P APER Received: 29.04.2021 Accepted: 26.05.2021 ON THE METHOD OF CALCULATION OF BUCKLING AND POST-BUCKLING BEHAVIOR OF LAMINATED SHELLS WITH SMALL ARBITRARY IMPERFECTIONS 1 2 1 Mykola Semenyuk , Volodymyr Trach , Natalia Zhukova S.P. Timoshenko Institute of Mechanics NAS Ukraine, Kyiv, Ukraine Institute of Civil Engineering, Warsaw University of Life Sciences – SGGW, Warsaw, Poland; National University of Water Management and Nature Resources Use, Rivne, Ukraine ABSTRACT In the present paper the variant of computational methods of stability and initial post-buckling behavior of isotropic shells was generalized with respect to laminated composite shells. Using methods of the asymp- totic analysis of the Timoshenko–Mindlin theory, the relationships for calculation of shells with the small geometrical imperfections of the different shapes have been produced. On the basis of obtained equations, the technique of calculation of a non-linear pre-buckling state, limiting loads and bifurcation, and also initial post-buckling behavior of laminated cylindrical shells at an axial compression and external pressure have been worked out. The results of calculation for the shells made of glass fiber-reinforced plastic and carbon fiber-reinforced plastic with multimodal imperfections and dimple imperfections have been presen- ted. The features of transformation of the interacting modes of imperfections have been researched. Key words: buckling, post-buckling, imperfection, laminated shell, mode interaction, Cauchy problem The stress vector σ for a shell with imperfections MODEL EQUATIONS ε uu , () is related to the deformation vector by the Let a shell be loaded by a system of forces proportional stiffness matrix for a laminated shell H to a certain parameter λ. According to a Timoshenko– –Mindlin theory, the generalized displacements of the σε =Hu(,u) (2) shell are described by the vector U = (u, ν, w, θ, ψ). Analogous vector notation for the generalized stresses The principle of virtual work takes the form and strains σ = (T , T , S, T , T , M , M , M , M ), 11 22 23 13 11 22 12 21 ′′ ε = (ε , ε , ε , ε , ε , k , k , τ , τ ) is introduced σε (,uu)δu−Δ λ (u,u)δu =0 (3) 11 22 12 23 13 11 22 1 2 here. If the field of initial imperfections is character- ized by the vector u , then the following expressions Though modified, Eqs. (1)–(3) enable solving a va- for the deformations and generalized displacement can riety of nonlinear problems on deformation of shells be derived with initial geometric imperfections. The asymptotic method (Byskov & Hutchinson, 1997) to the Eqs. ′′  ′′ εε (,uu )=+ (u)ε uu; Δ= (,uu) Δ(u)+Δ uu (1) (1)–(3) has been applied here. It has been assumed trach-vm@ukr.net © Copyright by Wydawnictwo SGGW Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 ′′ ′′ that the subcritical stress-deformed state of the shells ξλ−− λ σ ε uuδ Δ uuδ+ ii()( 0 i i ) is linear. For λ = 1, assume that the fields of displace- (12) ++ ξξLL () λ ξ ξ ξ +...=0 i j ij i j k ijk ments, deformations and stresses are characterized by the vectors u , ε ,σ . After linearization of equations in 0 0 0 where: the neighborhood of the bifurcation load in aim to de- termine the critical values of λ and the corresponding ′′′ ′′ ′′ L ()λσ=+ ε (0)δu σε u δu+λσ ε uuδ−λΔ uuδ ij ij i j 0 ij ij buckling modes u , one obtains a set of equations ′′ ′′ ′′ ′ Lu=+ σε δu σ εu δu+Hεuu ε (0)δu ijk i jk ij k i jk ′′′ ′′ σε (0)δuu+− λ σ ε δu λΔu δu= 0 (4) ii 0 i i i (13) σε = H (5) ii ′ A variational equation in the fields u , ε , σ can be εε== (0)ui; 1, ..., M (6) ii ij ij ij obtained from Eq. (12), when it has been taken into These M forms are orthogonal account that the variation δu in this case satisfies the orthogonality condition σε ′′uu −Δ ′′uu = 0, i ≠ j (7) 0 ij ij σε ′′uu δ −Δ ′′u δu = 0 (14) The amplitudes ξ of the modes u remain indeter- 0 ii i i minate during solution of the homogeneous problem Neglecting, in Eq. (12), terms of third or higher or- (4)–(6) and can be found only by solving the original der in ξ , the symmetry of the desired functions relative non-linear problem (1)–(7). The displacement vector to the indices i, j results in an equation can be represented as the asymptotic expansion ′′′ ′′ σε (0)δuu+− λσ ε δu λΔu δu= ij 0 ij ij uu=+ λξu +ξξ u (8) 0 ii i j ij (15) =− σε ′′uu δ +σ ε ′′u δu () ij j i Here and further the rule of summing over repeated indices is adopted. The smallness of parameters ξ fol- In contrast to Koiter’s classical approach (Koit- lows from the fact that ξ → 0 as λ → λ . The displace- i i er, 1963), which is based on the Lyapunov–Schmidt ments u are orthogonal to the buckling modes ij method, the method proposed by Byskov and Hutch- ′′ ′′ σε uu −Δ u u = 0; 0 ikl i kl inson (1977) does not use expansion of the coefficients i = 1, ..., M; k =1, ..., M; l = 1, ..., M (9) of L (λ) in Tailor series in the parameter (λ – λ ). As a ij j result, the solution of Eq. (15) depends on the load λ. Substituting (8) into (4), one obtains Koiter (1963) proposed a use of the smallest λ for the λ in Eq. (15). However, the problems under considera- λσ ε′′ (0)δuu −Δ λ (0)δ + tion make it necessary to determine this inaccuracy. ′′′ ′′ ++ ξσ ε (0)δuu λσ ε δu−λΔuδu+ ii[] 0 i i If the homogeneous boundary-value problem (4), (5) and inhomogeneous problem (11), (15) are resolved, ′′ ′′ ′′ ++ ξξ[] σ ε (0)δuu σ ε δu+ λσ εu δu−λΔu δu+ i j ij i j 0 ij ij then, for the amplitudes ξ from equation (12) (where δu = u is set), one can derive the following system of ++ ξξ ξ σ ε ′′uu δ σ ε ′′u δu+Hε ′′uu ε ′(0)δu+...= 0 i j k[] i jk ij k i jk non-linear equations: (10) §·λλ ξ 1−+ξξab +ξ ξ ξ =ξ ;r= 1, ..., M Here, terms of higher order in ξ have been omitted r i j ijr i j k ijkr r ¨¸ λλ ©¹rr and, moreover (16) σε==Hu ,0 ε ε′′+ ε′uu (11) ij ij ij () ij i j where: A B ijr ijr Because λσ ε’(0)δu – λΔ’(0)δu = 0, basing on (4), 0 ; a =− b =− ijr ijr 2 D the expression (10) can be reduced to the form 18 architectura.actapol.net Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 22 x, y – coordinate lines which coincide with a gen- ′′ ′′ ′′ ′′ Au=+ σε u 2, σ εuuD=λ σ εu−Δu ijr r i j i j r r() 0 r r eratrix and directrix of a cylindrical reduced surface, Bu=+ σε ′′u σ ε ′′uu+σ ε ′′uu ijkr i r jk ij k i r i jk T , M – efforts and moments, which are statically ij ij (17) equivalent to actual stresses. Equation (16) can be used to investigate nonlinear ** 1 TS =+Tωω ;T =S+ H+T 12 11 1 21 22 2 deformations of imperfect structures in the subcritical state, for computation of critical (bifurcation or limit- ** TT=+T θθ+S ;T =T +Sθ +T 13 13 11 1 2 23 23 1 22 ing) loads and for computation of the initial post-buck- ling behavior of the shells under discussion. The strains are given by In general, the representation (16) is uniformly valid whether the modes are simultaneous, nearly 11 22 2 2 εε=+ ω +θ ; ε =ε + ω +θ 11 1 1 1 22 2 2 2 () ( ) simultaneous, or well separated. This property of the Byskov–Hutchinson method εω=+ω +θθ 12 1 2 1 2 is used below for development of a technique of cal- culation of stability and post-buckling behavior of εω kk== ; k k+= ; k t+t+ 11 1 22 2 12 1 2 laminated composite cylindrical shells with multi- RR modal imperfections, in particular, by imperfections which are approximated by trigonometric Fourier εθ =+θ ; ε =ψ+θ 13 1 23 2 series. where: THE SOLUTION FOR THE CYLINDRICAL SHELLS ∂∂ uv ∂w ε==;;ωθ= 11 1 ∂∂ xx ∂x The relationships obtained have been used for buck- ling analysis of a multilayer cylindrical composite ∂∂ vw u ∂w v shells with geometrical imperfections. The imperfec- ε =−;;ωθ = = + 22 2 ∂∂ yR y ∂y R tions’ shape is described by trigonometric polynomials (fragmenting of Fourier series). According to the basic ∂∂ ψψ ∂∂ θθ kk== ;; t= ;t= 122 1 2 statements of the Byskov–Hutchinson method (1997) ∂∂ xy ∂x ∂y and Byskov method (2004) relationships will be pre- sented which are necessary for solving with use of the The variation of work of external loads λA has Timoshenko–Mindlin non-linear shell theory (Vanin a different kind depending on character of their distri- & Semenyuk, 1987). The expression of virtual work bution on a frontal or on a lateral surface. principle in this case is In case of a cylindrical shell, one of the most prac- tically interesting is the version of loading by longitu- LR 2 π ** dinal compressive forces – T TT δε++ δε T δω+T δω+ 11 11 1 22 2 12 1 21 2 ³³ 2 π R ** L ++ TT δθ δθ+T δθ+T δψ+ 13 1 23 2 13 23 0 δδ AT =− u dy ³ 11 (18) ψ++ Mkδδ M k+Mδt+ 11 1 22 2 12 1 or uniform external pressure with intensity q ++ Ht() δδt dxdy−δA= 0 LR 2 π where: δθ Aq=−ªº δu−θδv+() 1+ε+εδwdxdy 12 1 2 ³³¬¼ L, R – length and radius of a shell, architectura.actapol.net 19 Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 The initial imperfections are characterized by a deflection w (x, y). It is supposed, that the shell consists of N layers of a fibrous composite material which have a symmetrical structure with regard to a middle surface. The relationships of elasticity in this case may be written as TC=+εεC ; T =Cε +Cε ; S=Cε ; T =Cε ; T =Cε 11 11 11 12 22 22 12 11 22 22 66 12 13 55 13 23 44 23 MD=+k D k ; M =D k +D k ; H=D k 11 11 11 12 22 22 12 11 22 22 66 12 in which the rigidities C and D are given by ij ij NN ii 2i CC== ; D D+zC kl kl () ¦¦ i kl kl kl ii == 11 where z is coordinate of a middle surface of i-th layer. The linearized equations for a cylindrical shell may be expressed as LR 2 π ii i i 2 i i §· TS δε++ δω T δε +S+ H δω+T δθ+ δθ+ 11 2 2 ()1 ¨¸ 11 22 13 ³³ { ©¹ R ii i i 0 i i (19) ++ Hkδδ T ()θ+δψ+Mδk +Mδk +λ T ωδω+θδθ+ 12 2 11 22 i () 1 1 23 11 22 11 1 1 00 ii i i ++TS ωδω θ δθ+ θ δθ+θ δθ dxdy= 0 () 22 ( 2 1) ¾ 22 2 1 2 Variational equation (19) relative to variables of second order in the Timoshenko–Mindlin shell theory is represented as LR 2 π ij ij ij ij 2 ij ij ij ij ij §· TS δε++ δω T δε+S + H δω+T δθ+ δθ+T δθ+ δψ+M δk +M δk + 11 2 2 ()1 ()2 11 22 ¨¸ ³³ { 11 22 13 23 11 22 ©¹ R ij ij ij ij ij ij ij 00 0 ªº ++ Hk δ λ T ωδω+θ δθ+T ωδω+θ δθ +S θ δθ+θ δθ dxdy= 12 i 1 1 2 2 2 1 ¾ () ( ) ( ) 11 1 1 2 2 2 1 2 ¬¼ (20) LR 2 π ji j j j j j j 11 ª ii i i 1i i i ji =− TT ω + ω δω − T ω +T ω δω − T θ +S θ +T θ +S θ δθ − 12 1 () ( ) ( ) ³³ 11 1 11 1 22 2 22 2 11 1 2 11 1 2 22 ¬ 2 o 0 1 iijj j i ji º −+ T θθ S +T θ + Sθ δθ dxdy () 22 2 2 22 21 2 2 ¼ If the solutions of boundary value problems (4) and (15) are obtained, then the coefficients a and b of ijr ijkr system of Eq. (16) are determined by Eq. (17). The stated technique has been applied for the calculation of laminated cylindrical shells having a total thick- ness t at axial compression and under external pressure. The displacements are adopted as the resolvents. In 20 architectura.actapol.net Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 a solution of the homogeneous problem (4), the dis- The system of Eq. (16) is solved for a given value placements are represented by trigonometric series, of the load parameter λ Newton’s method. A stepped each term of which separately satisfies simple support procedure has been considered. The solution obtained conditions at the ends: in the i-th loading step is used as the initial approxi- mation for the next (i + 1)-st step. The following pro- uA = coslξφ cosn im ,n m i cedure (Bazhenov, Semenyuk & Trach 2010) is used to construct the solution at points where the Jacobian vB = sinlξφ sinn im ,n m i of the system (16) is equal to zero. We introduce an (M + 1)-dimensional vector X with components (21) wC = sinlξφ cosn im ,n m i (ξ , ..., ξ , λ) . The system (16) can be written in the 1 M compact form θξ =Dl cos cosnφ im ,n m i FX== 0; r 1, ...,M r () (23) ψξ =El sin sinnφ im ,n m i where: Differentiating Eq. (21) with respect to the param- eter ξ, which corresponds to progression along the mR π x l== ; ξϕ ; = . curve of solutions of the system, a system of M linear LR R ordinary differential equations at (M + 1) unknowns can be obtained The system of homogeneous algebraic equations is obtained by substituting (21) into (19). An exhaustive M +1 sequential inspection of the mode-generation param- dξ (24) Fr == 0; 1, ...,M rj , eters m and n yields the spectrum of eigenvalues λ and ds j=1 corresponding eigenvectors, which are normalized in such a way as to make Ci =1. mn , ªº ∂F where JF== is the Jacobian matrix of the The solution of the system of Eq. (12) with allow- [] rj , «» ∂ξ ¬¼ ance for the form of their right-hand sides and simple supports at the ends is written system (23). The Jacobian has the rank ªºJM = at ¬¼ regular and limit points. The solution of the system ij ij ªº (24) can be written in the form of a Cauchy prob- uA=− cosn nϕϕ +A cosn+n coslξ ij () i j ( i j) k kk ,1 ,2 ¬¼ lem: ij ij ªº vB=− sin()n nϕϕ +B sin(n+n) sinlξ ij i j i j k ¦ dX kk ,1 , 2 ¬¼ = ort J , Q () ds ij ij ªº basing on the initial condition Xs =X . ()00 wC=− cosn nϕϕ +C cosn+n sinlξ ij () i j ( i j) k kk ,1 ,2 ¬¼ The operator ort(J, Q) denotes the process of or- ij ij ªº thogonalization of row vectors of the matrix J and θ=− Dn cos nϕϕ +D cosn+n coslξ ij () i j ( i j) k kk ,1 ,2 ¬¼ k determination of the unit vector that completes the original basis to an (M + 1)-dimensional basis (Bazhe- ij ij ªº nov et al., 2010). The initial value Xs is found by () 0 ψ=− En sin()nϕϕ +E sin(n+n) sinlξ ij i j i j k kk ,1 , 2 ¬¼ k solving the system (23) according to Newton’s method for λ << λ (limiting value). (22) architectura.actapol.net 21 Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 For estimation of reliability of the results obtained NUMERICAL ANALYSIS has been assumed in Eq. (16) that r = 1 and the equi- With use of the above-mentioned calculation tech- librium curves have been compared with those that nique, some features of the nonlinear deformation correspond to the equations by Koiter (1963): of composite cylindrical shells with geometric im- perfection in the form of superposition of the eigen- §· λλ ξξ 1−+ b=ξ (25) ¨¸ modes of the initial linearized problem have been λλ ©¹cc investigated. Consider a shell of medium thickness (R/t = 30/1.6), medium length (L/R = 2), consisting Shown in Figure 1, Curves 1 and 2 are obtained by of 16 individual layers, which are reinforced fib- Eqs. (16) and (25) in the case of an axial compression erglass or carbon fiber. In the examples below, the of fiberglass shells. The closeness of these curves in- distribution of reinforcement directions through the dicates the possibility of calculation of post-buckling thickness is supposed as (0°, 45°, –45°, 90°, …). The behavior of the considered shells with use of Eq. (16) layer is assumed to have the following mechanical or (25), according to the shape of imperfections. characteristics: Taking into account the interaction of several − for carbon fiber: E = 0.161·10 MPa, E = modes, it has been examined the kind of equilibrium 11 22 5 4 = 0.115·10 MPa, G = 0.717·10 MPa, G = 12 13 §· 4 4 curve ξ at q = 1, depending on the number of = 0.717·10 MPa, G = 0.7·10 MPa, v = 0.349; ¨¸ 23 12 ©¹c 5 5 − for fiberglass: E = 0.454·10 MPa, E = 0.109·10 11 22 4 4 modes. The numbering of the curves in Figure 2 cor- MPa, G = 0.424·10 MPa, G = 0.436·10 MPa, 12 13 4 responds to the number of modes. G = 0.384·10 MPa, v = 0.26. 23 12 λ/λ 0,8 0,4 0,0 0,0 0,5 1,0 1,5 Fig. 1. The curves obtained by Eqs. (16) and (25) in the case of an axial compression of fiberglass shells λ/λ 7, 8 1,0 1, 2, 3 5, 6 0,5 0,0 0,0 0,5 1,0 1,5 Fig. 2. The kind of equilibrium curve depending on the number of modes 22 architectura.actapol.net Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 As it is seen, the additions of imperfections of the second and the third modes do not change Curve 1 ob- The function e is even. Its expansion in a Fou- tained only at presence of the first mode. The indicated rier series can be expressed as modes are passive with respect to the first mode. The shape of Curve 4 shows that fourth mode is active. a (27) + an cos ϕ Taking this mode into account essentially changes the form of the equilibrium curve. At consequent increase where: of the number of modes, their considerable activity 2 −1 also appears (Curves 5–7). ae =− ϕ 1 ; 01() Figure 3 shows the curves which illustrate one more feature of deformation of shells with multimodal 2 −1 ªº ae =+1snϕϕinn−cosnϕ n () 11 1 imperfections Number of curves in this figure corre- 22¬¼ 1+ n ϕ spond to the indexes of unknown amplitudes ξ in the system of Eq. (16). Basing on the expansion (26), the initial deflection In spite of the fact that the initial values of all am- can be written as plitudes ξ n a considered example are identical, their dependence on the load parameter λ has the brightly ww ξϕ , =ξ 0() ii expressed disproportionate character. They become essentially different with increasing number i. where ξ = εaw ; is i-th eigenmode, corresponding ii i Representation of solutions as a Fourier series in to the eigenvalue λ . If the initial values of amplitudes terms of the eigenmodes of the linearized problem has ξ and the critical parameters λ are known, then there i i been used to study the non-linear deformation of com- is a possibility to explore the process of deformation posite shells with local geometrical imperfections. It of the shell, which has a small dimple depth. has been supposed that the shell surface has the dim- ple shaped initial imperfections given by (Amazigo & It is necessary to note that the further computing Fraser, 1971) result are obtained basing on the 19 terms of a Fourier series in the expression (27). Figure 4 shows the equilibrium curves for a fiber- we ξϕ,s =ε inl ξ (26) glass shell of the assumed sizes at φ = π/9 under exter- 0() m nal pressure. Curve 1 is obtained at ε = 0.05, Curve 2 where ε is a small parameter, –φ ≤ φ ≤ φ ; m = 1, … at ε = 0.1, Curve 3 at ε = 0.5. 1 1 λ/λ 0,8 4 3 0,4 0,0 0,0 0,5 1,0 Fig. 3. The curves illustrating one more feature of deformation of shells with multimodal imperfections architectura.actapol.net 23 Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 §· λ dure by Byskov–Hutchinson is an effective mean of Figures 5 and 6 show the dependence for ¨¸ λ the solution of some non-linear problems. ©¹i the fiberglass (Fig. 5) and carbon fiber (Fig. 6) shells with the same dimples at axial compression. These CONCLUSIONS curves essentially differ from those shown in Figure 4, just as it takes place in the non-linear range of the Using methods of the asymptotic analysis of the theory of shells (Koiter, 1976). Timoshenko–Mindlin theory, the relationships for cal- It has been shown that the combination of an am- culation of shells with the small geometrical imperfec- plitude modulation by Koiter and asymptotic proce- tions of the different shapes have been produced. On λ/λ 0,8 0,4 0,0 0,0 0,5 Fig. 4. The equilibrium curves for a fiberglass shell of the assumed sizes at under external pressure λ/λ 0,8 2 0,4 0,0 0,00 0,25 0,50 Fig. 5. The dependence for the fiberglass shells λ/λ 0,8 0,4 0,0 0,00 0,05 0,10 0,15 Fig. 6. The dependence for the carbon fiber shells 24 architectura.actapol.net Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 initial imperfections. International Journal of Solids the basis of the obtained equations, the technique of and Structures, 7 (8), 883–900. calculation of a non-linear pre-buckling state, limiting Bazhenov, V. A., Semenyuk, N. P. & Trach, V. M. (2010). loads and bifurcation, and also initial post-buckling Neliniyne deformuvannya, stiykist i zakrytychna pove- behavior of laminated cylindrical shells at an axial dinka anizotropnykh obolonok [Nonlinear deforma- compression and external pressure has been worked tion, stability and postbuckling behavior of anisotropic out. The results of calculation for the shells made of shells]. Kyiv: Caravela. glass fiber-reinforced plastic and carbon fiber-rein- Byskov, E. (2004). Mode Interaction in Structures – An forced plastic with multimodal imperfections and Overview. In Proceedings of the Sixth World Congress dimple imperfections have been presented. on Computational Mechanics in conjunction with the Second Asian-Pacific Congress on Computational Me- Authors’ contributions chanics: Sept. 5-10, 2004, Beijing, China. Beijing: Tsin- Conceptualization: M.S. and V.T.; methodology: V.T.; ghua University [CD-ROM]. validation: M.S. and V.T.; formal analysis: M.S. and Byskov, E. & Hutchinson, J. W. (1977). Mode interaction in axially stiffened cylindrical shells. AIAA Journal, V.T.; investigation: V.T.; resources: M.S.; data cura- 15 (7), 941–948. tion: N.Z.; writing – original draft preparation: N.Z.; Koiter, W. T. (1963). Elastic Stability and Post Buckling Be- writing – review and editing: V.T. and M.S.; visualiza- haviour in Nonlinear Problems. In Nonlinear Problems. tion: N.Z.; supervision: V.T. and M.S.; project admin- Proceedings of a Symposium (pp. 257–275). Madison, istration: M.S.; funding acquisition: M.S. and V.T. WI: University of Wisconsin Press. All authors have read and agreed to the published Koiter, W. T. (1976). General theory of mode interaction in version of the manuscript. stiffened plate and shell structures (Report WTHD 91). Delft: Delft University of Technology. Vanin, G. A. & Semenyuk, N. P. (1987). Ustoychivost obo- REFERENCES lochek iz kompozitsionnykh materialov s nesovershenst- Amazigo, J. C. & Fraser, W. B. (1971). Buckling under ex- vami [Stability of composite shells with imperfections]. ternal pressure of cylindrical shells with dimple shaped Kyiv: Naukova Dumka. O METODZIE OBLICZANIA WYBOCZENIA I STANU POWYBOCZENIOWEGO DLA POWŁOK Z MAŁYMI IMPERFEKCJAMI STRESZCZENIE W niniejszym artykule uogólniono wariant metody obliczania stateczności i początkowego stanu powybo- czeniowego powłok izotropowych w odniesieniu do powłok kompozytowych. Z wykorzystaniem metody analizy asymptotycznej teorii Timoszenki–Mindlina stworzono związki do obliczeń powłok z małymi im- perfekcjami geometrycznymi o różnych kształtach. Na podstawie otrzymanych równań opracowano technikę obliczeń nieliniowego stanu przed wyboczeniem, obciążeń granicznych i bifurkacji oraz początkowego stanu powyboczeniowego warstwowych powłok cylindrycznych przy ściskaniu osiowym i ciśnieniu zewnętrznym. Przedstawiono wyniki obliczeń powłok wykonanych z tworzywa sztucznego wzmacnianego włóknem szkla- nym i włóknem węglowym z imperfekcjami wielopostaciowymi i wgłębieniami. Zbadano cechy przekształ- cenia współdziałających postaci wyboczenia. Słowa kluczowe: wyboczenie, stan powyboczeniowy, imperfekcje, powłoki warstwowe, współdziałanie postaci wyboczenia, problem Cauchy’ego architectura.actapol.net 25 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Scientiarum Polonorum Architectura de Gruyter

On the Method of Calculation of Buckling and Post-Buckling Behavior of Laminated Shells with Small Arbitrary Imperfections

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Acta Sci. Pol. Architectura 20 (2) 2021, 17–25 content.sciendo.com/aspa ISSN 1644-0633 eISSN 2544-1760 DOI: 10.22630/ASPA.2021.20.2.11 ORIGINAL P APER Received: 29.04.2021 Accepted: 26.05.2021 ON THE METHOD OF CALCULATION OF BUCKLING AND POST-BUCKLING BEHAVIOR OF LAMINATED SHELLS WITH SMALL ARBITRARY IMPERFECTIONS 1 2 1 Mykola Semenyuk , Volodymyr Trach , Natalia Zhukova S.P. Timoshenko Institute of Mechanics NAS Ukraine, Kyiv, Ukraine Institute of Civil Engineering, Warsaw University of Life Sciences – SGGW, Warsaw, Poland; National University of Water Management and Nature Resources Use, Rivne, Ukraine ABSTRACT In the present paper the variant of computational methods of stability and initial post-buckling behavior of isotropic shells was generalized with respect to laminated composite shells. Using methods of the asymp- totic analysis of the Timoshenko–Mindlin theory, the relationships for calculation of shells with the small geometrical imperfections of the different shapes have been produced. On the basis of obtained equations, the technique of calculation of a non-linear pre-buckling state, limiting loads and bifurcation, and also initial post-buckling behavior of laminated cylindrical shells at an axial compression and external pressure have been worked out. The results of calculation for the shells made of glass fiber-reinforced plastic and carbon fiber-reinforced plastic with multimodal imperfections and dimple imperfections have been presen- ted. The features of transformation of the interacting modes of imperfections have been researched. Key words: buckling, post-buckling, imperfection, laminated shell, mode interaction, Cauchy problem The stress vector σ for a shell with imperfections MODEL EQUATIONS ε uu , () is related to the deformation vector by the Let a shell be loaded by a system of forces proportional stiffness matrix for a laminated shell H to a certain parameter λ. According to a Timoshenko– –Mindlin theory, the generalized displacements of the σε =Hu(,u) (2) shell are described by the vector U = (u, ν, w, θ, ψ). Analogous vector notation for the generalized stresses The principle of virtual work takes the form and strains σ = (T , T , S, T , T , M , M , M , M ), 11 22 23 13 11 22 12 21 ′′ ε = (ε , ε , ε , ε , ε , k , k , τ , τ ) is introduced σε (,uu)δu−Δ λ (u,u)δu =0 (3) 11 22 12 23 13 11 22 1 2 here. If the field of initial imperfections is character- ized by the vector u , then the following expressions Though modified, Eqs. (1)–(3) enable solving a va- for the deformations and generalized displacement can riety of nonlinear problems on deformation of shells be derived with initial geometric imperfections. The asymptotic method (Byskov & Hutchinson, 1997) to the Eqs. ′′  ′′ εε (,uu )=+ (u)ε uu; Δ= (,uu) Δ(u)+Δ uu (1) (1)–(3) has been applied here. It has been assumed trach-vm@ukr.net © Copyright by Wydawnictwo SGGW Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 ′′ ′′ that the subcritical stress-deformed state of the shells ξλ−− λ σ ε uuδ Δ uuδ+ ii()( 0 i i ) is linear. For λ = 1, assume that the fields of displace- (12) ++ ξξLL () λ ξ ξ ξ +...=0 i j ij i j k ijk ments, deformations and stresses are characterized by the vectors u , ε ,σ . After linearization of equations in 0 0 0 where: the neighborhood of the bifurcation load in aim to de- termine the critical values of λ and the corresponding ′′′ ′′ ′′ L ()λσ=+ ε (0)δu σε u δu+λσ ε uuδ−λΔ uuδ ij ij i j 0 ij ij buckling modes u , one obtains a set of equations ′′ ′′ ′′ ′ Lu=+ σε δu σ εu δu+Hεuu ε (0)δu ijk i jk ij k i jk ′′′ ′′ σε (0)δuu+− λ σ ε δu λΔu δu= 0 (4) ii 0 i i i (13) σε = H (5) ii ′ A variational equation in the fields u , ε , σ can be εε== (0)ui; 1, ..., M (6) ii ij ij ij obtained from Eq. (12), when it has been taken into These M forms are orthogonal account that the variation δu in this case satisfies the orthogonality condition σε ′′uu −Δ ′′uu = 0, i ≠ j (7) 0 ij ij σε ′′uu δ −Δ ′′u δu = 0 (14) The amplitudes ξ of the modes u remain indeter- 0 ii i i minate during solution of the homogeneous problem Neglecting, in Eq. (12), terms of third or higher or- (4)–(6) and can be found only by solving the original der in ξ , the symmetry of the desired functions relative non-linear problem (1)–(7). The displacement vector to the indices i, j results in an equation can be represented as the asymptotic expansion ′′′ ′′ σε (0)δuu+− λσ ε δu λΔu δu= ij 0 ij ij uu=+ λξu +ξξ u (8) 0 ii i j ij (15) =− σε ′′uu δ +σ ε ′′u δu () ij j i Here and further the rule of summing over repeated indices is adopted. The smallness of parameters ξ fol- In contrast to Koiter’s classical approach (Koit- lows from the fact that ξ → 0 as λ → λ . The displace- i i er, 1963), which is based on the Lyapunov–Schmidt ments u are orthogonal to the buckling modes ij method, the method proposed by Byskov and Hutch- ′′ ′′ σε uu −Δ u u = 0; 0 ikl i kl inson (1977) does not use expansion of the coefficients i = 1, ..., M; k =1, ..., M; l = 1, ..., M (9) of L (λ) in Tailor series in the parameter (λ – λ ). As a ij j result, the solution of Eq. (15) depends on the load λ. Substituting (8) into (4), one obtains Koiter (1963) proposed a use of the smallest λ for the λ in Eq. (15). However, the problems under considera- λσ ε′′ (0)δuu −Δ λ (0)δ + tion make it necessary to determine this inaccuracy. ′′′ ′′ ++ ξσ ε (0)δuu λσ ε δu−λΔuδu+ ii[] 0 i i If the homogeneous boundary-value problem (4), (5) and inhomogeneous problem (11), (15) are resolved, ′′ ′′ ′′ ++ ξξ[] σ ε (0)δuu σ ε δu+ λσ εu δu−λΔu δu+ i j ij i j 0 ij ij then, for the amplitudes ξ from equation (12) (where δu = u is set), one can derive the following system of ++ ξξ ξ σ ε ′′uu δ σ ε ′′u δu+Hε ′′uu ε ′(0)δu+...= 0 i j k[] i jk ij k i jk non-linear equations: (10) §·λλ ξ 1−+ξξab +ξ ξ ξ =ξ ;r= 1, ..., M Here, terms of higher order in ξ have been omitted r i j ijr i j k ijkr r ¨¸ λλ ©¹rr and, moreover (16) σε==Hu ,0 ε ε′′+ ε′uu (11) ij ij ij () ij i j where: A B ijr ijr Because λσ ε’(0)δu – λΔ’(0)δu = 0, basing on (4), 0 ; a =− b =− ijr ijr 2 D the expression (10) can be reduced to the form 18 architectura.actapol.net Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 22 x, y – coordinate lines which coincide with a gen- ′′ ′′ ′′ ′′ Au=+ σε u 2, σ εuuD=λ σ εu−Δu ijr r i j i j r r() 0 r r eratrix and directrix of a cylindrical reduced surface, Bu=+ σε ′′u σ ε ′′uu+σ ε ′′uu ijkr i r jk ij k i r i jk T , M – efforts and moments, which are statically ij ij (17) equivalent to actual stresses. Equation (16) can be used to investigate nonlinear ** 1 TS =+Tωω ;T =S+ H+T 12 11 1 21 22 2 deformations of imperfect structures in the subcritical state, for computation of critical (bifurcation or limit- ** TT=+T θθ+S ;T =T +Sθ +T 13 13 11 1 2 23 23 1 22 ing) loads and for computation of the initial post-buck- ling behavior of the shells under discussion. The strains are given by In general, the representation (16) is uniformly valid whether the modes are simultaneous, nearly 11 22 2 2 εε=+ ω +θ ; ε =ε + ω +θ 11 1 1 1 22 2 2 2 () ( ) simultaneous, or well separated. This property of the Byskov–Hutchinson method εω=+ω +θθ 12 1 2 1 2 is used below for development of a technique of cal- culation of stability and post-buckling behavior of εω kk== ; k k+= ; k t+t+ 11 1 22 2 12 1 2 laminated composite cylindrical shells with multi- RR modal imperfections, in particular, by imperfections which are approximated by trigonometric Fourier εθ =+θ ; ε =ψ+θ 13 1 23 2 series. where: THE SOLUTION FOR THE CYLINDRICAL SHELLS ∂∂ uv ∂w ε==;;ωθ= 11 1 ∂∂ xx ∂x The relationships obtained have been used for buck- ling analysis of a multilayer cylindrical composite ∂∂ vw u ∂w v shells with geometrical imperfections. The imperfec- ε =−;;ωθ = = + 22 2 ∂∂ yR y ∂y R tions’ shape is described by trigonometric polynomials (fragmenting of Fourier series). According to the basic ∂∂ ψψ ∂∂ θθ kk== ;; t= ;t= 122 1 2 statements of the Byskov–Hutchinson method (1997) ∂∂ xy ∂x ∂y and Byskov method (2004) relationships will be pre- sented which are necessary for solving with use of the The variation of work of external loads λA has Timoshenko–Mindlin non-linear shell theory (Vanin a different kind depending on character of their distri- & Semenyuk, 1987). The expression of virtual work bution on a frontal or on a lateral surface. principle in this case is In case of a cylindrical shell, one of the most prac- tically interesting is the version of loading by longitu- LR 2 π ** dinal compressive forces – T TT δε++ δε T δω+T δω+ 11 11 1 22 2 12 1 21 2 ³³ 2 π R ** L ++ TT δθ δθ+T δθ+T δψ+ 13 1 23 2 13 23 0 δδ AT =− u dy ³ 11 (18) ψ++ Mkδδ M k+Mδt+ 11 1 22 2 12 1 or uniform external pressure with intensity q ++ Ht() δδt dxdy−δA= 0 LR 2 π where: δθ Aq=−ªº δu−θδv+() 1+ε+εδwdxdy 12 1 2 ³³¬¼ L, R – length and radius of a shell, architectura.actapol.net 19 Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 The initial imperfections are characterized by a deflection w (x, y). It is supposed, that the shell consists of N layers of a fibrous composite material which have a symmetrical structure with regard to a middle surface. The relationships of elasticity in this case may be written as TC=+εεC ; T =Cε +Cε ; S=Cε ; T =Cε ; T =Cε 11 11 11 12 22 22 12 11 22 22 66 12 13 55 13 23 44 23 MD=+k D k ; M =D k +D k ; H=D k 11 11 11 12 22 22 12 11 22 22 66 12 in which the rigidities C and D are given by ij ij NN ii 2i CC== ; D D+zC kl kl () ¦¦ i kl kl kl ii == 11 where z is coordinate of a middle surface of i-th layer. The linearized equations for a cylindrical shell may be expressed as LR 2 π ii i i 2 i i §· TS δε++ δω T δε +S+ H δω+T δθ+ δθ+ 11 2 2 ()1 ¨¸ 11 22 13 ³³ { ©¹ R ii i i 0 i i (19) ++ Hkδδ T ()θ+δψ+Mδk +Mδk +λ T ωδω+θδθ+ 12 2 11 22 i () 1 1 23 11 22 11 1 1 00 ii i i ++TS ωδω θ δθ+ θ δθ+θ δθ dxdy= 0 () 22 ( 2 1) ¾ 22 2 1 2 Variational equation (19) relative to variables of second order in the Timoshenko–Mindlin shell theory is represented as LR 2 π ij ij ij ij 2 ij ij ij ij ij §· TS δε++ δω T δε+S + H δω+T δθ+ δθ+T δθ+ δψ+M δk +M δk + 11 2 2 ()1 ()2 11 22 ¨¸ ³³ { 11 22 13 23 11 22 ©¹ R ij ij ij ij ij ij ij 00 0 ªº ++ Hk δ λ T ωδω+θ δθ+T ωδω+θ δθ +S θ δθ+θ δθ dxdy= 12 i 1 1 2 2 2 1 ¾ () ( ) ( ) 11 1 1 2 2 2 1 2 ¬¼ (20) LR 2 π ji j j j j j j 11 ª ii i i 1i i i ji =− TT ω + ω δω − T ω +T ω δω − T θ +S θ +T θ +S θ δθ − 12 1 () ( ) ( ) ³³ 11 1 11 1 22 2 22 2 11 1 2 11 1 2 22 ¬ 2 o 0 1 iijj j i ji º −+ T θθ S +T θ + Sθ δθ dxdy () 22 2 2 22 21 2 2 ¼ If the solutions of boundary value problems (4) and (15) are obtained, then the coefficients a and b of ijr ijkr system of Eq. (16) are determined by Eq. (17). The stated technique has been applied for the calculation of laminated cylindrical shells having a total thick- ness t at axial compression and under external pressure. The displacements are adopted as the resolvents. In 20 architectura.actapol.net Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 a solution of the homogeneous problem (4), the dis- The system of Eq. (16) is solved for a given value placements are represented by trigonometric series, of the load parameter λ Newton’s method. A stepped each term of which separately satisfies simple support procedure has been considered. The solution obtained conditions at the ends: in the i-th loading step is used as the initial approxi- mation for the next (i + 1)-st step. The following pro- uA = coslξφ cosn im ,n m i cedure (Bazhenov, Semenyuk & Trach 2010) is used to construct the solution at points where the Jacobian vB = sinlξφ sinn im ,n m i of the system (16) is equal to zero. We introduce an (M + 1)-dimensional vector X with components (21) wC = sinlξφ cosn im ,n m i (ξ , ..., ξ , λ) . The system (16) can be written in the 1 M compact form θξ =Dl cos cosnφ im ,n m i FX== 0; r 1, ...,M r () (23) ψξ =El sin sinnφ im ,n m i where: Differentiating Eq. (21) with respect to the param- eter ξ, which corresponds to progression along the mR π x l== ; ξϕ ; = . curve of solutions of the system, a system of M linear LR R ordinary differential equations at (M + 1) unknowns can be obtained The system of homogeneous algebraic equations is obtained by substituting (21) into (19). An exhaustive M +1 sequential inspection of the mode-generation param- dξ (24) Fr == 0; 1, ...,M rj , eters m and n yields the spectrum of eigenvalues λ and ds j=1 corresponding eigenvectors, which are normalized in such a way as to make Ci =1. mn , ªº ∂F where JF== is the Jacobian matrix of the The solution of the system of Eq. (12) with allow- [] rj , «» ∂ξ ¬¼ ance for the form of their right-hand sides and simple supports at the ends is written system (23). The Jacobian has the rank ªºJM = at ¬¼ regular and limit points. The solution of the system ij ij ªº (24) can be written in the form of a Cauchy prob- uA=− cosn nϕϕ +A cosn+n coslξ ij () i j ( i j) k kk ,1 ,2 ¬¼ lem: ij ij ªº vB=− sin()n nϕϕ +B sin(n+n) sinlξ ij i j i j k ¦ dX kk ,1 , 2 ¬¼ = ort J , Q () ds ij ij ªº basing on the initial condition Xs =X . ()00 wC=− cosn nϕϕ +C cosn+n sinlξ ij () i j ( i j) k kk ,1 ,2 ¬¼ The operator ort(J, Q) denotes the process of or- ij ij ªº thogonalization of row vectors of the matrix J and θ=− Dn cos nϕϕ +D cosn+n coslξ ij () i j ( i j) k kk ,1 ,2 ¬¼ k determination of the unit vector that completes the original basis to an (M + 1)-dimensional basis (Bazhe- ij ij ªº nov et al., 2010). The initial value Xs is found by () 0 ψ=− En sin()nϕϕ +E sin(n+n) sinlξ ij i j i j k kk ,1 , 2 ¬¼ k solving the system (23) according to Newton’s method for λ << λ (limiting value). (22) architectura.actapol.net 21 Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 For estimation of reliability of the results obtained NUMERICAL ANALYSIS has been assumed in Eq. (16) that r = 1 and the equi- With use of the above-mentioned calculation tech- librium curves have been compared with those that nique, some features of the nonlinear deformation correspond to the equations by Koiter (1963): of composite cylindrical shells with geometric im- perfection in the form of superposition of the eigen- §· λλ ξξ 1−+ b=ξ (25) ¨¸ modes of the initial linearized problem have been λλ ©¹cc investigated. Consider a shell of medium thickness (R/t = 30/1.6), medium length (L/R = 2), consisting Shown in Figure 1, Curves 1 and 2 are obtained by of 16 individual layers, which are reinforced fib- Eqs. (16) and (25) in the case of an axial compression erglass or carbon fiber. In the examples below, the of fiberglass shells. The closeness of these curves in- distribution of reinforcement directions through the dicates the possibility of calculation of post-buckling thickness is supposed as (0°, 45°, –45°, 90°, …). The behavior of the considered shells with use of Eq. (16) layer is assumed to have the following mechanical or (25), according to the shape of imperfections. characteristics: Taking into account the interaction of several − for carbon fiber: E = 0.161·10 MPa, E = modes, it has been examined the kind of equilibrium 11 22 5 4 = 0.115·10 MPa, G = 0.717·10 MPa, G = 12 13 §· 4 4 curve ξ at q = 1, depending on the number of = 0.717·10 MPa, G = 0.7·10 MPa, v = 0.349; ¨¸ 23 12 ©¹c 5 5 − for fiberglass: E = 0.454·10 MPa, E = 0.109·10 11 22 4 4 modes. The numbering of the curves in Figure 2 cor- MPa, G = 0.424·10 MPa, G = 0.436·10 MPa, 12 13 4 responds to the number of modes. G = 0.384·10 MPa, v = 0.26. 23 12 λ/λ 0,8 0,4 0,0 0,0 0,5 1,0 1,5 Fig. 1. The curves obtained by Eqs. (16) and (25) in the case of an axial compression of fiberglass shells λ/λ 7, 8 1,0 1, 2, 3 5, 6 0,5 0,0 0,0 0,5 1,0 1,5 Fig. 2. The kind of equilibrium curve depending on the number of modes 22 architectura.actapol.net Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 As it is seen, the additions of imperfections of the second and the third modes do not change Curve 1 ob- The function e is even. Its expansion in a Fou- tained only at presence of the first mode. The indicated rier series can be expressed as modes are passive with respect to the first mode. The shape of Curve 4 shows that fourth mode is active. a (27) + an cos ϕ Taking this mode into account essentially changes the form of the equilibrium curve. At consequent increase where: of the number of modes, their considerable activity 2 −1 also appears (Curves 5–7). ae =− ϕ 1 ; 01() Figure 3 shows the curves which illustrate one more feature of deformation of shells with multimodal 2 −1 ªº ae =+1snϕϕinn−cosnϕ n () 11 1 imperfections Number of curves in this figure corre- 22¬¼ 1+ n ϕ spond to the indexes of unknown amplitudes ξ in the system of Eq. (16). Basing on the expansion (26), the initial deflection In spite of the fact that the initial values of all am- can be written as plitudes ξ n a considered example are identical, their dependence on the load parameter λ has the brightly ww ξϕ , =ξ 0() ii expressed disproportionate character. They become essentially different with increasing number i. where ξ = εaw ; is i-th eigenmode, corresponding ii i Representation of solutions as a Fourier series in to the eigenvalue λ . If the initial values of amplitudes terms of the eigenmodes of the linearized problem has ξ and the critical parameters λ are known, then there i i been used to study the non-linear deformation of com- is a possibility to explore the process of deformation posite shells with local geometrical imperfections. It of the shell, which has a small dimple depth. has been supposed that the shell surface has the dim- ple shaped initial imperfections given by (Amazigo & It is necessary to note that the further computing Fraser, 1971) result are obtained basing on the 19 terms of a Fourier series in the expression (27). Figure 4 shows the equilibrium curves for a fiber- we ξϕ,s =ε inl ξ (26) glass shell of the assumed sizes at φ = π/9 under exter- 0() m nal pressure. Curve 1 is obtained at ε = 0.05, Curve 2 where ε is a small parameter, –φ ≤ φ ≤ φ ; m = 1, … at ε = 0.1, Curve 3 at ε = 0.5. 1 1 λ/λ 0,8 4 3 0,4 0,0 0,0 0,5 1,0 Fig. 3. The curves illustrating one more feature of deformation of shells with multimodal imperfections architectura.actapol.net 23 Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 §· λ dure by Byskov–Hutchinson is an effective mean of Figures 5 and 6 show the dependence for ¨¸ λ the solution of some non-linear problems. ©¹i the fiberglass (Fig. 5) and carbon fiber (Fig. 6) shells with the same dimples at axial compression. These CONCLUSIONS curves essentially differ from those shown in Figure 4, just as it takes place in the non-linear range of the Using methods of the asymptotic analysis of the theory of shells (Koiter, 1976). Timoshenko–Mindlin theory, the relationships for cal- It has been shown that the combination of an am- culation of shells with the small geometrical imperfec- plitude modulation by Koiter and asymptotic proce- tions of the different shapes have been produced. On λ/λ 0,8 0,4 0,0 0,0 0,5 Fig. 4. The equilibrium curves for a fiberglass shell of the assumed sizes at under external pressure λ/λ 0,8 2 0,4 0,0 0,00 0,25 0,50 Fig. 5. The dependence for the fiberglass shells λ/λ 0,8 0,4 0,0 0,00 0,05 0,10 0,15 Fig. 6. The dependence for the carbon fiber shells 24 architectura.actapol.net Semenyuk, M., Trach, V., Zhukova, N. (2021). On the method of calculation of buckling and post-buckling behavior of laminated shells with small arbitrary imperfections. Acta Sci. Pol. Architectura, 20 (2), 17–25. doi: 10.22630/ASPA.2021.20.2.11 initial imperfections. International Journal of Solids the basis of the obtained equations, the technique of and Structures, 7 (8), 883–900. calculation of a non-linear pre-buckling state, limiting Bazhenov, V. A., Semenyuk, N. P. & Trach, V. M. (2010). loads and bifurcation, and also initial post-buckling Neliniyne deformuvannya, stiykist i zakrytychna pove- behavior of laminated cylindrical shells at an axial dinka anizotropnykh obolonok [Nonlinear deforma- compression and external pressure has been worked tion, stability and postbuckling behavior of anisotropic out. The results of calculation for the shells made of shells]. Kyiv: Caravela. glass fiber-reinforced plastic and carbon fiber-rein- Byskov, E. (2004). Mode Interaction in Structures – An forced plastic with multimodal imperfections and Overview. In Proceedings of the Sixth World Congress dimple imperfections have been presented. on Computational Mechanics in conjunction with the Second Asian-Pacific Congress on Computational Me- Authors’ contributions chanics: Sept. 5-10, 2004, Beijing, China. Beijing: Tsin- Conceptualization: M.S. and V.T.; methodology: V.T.; ghua University [CD-ROM]. validation: M.S. and V.T.; formal analysis: M.S. and Byskov, E. & Hutchinson, J. W. (1977). Mode interaction in axially stiffened cylindrical shells. AIAA Journal, V.T.; investigation: V.T.; resources: M.S.; data cura- 15 (7), 941–948. tion: N.Z.; writing – original draft preparation: N.Z.; Koiter, W. T. (1963). Elastic Stability and Post Buckling Be- writing – review and editing: V.T. and M.S.; visualiza- haviour in Nonlinear Problems. In Nonlinear Problems. tion: N.Z.; supervision: V.T. and M.S.; project admin- Proceedings of a Symposium (pp. 257–275). Madison, istration: M.S.; funding acquisition: M.S. and V.T. WI: University of Wisconsin Press. All authors have read and agreed to the published Koiter, W. T. (1976). General theory of mode interaction in version of the manuscript. stiffened plate and shell structures (Report WTHD 91). Delft: Delft University of Technology. Vanin, G. A. & Semenyuk, N. P. (1987). Ustoychivost obo- REFERENCES lochek iz kompozitsionnykh materialov s nesovershenst- Amazigo, J. C. & Fraser, W. B. (1971). Buckling under ex- vami [Stability of composite shells with imperfections]. ternal pressure of cylindrical shells with dimple shaped Kyiv: Naukova Dumka. O METODZIE OBLICZANIA WYBOCZENIA I STANU POWYBOCZENIOWEGO DLA POWŁOK Z MAŁYMI IMPERFEKCJAMI STRESZCZENIE W niniejszym artykule uogólniono wariant metody obliczania stateczności i początkowego stanu powybo- czeniowego powłok izotropowych w odniesieniu do powłok kompozytowych. Z wykorzystaniem metody analizy asymptotycznej teorii Timoszenki–Mindlina stworzono związki do obliczeń powłok z małymi im- perfekcjami geometrycznymi o różnych kształtach. Na podstawie otrzymanych równań opracowano technikę obliczeń nieliniowego stanu przed wyboczeniem, obciążeń granicznych i bifurkacji oraz początkowego stanu powyboczeniowego warstwowych powłok cylindrycznych przy ściskaniu osiowym i ciśnieniu zewnętrznym. Przedstawiono wyniki obliczeń powłok wykonanych z tworzywa sztucznego wzmacnianego włóknem szkla- nym i włóknem węglowym z imperfekcjami wielopostaciowymi i wgłębieniami. Zbadano cechy przekształ- cenia współdziałających postaci wyboczenia. Słowa kluczowe: wyboczenie, stan powyboczeniowy, imperfekcje, powłoki warstwowe, współdziałanie postaci wyboczenia, problem Cauchy’ego architectura.actapol.net 25

Journal

Acta Scientiarum Polonorum Architecturade Gruyter

Published: Jun 1, 2021

Keywords: buckling; post-buckling; imperfection; laminated shell; mode interaction; Cauchy problem

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