Analytical Calculation of Cylindrical Shells in the Form of Second-Order Algebraic Surfaces

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1. Introduction Various analytical, semi-analytical and numerical methods for the calculation of thin- and thick-walled, single- and multi-layer shells made of physically linear and nonlinear structural materials have been developed in response to the demands of practice. To assist the design engineer, the corresponding computational software systems have been created. Various optimality criteria have been developed for the design of thin shells. A number of review papers [1-4] show that shells of rotation, translation, cylindrical and conical shells are among the most popular thin and thick shells that can satisfy a variety of practical and scientific demands arising in different time periods. The greatest attention has been given to shells of rotation. Twenty three optimality criteria were proposed for them [5]. V.V. Novozhilov et al. [6] showed that, in terms of stress magnitude, a parabolic dome is the most efficient, but it requires the largest area of the supporting ring and is the least efficient from this standpoint. In [7], the stress-strain state of the rotation shells with holes in the apex and the same overall dimensions, but different meridians, was compared. Comparative calculations of domes were performed to determine the strength parameters under external static [6], dynamic and explosive [8] loads. The choice of an optimal velaroidal shell on a rhombic flat base, the middle surface of which is formed by the motion of different plane curves with variable curvature, was studied in [9]. However, such comparative calculations have not been performed for hyperbolic, parabolic, elliptic and circular cylindrical roofs, although architects have tried to expand the list of surfaces of zero Gaussian curvature on a rectangular base by including developable surfaces [10]. Materials for analytical calculation of cylindrical shells defined by algebraic surfaces are considered primarily in this paper. At present time, virtually all problems of structural mechanics are solved using numerical methods. But there is another standpoint, such as that of the authors of monograph [6]: “In a reasonable combination of analytical and numerical methods with an understanding of the mechanical side of the problem under consideration”. Purpose of the study. To determine the strength parameters of two elliptic, circular, hyperbolic and parabolic cylindrical thin shells, having the same span and height, constant thickness, physical and mechanical characteristics of the structural material and subjected to the same static external load using the analytical calculation technique of the momentless shell theory. By comparing the obtained approximate calculation results, conclusions need to be drawn regarding the choice of an optimal cylindrical shell in the form of a second-order algebraic surface and regarding the applicability of the momentless shell theory to the calculation of the considered shells. It is also necessary to determine the position of the shells in the form of second-order cylindrical surfaces in the modern construction practice and in the solution of new problems, arising in the design of these shells for various needs of society. 2. Explicit and Parametric Equations of Second-Order Algebraic Cylindrical Surfaces and Their Coefficients of Fundamental Quadratic Forms Second-order algebraic surfaces include 3 degenerate surfaces: parabolic (Figure 1), elliptic (Figure 2) and hyperbolic cylindrical surfaces. These three surfaces are well known to geometers, architects [11], and engineers. Based on these surfaces, two more surfaces can be derived: circular cylindrical surface (Figure 3) and cylindrical surface with incomplete ellipse in cross-section (Figure 4). 2.1. Parabolic Cylindrical Thin Shell The canonical explicit equation of parabolic cylindrical surface (see Figure 1): x2 y y x= ( )=h 1-a2 , z z= . The parametric equations of this surface may be expressed as x = x u( )= au, y = y u( )= h(1 - u2), z = z v( )= lv, where u, v are the dimensionless independent parameters, -1≤ ≤u 1, 0 ≤ ≤v 1. Then, the coefficients of the first (A, F, B) and the second (L, M, N) quadratic forms of the surface will be the following: A2 = +a2 4h u2 2, F = 0, B = l L, = 2ah A/ , M = 0, N = 0, and the radius of curvature of coordinate line u will be R1 = A2/L = A3/(2ah). Hence, hereinafter, a coordinate grid in the lines of curvature is used, with coordinate line u coinciding with the directing parabola of the cylinder, and coordinate lines v coinciding with the rectilinear generators of the cylinder(Rv =R2 =∞). 2.2. Elliptic Cylindrical Thin Shell The canonical explicit equation of elliptic cylindrical surface: x2 y2 2 + 2 =1, z = z, a h where a, h are the lengths of the semi-axes of the ellipse. 2.2.1. Cylindrical Thin Shell With Semi-Ellipse in Cross-Section (Figure 2) The parametric equations of this surface can be represented as x = x u( ) = au y, = y u( ) = h(1- u2 )1/2 , z = z v( ) = lv, where u, v are the dimensionless independent parameters, -1≤ ≤u 1, 0 ≤ ≤v 1. The coefficients of the first (A, F, B) and the second (L, M, N) quadratic forms of the surface will be A2 = a2 + h u2 2/ 1 -( u2 ), F = 0, B = l L, = ah / A(1 - u2 )3/2 , M = 0, N = 0, and the radius of curvature of coordinate line u will be R1 = A L A2/ = 3 (1 - u2 3) /2/(ah). 2.2.2. Cylindrical Thin Shell with Incomplete Ellipse in Cross-Section The canonical explicit equation of the considered cylindrical surface (see Figure 4): 2 x2 y a2 a12 + h1 + -1 a12 =1, z = z. In the considered case it is necessary to set the length of the semi-axis of the full ellipse a1 ≥ a, and then to determine the length of the other semi-axis of the full ellipse h1: h1 = h . The parametric equations of the considered cylindrical surface can be represented in the following form: x = x u( )= au, z = z v( )= lv, a2 a2 y y u= ( )=h1 1-u2 a12 - -1 a12 . The coefficients of the first (A, F, B) and the second (L, M, N) quadratic forms of the surface will be A2 = +a2 h a u12 4 2 , F = 0, B = l, a14 1- a22 u2 a1 L = a h a3 1 1/ A a( 12 - a u2 2 )3/2 , M = 0, N = 0, and the radius of curvature of coordinate line u will be 3/2 = R1 = A aa h33 121 1 aa21 u2 . Ru - 2 2.2.3. Circular Cylindrical Thin Shell (See Figure 3) The formulas obtained in section 2.2.2 may be used for the circular cylindrical shell, but one must set a1 = R, h1 = R, then x2 + y + a22-hh2 2 = (h2 + a22 )2 = R2, 4h x = x u( )= au, z = z v( )= lv, = ( )= R2 -a u2 2 -a2 -h2 . y y u 2h The coefficients of the first (A, F, B) and the second (L, M, N) quadratic forms of the surface will be A2 = a R2 2 , F = 0, B = l, (R2 - a u2 2) a R3 2 a R2 L = A R( 2 - a u2 2 )3/2 = R2 - a u2 2 , M = 0, N = 0, and the radius of curvature of coordinate line u will be Ru = = =R1 R a2 +h2 . 2h The radius of curvature of coordinate line v will be Rv = R2 = ∞. 2.3. Hyperbolic Cylindrical Thin Shell The implicit equation of one branch of a hyperbolic cylindrical surface can be written in the following form (Figure 5): y2 x2 Figure 5. One branch of the hyperbolic cylindrical shell S o u r c e: made by S.N. Krivoshapko taking into account the introduced geometric notation shown in Figure 5. The parametric equations of the considered cylindrical surface can be represented in the following form: - =1, 2 2 c b where for a single considered branch b2 = a c2 2 . h(2c+ h) In this case, the explicit equation of the hyperbolic cylindrical surface defined in the xOy axes is written as 1/2 y = h + c - c2 + x h2 · 2( c + h) / a2 , x = x(u) = au, z = z(v) = lv, y = + -c h c2 + hu2(2c + h). The coefficients of the first (A, F, B) and the second (L, M, N) quadratic forms of the surface will be A2 = +a2 ch22+(hu2c2+(h2)c2+uh2), F = 0, B = l, ahc2 (2c + h) L = 3/2 , M = 0, N = 0. A c 2 + hu2 (2c + h) The radii of curvature of coordinate lines u and v are R1 = A2/L, Rv = R2 = ∞. 3. Three Groups of Governing Equations of Linear Theory of Thin Shells in Lines of Principle Curvatures Taking into account that the coefficients of the first fundamental form for cylindrical surfaces are B = l, A = A(u), then дB/дu =дB/дv = 0, дA/дv = 0, Rv = =R2 ∞, and therefore the equations of the general linear theory of thin shells will be simplified in some way. Equilibrium equations: ∂Nu + A∂S - Al Qu + AlX = 0, l ∂u ∂v Ru ∂S + A∂Nv + AlY = 0, ∂u ∂v Al Nu +l ∂Qu + A∂Qv - AlZ = 0, Ru ∂u ∂v ∂H - A∂Mv - AlQv = 0, l ∂u ∂v ∂Mu - A∂H + AlQu = 0, (1) l ∂u ∂v where the positive directions of bending and twisting moments, internal normal and shear forces, external surface load are shown in Figure 6. Figure 6. Positive directions of forces (a) and moments (b) S o u r c e: made by S.N. Krivoshapko Constitutive equations: Nu = C (ε +νεu v ), Nv = C (ε +νεv u ), S = C (1 - ν ε) uv/2 , Mu = -D(κ +νκu v ), Mv = -D(κ +νκv u ), H = -D(1 - ν κ) uv, (2) where С= Eh / 1-( ν2 ), D = Eh3/ 12 / 1- ( ν2 ) , h is the shell thickness, ν is the Poisson’s ratio, E is the modulus of elasticity of the shell material. εu, εv, εuv, κu, κv, κuv are the components of the membrane and flexural strains. The contradictions introduced by formulas (2) into the shell theory were first pointed out by V.Z. Vlasov. However, in the opinion of V.V. Novozhilov, all these contradictions do not exceed the errors introduced into the shell theory by the initial assumptions of the thin shell theory. Geometrical equations: ε =u 1 ∂uu - uz , ε =v 1∂uv , εuv = 1 ∂uv +1∂uu , A ∂u Ru l ∂v A ∂u l ∂v κu = 1 ∂ 1 ∂uz + uu , A u∂ A ∂u Ru 1 ∂2 κv = 2 ∂vu2z , l 1 ∂ 2 ∂u u 2κuv = z + u , l v∂ A ∂u Ru (3) where uu, uv, uz are the displacement components. 4. Determination of Internal Forces of Thin Cylindrical Shells in the Form of Second-Order Algebraic Surfaces Using Momentless Shell Theory In this section, the linear theory of thin rigid shells will be considered in the approximate momentless setting, when it is assumed that internal bending and twisting moments, and hence shear forces, can be neglected. The assumption of uniform stress distribution over the thickness of the shell led to the emergence of the momentless theory of rigid shells. Those who are interested in the momentless thin shells that only accept tensile internal forces and change their shape under external forces can explore the materials of review article [12] with 80 sources used. Assuming that a rigid shell satisfies the requirements of momentless state, then the first three general equations of equilibrium (1) of the shell can be used excluding the shear forces: ∂Nu + A∂S + AlX = 0, l ∂u ∂v ∂S + A∂Nv + AlY = 0, l ∂u ∂v Nu - Z = 0, Ru from which it follows that Nu = ZRu, S = -l 1A ∂∂Nuu + X dv + f u1( ) =- lv 1A ∂∂Nuu + X + f u1( ), Nv = - l ∂S dv - lYv + f2( )u . (4) A ∂u Thus, the problem of determining the internal normal and shear forces by the approximate momentless shell theory is a statically determinate problem for all shells in the form of second-order cylindrical surfaces. Arbitrary functions of integration f1(u), f2(u) and df1(u)/du are determined by satisfying boundary conditions according to the momentless shell theory. In further calculations of all five cylindrical shells defined by second-order algebraic surfaces, it is assumed that a = 5 m, h = 4 m, and the external surface load components X, Y, Z are X = X u( ) = qsinφ= q tgφ , Y = 0, (1+ tg2φ) Z = Z u( ) =-qcosφ=-q 1 , (1+ tg2φ) where q is the distributed load of self-weight kind, φ is the Figure 7. Geometrical parameters, external loading, angle between the tangent to coordinate line u and the fixed and hinge supports of the considered shells Ox axis (Figure 7). S o u r c e: made by S.N. Krivoshapko It is assumed that the hinges are connected to an absolutely rigid in-plane and flexible out-of-plane support at the ends v = 0, v = 1. For open cylindrical shells it is impossible to satisfy the boundary conditions on the straight edges of the contour. Then, formulas (4) give Nv = l v2 2 d 1 dNu + X - l v df1 + +C f2( )u , 2A du A du A du from where С = 0, f2 (u) = 0, since Nv = 0 at v = 0. Then, from the boundary condition of Nv = 0 at v = 1, the following is obtained: df1( )u = l d 1 dNu + X , du 2 du A du from where f1( )u = l 1 dNu + X . 2 A du By substituting the obtained expressions for f1(u), df1/du and f2 (u) into formulas (4), the final formulas for calculating the internal membrane forces under the considered boundary conditions are written: Nu = ZRu, S = l 1 dNu + X (1- 2v), 2 A du l v d2 1 dNu Nv =- + X (1-v). (5) 2A du A du 4.1. Parabolic Cylindrical Thin Shell The following was obtained previously for the cylindrical parabolic surface: A2 = a2 + 4h2u2, R1 = Ru = A3/(2ah). In the previous section, the formuals for calculating sinφ, cosφ were presented, thus X = 2hu q, Y = 0, Z = - qa. A A According to formulas (4): q a( 2 + 4h u2 2 ) The boundary condition of Nv = 0 at v = 0 gives f2(u) = 0. The boundary condition of Nv = 0 on the opposite end of the shell v = 1 gives df1 ( )u =-lqha2 1 , du (a2 + 4h u2 2 ) from where u f1( )u =-lqh+C, a2 + 4h u2 2 where С = 0 when S(u = 0) = 0. Finally, for the parabolic cylindrical shell: q a( 2 + 4h u2 2 ) Nu =- , 2h lqhu(2v -1) S = , a2 + 4h u2 2 l qha v2 2 (1-v) Nv = 2 . (a2 + 4h u2 2 ) 4.2. Elliptic Cylindrical Thin Shell 4.2.1. Cylindrical Thin Shell with Semi-Ellipse in Cross-Section All geometric parameters of the elliptic cylindrical middle surface with a semi-ellipse in the crosssection (see Figure 2) are given in Section 2.2.1. In addition to this X = q uh , Y = 0, Z =-q a . A 1-u2 A According to formulas (5): 3 N = - q A2(1-u2)2 , u h Nv = ql v v 2 ( -1() 1-u2) 2 3a4 -9a u2 2 (a2 - h2 )- a h2 2 + 6u4 (a2 - h2 )2 . 2h a2 -u2 a2 -h2 4.2.2. Cylindrical Thin Shell with Incomplete Ellipse in Cross-Section All geometrical parameters of the elliptic cylindrical middle surface with an ellipse fragment in the crosssection (Figure 4) are given in Section 2.2.2. In addition to this Z =-q, Ru = A3 (1-βu2 )3/2 . h a1 β According to formulas (5): 2 2 Nu =- a2 +β βu2 ( h12 - a2 ) , βh1 βh1 S = lqu(1--2βvu)2 3A2(1-βu2) -βh12 = 2h A1 1 = 2h1 a2lqu+β β(1u2-(2vh)12 - a2 ){3 a2 +β βu2 ( h12 - a2 )-βh12 }, Nv = ql v v22h A(1 -1) 3 aa22++2β ββ βuu22(( hh1212--aa2)) - a β(a h2 212 ) 3/2 . 2 2 +β βu2 h1 -a2 The obtained formulas allow to the calculate the elliptic cylindrical shell with a semi-ellipse in the crosssection (see Figure 2). For this purpose, in the formulas of Section 4.2.2, it is necessary to take β = 1, a1 = a, h1 = h. 4.2.3. Circular Cylindrical Thin Shell In this case, it is necessary to take R = 5.125 m, dy = tgφ= x , sinφ= x = au , cosφ= R2 - x2 = R2 -a u2 2 . dx R2 - x2 R R R R The boundary condition of Nv = 0 at v = 0 gives f2(u) = 0. The boundary condition of Nv = 0 on the other end of the shell v = 1 gives df1/du = alq/R, and f1(u) = alqu/(R) + C. Let S = 0 at u = 0, then С = 0. Substituting the obtained results into the equations for determining the internal forces according to the momentless theory allows to write the following: Nu =-q R2 -a u2 2 =-qRcosφ, S = laq u(1- 2v), R Nv =l q2 R2 -2a u2 2 v v( -1 .) R For the calculation of long circular cylindrical shells, it is advisable not to use the momentless theory. When using the momentless theory, it is necessary to fulfill the following relation [18; 12]: l R R = h . 4.3. Hyperbolic Cylindrical Thin Shell All geometrical parameters of the hyperbolic cylindrical middle surface (see Figure 5) are given in Section 2.3. It is necessary to take into account that huγ a c2 +γhu2 X = q , Y = 0, Z =-q, a c2 2 +γhu2(a2 +γh) A3 (c2 +γhu2 )3/2 A = , γ= 2c + h R, u = 2 , ah cγ where с > 0 is an arbitrary number; 0 < cosφ ≤ 1, i.e. 0о ≤ φ < 90о. According to formulas (5): N =- qaA =- q c2 +γhu2 a c2 2 +γhu2 (a2 +γh) , u γ 2 L h c S =- 2luqc2 (1- 2v)3 a c2 2a c2 2+γhu+γ2hu(a22(+aγ2h+) γh+)c2γh, Nv = q l v2 (1-v) c2 +γhu2 2 a c2 4 (3a2 +γh)+ 9γha c u2 2 2 (a2 +γh)+ 2c2 a c2 2 +γhu2 (a2 +γh) + γ6 2 2 4h u (a2 +γh)2 . 5. Comparison of Results of Momentless Analysis of Five Cylindrical Thin Shells in the Cross-Sections of the Shells (in the Longitudinally Middle Section of the Shells, i.e. at v = 0.5) As noted earlier, all shells are assumed to have the same overall dimensions, i.e. h = 4 m, a = 5 m, l = 20 m. In this case, the formulas for calculating the internal membrane forces per unit length of the coordinate line take the following form: - parabolic cylindrical thin shell: Nu =-8q(0.39+u2 ), S= 10uq v(2 -21), Nv = (9.77v(1-2v)2) q; 0.39+u 0.39+u - cylindrical thin shell with semi-ellipse in cross-section: Nu =-2.25q(2.78-u2) 1-u2, = 49.2 - 22.5u2 qu(1- 2v), qv v( -1) 1-u ( 4 2 ) (2.78-u2) - cylindrical thin shell with incomplete ellipse in cross-section (a1 = 5.98 m, h1 = 8.84 m, β = 0.7): Nu =-q(4.04+3.36u2) 1-0.7u2, S = 14.1qu(1- 22v) 0.37( +u2 ), 1+ 0.83u v = q18.35v v( -1) 12-)20.7u2 (1+9.22u2 +5.1u4); N (1+ 0.83u - circular cylindrical thin shell (R = 5.125 m): Nu =-q R2 - 25u2 =-qRcosφ, S= 1000qu(1- 2v) v / R2; - hyperbolic cylindrical thin shell (c = 2 m, γ = 8 m, b = 0.88 m; 0.68 ≤ cos φ ≤ 1, i.e. 0о ≤ φ ≤ 47о): Nu =-1.56q 1+8u2 1 18.24+ u2 , ( )107 1( + 0.17u2) S =-qu 1- 2v , 1 18.24+ u2 Nv = 107q 1+8u22 v(1-v)(1+ 38.36u2 + 466u4 ). (1 18.24+ u2 ) The values of shear and normal forces per unit length of coordinate lines u and v in the z = l/2 section of cylindrical shells are presented in Tabl. Momentless analysis results v = 0.5 (z = l/2 = 10 m section) No. Shell type Force/q u = 0 u = ± 0.4 u = ±1 (support along the length of the shell) 1. Parabolic cylindrical thin shell Nu/q S/q Nv/q -3.12 0 16.05 -4.4 0 8.07 -11.12 0 1.27 2. Cylindrical thin shell with semi-ellipse in cross-section Nu/q S/q Nv/q -6.25 0 -29.4 -5.42 0 -24.05 0 0 0 3. Cylindrical thin shell with incomplete ellipse in cross-section Nu/q S/q Nv/q -4.04 0 -4.59 -4.3 0 -8.8 -4.05 0 -11.6 4. Circular cylindrical thin shell Nu/q S/q Nv/q -5.125 0 -19,51 -4.72 0 -18 -1.13 0 -4.28 5. Hyperbolic cylindrical thin shell Nu/q S/q Nv/q -1.56 0 26.75 -9.23 0 50.14 -90 0 109 S o u r c e: made by S.N. Krivoshapko Using the momentless theory, normal forces Nu in a hyperbolic cylindrical shell are determined by the first formula (5): Nu = ZRu = -qcosφ Ru, where the radius of curvature Ru = 1.56 m at u = 0, but Ru = 131.8 m on the contour u = ±1, therefore this normal force takes on a larger value Nu = -90q. The momenless theory does not allow to specify a boundary condition on these edges. Taking into account that at the apex u = 0, φ = 0o, cosφ = 1, this results in Nu/q = Ru, i.e. the values in the u = 0 column of Table in the first row for Nu show the radius of curvature of a coordinate line Ru in its apex. All the above formulas in Sections 4 and 5 should be used with caution. Biderman V.L. [14] found that for open circular cylindrical shells, the momentless theory can be used in exceptional cases if the membrane boundary conditions on longitudinal edges are automatically satisfied. The scope of application of the momentless theory is limited to short shells with respect to shell radius R (see Section 4.2.3). Apparently, for non-circular cylindrical shells, the applicability of the momentless theory, by analogy with circular shells, can be limited by the relation: l a a = h. Although in some cases the formulas for the momentless theory provide acceptable results, and the possibilty of their application needs to be confirmed by more accurate calculation methods. In many cases, it is possible to incorporate the solutions of the momentless theory equations into the approximate solution of the general equations by splitting the latter into the momentless stress state equations and the edge effect. In monograph [6], a similar analysis is performed on four cylindrical roof shells under dead load according to the momentless theory and it is concluded that the most optimal shell is the cylindrical shell with an incomplete ellipse in the cross-section, since this shell transfers a relatively small normal load to the supporting beams. 6. The System of Design Equations of the General Theory of Thin Cylindrical Shells in Terms of Displacements Five equations of equilibrium (1), six constitutive equations (2) and six geometric equations (3) were derived for the calculation of thin elastic cylindrical shells using the general linear theory. Substituting the geometric equations (3) into the constitutive equations (2), the following is obtained: 1 ∂uu Nu = C - uz + ν ∂uv ; A ∂u Ru l ∂v Nv =C 1∂uv -ν uz +ν∂uu ; l ∂v Ru A ∂u S = 1-ν2 C 1A ∂∂uuv +1l ∂∂uvu ; Mu = -D 1A u∂∂ 1A ∂∂uuz + uRuu + lν2 ∂∂2vu2z ; Mv = -D 1 ∂2uz + ν ∂ 1 ∂uz + uu ; l2 ∂v2 A u∂ A ∂u Ru H =-D(1-ν) z + u . l v∂ A ∂u 2Ru The values of the shear forces are determined from the last two equilibrium equations (1): (6) Qu = 1 ∂H - l ∂Mu , Qv = 1 ∂H - A ∂Mv . (7) 1 ∂ 1 ∂u u l ∂v A ∂u A ∂u l ∂v From the last three equations (6) and equations (7), it can be seen that internal moments Mu, Mv, H and shear forces Qu and Qv do not depend on displacement uv. Substituting the expressions for internal forces and moments into the first three equilibrium equations (1) results in a system of three partial differential equations in terms of displacements of the middle surface of the eighth order, which are usually written in a reduced form: L u112 u + L u122 v + L U133 z + AlX = 0 , L u221 u + L u222 v + L U123 z + AlY = 0 , L331uu + L u132 v + L U433 z - AlZ , where … are differential operators, the upper index of which shows the maximum order of the corresponding partial derivative. In a more extended form these three equations can be expressed as ∂∂u Al ∂∂uuu +ν∂∂uvv -l Ruzu + (1-2ν) ∂ ∂∂v ∂uuv + Al ∂∂uvu + +12hR2u (1-l2ν) ∂ ∂∂u v3uz2 + (1-2νl)A uRuu -l ∂∂u A u 1 ∂∂ 1A ∂∂uuz + Ruuu + AlC X = 0, 12A l ∂uu + ν ∂uv -l uz + h R2 u A ∂u ∂v Ru +∂∂u A u A ul ∂∂ 1 ∂∂ 1A ∂∂uuz + Ruuu + lAν ∂3uz2 -(1-l ν) ∂∂v22 A2 ∂∂uuz + Ruuu + ∂ ∂u v +1 ∂∂22 A2 ∂∂2vu2z + ν ∂∂u 1A ∂∂uuz + Ruuu - AlD Z = 0, l v l (1-ν)l ∂ 1 ∂uv + ∂ A ∂uv + (1+ν) ∂uu -νA uz + AlY =0. (8) 2 ∂u A ∂u ∂v l ∂v 2 ∂u Ru C Thus, the equilibrium equations (1) of cylindrical shells are expressed in terms of displacements uu, uv, uz of the middle surface. The system of three partial differential equations (8) with variable coefficients is obtained. The system is of the eighth order. The use of geometric equations (3) guarantees the satisfaction of the strain compatibility conditions in the mid-layer of the shell. The system of three differential equations in displacements for developable shells, i.e., for shells of zero Gaussian curvature defined in a non-orthogonal conjugate coordinate system, is given in monograph [15], but for the case of zero Poisson's ratio (ν = 0). Taking ν = 0 and the average coefficient of the first quadratic form F = 0, equations (8) can be reduced to the form presented in monograph [15]. In general form, equations (8) will probably never be applied to the calculation of cylindrical shells defined in lines of curvature. These equations are derived here only to illustrate the displacement method for cylindrical shells in the form of analytic surfaces. Their application to circular cylindrical shells, the geometry of which is described in Section 2.2.3, will be shown only in Section 7. 7. Existing Approaches to the Calculation of Thin Elastic Cylindrical Shells Using Analytical Methods During the “Golden Age of Shells” It is commonly believed that the “Golden age of shells” fell on 1924-1970s. It was at this time that reliable analytical methods for the calculation of thin elastic shells emerged, which are related to the requests of the construction practice for shell design and making it possible to describe the behavior of shells under external static loads more or less accurately. In addition to specific theories of cylindrical shells (V.Z. Vlasov, L. Donnel, A.A. Umansky, H.M. Mushtari, S.M. Feinberg), V.V. Novozhilov [16], at that time, proposed a complex resulting equation that includes all specific theories of cylindrical shells, including those strenghthened by stiffeners. 7.1. Circular Cylindrical Thin Shell The largest number of analytical methods for the calculation of thin cylindrical shells based on the general linear theory of elastic shells has been proposed for circular cylindrical shells. The equations and formulas of the theory of circular cylindrical shells are mainly written in terms of z and s or ξ = z/R and θ = s/R, where s is the distance along the middle surface from the initial position to the corresponding point. As a result, A = B = 1 in equations (1) - (3), and Ru = R = const. V.Z. Vlasov [17] introduced function Ф(ξ,θ), through which displacements uu, uv, uz are expressed. This allowed to turn the first two equations of equilibrium (1) into identities, and the third equation of equilibrium took the form of a partial differential equation with respect to the introduced function Ф(ξ,θ). This function was expressed as a double trigonometric sine series, which corresponds to a shell with hinged supports along the contour. The extension of the solution to shells with other boundary conditions is possible, but requires cumbersome calculations [18]. In the calculation of open circular shells of length l, the load and displacements were decomposed into Fourier series with period l: ∞ n zπ ∞ n zπ uv = uvn ( )s cos ; uu = uun ( )s sin ; n=0 l n=1 l ∞ n zπ uz = uzn ( )s sin ; n=1 l ∞ m zπ Y = Ym( )s cos ; m=0 l ∞ n zπ ∞ n zπ Xm = Xm( )s sin ; m=1 l Z = Zm ( )s sin , m=1 l where the known decomposition coefficients Xm, Ym, Zm can be determined using the formulas given by V.Z. Vlasov [17]. Substituting these values into equations (8) for open circular cylindrical shells, three simple differential equations with respect to uun, uvn, uzn are obtained. In z = 0 и z = l planes, only the free support conditions will be satisfied, i.e., uu = uz = 0 and Nv = Mv = 0. For a closed circular cylindrical shell loaded with axisymmetric load, the solution can be represented as a single cosine series, which leads to an eighth order ordinary differential equation with respect to variable ξ = z/R. Its solution is sought in the form of the sum of hyper-geometric functions of ξ. Lurye A.I. [19] proposed the calculation of the optimal strenghthening of a circular hole in a circular cylindrical shell. In [20], solutions in the framework of technical and semi-integral shell theories using the method of separation of variables are demonstrated. The solution methods and simplifying assumptions for closed and open circular shells are different. In this work, it is noted that in many cases calculations for building shells give excessive accuracy from the practical standpoint, therefore it was possible to introduce a number of simplified versions of the shell theory, which allowed to solve a certain class of problems at that time. For example, V.Z. Vlasov [17] and his students introduced additional static and geometric hypotheses, and A.L. Goldenweiser [21] and V.V. Novozhilov [22] and their followers revealed conditions under which some terms of the design equations turn out to be insignificant. In the semi-momenltess theory of cylindrical shells, it is assumed that Mv = H = Qv = 0 and εu = εuv = 0, due to which the design equations of shells are simplified. The semi-momentless theory is discussed in more detail in [20; 23]. In [20] it is argued that one can introduce additional assumptions that do not have a noticeable effect on the behavior of closed cylindrical shells, in particular, one can assume that the Poisson’s ratio is zero (ν = 0) or one can discard relatively small values compared to a unit. W. Flügge [23] gave recommendations for the calculation of multi-wave reinforced concrete circular cylindrical shells with flexible or high sides. Different assumptions for the calculation of shallow and non-shallow short open shells and mediumlength and long shells were proposed. During this period, many researchers studied buckling of circular cylindrical shells with closed contour under compressive and shear loads applied at the ends [23] and under pure shear [23]. The theory of multilayer shells also appeared at this time. The main directions in the development of the theory of multilayer shells were presented in 1972 by E.I. Grigolyuk and F.A. Kogan, as described in article [24]. In this article, the first works on the calculation of cylindrical shells are mentioned. 7.2. Elliptic Cylindrical Thin Shell In [20], it is stated that cylindrical thin shells of non-circular cross-section were generally supported by flexible diaphragms at the transverse ends and the longitudinal edges were strenghthened with edge elements. The performance of such a system depends largely on the l/(2a) ratio. In long shells (l/(2a) > 4), the external load is taken up almost exclusively by the Nu, Nv, S forces. In medium-length shells (4 > l/(2a) > 1), in addition to these membrane forces, bending moment Mu is significant, while bending moment Mv and torque H play a minor role and may not even be taken into account. In short shells (l/(2a) < 1) the effect of bending moment Mu is relatively small, but the role of bending moment Mv increases. The mentioned features of shell behavior allow simplifying their calculation by replacing the actual structure with a simplified model. For example, short shells can be calculated as beams of length l. In contrast to circular cylindrical shells, the exact calculation of non-circular shells was extremely difficult in the mentioned period of time [20]. Most of the approximate methods were based on the idea of replacing the shell by prismatic folds. Basically, the methods differ from each other by the hypotheses about the degree of influence of certain internal forces, moments, or strains on the behavior of the considered shell. Studies on the use of computers for the calculation of the approximating folds began to appear [25]. 7.3. Parabolic Cylindrical Thin Shell The contents of the first two paragraphs of the previous Section 7.2, which deals with elliptic cylindrical shells, are fully consistent with parabolic cylindrical shells as well. In the considered time interval, studies of multilayer shallow parabolic cylindrical shells, taking into account plastic deformations, have begun, e.g. [26]. 7.4. Hyperbolic Cylindrical Thin Shell Hyperbolic cylindrical surface has been known for a long time, but only geometers have used it in their research. Not a single structure has been built. Not a single scientific article devoted to the strength analysis of a shell in the form of this surface has been found. 7.5. Conclusion of Section 7 E. Freyssinet, a French bridge engineer, designed and built the first parabolic cylindrical shell, which is technically a system of parabolic arches, to cover a 30-meter span of a factory in Monluson (France) in 1905. In Russia, long and short cast-in-situ cylindrical shells have been used in industrial buildings since 1928. For example, the first reinforced concrete cylindrical shell was erected under the direction of Prof. M.A. Novogorodsky over a water tank in Baku in 1925. After, cylindrical shells were used for the building of the Kharkov post office (1928), Moscow automobile depot (1929) and Rostov agricultural machinery plant (1931) [27]. A large number of circular and elliptic cylindrical shells made of reinforced concrete and brick constructed for the roofs of industrial buildings in 1930-1970 in Italy are presented in review article [28]. The advent of reinforced concrete and public demand for large-span structures for warehouses, hangars and public buildings led to the need of static calculation methods for these rigid shells. It was during the “Golden age of shells” that the fundamental discipline - shell theory - emerged, initial hypotheses and assumptions were accepted, resolving equations adequately describing the behavior of shells were formed, simplifications of the equations of the shell theory were proposed, and qualitative studies of a number of specific problems appeared. 8. Information on Publications After 2000 about Calculation of Cylindrical Shells with Second-Order Algebraic Middle Surfaces and Their Application in Architecture and Construction 8.1. Application of Cylindrical Shells with Second-Order Algebraic Middle Surfaces Many review [3] and scientific [11] papers, monographs [29] and reference books [30] are devoted to the application of cylindrical shells, structures and cylindrical surfaces in architecture, construction and mechanical engineering. In all these works, there are sections on structures and products in the form of second-order cylindrical algebraic surfaces. 8.2. Information on Publications After 2000 About Calculation of Cylindrical Shells With Second-Order Algebraic Middle Surfaces As already mentioned in the Introduction, in the last quarter of a century, almost no studies have been conducted to determine the optimal cylindrical shell among the five cylindrical shells with the same overall dimensions and boundary conditions and subjected to the same external load. There is only paper [32], where the stability of the computational algorithm is studied when one type of cylindrical surface is replaced with another, which can simplify the computation. It is argued that the replacement of hyperbolic and elliptic cylindrical shells with a parabolic shell simplifies the calculations. All other scientific articles are devoted to studies on the calculation of specific cylindrical shells of the same shape for strength, stability, and dynamics, i.e., each article is devoted to only one type of closed or open, thin or thick cylindrical shells. The most complete review of works on the calculation of three degenerate cylindrical shells in the form of second-order algebraic surfaces is given in article [11], which contains 14 references on the subject. Almost all new studies are devoted to circular cylindrical shells. It is noted in [33] that, despite the extensive literature on the development of shell theory, the comparative analysis of analytical and numerical solutions is not sufficiently covered. One of the first works devoted to this direction in the study of natural frequencies and vibration modes of closed circular cylindrical shells is [33]. Very interesting results have been obtained in the experimental study of the behavior of masonry cylindrical roofs under the settlement of one of the longitudinal rectilinear supports, due to which longitudinal cracks appear [34]. Many papers claim that the momentless theory provides acceptable results for thin shells with a ratio of thickness to internal diameter equal to 0.1 and having the shape of shells of rotation [13]. Examples of using the momentless theory for cylindrical shells subjected to axisymmetric loading are given mainly in textbooks for students of various fields [13; 35; 36]. In monograph [6], the maximum stresses and displacements in an elliptical tube caused by uniform normal pressure are determined. It is found that the maximum normal stresses in the direction of the ellipse will occur at the ends of the minor semi-axis, the shear stresses will reach the highest value at the shell supports. The normal axial stresses reach their maximum in the middle of the span. The authors of monograph [6] point out that the results obtained by the momentless shell theory are not applicable to long pipes. Similar conclusions were made in Section 5 and in the “Results” section of the presented paper. With a major excess of studies on open or closed circular cylindrical shells, there are works whose authors believe that non-circular shells are an interesting object for research, for example, elliptic, in which the values of semi-axes Rx and Ry act as variable parameters. At their specific values, it is possible to achieve a decrease or increase in the minimum natural frequency relative to the cylindrical configuration without changing the mode of vibration. Paper [37] shows the variation of natural frequencies of vibration depending on the type of boundary conditions set at the edges of an elliptic shell and the frequencies of vibration of a cantilever steel non-circular shell at different lengths of one semi-axis. The aeroelastic stability of closed cylindrical shells with an elliptical cross-section streamlined from the outside by a supersonic gas flow is investigated in article [38] in a three-dimensional setting using the mathematical model and its numerical realization based on the finite element method. Using the developed program, the influence of kinematic boundary conditions, dimensions, and the ellipticity parameter on the critical characteristics of buckling in the form of flutter was analyzed. It is demonstrated that in the cantilever case, there exist such ellipse semi-axis ratios that provide higher aeroelastic stability boundaries as compared to a similar circular configuration. Finite element modeling and buckling analysis of a composite closed elliptic cylindrical lattice shell under axial compression and lateral bending were studied for spacecraft in [31]. Here, the natural vibration modes of the considered product are obtained and it is found that the structure and geometry of the object affect the magnitude of the critical load and the natural vibration modes. There are studies on the determination of strength of parabolic cylindrical shells. For example, FEM was used to numerically investigate the behavior of a straight parabolic cylindrical shell with supports at four corners, subjected to dynamic loading, when increasing the shell height or its thickness [39]. The tabular and graphical results of the study were obtained using the standard SAP 2000 software. A numerical study, performed together with the experimental one, proves the effect of cracks in parabolic cylindrical reinforced concrete shells on the static and dynamic behavior of the studied shells [40]. The obtained results will help to determine the choice of shell repair times. In [41], it is argued that parabolic cylindrical panels made of memory polymers are very useful in spacecraft that are subjected to dynamic (vibration) loads. One of the first attempts to select an optimal shell among shells with different cross-sections was made in [42], where single-layer shells with a triangular lattice with circular, catenary and parabolic cross-sections under a distributed load such as self-weight were considered. It is established that the shells of greatest mass will have a cross-section in the form of a catenary curve. The same article presents graphs of the relationship between the mass of the three considered lattice shells and the height. The parabolic shell will be the most economical in terms of material. If the maximum deflection of the structure is taken as the main criterion, the shell in the shape of a catenary curve has an advantage over the circular and parabolic shapes. An interesting approach to the calculation of long open cylindrical building structures of non-circular cross-section is proposed in [43], where the stress-strain state of a structure obtained using a beam model with supports at two ends and using the equations of linear shell theory is studied. 8.3. Conclusion of Section 8 The presented brief review of the studies of cylindrical roofs and closed circular and elliptic shells carried out over the last 25 years shows a decline in interest in their investigation. The only exceptions are closed circular cylindrical shells, which have found applications in underground structures [44] and in mechanical engineering [45]. However, there are still unsolved problems. For example, as mentioned above, there is still no answer to the question of the optimal shell with respect to the five cylindrical shells shown in Figures 1-4 and Figure 5, which are subjected to some static or dynamic load. There are practically no studies on the determination of radial displacements of a hollow cylinder under the combined action of internal pressure, centrifugal forces, and temperature effects, which is very important in the manufacture of products in the field of mechanical engineering [46]. To determine the strength and dynamic characteristics of cylindrical objects, numerical methods and standard computer systems are almost always used. Analytical methods are developed only for individual cases. The form of cylindrical shells, shell structures and cylindrical external outlines of buildings, defined by second-order algebraic surfaces, is used very widely by architects and civil and mechanical engineers in the XXI century. A brief review of studies on the strength, stability, and dynamics of cylindrical shells conducted over the last 25 years confirms the conclusions of V.V. Novozhilov and other scientists in the field of mechanics [6] that the “Shell Theory” fundamental discipline becomes one of the sections of applied mathematics. In this case, the theory is used only to write out the initial system of equations. An investigation of recent publications shows that the calculation of cylindrical shells is currently associated with the consideration of multilayered walls, geometrical and physical nonlinearities, new structural materials [41] and new types of external static, dynamic [47] and thermal [46] loads, sometimes after the adoption of refining hypotheses [48]. After circular cylindrical shells, the most popular are parabolic cylindrical shells, especially as roofs for warehouses, industrial and exhibition halls, and elliptic cylindrical shells for mechanical engineering projects. Open cylindrical shells play a much more important role in construction. Section 8 summarizes only the main directions of research on cylindrical shells in the form of secondorder algebraic surfaces and which have been presented in publications over the last 25 years. Additional information on the topic can be found in review and scientific articles [11; 49; 50; 51]. 9. Results A comparative analysis of five thin shells under a static load of self-weight type according to the momentless shell theory has been performed. 1. Three formulas (4) in explicit form are obtained for the determination of internal membrane forces in the five shells defined by second-order algebraic surfaces with boundary conditions in terms of forces, acceptable for the momentless theory of thin shells. 2. Three formulas (5) in explicit form are obtained for the determination of internal membrane forces in the five shells defined by second-order algebraic surfaces with two specified boundary conditions in terms of forces at two opposite ends for normal forces in the form Nv = 0 involving the momentless theory of shells. 3. The main geometric characteristics of the five second-order middle algebraic surfaces are presented. 4. It is shown that in the framework of the approximate momentless theory of shells, the parabolic cylindrical thin shell (Figure 1) and the cylindrical thin shell with an incomplete ellipse in the cross-section (Figure 4) perform best under external load similar to self-weight, and the greatest internal forces arise in the cylindrical thin shell with a semi-ellipse in the cross-section (Figure 2). 5. It has been established that in cylindrical shells, the axial normal forces Nv determined using the momentless theory depend significantly on the square of the shell length l, which is clearly seen from the third formulas of systems (4) and (5). This is contrary to common sense, so it is necessary to agree with the conclusions of many scientists in the field of mechanics and the author, that the application area of the momentless theory is limited to short cylindrical shells, or to take into account that long shells behave according to the beam model with hinged supports at the ends [23]. However, cylindrical shells cannot be calculated as beams with curvilinear contour, because they generally do not follow the law of plane sections and do not obey the hypothesis of invariability of the cross-sectional shape. Deplanation of cross-sections occurs. 6. The known statements of the momentless shell theory that the forces determined in the shell regions farthest from the supports, for which boundary conditions cannot be set, are the closest to the true internal forces in shells, are confirmed. In the considered cases, these regions are located around the coordinate line u = 0. The following has been done with respect to the general linear theory of shells: 7. A system of three partial differential equations (8) of the eighth order with respect to the displacements of the middle surface of a cylindrical shell defined in the curvilinear coordinate system u (-1 ≤ u ≤ 1), v (0 ≤ v ≤ 1) is derived for the first time for possible future application. 8. It is shown that the internal bending moments Mu, Mv and torque H as well as shear forces Qu, Qv do not depend on displacements uv along the straight cylinder generators. 9. It was found that no studies on the calculation and application of hyperbolic cylindrical roofs have been published after 2000. 10. Conclusion With the modern development of numerical calculation methods and computer technology, most approximate methods of calculation of curvilinear civil and mechanical engineering structures of shell kind, including the momentless theory of shells, are practically not used. However, virtually all textbooks on structural mechanics of shells have sections devoted to the momentless theory of shells. The materials contained in this article can be used in the teaching process to illustrate the applicability of approximate methods of shell analysis. In the practice of thin shell design, the above materials are unlikely to be used, if only for preliminary assignment of thicknesses of cylindrical shells. All well-known scientists in the field of mechanics related to the design of thin shells state that when selecting the shell shape, one should strive for the shell behavior to be close to momentless. Having the formulas for calculating the values of internal forces, it is possible to determine the boundary conditions at the edges of the shell in terms of displacements and forces, by satisfying which it is possible to obtain a momentless stress-strain state in the considered shell. However, the required boundary conditions can be created only theoretically, but it is necessary to strive for it. It may seem that the momentless theory of cylindrical shells is practically useless due to a large number of constraints. However, in a number of cases it allows to obtain simple and sufficiently accurate solutions. In particular, it is applicable to the calculation of shells supported by spandrels, since the length of the compartments between them is set relatively small. The momentless stress state is taken as the basic stress state, and the stability of cylindrical shells in the basic stress state is investigated.

About the authors

Sergey N. Krivoshapko

RUDN University

Author for correspondence.
Email: sn_krivoshapko@mail.ru
ORCID iD: 0000-0002-9385-3699
SPIN-code: 2021-6966

Doctor of Technical Sciences, Professor of the Department of Construction Technology and Structural Materials, Academy of Engineering

Moscow, Russia

References

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