Analytical Calculation of Momentless Conical Shell with Elliptical Base

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1. Introduction The momentless theory of the analysis of rigid thin shells is an approximate theory, but in some cases it gives fairly accurate values of tangential internal forces, which can be used for preliminary analysis of the stress state of a thin shell [1]. This data can be useful, for example, when assigning the thickness of the shell. It has been established that the momentless theory of shells yields reasonable results in comparison with exact results when fulfilling well-known requirements for supports, the type of external loads, boundary conditions nd the shape of the shell [2; 3]. Research on the momentless theory of rigid thin shells was being actively developed especially until the 1980s [4]. Then came the fascination with numerical methods for calculating moment shells using linear and physically or geometrically nonlinear calculation theories. However, in many monographs of famous scientists of the last years of the 20th century, there was a necessary chapter devoted to the momentless theory of calculating thin shells [1-3; 5], and textbooks contained information about the application potential of this theory [6]. The momentless theory is used in the study of stability of thin shells [7]. At the present day, the amount of studies on the momentless theory of shells has been significantly reduced, but they are available [8]. Generally, a comparative analysis of the results obtained by the momentless theory and by more precise methods using numerical methods is carried out [9]. There are studies containing comparative analysis of the calculation results obtained using the momentless theory of shells and the results obtained experimentally, for example, in the process of designing the conical shell foundations [10]. The conical shape of shells is currently widely used in civil [11] and mechanical engineering [12]. Shells in the form of right thin (Figure 1) and thick [13] circular cones have been applied in most cases, but conical shells with elliptical base are also used [14-16]. Moreover, truncated conical shells with elliptical base have found application even in medicine [17]. The purpose of this work is to obtain analytical expressions for determining tangential internal forces in an elliptical conical shell according to the momentless theory. The symmetric equation of a right elliptical conical surface may be expressed in the following form: x2 y2 z2 2 +W 2 - =T 2 0, L where T is the height of the conical surface. Parametric equations of the elliptical cone (Figure 2) are known [18]: 2 1/2 x = x u( ,ν)= vL 1 -u , y = y u( ,ν)= W 1 -u 1 - ν , z = z u( ) =uT, (1) where -L ≤ x ≤ L, -W ≤ y ≤ W, 0 ≤ z ≤ T; 2L, 2W, T are the dimensions of the surfaces under consideration. Figure 1. Circular cone, “City of Arts and Sciences”, Valencia, Spain Figure 2. Right elliptical conical surface S o u r c e : photo by S.L. Shambina S o u r c e : compiled by S.N. Krivoshapko РАСЧЕТ ТОНКИХ УПРУГИХ ОБОЛОЧЕК Structural Mechanics of Engineering Constructions and Buildings. 2024;20(3):265-275 Lateral area of an elliptical cone can be calculated with the following formula [19]: Acone π2 W L2 +T 2 + L W 2 +T 2 . Volume of an elliptical cone is determined by the following formula: Vcone = πTLW. 3 The surface depicted in Figure 2 is generated by a family of z = const sections (ellipses). By adopting a new variable parameter 0 ≥ β ≥ 2π, so that ν = sinβ, 1 - ν2 = cos β,2 parametric equations (1) can be expressed in the following form: x = x u( ,ν)= L 1 -u sinβ; y = y u( ,ν)= W 1 -u cosβ; (2) z = z u( ) =uT. Curvilinear coordinates u, ν of the elliptical cone defined by parametric equations (1), (2) are nonorthogonal and conjugate [18]. Coordinate lines u are straight generatrices of the cone, lines ν are ellipses lying in parallel planes. Coordinate lines v intersect coordinate lines u at right angles only along the straight generatrices ν = 0 and ν = ±1. 2. Momentless Theory of Calculating Right Conical Surface with Elliptical Base Paper [18] contains the derivation of analytical equations for determining the normal and tangential internal forces of a thin shell, the middle surface of which is defined by equations (1). Using the momentless shell theory, the system of three equilibrium equations of a shell element is obtained from the general equations of equilibrium for shells defined in curvilinear non-orthogonal conjugate coordinates [5]. The equations for determining normal forces Nu, Nν and tangential forces Su ≠ Sν per unit length of the corresponding coordinate lines, obtained in article [18], can be expanded in the form, convenient for computer-assisted calculations: Nv = 1 -( u f) 5 (ν ;) ((1L-u-) f8 3ν 2 W 2)V1 + dVdν1 + 2 1( A-u)V2 + -(1 u f) 5; Nu = 2 W2) ( f9 - f10)+ 2 1( -u)2 f0 A2 (L - V1(ν) Sv = -(1 u f) 6 (ν)+ 2 ; (3) (1-u) (1- ν2) f V Su = -(1 u)(1- ν2) νf10 + 2 ( 8 12) A2 - 2νA22 (L2 -W2)2V1 - 2ν (L2 -W2) dVdν1 - (1-u) f0 1- ν (L - - ν 11(-u Af-)ν2)0 2 W2)V2 (ν ;) ( where fi = fi(ν) are known values, V1(ν), V2(ν) are arbitrary functions of integration, which are determined by satisfying boundary conditions defined in forces. Кривошапко С.Н. Строительная механика инженерных конструкций и сооружений. 2024. Т. 20. № 3. С. 265-275 Equations (3) contain constant geometric dimensions L, W, T of the shell middle surface, mentioned in the comments to equations (1). In addition, new constants K and R have been introduced, which allowed to slightly reduce the formulas given below for expressing the known functions fi(ν): K L2 W 2, R W 2 T 2 T 2 L2 . The known functions fi(ν), contained in equations (3), obtained by integration of the three equilibrium equations in paper [18], can be written as follows: f3 ν A2 T 2 ν2L2 W 2 1 ν2 ; A2 A2 ν ν2L2 W 2 1 ν2 T 2; f0 ν L2 ν2K; f5 ν q A2 T 2 L W2 2LTWA T 2 3 ν2T 2(L2 W 2) q A2LTWA T f2 38 ν ; 1 f6 ν 1 ν2 2 νf9 ν ; 1 f7 ν 1 ν2 2 vf10 ν ; f8 ν L W2 2 T 2 ν2T 2 L2 W 2 ; f9 ν 3ALTWqKA2 T 2 R 4TA22 3 T 4 ; f10 ν 2Af1 ν 2 f f8 9A 3LTWfqK23 ν 1 ν2 fν322 2 Lν 4RTA22 3R T 4 0 1 ν2 A84νRT K2 f8A ν K f5 ν qT . (4) In equations (4), q denotes the external surface load, such as self-weight, in the direction opposite to the fixed coordinate axis z. Thus, the momentless shell theory allows to obtain approximate values of normal forces Nu, Nν and tangential forces Su, Sν using analytical equations (3). 3. Truncated Elliptical Conical Shell with Free Upper Edge Let the upper edge u = u0 of the thin shell be free and the lower edge u = 0 be simply supported, with the direction of the supports coinciding with the direction of the straight generatrices of the cone. The shell is smooth, without fractures, and of constant thickness. External load q = const is a constant distributed load, such as self-weight. Thus, all the requirements for the application of the momentless theory for the shell are fulfilled. РАСЧЕТ ТОНКИХ УПРУГИХ ОБОЛОЧЕК Structural Mechanics of Engineering Constructions and Buildings. 2024;20(3):265-275 Since the upper edge u = u0 is free, the following two boundary conditions can be defined at this edge: Su = 0, Nu = 0 at u = u0. The expression for normal force Nu is taken from paper [18] and is equated to zero: Nu Su Sν Nν 0, 1 ν2 ν L2 W 2 (5) and by considering that Su = 0, the presented condition is simplified: Nu Sν Nν 0. By substituting the second and the fourth formula from the system of equations (3) into the last expression, it is possible to determine integration constant V1(ν): V1 ν q 1ALTW u0 3 LA22 WT 2 v 1 v2 12 A2 2T 2 f8 ν 13 R 4TA22 3 T 4 A ν 1 ν2 1 u0 3 f9 ν qKLTWAA2 3T 2 f8 ν v 1 ν2 12 1 u0 3 V01 ν . (6) The second integration constant V2(ν) is determined from the fourth formula of the system of equations (3) by satisfying the boundary condition of Su = 0: v 1 Afv2 KV v 1 u 2 v 1 ν2 1/2 f 1 u10 f0 V1A 2ν f8 ν2 1 2ν2 K 2 2 0 10 0 ν 1 ν2 K dV1 ν . (7) Кривошапко С.Н. Строительная механика инженерных конструкций и сооружений. 2024. Т. 20. № 3. С. 265-275 Normal force Nν is calculated with the first formula of the system of equations (3). Normal force Nu is determined according to formula (5): (1-u)3 - -(1 u 8 0)3 qK (1-u Nu = K 2 f9 - 02)3 3 A2 -T2 f8 -(1-u)(21- --(u1) u0)2 f10 - (1-u) (1-u) A LTW - (1-u)-(1-u20 ) (1-u0 )2 V01 Af82 + K (1- ν22)A(W2 2 +T 2 ) - K2ν2 + (1-u) f0 + ν 1( - ν2)(u -u20)(1-u0)2 K dVdν01 +(1-u) q A3 2 -T2 f832. (10) 2 1( -u) f0 A LTW Coefficients fi = fi(ν) are defined by equations (4); coordinate u varies in the range of 0 ≤ u ≤ u0, and u0 < 1; coordinate v varies from -1 to +1 (Figure 3). It should be noted that truncated conical shells are widely used in mechanical engineering (Figure 4, a) and civil engineering (Figures 4, b, c). The momentless shell theory can be useful in studying the stability of truncated conical shells [20]. Figure 4. Truncated conical shells in mechanical and civil engineering: Figure 3. Truncated elliptical а - metal product; b - conical roofs at the Bundeskunsthalle in Bonn Germany; conical shell c - water tower in Midrand, South Africa S o u r c e : compiled S o u r c e : а - http://molodec-kyznec.ru/market/ by S.N. Krivoshapko izgotovlenie_obechaek/val_covka_konusov/; б - [11]; c - [11] 3.1. Example of Calculation As an example, let us determine the internal tangential forces along the straight generatrices of the truncated conical elliptical shell considered above. Let us examine the straight generatrices coinciding with coordinate lines ν = 0 and ν = ±1. For these lines, parameters fi(ν) are calculated and summarized in Table. It was noted earlier that the curvilinear coordinate system u, v is non-orthogonal conjugate, so Sν ≠ Su. Coordinate lines u = const intersect coordinate lines v at right angles only at ν = 0 and ν = ±1. Therefore, due to the law of reciprocity of tangential stresses according to equations (8) and (9), it turns out that Sv = Su. РАСЧЕТ ТОНКИХ УПРУГИХ ОБОЛОЧЕК Structural Mechanics of Engineering Constructions and Buildings. 2024;20(3):265-275 The straight generatrices coinciding with coordinate lines v = 0 and ν = ±1 Normal forces Nν along coordinate lines u, i.e. at ν = 0 and ν = ±1, are calculated according to equations presented in Table. Normal forces Nu along the same coordinate lines can be calculated using formula (10) at ν = 0 and ν = ±1. For example, formula (10) takes the following form when evaluated at ν = 0: Nu ν 0 6LW T2q 2 1 u T L2 4 7LW T2 2 2 3LW2 4 3W T4 2 11 uu0 23 T 2 L2 3W 2 5L2 W 2 1 1 uu0 2 [4T L2 4 3LW2 4 9LW T2 2 2 . (11) One can perform a check that normal force Nu, calculated with formula (11), must be zero at the free edge u = u0 (Figure 3). Кривошапко С.Н. Строительная механика инженерных конструкций и сооружений. 2024. Т. 20. № 3. С. 265-275 3.2. Right Circular Conical Surface A right circular conical surface may be analysed using the formulas presented above by setting K = L2 - W2 = 0 and L = W = r, where r is the radius of the base of the cone. Thus, the following may be obtained for a truncated circular conical shell with a free upper edge: Su = Sv = 0; Nν [4] u qr[5] ; T Nu 6qT 3 1 u r2 T 2 8 11 uu0 2[6] T 2 1 1 uu0 2 3r2 [7]T 2 . 4. Results and Discussion It is known that the differential equilibrium equations of the momentless shell theory are most easily integrated for shells of zero Gaussian curvature, the middle surfaces of which are defined in lines of curvature, e.g., cylindrical and right circular conical shells. The problem is more difficult to solve for shells of zero Gaussian curvature defined in arbitrary coordinate lines [1]. This is further confirmed by the example of an elliptical conical shell. The main difficulty in the calculation was the application of a curvilinear non-orthogonal conjugate coordinate system, in which the middle surface of the thin shell under consideration is defined. The equilibrium equations of the shell element defined in an arbitrary curvilinear coordinate system were used for the analysis. These equations were obtained by the author earlier.

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 Civil Engineering, Academy of Engineering

Moscow, Russia

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