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Строительная механика инженерных конструкций и сооружений Structural Mechanics of Engineering Constructions and Buildings
2024. 20(3). 265-275 ISSN 1815-5235 (Print), 2587-8700 (Online)
http://journals.rudn.ru/structural-mechanics
РАСЧЕТ ТОНКИХ УПРУГИХ ОБОЛОЧЕК ANALYSIS OF THIN ELASTIC SHELLS
DOI: 10.22363/1815-5235-2024-20-3-265-275 UDC 624.04:624.074
EDN: RDDWZD Research article / Научная статья
Analytical Calculation of Momentless Conical Shell with Elliptical Base
Sergey N. Krivoshapko
RUDN University, Moscow, Russia H [email protected]
Article history
Received: February 28, 2024 Revised: April 17, 2024 Accepted: April 26, 2024
Conflicts of interest
The author declares that there is no conflict of interest
Abstract. Differential equilibrium equations of the momentless shell theory are very easily integrated in cases of cylindrical and right circular conical shells. Shells of zero Gaussian curvature defined in arbitrary curvilinear coordinates are more difficult to analyze, which was reaffirmed by the case of elliptical conical shells. For the first time, analytical expressions of normal and tangential internal forces in a momentless right elliptical conical shell defined in non-orthogonal conjugate system of curvilinear coordinates are obtained. The derived results can be used for approximation of the stress state of thin conical shells with elliptical base and also for the investigation of stability of these shells. Four internal tangential forces obtained by integration of the system of four equilibrium equations of a shell element contain two unknown integration functions, which are determined by satisfying given boundary conditions. The application of obtained analytical equations is demonstrated by an example of analysis of a truncated elliptical conical shell with free upper edge. A uniformly distributed surface load in the direction of the vertical axis of the shell was assumed as external load. The presented formulae are easily adapted for the analysis of a right circular conical shell.
Keywords: elliptical cone, momentless shell theory, non-orthogonal curvilinear coordinates, truncated elliptical cone
For citation
Krivoshapko S.N. Analytical calculation of momentless conical shell with elliptical base. Structural Mechanics of Engineering Constructions and Buildings. 2024;20(3):265-275. http://doi.org/10.22363/1815-5235-2024-20-3-265-275
Sergey N. Krivoshapko, Doctor of Technical Sciences, Professor of the Department of Civil Engineering, Academy of Engineering, RUDN University, Moscow, Russia; eLIBRARY SPIN-code: 2021-6966; ORCID: 0000-0002-9385-3699; E-mail: [email protected]
© Krivoshapko S.N., 2024
i This work is licensed under a Creative Commons Attribution 4.0 International License K^KIZI^M https://creativecommons.Org/licenses/by-nc/4.0/legalcode
Аналитический расчет конической оболочки на эллиптическом основании по безмоментной теории
С.Н. Кривошапко
Российский университет дружбы народов, Москва, Россия Н [email protected]
Аннотация. Дифференциальные уравнения равновесия безмоментной теории оболочек легче всего интегрируются для цилиндрических и прямых конических круговых оболочек. Труднее задача решается для оболочек нулевой гауссовой кривизны, заданных не в линиях кривизны. Это еще раз подтверждено на примере конической эллиптической оболочки. Впервые получены аналитические формулы для определения нормальных и касательных внутренних усилий в прямой конической эллиптической оболочке по безмоментной теории оболочек, заданных в неортогональной сопряженной системе криволинейных координат. Полученные результаты могут быть использованы для приближенной оценки напряженного состояния тонких конических оболочек на эллиптическом основании, а также при исследовании устойчивости этих оболочек. Четыре внутренних тангенциальных усилия, полученные интегрированием системы четырех уравнений равновесия элемента оболочки, содержат две неизвестные функции интегрирования, которые находятся при выполнении поставленных граничных условий. Использование полученных аналитических формул проиллюстрировано на примере расчета усеченной конической эллиптической оболочки со свободным верхним краем. Внешняя нагрузка — поверхностная равномерно распределенная нагрузка в направлении вертикальной оси оболочки. Приведенные формулы легко адаптируются для случая расчета прямой круговой конической оболочки.
Ключевые слова: эллиптический конус, безмоментная теория оболочек, неортогональные криволинейные координаты, усеченный эллиптический конус
Для цитирования
Krivoshapko S.N. Analytical calculation of momentless conical shell with elliptical base // Строительная механика инженерных конструкций и сооружений. 2024. Т. 20. № 3. С. 265-275. http://doi.org/10.22363/1815-5235-2024-20-3-265-275
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
Кривошапко Сергей Николаевич, доктор технических наук, профессор, профессор департамента строительства, инженерная академия, Российский университет дружбы народов, Москва, Россия; eLIBRARY SPIN-код: 2021-6966; ORCID: 0000-0002-9385-3699; E-mail: sn_krivoshapko@ mail.ru
История статьи
Поступила в редакцию: 28 февраля 2024 г. Доработана: 17 апреля 2024 г. Принята к публикации: 26 апреля 2024 г.
Заявление о конфликте интересов
Автор заявляет об отсутствии конфликта интересов.
and 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:
where -L < x <L, - W<y < W, 0 < z < T; 2L, 2W, T are the dimensions of the surfaces under consideration.
where T is the height of the conical surface.
Parametric equations of the elliptical cone (Figure 2) are known [18]:
r 2 ~l1/2
x = x(u, v)= vL [1 - u], y = y(u,v)= ± W [1 - u] 1 - |v| , z = z (u) = uT, (1)
2
I
Figure 1. Circular cone, "City of Arts and Sciences", Valencia, Spain S o u r c e : photo by S.L. Shambina
Figure 2. Right elliptical conical surface S o u r c e : compiled by S.N. Krivoshapko
Lateral area of an elliptical cone can be calculated with the following formula [19]:
W^jl: + T2 + L^W2 + T2
Volume of an elliptical cone is determined by the following formula: n
V = - TLW.
cone 3
The surface depicted in Figure 2 is generated by a family of z = const sections (ellipses).
By adopting a new variable parameter 0 > P > 2n, so that v = sinp, 1 - v2 = cos2p, parametric equations (1) can be expressed in the following form:
x = x (u, v )= L [1 - u ] sinP;
y = y (u, v )= W [1 - u ] cosp; (2)
z = z (u) = uT.
Curvilinear coordinates u, v 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 v are ellipses lying in parallel planes. Coordinate lines v intersect coordinate lines u at right angles only along the straight generatrices v = 0 and v = ±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, Nv and tangential forces Su fSv per unit length of the corresponding coordinate lines, obtained in article [18], can be expanded in the form, convenient for computer-assisted calculations:
N = (1 - « ) f ( v ) ;
1
n =(' - « (f - f
N« (Î - W2 )(/' J" )+ 2 (1 - « )2 f
4 ( L - W 2 + dV + 2ÛIÏ) V
A2y n dv A 2
+ (1 - «) fs;
Sv =(1 - « ) f ( v) +
V ( v ) .
(1 - « )2 ;
(3)
1
S« =(1 - « )(1-v2 )2 f +
(1 - v2 )
(1 - « )2 f0
fV
(1 - v2 ) A2 2A7
(L2 - W2 )2 V - v (L2 - W2 )
dy
dv
v (1 - v2 )
(1 - « ) Af0
( L2 -W2 ) V ( v ) ;
where ft = fi(v) are known values, Fi(v), V2(v) are arbitrary functions of integration, which are determined by satisfying boundary conditions defined in forces.
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(v):
K = L2 -W2, R = (W2 + T2)(T2 + L2).
The known functions fi(v), contained in equations (3), obtained by integration of the three equilibrium equations in paper [18], can be written as follows:
f3 (v) = >/A2 -T2 = ^v2L2 + W2 (l - v2); A2 = A2 (v) = v2L2 + W2 (l - v2) + T2; f (v)= L2 - v2K;
3
fs ( V ) =
qJAF-T2 + L2 (W2 + T2 )- v2T 2(L2 - W2)]2 qJA
q4A2 - T2 f82 ( v) ;
LTWA3
1
f6 ( v )=(! - v2 )2 vf ( v ) ;
1
f7 ( v )=(l - v2 )2 f ( v ) ;
f8 (v) = L2 (W2 + T2 ) - v2T2 (L2 - W2 ) ; qK
LTWA3
f ( v )"
fl0 (V)"
3ALTWyjA2 - T
1
R
i T2 ^
4 A2 -3 0
2 Afo ( V )
f qK
A 3LTWf (v)
- t 4
l - v2 -
v2 L2
f32 ( v ).
^ T2
4R—- 3R - T4
A2
(l - v2 ) 8vRT2 K
к[f5 (v) + 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, Nv and tangential forces Su, Sv using analytical equations (3).
3. Truncated Elliptical Conical Shell with Free Upper Edge
Let the upper edge u = uo 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.
3
4
edge:
Since the upper edge u = uo is free, the following two boundary conditions can be defined at this
Su = 0, Nu = 0 at u = uo.
The expression for normal force Nu is taken from paper [18] and is equated to zero:
N = -( Su - S )-
-+N, = 0,
(1 -v2)2 v(L2-W2)
and by considering that Su = 0, the presented condition is simplified: f_
(5)
Nu = S,
u v
1
(1 - v2 )2 v ( L2 - W2 )
- + Nv = 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 Vi(v):
(1 - u0 )3 ( L2 - W2 ) , .if a2 - T2 1
V (v) = V—0,\ ' v(1 - v2)2 7 iL-L f (v) +1 1W ALTwJÄrT y ' 7 A ^ ' 3
R
= - v 1 - v
"- v (1 - v2 )2 (1 - Uo )3
f9 ( v ) +
qKyjA2 - T2 LTWA3
f ( v )
/ T2
A-j - 3 A2
- t 4
-v (1 - v2 )2 (1 - Uo )3 V01 ( v).
(6)
The second integration constant V2(v) is determined from the fourth formula of the system of equations (3) by satisfying the boundary condition of Su = 0:
(1-v2)K ^2 / 2\1/2 1
1 ^ V2 ( v ) = (1 -U0)2 v (1 -v2) fw +
'V ( v )
Afo
(1 - U0 ) f 8 A2
f8 -
;(1 - v2 ) K2 2
v (1 - v2 ) K dV1
1( v)
2
dv
(7)
Now, by using equations (3), internal tangential forces per unit length of coordinate lines can be calculated:
Sv =■
v (1 - v2 )
(1 - u Y
Su = v (1 - v2 )
(1 V (1 ^3 * f qK (1 - Up )VA2 - T
(1 -u) -(1 -u0) f9(v)--V ,3,7W-f8 (v)
A LTW
(8)
(1 - u )2- "0 - u0 )2
#1 -u)
f10 +
Iu -U0)(1 -U0
f8 ( v )
K(1 - v2) (W2 + T2 )
K
2A2
V01 ( v )— v V01 ( v ) +
(1 - u)2 f ( v)
Kv (1 - v2 ) dVm ( v )
V01 ( v ) +
dv
(9)
2
2
Normal force N is calculated with the first formula of the system of equations (3). Normal force Nu is determined according to formula (5):
N =
J [[(1--(1-U)3
K
(1-u )2
qK(1 -Uo)4A2 -T2 (1 -u)2-(1 -Uo)2
J 9 ,2 л_____ J8 \ J1o
(1-u )2 A3 LTW
(1 - U
[(1 - u )-(1 - Uo )](1 - Uo )2 (1 - U)2 Jo
V
o1
ft K (1 - v2 )(W 2 + T 2) Kv2
A2
2 A2
+ -
'(1 - v 2 )(u - uo )(1 - Uo)^K dVo1
2 (1 - u) fo
dv
( ,Wa2 - T + (1-u)
A3 LTW
2 3
J 2
J 8 •
(10)
Coefficients f = f(v) are defined by equations (4); coordinate u varies in the range of 0 < u < uo, and uo < 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 3. Truncated elliptical conical shell S o u r c e : compiled by S.N. Krivoshapko
a 6 c
Figure 4. Truncated conical shells in mechanical and civil engineering: a — metal product; b — conical roofs at the Bundeskunsthalle in Bonn Germany; c — water tower in Midrand, South Africa S o u r c e : a — http://molodec-kyznec.ru/market/ izgotovlenie_obechaek/val_covka_konusov/; 6 — [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 v = 0 and v = ±1. For these lines, parameters f(v) are calculated and summarized in Table.
It was noted earlier that the curvilinear coordinate system u, v is non-orthogonal conjugate, so Sv ^ Su. Coordinate lines u = const intersect coordinate lines v at right angles only at v = 0 and v = ±1. Therefore, due to the law of reciprocity of tangential stresses according to equations (8) and (9), it turns out that Sv = Su.
The straight generatrices coinciding with coordinate lines v = 0 and V = ±1
v = 0 (Figure 3)
v = ±1 (Figure 3)
f = L
f = (A2 -T2)1/2 = W
A2 = W2 + T2 f = qL2/t
f = o
f7 = 0
f8 = L2 (W2 + T2)
= W2
(A2 - T2 )1/2 = L
f = f3
A2
f = q^ / T
f = f =
f = W2 ( L2 + T2)
= L2 + T2
t2
W
0 0
f =
qK
T2L2 - 3W2T2 - 3W2L2
3LTW 2V W2 + T2
/9 =
./10
qK ' L2T2 ( L2 + W2) - 3W4 ( L2 + T2)]
6L3TW 2VW2+TI qKT ( L2 - 3W2)
V = _
01 3LW2 ^|W2+T2
f10
V01 =
qK [T2W2 - 3W2L2 - 3T2L2 ] 3WTL2V L + T2
qK [W2T2 (L2 + W2) - 3L4 (W2 + T2)
6L2TW 3V L2 + T2
qKT (W2 - 3L2)
3WL2V L2 + T2
=(1 - u)
— ^V — 0
qL2
Nv =(1 - u)
— Sv — 0
qW2 T
S o u r c e : compiled by S.N. Krivoshapko
Normal forces Nv along coordinate lines u, i.e. at v = 0 and v = ±1, are calculated according to equations presented in Table. Normal forces Nu along the same coordinate lines can be calculated using formula (10) at v = 0 and v = ±1. For example, formula (10) takes the following form when evaluated at v = 0:
q
N (v-0)= q 2 <(1 -u)(T2L4 -7L2W2T2 + 3L2W4 + 3WT2)-6L W T" y '
(!- u0)
[T2 (L2 - 3W2)(51? - W2 )l + i!-U^2 (1 -u)2 ^ )] (1 -u)
[4T2L - 3L2W4 - 9L2W2T2 :.
(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).
3
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;
N =(1 - u $
qr —;
N "q u 6T
3(1 -u$(r2 -T2) + 8^^T2-iL^L(3r2 + 5T2$ V Л ' (1 - u $2 (1 - u $ 1 '
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.
1. For the first time, analytical expressions for the determination of normal and tangential internal forces in a right elliptical conical shell according to the momentless theory of shells defined in a nonorthogonal conjugate system of curvilinear coordinates are obtained. The obtained approximate results can be used to evaluate the stress state of thin conical shells with an elliptic base, as well as to study the stability of these shells.
2. The application of the formulas obtained for the first time for the calculation of tangential forces according to the momentless shell theory is demonstrated by the example of a truncated elliptical conical shell with a free upper edge subjected to a distributed load such as self-weight.
3. It is shown that the obtained equations can be used for the analysis of right circular conical shells according to the momentless theory.
5. Conclusion
Despite the development of powerful computer hardware and the creation of powerful software systems capable of calculating the stress-strain state of complex thin-walled spatial structures on the basis of the moment theory of shells, the momentless theory has not lost its relevance. Computational systems are most often based on the application of numerical methods of calculation. The possibility of obtaining analytical solutions is always preferable, as it allows to review the obtained solution. The momentless stress state occupies an honorable place in the calculation of shells, being the ideal to which one should strive when designing shells. The conditions of the momentless stress state cannot always be fulfilled structurally. Even so, the results of the calculation, at some distance from problematic areas, may be acceptable for preliminary design decisions.
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