DEFORMATION OF A CYLINDRICAL SHELL BY EXTERNAL PRESSURE Текст научной статьи по специальности «Строительство и архитектура»

прогнозную модель снежно-ледяных образований на горных дорогах [5]. Применение модели позволит разработать предупредительные мероприятия по снегозащите инженерных сооружений, а также принять оперативные решения по расстановке снегоуборочной техники и оптимальному ее использованию, что сократит расходы на зимнее содержание автомобильных дорог.

Выводы

Таким образом можно сформулировать следующий вывод.

Выявлены природные и техногенные факторы, влияющие на эффективность транспортного обеспечения: природные факторы - климатические условия, рельеф местности, характер растительного покрова, свойства снега; техногенные - расположение дороги по отношению к направлению господствующих ветров; особенности конструкции дорожного полотна (в выемках, на нулевых отметках, в насыпях) и приуроченность дороги к различным

геоморфологическим элементам, что служит основой для построения оценочно-прогнозной модели снежно-ледяных образований на горных дорогах.

Литература

1. Подольский Вл.П., Самодурова Т.В., Федорова Ю.В. Экологические аспекты зимнего содержания дорог. - Воронеж: ВГАСА, 2000. - 152 с.

2. Бутюгин В.В. Инженерная геодинамика снега. Норильский регион: монография - СПБ. ГУАП, 2008. - 200с.

3. ВСН 24-88. Технические правила ремонта и содержания автомобильных дорог/ Минавтодор РСФСР. - М.: Транспорт, 1989. -198 с.

4. Природные ресурсы республики Северная Осетия-Алания: т.7. Климат. -Владикавказ: Проект-Пресс. - 2002 - 224 с.

5. Лолаев А.Б., Масланов Н.А. Оценочно-прогнозная модель формирования снежно-ледяных образований на горных автомобильных дорогах // Устойчивое развитие горных территорий. 2011, № 3 с.59-64

DEFORMATION OF A CYLINDRICAL SHELL BY EXTERNAL PRESSURE

Fidrovska N.

Kharkiv national automobile and road university, Professor

Slepuzhnikov E.

National University of Civil Defence of Ukraine, Lecturer

Perevoznik I.

Ukrainian Engineering and Pedagogical Academy, postgraduate student

Khursenko S.

Ukrainian Engineering and Pedagogical Academy, postgraduate student

Kharkiv, Ukraine

ABSTRACT

The article considers the problem of determining the stress-strain state of a thin-walled cylindrical shell under the action of external pressure. Such structures are used in many machines, for example, aircraft casings, submarine casings, drums of hoisting machines.

Equations were derived for circumferential, longitudinal and axial displacements depending on the geometric dimensions and elastic properties of the rope and drum. At the same time, the hypotheses about the non-tightness of the shell in the circumferential direction and the absence of displacement were not accepted. This brings the design scheme much closer to the actual operating conditions of the cylindrical shell.

Keywords: cylindrical shell, external pressure, stress-strain state, rope drum.

Cylindrical shells are the most common forms of closed structures. Therefore, the determination of their stress state under the influence of external pressure is a very urgent task. Different load schemes of structures require different calculation methods. A large number of different shell designs, different conditions of their loading and operation cause a certain difficulty in the analysis of the stress-strain state. The rope drum is considered as a cylindrical shell with variable external pressure, which arises under the action of the rope, which is wound on the drum.

When designing thin-walled structures, the results of theoretical research in the field of structural mechanics and the theory of elasticity are used [1, 2]. But the calculation methods must correspond to the design features for which they will be applied [3-5]. For cylindrical shells, the ratio of the wall thickness to its radius is

of great importance, which is not always taken into account when choosing a calculation method [6, 7].

The transverse bending of a cylindrical shell is a rather complex problem, on the solution of which many researchers have worked [8-10].

In work [11], a theory for the calculation of spatial systems was developed, which makes it possible to determine the stress state of various structures. The shell is considered as a multiply statically undefined system. But if we consider it as a system with a stressed state that arises in a thin-walled bank with an under-formed period, then it is possible to determine the normal stresses and shear forces in the elements of cross-sections. In addition, elementary rings can be considered, which are distinguished by planes perpendicular to the axis of the system, while internal forces in longitudinal sections are determined.

That is, the cylindrical shell is considered in the form of a cross system consisting of longitudinal beams-strips and transverse frames-rings. It is clear that the lateral load will simultaneously bend the strip beams and the ring frames.

The deformation compatibility conditions are fulfilled taking into account additional internal forces acting in both transverse and longitudinal directions.

In work [12], a simplification of the solution of the problem is proposed if all unknown internal forces are expressed in terms of the total bending moments: m = mf0+ mfd, or through the total radius displacement w.

(1)

This calculation method is based on the application of Hooke's law and two hypotheses.

In the first hypothesis, it is assumed that there are no shifts in the middle surface of a thin-walled structure. The second hypothesis assumes that the shell is not tight in the circumferential direction.

The presences of tangential forces qa and hoop normal stresses ava, which are determined by the equilibrium equation, indicate a very close proximity of these hypotheses. This is also indicated by the results of experiments [12].

With a symmetric load of the shell relative to the x axis, its individual elements have radial displacements w, longitudinal u and circular v. Consider an element of a cylindrical shell (Fig. 1), where:

b

Fig. 1. Deformation of the middle surface of the shell - radial displacements, b - circumferential displacement

Let us consider a rope drum as a cylindrical shell, loaded with an asymmetric external pressure, which arises from the action of the wound rope. The rope is wound on a drum along a helical line. In most cases, the

drum is made profiled. If we decompose the rope tension force into two components - longitudinal and transverse (Fig. 2), then we can write:

T1 = mcosp, (2)

t2 = msinp, (3)

where [1 is the deviation angle.

r

\ \ T1 \ \ 1 \ \

T

Fig. 2. Design scheme

These forces will act on the crest of the profiled drum; they will be less in the wall itself. That is, if we assume that the forces Nv and Nx are equal to Ti and T2, then this error will go into the safety margin of the drum.

Then we have:

ES fdu v (w —

1-v2 Ydx R ( dy)\ =

Suppose that the radial displacements change according to the following law:

w = f(x) • coswp, (5)

where f (x) is a function that is variable along the length of the shell;

y is the angle that is measured from the vertical y - axis;

n - 2, 3, 4 - natural numbers.

a

a

The rope load will not be constant, it changes de- ^ = E^Fk , (7)

pending on many factors: the winding angle p, the fric- . „ . ^ . . 8E •„_

tion coefficienty , the geometric and elastic properties where Ek is the modulus of eJasticity of the rope;

ofthc rope and drum. FK " area of t*. rope;

Weaccept the following law of rope tension l is leng? of wiling; A

, i is the coefficient of friction between the rope and

: _Ml=xL2„ the drum;

T = T0 • e h n. (6) h - pitch of rope winding.

where Do is the maximum rope tension (at the We substitute expressions (5) and (6) into the sys-

point where the rope coincides with the drum); tem of equations (4). After solving, we get:

k - coefficient depending on the geometric and elastic properties of the rope and drum;

v = To ek"J27lR • (sinP - vcosft -to-")- (8)

. 1-271

k^~ir r „ 1 i 27-k-H

■ [c osß (sinß - vcosß)] • (Inh ß - 1). (9)

u = To(1 v)'h'e-—\cosß--—(sinß -vcosß )M l h

27T-E-k-^-S

Thus, we will receive the formulas of all three dis- The work of internal and external forces, which is

placements depending on one functionf (x). To find it, performed during the transition of the system from the we use the theorem of the minimum potential energy. deformed to the initial state, will have the form:

r = § [1 • m<pX<p +1 • mxdxX!p + ^ • axdex + ^ • a^e^ - m^x^] Rdcp, (10)

where

xxv - relative twist angle of the element,

d2™ 1 \

= (11)

1 id2w ,

W

R2\dx2

), (12)

R\dx dpdxj'

ev - relative deformation in the circumferential direction,

w , 1 dv

= 1(dv d2w) (13)

R (dx dpdx)' ( )

ex - relative deformation along the generatrices, &xd - additional normal stresses, a< - ring normal stresses,

where Qxd and Q< are shear forces,

= - + ^•1^, (14)

R R dp V ' du

£x=dU' (15)

= !R.(dQM + ddSv\ + pRR

S \ dx Rdy) S '

(17)

Q + (17)

^xa dx Rdcp v '

Q ^ + zJnxAdXi (18)

p R d p d x

where mxd - additional longitudinal bending moments,

mxa = D-(xx + v- Xp). (19)

mx<pd and mpxd additional torques of individual shell elements,

mpxa = mxpa = D • (1-v) • Xxp. (20)

p - radial load,

T

P = RT, (21)

Let us write down the Euler equation for the variational problem:

+ = (22)

df(x) ax (af(x)) ax2 (df"(x) ). ( )

Then we obtain the differentiated equations of the fourth degree:

f'v - a1f(x) + a2f(x) = a3rkli~ . (23)

where

1 \3(n2-l) (1-v)^(n-2)-n3 + 1

ai=^_^+i1-v)<n-2yn3+i]t (24)

R 0.2 = ^ n (25)

_ 2(1-v) T0.k2 pL2 4n2 (sinp-vcosP) (n2-l)

a3 = ETh2R . (26)

Taking into account that a2 < a: and a3, we obtain the solution of equation (23) in the form:

i-x _ ^_

f(x) = A • ek^~2n + c^e^ + c2e^ + c3x + c4. (27)

p

a

p

where

A =■

a3h4

The coefficients ci, C2, c3 and C4 are found from the initial conditions:

f(x)L=0,L = 0, d2f\

dx2x=0,L

= 0.

Then the calculated system of equations will have the form:

Ae-k^h2n + ci + c2 + c4 = 0

, ll-LI _

Ae-k^~T2n + c1l^aiL + c2e ^aiL + c3L + c4 =

|,2..2„_2 . I

■ ek^h2n + a1(c1+c2) = 0

h A

2 2 2

•JOÎL +

0

h2

We solve this system and get:

A'-^e-^2" + ai (Cle^ + c2lla^

=0

,M-L In

a1h2

e/âlL-e JcÏL

C=

С? =-<

Ak2p24n2(e-k^^2n)

h2a1(e^K1L-e-/J1L)' ( , U-iL_ , i-

ie

e k^n2n) Il-Ll

k2jl24TT a1h2

екП2п (e^ -l)- e-kfj}±-h

c^A^-m2-!).

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

e

A

L

Conclusion. The article considers the rope drum as a cylindrical shell with variable external pressure. The pressure is generated by the rope that is wound around the drum.

When calculating the drum, the friction force acting between the rope and the drum wall is taken into account. When calculating the drum, the hypothesis of the non-tightness of the shell in the circumferential direction and the absence of displacement was not accepted.

Equations are derived for circumferential, longitudinal and axial displacements depending on the geometric dimensions and elastic properties of the rope and drum.

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