CC BY
13
UDK 539.3
Levitskaya T. I., Ph.D., Pozhuyeva I. S., Ph.D.
Zaporozhye National Technical University
RESEARCH LOADED SHELLS OF REVOLUTION SUPPORTED BY
RIBS OF DIFFERENT SHAPES
The usage of superelement method for composite shells of rotation enabled to work out an effective algorithm of calculation of their stress-strain state. For the element of ring-type the local stiffness matrix has been made, thus giving the possibility to observe shell-type ribs as well as ring-type ones, which cross-section sizes are smaller than their radius. A number of calculations have been made to compare the impact of the rib form upon the tense state of shells.
Key words: shell rotation, superelement, stiffness matrix, the stress-strain state, rib.
To calculate the stress state of closed shell of rotation at axis-symmetrical loading we use the system of differential equations [1], [5]:
d(rNs ) - sin toN --Qs + rqs = 0 ds —
d(-Qs) + -NS + cos ^Ne - rq- = 0 ds —
d
ds
(rMs )- sin q>M e + rQs = 0,
(1)
where Ns = Dn (s s + vse ^ Ms = Dm (X s + vXe) Dn = Eh/ (1 -v2), Ne = Dn (vs s +Se),
3 /
Me = Dm (vxs + Xe), Dm = Eh3/(12 • (1 -v2)) ;
dU W
" sin toTr cos mT„
ss + —, se =--U +-^W , (2)
ds r
r
r
dy, sin to, n 1 sin2 m L 2\
=---(1 - v))i +- y2 dn (1 -v2 ))4 +
ds r r1 r
sin to cos ton/i 2 \
--2-dn (1 -v ))5 - qs
r
dy
1 v cos to
2| 1 v cos to I sin to cos to sin ton A 2 \ 2 = 1----i y--y2----dn (1 -v ))4 -
ds [ r1 r I r r
cos2 to 2
--^ Dn (1 -v2 ) + qr
r
dy3 sintou \ sin* ton A 2\
"f3 = -y2--21(1 -v)y3--r^DM(1 -v l
ds
dy4
ds D 1
1 sin to ( 1 v cos to 1
-y1 - v-y4 -1 - +-Iy5
r I r r
dy5
"T5 = - y4 + y6 ds r
dy6
ds
D
1 sin to
y3 -v-y6-
(3)
The middle surface of the shell should be in parametric form:
(x = x(t)
|y = y(t),
(to < t < tj).
(4)
des sin toQ A Xs =~r, Xe^-^ös , es =--- + —.
ds
dW U -+ —
ds r1
In formula (2) E is modulus of elasticity, v is Poisson's ratio, h is thickness of shell, r1 is curvature radius of meridian line.
Let's introduce unknown values y1 = NS , y2 = QS ,
y3 = MS, y4 = U, y5 = W, y6 = -9s .
By this the system (1) is brought to the normal system of ordinary differential equations relative to unknown functions, that will become:
Functions, included into the system (3), are determined according to the formula
sin to =
4
,r-+y-
, cos to =
y
i
x '2 + y '2;
x y - y x
(2 + y '2
ds = Jy
= Vx'2 + y'2 dt. (5)
Let's use the method of superelements based on the finite element method, but allows observation of more extensive shells. To build local stiffness matrix of
r
x
1
© Levitskaya T. I., Pozhuyeva I. S., 2015
104
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superelements we use Godunov's method of differential equation system solution, that has higher accuracy of solution [2]. It was reachable due to ortogonalization of intermediate solutions.
To make the local stiffness matrix of superelement, it is necessary to solve a series of two-pointed edging problems for a homogeneous system
1 dy .
--— = Ay
B dt
(6)
1)
Here yT = (y ••• y6), B = x'2 + y'2 , A- is a linear system operator (3).
The first column of the local stiffness matrix k e will be full as a vector of solution for the first variant of initial data, the second as for the second variant of initial data, ... , the sixth column will correspond to the 6th variant of problems (6), (7).
By solving non-homogeneous problems
1 dy .
--— = Ay + q
B dt
(8)
at zero boundary conditions y4(t0) = 0, y5(t0) = 0,
•MO = 0, MO = 0, MO = 0, y6(tl) = 0 the vector of forces, acting at the left and right edges in the local system of these edges coordinates is found:
fe=( it n i: fe /: )
(9)
where qT = (- qS qr 0 0 0 0) is an acting loading.
To make the global stiffness matrix and vectors of force loading we should build the local stiffness matrices and force in junctions for global coordinate system. The transformation is made according to the formula:
ke = tt - ke - t-e
fe = -Tt - fe - e,
(10)
T =
cos 90 sin 90 - sin 90 cos 90
cos 91 sin 91 sin 91 cos 91
yA)=1 yA)=0 y:(t0) = 0 (
ysCO=0 ys(t0) = 1 ys(t0) = 0
yA)=0 2) y6(t0) = 0 ) y&)=0 3) y6(t0) = 0 ) y&)=0 e = 2n
y4(t1)=0
ysft)=0 ys(1)=0 ysCD=0
yA)=0, yA)=0, yA)=1. (7) V
(- R
R
R
R
- R
- R
(11)
1 y
Here cos90, sin, cosq)j, sinq>j, R0 = y(t0) ,
R = y(tj) - are functions, formed according to the formula (4), (5) for initial and final focal points of superelement.
Matrix 9 is formed as consequence of consideration of superelement reaction at the edges. The system of linear algebraical equations for finding of the displacements in focal points of shell superelement is:
K-0 = f,
where K = ^ °ke is a global stiffness matrix,
(12)
f _ ^ © fe
J = 2 J is a global vector of force,
e
5T = (Ul,Wl-9sl,...,Uk,Wk,-9sk) - are the global
displacements in focal points. Sign 2 means the ensemble process used in the finite element method [3].
This methods has allowed to consider ribs as shell constructions. In case ribs sizes are small compared to their radiuses, it is possible to use local focal matrices of stiffness for consideration of their impact. These matrices were built with the use of the results [4] at the observation of z-shaped rib. In the global coordinate system they are:
( 0 0
k = 2n-
where
0
A
0 -
EF R
aFE R
aFE R
EJy + a2 FE R
(13)
0
1
0
1
0
0
e
0
ISSN 1607-6885 Hoei Mamepianu i техноnогn e меma.nургiï ma MawuHoôyàyeaHHi№2, 2015
105
where E is modulus of elasticity, F is area of section, R is radius of the shell in the focal point, Jy is a moment of inertia with respect to an 0y axis, going through the center of section mass, a is deviation of focal point from the center of rib mass.
Local matrix kr is added to the matrix K by method of ensemble.
Presence of symmetry in the local stiffness matrices ensures the symmetry of global matrix and we can use matrix operations with banded matrices for solution (12). After solution there will be displacements in the global system of coordinates in the focal points of superelements. To find the stress state of superelement from the global
solution 5 the displacements are chosen at its edges 5e and then they are recalculated into the displacements in the local system of coordinates
5e = T -5e
(14)
Then the corresponding non-homogeneous two-pointed edging problem (6) is solved with the following initial data:
Then the corresponding non-homogeneous two-pointed edging problem (6) is solved with the following initial data:
y*(0 = 84, y5(ti) = 85 , y6(tx) = 56 , (15)
where 5e 81 are 5e matrix elements.
i' '6
A set of applied programs for personal computer was worked out for the investigated methods.
There were made calculations of stress-strain state of cylindrical shells for comparative analyses: 1. Without ribs, 2. With plate ring element of shell-type, 3. With rectangular rib of ring-type, that has characteristics of the case No2, 4. Z-shaped rib. The type of ribs and their geometrical characteristics are given in Figure 1. Cylindrical shell and ribs are made of material with E = 2.1-105 MPa - modulus of elasticity, n = 0.3 - is Poisson's ratio. Thickness of shell h = 0.03 m. The left edge of the shell is free, the right is fixed. The shell is loaded with normal pressure qr = 200 H/m2.
Given above variants are calculated for the data:
l = 0.5m, h = 0.03m, a = -0.0275m, b = 0.2m, c = 0.14m, d =0.12m.
Z-shaped rib has the same area of cross-section as ribs 2, 3.
The results of the W deflection calculations, QS shearing force and MS curving moment depending on the point distance of the middle surface from the left butt-edge of the shell are given in fig.2.
a b
Fig. 1. Geometrical characteristics of cylindrical shell with ribs: a is ring shaped rib (cases 2,3), b is z-shaped rib (4)
Fig. 2. The results of calculation of W(s), Qs(s) and Ms(s) for different types of ribs
МОДЕЛЮВАННЯ ПРОЦЕС1В В МЕТАЛУРПТ ТА МАШИНОБУДУВАНН1
It is visible from comparison that the results of the calculations for the shell construction (2) and rectangular rib of ring-type (3) coincide. Thus there is an opportunity to calculate the shells with the ribs of the arbitrary section, for example, case 4, that are closely attached to the shell surface.
Literature
1. Grigorenko Y.M. Isotropic and anisotropic layer shells of rotation of variable stiffness. - Kiev : Naukova Dumka. -1973. - 212 p.
2. Myachenkov V.I., Maltsev V.P. Methods and algorithms of calculation of the space structures on the electronic computer. - Moscow: Mashinostroyeniye. - 1984. - 280 p.
3. Zenkevich O. Method of Finite Elements in Technology. -Moscow : Mir. - 1975. - 544 p.
4. Avdonin A.S. Applied Methods of Shell Calculation and thin-walled structures. - Moscow: Mashinostroyeniye. -1969. - 403 p.
5. Flbgge W. Stresses in shells. - Berlin : Springer-Verlag. -1967.
Одержано 16.12.2015
Левицька Т.1., Пожуева 1.С. Дослщження навантажених оболонок обертання пщкр1плених шпангоутами р1зноТ форми
Застосування методу суперелементгв для складених оболонок обертання дозволилорозробити ефективний алгоритм знаходження Их напружено-деформованого стану. Побудова локальноI матрицI жорсткостг для елемента кшьцевого типу дало змогурозглядати як шпангоути оболонкового типу, так I шпангоути юльцевого типу, розмгри поперечного перергзу яких малI поргвняно з Их радгусом. Проведено рядрозрахунюв для поргвняння впливу форми шпангоутгв на напружений стан оболонок.
Ключовi слова: оболонки обертання, суперелемент, матриця жорсткостг, напружено-деформований стан, шпангоут.
Левицкая Т.И., Пожуева И. С. Исследование нагруженных оболочек вращения подкрепленных шпангоутами разной (формы
Применение метода суперэлементов для составных оболочек вращения позволило разработать эффективный алгоритм нахождения их напряженно-деформированного состояния. Построение локальной матрицы жесткости для элемента кольцевого типа дало возможность рассматривать как шпангоуты оболочечного типа, так и шпангоуты кольцевого типа, размеры поперечного сечения которых малы по сравнению с их радиусом. Проведен ряд расчетов для сравнения влияния формы шпангоутов на напряженное состояние оболочек.
Ключевые слова: оболочки вращения, суперэлемент, матрица жесткости, напряженно-деформированное состояние, шпангоут.
ISSN 1607-6885 Hoei Mamepitrnu i технологи в металурги та машинобудувант №2, 2015 107
