Z E S Z Y T Y N A U K O W E P O L IT E C H N IK I ŚLĄSK IEJ 1993
Seria: M E C H A N IK A z. 116 N r kol. 1231
N atalija P A N K R A T O W A In stitu te o f M echanics
U k rain ian A cadem y o f Sciences
ST A N N A P R Ę Ż E Ń G R U B O Ś C IE N N Y C H E L E M E N T Ó W S T O Ż K O W Y C H
S treszczenie. R o zp atru je się pew ne podejście do rozw iązania zad ań o stanie n a p rę ż e ń w stożkowych elem en tach konstrukcji, form ow anych n a d ro d ze zbrojenia przez naw inięcie. M e to d a je s t o p a rta na racjonalnym połączeniu rów nań teo rii sprężystości n iejed n o ro d n eg o ciała anizotropow ego, przekształcenia analitycznego i na m eto d ach analizy num erycznej. R ozpatryw ana je st postać anizotropii, gdy m ateriał m a je d n ą płaszczyznę sym etrii sprężystej, p ro sto p a d łą do pow ierzchni stożka. O bliczenia pozwoliły ujaw nić efekty, u w arunkow ane niejednorodnością i anizotropią m ateriału,rodzajam i zbrojenia i obciążeń.
S T R E S S E D S T A T E O F T H IC K -W A L L E D L A M IN A T E D C O N IC E L E M E N T S
Sum m ary. A n ap p ro ach is p re se n te d to solution o f p roblem s o f stressed state of an iso tro p ic hollow conic elem ents w ith one plane o f elastic sym m etry fab ricated by winding.
C onstitutive relations in this ap p ro ach are equations o f three-dim ensional p roblem of elasticity theory. By m eans o f various analytic al transform ations three-dim ensional p roblem is accu rately red u ced to o n e dim ensional problem which is solved by stab le num erical m eth o d . C o m p u tatio n s allow ed to find effects p reconditioned o f non-hom ogeneity and an iso tro p y o f elastic p ro p ertie s o f m aterials and types o f ap p lied loads.
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162 N. P an k rato v a
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1. IN T R O D U C T IO N
W h en d e term in in g stress level an d deform ability o f conical stru ctu ral elem en ts of co m p o site m aterials allow ance should be m ade for th eir characteristics, as fo r exam ple, an iso tro p y a n d non-hom ogeneity o f m echanical p a ra m e te rs, th e m a n n e r o f layers conjugation, types o f loading, etc. A m ong th e available m eth o d s o f solving th e p ro b lem s we should distinguish th o se on defo rm atio n o f elem ents o f aniso tro p ic m aterials having one p lan e o f elastic sym m etry. O f practical significance is th e fact th a t th e solution o b tain ed for th o se p ro b lem s can b e used for th e calculation o f hollow conical stru ctu ral e lem en ts o f an an iso tro p ic m aterials w hen th e m ain directions o f elasticity do n o t coincide w ith those o f co o rd in ates. T his tak es place fo r stru ctu ral elem ents m ad e by w inding. F o r thick- w alled conical e lem en ts it is im p o rtan t to find a rational schem e o f rein fo rcem en t. O n e can p urposefully influence th e strength and stiffness o f stru ctu re changing th e schem e of re in fo rc e m e n t by ang u lar and radial coordinates. O n th e basis o f th e d ev elo p e d ap p ro ach to th e d e te rm in a tio n o f th e stress-strain sta te o f aniso tro p ic cone, th e a u th o r studies the influence o f rein fo rcem en ts angle on th e stress level an d deform ability o f a specific conical stru ctu ral.
2. IN V E S T IG A T IO N O F T H E IN F L U E N C E O F T H E W IN D IN G A N G L E O N T H E ST R E S S -S T R A IN O F A H O L L O W L A M IN A T E D C O N E
2.1. A c alc u latio n m odel fo r an iso tro p ic cones
A m o d el fo r stress-strain state calculation an anisotropic infinite m ultilayered cone is p ro p o se d . B o unding surfaces an d surfaces o f the conjugation layers a re coaxial conical
S tressed sta te o f thick-w alled .. 163 having a single vertex. T h e a u th o r studies the influence o f the w inding angle on stress and d isplacem ents th ro u g h th e thickness o f lam inated anisotropic cones in th e context o f a th ree-d im en sio n al p ro b lem o f th e elasticity theory. T h e cone is assigned to spherical c o o rd in ate system <p, r, 0 w hereas th e solution is sought in th e a re a <pe[(p0,<pn ], re[0,27r], w h ere <pQ, <pn a re conical bounding surfaces. C onsider th e case w hen in each po in t o f the given layer o f non-hom ogeneous cone th e re is one plane o f elastic sym m etry p erp en d icu lar to th e conical surface g enerating lines.
L e t us re p re se n t th e equations o f th e generalized H ookers law fo r th e i-oh layer as [1]
e ‘ = B ‘ a ‘ + /', e ‘ = {e^, ee‘, e ‘ , e/„, e ^ , e ^ ) ;
B i = \K (if)\\, b/t = bi1 = (.k,l= 1 , 2...6), b*s = bm6 = b5„ = 6m = 0 (m = 1,2, 3 , 4 ) ;
V 1 = <«V ° ‘r> °8. < > u , S a b T - < «¿T1, a'r, a % a^T, a^T, a ' ^ T ) .
H e re e ^ 1, e g 1 ,..., e^G1 - are th e strain tensor com ponents, c t^ 1, a g1 t ^ q1 - are the stress te n s o r com ponents. E lastic constants b ^ 1, coefficients o f linear th erm al expansion a ^ 1, ag1, ar* in th e direction cp, 0, r,coefficients o f te m p e ra tu re shift a *, a rg‘, a ^ g 1 are the functions o f th e co o rd in ate <p, this m akes it possible to take into account a rb itra ry variation o f m aterial p ro p ertie s,th ro u g h th e cone thickness.
R elatio n s (1) also hold fo r an o rth o tropic cone w hose m ain directions o f elasticity are tu rn e d a ro u n d th e norm al to th e surface r = const by th e angle J3. In this case elastic constants b^ 1 a re defined from corresponding characteristics o f an anisotropic m aterial by eq u atio n s [1],
i*1 I i+1 i _ i+l
% ’ T = T
rip "rip > ^<p0 — ^<p0
i i+l i i*1
^ ip » ur = u r , Uq= Uq
.
T w o m odels o f conjugation layers into a single p ack et are considered. In m ost cases th e re is a rigid contact w hen the cone layers are d eform ed w ithout slipping a n d tearing.
164 . N. P an k rato v a In many cases those conditions may be violated and desalination zones may appear.
W hen friction forces are small not taken into account it is possible to form ulate a m odel for the ideal slippage o f layers in those zones
i i+1 I i+1 i i+1 n
0 <P “ av ’ ~ T"P - _ - ^2^
t ¡»1 i ¡*1 i . ¡*1
= % , u, * ur . “a * “e ■
On conic bounding surfaces the loads applied are given by polynom ial laws
o ' ( < p p, r, e ) = r"-l a pv ( Vp), (<p^ r, e ) = r n' lxfv (<pp\
< e(«P ^ »•. 0 ) = ^ ‘ < e ( ‘p p (P = 0, N; n =1, 2, ...)
Take into consideration the equations o f equilibrium, the expressions for deformation by displacem ents, the H ook law for non-hom ogeneity anisotropy body the resolving system o f differential equations for definite o f stressed state o f laminated hollow cones is received.
T he resolving functions are taking as basis with the help o f which we can form ulate the conditions on limiting surfaces <p=<p0 , <P = <Pn ar>d surfaces o f conjugation o f layers (2-4).
Making a series transformation a system o f resolving equations in partial derivatives for i-oh cone layer is obtained
^ = £ c x , + * ' c ‘ = i c > p ) i , m-1
(5)
— j / l l I I I l\ —i / I I l\
O ' = i O v , T r v , X v 6 , U v , U „ Uq }, g ' = ( g „ g 2 , . . . , g 6 i ,
- i - i - i d o ' - t d a 1 - i - i c P o * - i cPo' O1 — 0 ,Oo — — , Oi = ---1 O 1 — — — -, Or — ----y Or —---
1 2 d r 3 dd 4 d rd d 5 d r2 dd2
( <PJ_1 s i p s i = 1, 2,..., N; p , q, m = 1, 2,..., 6)
The vector com ponents g1 are defined by the tem perature field and depend on the properties o f the layers material as well.
By representing the unknown functions and acting loads as expansion in terms of orthogonal trigonometric functions, on separating the variable in (5) w e get a system of ordinary differential equations for each layer i and the magnitude o f the harmonic number
S tressed s ta te o f thick-w alled .. 165 for the case o f axially symmetric deformation
— = i4(cp)o + / , d o a = { a v, xr<p, uv, ur, ue },
A , * w / ' • " ~ T « ' (O' " r * " V • '
d(f) y (6)
7 - </,. /2,.., /4}. ¿(<p) =
|a„(«p)l, P,
9 = 1, 2 6H ere nonzero elem ents o f matrix A and vector f take the form [2],
The boundary problem described by a system o f differential equations (6) is solved by a stable numerical method that makes it possible to solve one-dim ensional boundary problem s with a feasible range o f accuracy.
2.2. C alculation o f the stressed state o f a laminated hollow cone
The above approach to handling a problem in three- dimensional formulation is used to study the influence o f change in the winding angle through thickness on the stress level and deformability o f a conical elem ent made o f material with elastic constants E (() = 1.63Eq, E 0 =1.6Eq, E (p= 2 0 .1 E o , v (p0=O.543, v r(p=0.324, v 0r=O.O24, G (pr= G (p0=O.O878Eo.
A hollow cone is subjected to axially symmetric pressure CT=a0rn applied to the surface
<P = <Pq=tt/18.
Consideration is being given to single-, two-, three-, five- layers cones w hose layers are reinforced with Fibers at equal and opposite angles with the axis r, i. e. the main elasticity directions are turned relative to the coordinate lines <p and 0 by the angle B in layer 1, 3, 5 and by the angle -B in layers 2 and 4.
The calculations were performed with the following initial data n i l <<pN <nl4, B0=rr/12, B jq =7r/6, ;r/4, tt/3, 5ir/12, n/2; n = 2 , 7.
Figure 1 gives the distribution o f stress in the vicinity o f conjugation surfaces o f the first and second layers (by solid lines) and on the surface o f applied loads (by one-dashed lines) o f five layers cone for various values o f parameters B, which characterizes the variability of winding. A s evident from the solution as the thickness cpN and the angle B increase so does the contribution o f stress and displacements induced by the material anisotro*^ whereas for winding angles considered above this is due to the discrepancy betw een the main directions o f elasticity and the directions o f coordinate lines. When the angle B is close to zero and n i l
166 N. P an k rato v a the influence o f Ug, r rg, r ^ g on other factors o f the strain-stress state is insignificant. A change in the angle J1 through the thickness o f the cone may lead to o f stress qualitatively different distribution o f the stress-strain state.
The investigation perform ed revealed that when determ ining the stress-strain state in cones o f advanced materials account must be taken o f specific features due to material non-hom ogeneity and anisotropy.
R E F E R E N C E S
[1] Lehnitskij S.G.: Theory o f elasticity o f anisotropy body. Nauka, Moskwa, 1977. In Russia.
[2] Pankratova N.D.: The solution o f problems o f the stressed state o f anisotropic laminated cone. D okladu AS Ukraine, N10, 1988. In Russia.
[3] Grigorenko Y.N., V asilenko A.T., Pankratova N.D.: Problems o f elasticity theory o f non
hom ogeneity bodies. Nauk. Dumka. Kiev, 1991. In Russia.
R ecenzent: Prof. dr hab. inż. Eugeniusz Świtoński W płynęło do Redakcji w grudniu 1993 r.
Streszczenie
Motywacją techniczną do opracowania m etod rozwiązania problem ów deformacji elem entów wykonanych z materiałów anizotropowych o jednej płaszczyźnie symetrii sprężystej jest potrzeba dokonywania obliczeń konstrukcyjnych elem entów wytworzonych z m ateriałów ortotropowych m etodą nawijania.
F ig.l. Distribution o f stress R ys.l. Rozkład naprężeń
S tressed sta te o f thick-w alled .. 167 W przypadku grubościennych elem entów stożkowych ważnym zagadnieniem jest znalezienie racjonalnego schematu zbrojenia. Sterując tym schem atem w kierunku promieniowym i obwodowym można wpływać na wytrzymałość i sztywność konstrukcji.
Zaproponowana m etoda rozwiązywania zagadnień dotyczących stanu naprężeniowo-odkształ- ceniow ego elem entów stożkowych wykonanych z materiału, który w każdym punkcie posiada jedną płaszczyznę symetrii sprężystej prostopadłą do powierzchni stożka, m oże służyć jako podstawa badań wpływu kąta zbrojenia na naprężenia i odkształcenia.
