THERMAL STRESS STATE IN THE END REGIONS OF MULTILAYERED SHELLS A. M. Butko
UDC 539.3
A broad range of studies has been devoted to the design of laminated shells for static, thermal, and dynamic loading. An example of such a study is [4]. Of the many theories proposed for the design of laminated shells, we note the theory developed in [i, 2]. This theory makes it possible to analyze the stress--strain state in a monolayer with allowance for its metric. The theory in [i, 2] was used in [5] to study thermoelastic edge effects in a cylindrical shell having two stiff layers. We should also point out certain new approaches taken in [6-8] to the design of multilaminate shells. Here, we propose a method of studying the deformation of the end regions of multilaminate cylindrical shells from thermal loading on the basis of the shell theory in [i, 2] and the method of discrete orthogonal trial runs developed by Godunov [3]. Different types of end fastenings are examined for a cylindrical shell with five stiff layers. I. Theoretical Model of Multilaminate Shells. We will examine a multilaminate shell of regular structure consisting of stiff layers with a thickness h and more flexible layers with a thickness s. We will henceforth refer to these two types of layers as stiff (or bearing) and flexible (or binding) layers. By regularity of the shell, we mean that the thermophysical and elastic constants of the material of the layers and the thicknesses of the layers are independent of the number of layers in a packet k. We will refer the laminated shell to a curvilinear coordinate system connected with the middle surface of one of the stiff layers. Also, we will assume that the principal axes of the tensors of the elastic and thermophysical constants coincide. Let the shell be located in a temperature field. Here, we designate the temperature of the external medium as T+(x~, x2, t), and the temperature of the internal medium as T_. The adjacent layers are assumed to be in ideal thermal contact. Heat transfer takes place by Newton's law on the shell surfaces in contact With the external and internal media. We adopt the hypothesis that the temperature field causes small elastic strains. Also, we will ignore the inverse effect of the strain field on the temperature field. The stiff layers are assumed to be thin elastic anisotropic shells subject to the Kirchhoff--Love hypotheses. The flexible layers are assumed'to be transversely flexible in the sense that only the transverse shear strains and the extensions of the normals are important for these layers. Here, we adopt a linear law of change in the displacements through the thickness. The components of the displacement and strain tensors and the changes in the curvature of the stiff layer are written as follows:
----
+ --
= [r~)- ~k) "'g' =
T
Ivy)
-
--~
~v
n,
(i.i)
_~k)V(k),,~+ b~(k); +
-
= ""
l a y e v~k) Herer ~(k i w(k) a r e t h e d i s p l a c e m e n t s of p o i n t s of the m i d d l e s u r f a c e of t h e k - t h s t i f f I are mixed components of the second principal tensor of the same surface; Ilk) is the symbo~l for covariant differentiation; 68 is the Kronecker symbol (a, 8, 7 = 1 2); z' is the normal coordinate. The important components of the strain tensor of the k-th flexible layer have the form
Moscow Institute of Water Management. Translated from Prikladnaya Mekhanika, Vol. 24, No. 5, pp. 68-75, May, 1988. Original article submitted June 17, 1986.
492
0038-5298/88/2405-0492512.50
9 1988 Plenum Publishing Corporation
e~3tkl = [c~7tkl (w(k+,) + w (k)) + v~ +1) - - o=" (~) - - ~,.m'tk]~.'~(v(~+,) . + v(~))l/2h; 8[kl
(w (k+b
w (k) - - a'T[~]h)/s;
c
(h "b s)/2.
(1.2)
The superscript in the brackets denotes a relationship to the flexible layer ; ~' is the coefficient of thermal expansion of the material of the stiff layer in the normal direction. Single primes henceforth denote that quantities pertain to the stiff layers, while double primes correspond to the flexible layers. It should he noted that the quantity e33[ k] was found with allowance for the thermal expansion of t h e a d j a c e n t stiff layers. With allowance for the temperature terms, we use the following formulas to find the forces and moments in the stiff layers and the normal and shearing forces in the flexible layers
N(~)~I~= h ~ ' ~ v ~ . (ev6 (~)
-
-
a$~T(~));
M(~)~ = ~ h~ ~,. "~l~v~~. (~) + ~$~oi,>); ~,,0
(1.3)
N [~1 = sE" ~3~~ -- oCT[~]): q[~]~ = 2sat~l~'~G.et~]. Here, k'~6~ 6 are contravariant components of the elastic-constant tensor; ~ 6 are con))Y travariant components of the tensor of the coefficients of thermal expansion; E and G are the transverse modulus and shear modulus of the material of the flexible layer; a[k]~6 is the metric tensor of the middle Surface of the flexible layer. Use of the principle of the free energy minimum [2~ makes it possible to obtain finitedifference differential equations for thermoelastic equilibrium
--2s-'cbtk]~(t~Q tkl'~ + t~q tk-q~
- t (t~N[~]
t"kN[k-q)
--- O;
(1.4)
O.
In Eqs. (1.4), we introduced the notation:
. 111
112
where Ha are the Lame constants. To determine the temperature parameters in Eqs. (1.3), we adopt a heat-conduction system which is consistent with the theoretical shell model. A linear approximation can be used for thin shell layers, a sufficiently slow change in temperature with respect to the coordinates of the middle surface, and a sufficiently low frequency of temperature variation over time. For example, such an approximation for the stiff layers can be written as
T (k) ---- T~ok~ -b z;Oik).
(1.6)
Here, T~ k) is the temperature of the middle surfaces of the k-th stiff layer. We use the conditions of ideal thermal contact to obtain relations linking the characteristics of the temperature distribution in the flexible layers with the corresponding functions for the stiff layers. These relations are simplified considerably if we use a
493
Is II
k
Wk
25O
/
0,80
I
0/'0
- 250
0
-o,,+o
-
o
o, I0
o,30
qm
x,
500 0
Fig. 1
0,02
0,0~,
0,0~
x~
Fig. 2
linear approximation through the thickness of both the stiff and flexible layers. This results in a situation whereby the gradients are the same in all of the layers and can be used to determine the temperature in any stiff layer. Then it is necessary to know only two functions -- To (I) = T and 0'(i) = O, for example -- to find the temperature distribution for the entire shell. As an illustration, let the number of stiff layers be odd n = 2m + i. We give the middle layer the subscript "0" and the remaining layers the subscripts • • ..., • Then
(1.7)
TOo~ = T + kh(1 -F ~,) O, where I = l's (~"h)-1; %', i" are the thermal conductivities and flexible layers in the direction of the common normal.
for the materials of the stiff
The heat-conduction equations are obtained on the basis of V. V. Bolotin's variational principle of heat conduction [2]. In cases where the change in the metrix for the temperature distribution can be ignored, the equations of nonsteady heat conduction have the form 0T
T) + (•
Zr - 3 7 - - - H ~ ( ~ ' ~ O0
+ ~t_)T + ( • 2 1 5
hTIO = x+T+ + x - T _ '
h2
xs--~--- H - ~ V- (X~~V ~0) + ;%HO + h~ (x+ -- • 1
q- ~ h ~q(•
+ •
T+
(1.8)
0 = h~q(•
We used the notation from [2] in Eqs. (1.8) and we used • to denote the heat-transfer coefficients of the material of the stiff layers in the external and internal media. 2. Analysis of the Thermal Stress State of a Cylindrical Shell in the End Region by the Method of S. K. Godunov. We will examine a multilayered cylindrical shell in a nonuniform steady-state temperature field. W w will study the stress--strain state which develops in the most hea~ily stressed regions of the shell -- near the ends. Let the external temperature field be axisymmetric. This allows us to solve the system of thermoelastic equations for multilayered cylindrical shells in an axisymmetric formulation. Inserting Eqs. (1.3) into Eqs. (i.4) with allowance for (i.i) and (1.2), w e obtain the system of thermoelastic equations for multilayered cylindrical shells in displacements. In canonical form, this system can be represented as :
d-Y(n) = Aln)7 dx~
Y(") + aC,,~'
(2.1)
-.).
Here, we have introduced a 6.n-dimensional vector Y(n)"
The elements of this vector are
~,n, ----(ul' u~, w1, w~. w~, w~"..... W~)r.
494
(2.2~
-f 33
!
o,~o q
7,5
ojo
5,o
o
z,5
-07o
o
o, to
o,2o
Fig.
o
o,3o
~2
/
"-5
o,oz
X t.
3
06
~o~
Fig.
4
The primes denote d i f f e r e n t i a t i o n with respect to the longitudinal longitudinal displacements of the k-th stiff layer. The nontrivial expressions
elements
of the matrix
of the coefficients
a~+i,2+i=a3+i,4+i=a4+i,5+i.=as+i,~+i a2+i.l+i__=
rh-}-rk-I 9~
R~x;
a6+1.2+i =" bk; rh . 2Oh
as+i,8+i =
u k denotes from the
l;
a2+i,4+i=Ph; rk
rk R*ex ; 9h
coordinate;
A (n) are found
:
- - - -
a2-ki,7+i ~
xt
a2+iJo+i = = -
a6+i,a+i = ch;
.
,
29h
(2.3)
a6+i,s+l = dh; pO
a6+i,9+i
'
P~ ~;
=
as+~j +i =
r k - I R~x ; Ph
a12+t,2+~ =
rk_l ; 2? h
as+~,4+,"=
al2+~,s+t
al2+i,5+i =
Here, we have introduced the following k = 2, 3, ..., n.
as+i.n+l
~t~
notation:
In the case of a steady-state temperature through the thickness, the nontrivial elements
%;
-=
rk_l ; 29h
9~ = - ?k
~;
P~ l 4? k %.
j = 6(k -- i), k = i, 2,. .... n; i = 6(k -- 2), field in the l~yers w h i c h is n o n u n i f o r m of the v e c t o r R(n ) are found from the formula
R~-~ = I.T ~k~-- x'l~R~.x (T[% ~
T[~-']O~-,)I Vr'.
(2.4)
Equation (2.4) is valid for the case when the temperature field is independent of the and the longitudinal coordinate and is axisymmetric. In this case, T [k ] _- (T( k+ I ) + T( k )/2 corresponding'temperature parameters are determined from (1.8) w i t h allowance for (1.6) and (1.7). We used
the following
dimensionless =
_
E ' . _ E"'
~z" ~*=--" ="
G" ~ = -'-Ew- • , =
parameters
in (2.3):
h ~ = - - ; h +s
* Rex
Rex h + s
m,(l --~)
+__~__~.
1 +v'
'
•
ra~
Rin . ' Rex
(2.5)
n1+ 1--m o ___
1 - - v 12 . /~(1 --~)'
495
P; R~
=
(R~ + R~,+0/2;
pO
=
?~ = Ph
pO
0;
=
/12;
0
=
*
pk%Rex;
p~ = (rk-1 - - r ~ -
bk = (rk-I - - rh - - 2v')/2?h; ck =
r~
(pO~ +
x
rD*2 9 =
~
,, ex,
2v')/2p~;
dh = (9 ~ + 9~
%/4~'k;
p~_~x + pZ~)/W.
Here, we examined the case when both the stiff and the flexible layers of the shell are isotropic. In these equations, E' and v' are the elastic modulus and Poisson's ratio of the material of the stiff layers; To is the reference temperature; Rex and Rin are the external and internal radii of the shell; Rk' and Rk" are the radii of the middle surface of the k-th stis and flexible layers. System (2.1) is closed by boundary conditions on the end of the shell. can be written in matrix form
BYcn) = g.
These conditions
(2.6)
§ Here, B is the coefficient matrix, with the dimensions ( n x n); g is a vector with n/2 dimensions. To analyze the stress--strain state of multilayered cylindrical shells, we developed a software package oriented toward the ES-series computers. The main subset of the package consists of programs which perform orthogonal trial runs of the type devised by S. K. Godunov. We examined shells having five stiff layers. system (2.1) was 30.
In other words, the order of canonical
As an example, we will look at shells with the main parameters: /:=10; mo = 0 . 9 ;
n----5;
=,=10;
v'=0.3;
ap=0.2;
(2.7)
v"=0,25;
l,----0.4,
where ~, is the region of the edge effect at the end of the shell. It was assumed in the numerical calculations that the temperature field is stationary and uniform (To = T+ = T_). We varied the types of end fastenings. The following possible variants of fastening of the shell in the section xl = 0 were examined. Rigid fastening of each stiff layer of the packet. In this case, the nontrivial (equal to unity) elements of the boundary conditions matrix will be as follows: b1~; b2~; b3~; b~+3Z,7+61;bs+3~,9+61;b6+~ij1o+~l; ~ = k -- 2; k = 2, 3, 4. Rigid fastening of the first four stiff layers, with the external layer (k = 5) being free of restraints and loads. In this case, for the external layer the longitudinal force, the moment, and the Kirchhoff generalized shearing force ar4 equal to zero. The latter force is determined from the expression
[ 0"~ = O dx~
do,k,
Fc u~k+l> - - u (k) + c
dx""'-~ q- 2 ~
+
dx'-----~ ;
(2.8) E'h s
D = 12 (1-- ~'2)'
G"
F=--.s
A rigid ring was installed in the section xl = 0 in the third variant. The ring was rigidly fastened to the external and internal layers. Here, we assumed that a system of rings is located regularly along the shell at distances significantly greater than the size of the region of the edge effect. Then the following conditions are established for the internal layers: 496
M•,
IOJ
0,25
k I
0
,4
\
3
-O,Z5
\/'%
-O,50
/\
-0,75
Fig. 5
uh=O; w~=O; Q;=O.
(2.9)
In the fourth variant, all of the stiff layers of the shell are rigidly attached to a flange. The flange, located at the level of the central layer, is connected to a hinge. In this case we have
wk=O;
u3=O;
Uh--w~Zk=O; =
(k =
(2.10) 2, 3, 4, 5),
Here, Zk is reckoned from the central layer. It is also necessary to require that the central bending moment be equal to zero. This moment is written with allowance for the longitudinal forces in the stiff layers and the metric of the flexible layers. Figure i shows the change in the normal displacements of the layers in the end region in the case when rigid fastening is realized. The numbers of the curves in Figs. I and 2 correspond to the number of the layer k (Figs. i and 2 show dimensionless relations). The effect of the parameter a, is shown in Fig. i. This parameter characterizes the ratio of the coefficient of linear expansion of the material of the flexible layer to the same for the stiff layer. The solid curves correspond to parameters (2.7), while the dot=dash curves correspond to a, = 20. Thus, an increase in the ratio of the coefficients of linear expansion of the flexible and stiff layers is accompanied by a substantial increase in the normal displacements. Here, the internal layer has a deflection toward the center of the shell. The deflection is greater, the greater the value of the parameter a*. The dashed curves in Fig. i correspond to the second type of fastening of the shell end. Figure 2 shows curves corresponding to the second derivative of the normal displacements of the layers relative to the longitudinal coordinate~ It should be noted that the first two layers have a negative curvature at xl = 0 in relation to the next three stiff layers. The results here pertain to the first type of end fastening. The values on the given curves are proportional to the values of the corrected bending moment in the stiff layers determined from the formulas
M~ =--
M~ =
dx~
'
(2.11)
12(1--v ,2 ) A,r E'R~x =T~ "'h"
Figure 3 shows the change in the elements of the vector of the solutions Y(n) in the end region with the third type of fastening. Curve i corresponds to normal displacements in the second stiff layer; curve 2 corresponds to normal displacements in the third layer; curve 3 corresponds to normal displacements in the fourth layer. Curve 4 corresponds to the first derivative of the longitudinal displacement in the third layer. We note the presence of negative normal displacements in the second stiff layer of the shell at xl = 0.
497
Figure 4 shows relations for the corrected strains in the flexible layers, calculated from the formula
e[k]__ Rex [ 93s-- l--~
~; .] wk+1--~v~--R~x( I q- v') '
(2.12)
in the end region for the fourth type of fastening. The numbers of the curves correspond to the numbers of the layers. The dashed curve corresponds to the second type of end fastening. Figure 5 shows values of the corrected bending moments in the section x~ = 0 in the stiff layers of the shell. Here, the numbers of the shells correspond to the numbers of the variants of end fastening. Analyzing the relations in Fig. 5, we can conclude that the thermal stress state in the end region is most nonuniform when the first and fourth types of end fastening are realized. LITERATURE CITED i. 2. 3. 4. 5. 6. 7. 8.
498
V. V. Bolotin, "Plane problem of the theory of elasticity for parts made of reinforced plastics," Raschety Prochn., No. 12, 3-31 (1966). V. V. Bolotin and Yu. N. Novichkov, Mechanics of Multilayered Structures [in Russian], ~shinostroenie, Moscow (1980). So K. Godunov, "Numerical solution of boundary-value problems for systems of linear ordinary differential equations," Usp. Mat. Nauk, 16, No. 3, 171-174 (1961). A. A. Dudchenko, S. A. Lur'e, and I. F. Obraztsov, "Anisotropic multilayered plates and shells," Itogi Nauki Tekh. Mekhaniki Deform. Tverd. Tela, 15, 3-68 (1983). Yu. N. Novichkov and A. M. Butko, "Thermoelastic edge effects in multilayered cylindrical shells," Raschety Prochn., 17, 76-85 (1976). B. E. Pobedrya, Mechanics of Composite Materials [in Russian], MGU, Moscow (1984). Ya. S. Podstrigach, V. A. Lomakin, and Yu. M. Kolyano, Thermoelasticity of Bodies with a Nonuniform Structure [in Russian], Nauka, Moscow (1984). L. P. Khoroshun, "Method of constructing equations of a shear theory of thermoelasticity of laminated plates and shells," Prikl. Mekh., 16, No. I0, 21-30 (1980).