Stresses and deformations in thick, short, composite cylinder subjected to axially varying radial pressure are studied. Effect of slippage at the interface is examined. In the NASTRAN finite element model, multipoint constraint feature is utilized. Results are compared with theoretical analysis and SAP-IV computer code. Results from NASTRAN computer code are in good agreement with the analytical solutions. Results suggest a considerable influence of interfacial slippage on the axial bending stresses in the cylinder

Patel, Y. A.

English

NASA Technical Reports Server

DEFORMATIONS OF THICK TWO-MATERIAL CYLINDER
UNDER AXIALLY VARYING RADIAL PRESSURE
Yakub A. Patel
Sargent & Lundy
SUMMARY
Stresses and deformations in thick, short, composite cylinder subjected to
axially varying radial pressure are studied. Effect of slippage at the inter-
face is examined. In the NASTRAN finite element model, multipoint constraint
feature is utilized. Results are compared with theoretical analysis and SAP-IV
computer code. Results from NASTRAN computer code are in good agreement with
the analytical solutions. Results suggest a considerable influence of inter-
facial slippage on the axial bending stresses in the cylinder.
INTRODUCTION
Th6 analysis of composite (multilayered) cylinders and other axisymmetric
bodies with shrink (or press) fit forms an extremely useful class of problem
in the design of pressure vessels and various machine components. For example,
Figure 1 illustrates a special slipping pipe anchor (Ref. i) consisting of the
clamped (or press fit) support. Figure 2 illustrates a typical bar drawing die
composed of an outer hardened steel cylinder press fitted on the inner carbide
insert. The loading, in both cases, on the inside surface of the cylinder is
variable axial pressure dependent on the seismic motions (Fig. i) and percent-
age reductions (Fig. 2) respectively. This paper attempts to present the prob-
lem in a generalized way and indicates, in quantitative terms, the effect of
interfacial slippage on the various stress components.
SYMBOLS
u r
u
z
L
Young's modulus of elasticity, MPa
Poisson's ratio (Note: Suffix 1 refers to inner cylinder and 2 refers
to outer cylinder)
Radial displacements, cm.
Axial displacements, cm.
Length of the cylinder, cm.
161
https://ntrs.nasa.gov/search.jsp?R=19770013551 2020-03-20T07:32:23+00:00Z
Zy,r
P
k
O
zg
_e6
CI
rr
a
Axial coordinate
Radial coordinate
Internal pressure, MPa
Scale factor for pressure
Axial bending stress, MPa
Hoop stress, MPa
Radial stress, MPa
Inside radius, cm.
Interface radius, cm.
c Outside radius, cm.
A I = (a2b 2) / (b2-a 2)
A 2 = b2c2/(c2-b 2)
BI = (b2+a2) / (b2-a 2)
B 2 = c2+b2/(c2-b 2)
CI = AI/2 (i-_ I)
N I = BI/(i+_i) 16
M I (a2+b2)Nl + AICI b= in-
2 a
a
M
c
= Residual moment for inner cylinder
i+i[i(11)All
E 1 - 2 i + _i Blb--b-
l+_l[ l(l-_l) ]1
_2 E1 _ _ b
_5 - E 2 - _ 1 7 B2b--b--
1+_2[ i(i-_2) ]q
i _2_6 E2 _ _ b
]
162
ANALYTICALMODELS
Figure 3 shows the NASTRAN(Ref. 2) finite element model of the composite
cylinder. The inner cylinder is of carbide and the outer cylinder is of steel.
The model consists of 210 nodes and 168 axisymmetric trapezoidal ring (CTRAPRG)
elements. Dimensions and material properties are typical of a bar drawing die
used in actual steel drawing process.
The loading consists of an axially varying radial pressure defined by the
equation P = k(Lz - z2). This pressure was input as discrete load on the in-
side surface using the 'FORCE'card of NASTRAN.
Nodes 91 through 120 are nodes along the interface of the carbide and
steel. This interface needs special consideration in the finite element model.
In practice, the case is shrinkfitted on the carbide insert, and a large amount
of compressive preloading is generated in the relatively brittle carbide insert.
This compressive preloading reduces the probability of cracking of the insert
during operation.
Since there is no welding (or permanentconnection) along the interface,
during nonuniform loading, slippage will occur along the mating surfaces. At
the interface, compatibility of the radial displacement provides:
Thus
u ) = u ) for O < Z <L_ _r carbide r steel
The axial displacement could vary depending on the amount of slippage.
u)z carbide z steel
NASTRAN (Ref. 3) has a very useful option available through the MPC command.
It defines the multipoint constraint of the form:
EA.u = 0 where Aj's are the real coefficients and u. are the nodal dis-
J ] J placements. J
Using this option, the present analysis was carried out on NASTRAN level
15.5 through the UCC 1108 System.
Figure 4 shows the finite element model for the SAP-IV computer code
(Ref. 4). This model is similar to the NASTRAN model - however, no slippage
can be allowed along the interface. Due to this, the compatibility of defor-
mation along the interface does not represent the actual deformation mode.
THEORETICAL ANALYSIS
A theoretical analysis of this problem is carried out by the author*. In
this analysis, the governing equations of three dimension elasticity are
*Being published as a separate paper. 163
satisfied on the curved surfaces using the series approach of Lee (Ref. 3).
The residual elasticity problem is then solved to partially satisfy the
boundary conditions on the ends of the cylinder. The compatibility of dis-
placements along the interface is included in such a manner that axial slip
is allowed during deformation, The resultant stress componentsderived are
as follows:
O
ZZ
carbide = (-2k I + 2k) [4Nlr2 + (C1 - 2MI) + 2Clln r]
_ee
- (2k I - 2k) 12 (a+br)Mc8(1 + _i [2r2 (a2 + b2)]+ (b a) 3 2
B1 All (Lz-z2)]
3Nlr2 + (CI M I)(M 1 a ml) _ + Clln
3r 2 - (a2 + b 2) - _ + E 1 + _lOzz16(1 + _i )
(kI - k)(Lz - z2)
+Vl_ (-2k I + 2k)
g
I
rr
where: kI =
[(k I - k) (Lz-z2)] [- BI_-+ r--_AI]--_'I [(k I + k)(Lz-z2)]
+Ik+k[rM
 11°r]
k(_ I - e2 )
_i + _2 + e5 - _6
2
a
+ (M1 - a2Nl ) r2
2kl + 2k [ a2b2 ]]r2 _ (a2 + b 2) + r---_
16(1 + _i )
RESULTS AND DISCUSSIONS
Fig. 5 shows the variation of axial stresses (Ozz) at r = a,b along the
length of the inner cylinder. The axial stresses reach their maximum value at
the center of the cylinder and the variation is symmetrical about Z = L/2 due
to the symmetry of the loading. The results from the NASTRAN model are in
good agreement with the analytical solution. Further, it may be noted that
the results are in better agreement on the inside surface than the outside sur-
face of inner cylinder - the maximum deviations are 2.8% and 8% respectively.
This is due to the fact that the inner surface is farther away from the slip -•
ping interface thereby less sensitive to the boundary effects.
164
Variations of the hoop stresses (Figure 6), radial stresses (Fig. 7) and
deflections (Fig. 8-9) also indicate good agreement between the analysis and
NASTRAN model. At the interface, in Fig. 7, the radial stresses on the out-
side surface of carbide and inner surface of steel are identical from the
analytical solution. In Figure 9, the length is twice the value given in
Fig. 3 while the remaining dimensions are the same as Fig. 3.
The effect of slippage at the interface, in quantitative terms, is demon-
strated by the values of stresses from SAP IV, NASTRAN and the analytical solu-
tion (Table I). As is indicated by this table, the effect is more pronounced in
the values of o - since the slippage makes the bending process more symmet-
zz
rical.
CONCLUDING REMARKS
The stresses and deformations in composite cylinder subjected to axially
varying radial pressure are examined using the NASTRAN finite element models.
The MPC feature of NASTRAN is very useful in simulating accurately the slipping
interface between carbide and steel. Results from the NASTRAN finite element
models and the analytical solutions are found to be in good agreement.
TABLE 1
Stresses in Carbide (Inner) Cylinder (Units: MPa)
Analytical NASTRAN SAP IV
Description Solution F.E. Model F.E. Model
At radius = 5.08 cm.
o
zz
o08
o
rr
-124.6 -121"0 -136.2
362.3 366-2 355.0
-223.4 -221.8 -220.6
At radius = 6.99 cm.
26.2 (Regular)
o 110-93 102.0 87.5 (Modified*)
zz
o88 279.7 275.2 279.2
-69-9 -74-4 -72.4
rr
(*Modified using, in steel at interface, one layer of a material having low
modulus of elasticity in z direction)
165
REFERENCES
o
#
o
Patel, Y. A.; Cho, F. L.; Dimopoulos, A. P.: Slipping Pipe Anchor - A Novel
Design for Piping in High Seismic Intensity Zones - ASME Paper 75-PVP-58,
AS_IE Conf. on Pressure Vessels and Piping, June 1975.
C. W. McCormick (Edit.) - The NASTRAN User's Manual - NASA SP-222(01) -
National Aeronautics and Space Administration, June 1972.
Lee, C. W.: A Theory of Thick Walled Cylinder Under Axisymmetric Loading -
Proceedings of the Fourth U. S. National Congress of Applied Mechanics,
(ASME) N.Y., 1962, p. 662.
Bath, K. J.; Wilson, E. L. and Peterson, F. E.: SAP-IV-A Structural
Analysis Program for Static and Dynamic Response of Linear Systems.
EERC Report 73-11, University of Calif., Berkeley, June 1973.
166
a, a 
a a 
167 
Q) 
a n  
Q) 
a .- - T I  .- 
L 
0 
3 
a c 
KJ 
3 
KJ 
Ll a 
KJ 
Iu 
0 
U 
V 
al cn 
168 
I 
II II 
N N  
ON ON 
aJ u 
d 
rn 
0 a 
0 
U 
a 
al 
5 
169 
I 
Z 
0 L ] t = 9.14cm. 
a = 5.08cm. c = 9, Icm. 
cm. 
Fig. 4 SAP I V  f i n i t e  e lemenr model of t h e  compos i t e  c y l i n d e r .  
170 
140.
105.
70.
I I I
• DNastran F. E.
--Analytical
- /.----R = b outside surface
of inner shell
I
Model
I I I I
2 4 6 8
Axial Distance, z, cm.
= a inside surface
of inner shell
I I I I
Fig. 5 Variation of axial stresses.
171
420.
280.
70.
0
I I
• [] Nastran F.E.
m Analytical
Model
[]
[]
[] "-_R - a inside surface
of inner shell
- b outside surface
of inner shell
R - b inside surface
of outer shell
"
outside surface
of outer shell
I I I I
2 4 6 8
Axial Distance z, cm.
Fig. 6 Variation of hoop stresses with axial distance.
172
I I I
• nNastran F.E. Model
-- Analytical
-30
Q_
(/)
-I0
I
rr
[]
-a inside surface of inner shell
[] R-- b outside surface of inner shell
m
_---R - b inside surface
of outer shell
[]
R = b interface
[] Axial Distance, z, cm.
10 ' ' '
8
Fig. 7 Variation of radial stresses with axial distance.
173
40 I I I I
•" 35
E
0
5=..
0
30
0
iB
0
25
CD
m
mm
15
R = b deflection in outer shell ..
I"
/
R = b deflection in inner shell
[]
,o Nastran Model
-- Analytical
I I I
2 3 4
Axial Distance,z, cm.
5
Fig. 8 Variation of radial deflections with axial distance z.
174
175
125
75
2
[]
! I I
R - b interface inner shell
[]
[]
R- b interface outer shell
[]
.... .o Nastran F.E. Model
--- Analytical
I I I
4 6 8
Axial Distance, cm.
10
Fig. 9 Variation of radial deflection.
175


Deformations of thick two-material cylinder under axially varying radial pressure

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