AD TECHNICAL REPORT ARCCB-TR-90016 A PLANE-STRAIN ELASTIC STRESS SOLUTION FOR A MULTIORTHOTROPIC-LAYERED CYLINDER 0 o MARK D. WITHERELL 0 Lfl N DTIC rE LECTE It AGO81S)I JUNE 1990 US ARMY ARMAMENT RESEARCH, DEVELOPMENT AND ENGINEERING CENTER *CLOSE COMBAT ARMAMENTS CENTER BEN9T LABORATORIES WATERVLIET, N.Y. 12189-4050 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED i i I I i !1 (l Ii
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AD
TECHNICAL REPORT ARCCB-TR-90016
A PLANE-STRAIN ELASTIC STRESS SOLUTION
FOR A MULTIORTHOTROPIC-LAYERED CYLINDER
0o MARK D. WITHERELL0Lfl
N DTICrE LECTE It
AGO81S)IJUNE 1990
US ARMY ARMAMENT RESEARCH,DEVELOPMENT AND ENGINEERING CENTER
*CLOSE COMBAT ARMAMENTS CENTERBEN9T LABORATORIES
WATERVLIET, N.Y. 12189-4050
APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED
i i I I i !1 (l Ii
DISCLAIMER
The findings in this report are not to be construed as an official
Department of the Army position unless so designated by other authorized
documents.
The use of trade name(s) and/or manufacturer(s) does not constitute
an official indorsement or approval.
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SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)
READ INSTRUCTIONSREPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER
ARCCB- TR- 90016
4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED
A PLANE-STRAIN ELASTIC STRESS SOLUTION FOR A FinalMULTIORTHOTROPIC-LAYERED CYLINDER 6. PERFORMING ORG. REPORT NUMBER
7. AUTHOR(@) 8. CONTRACT OR GRANT NUMBER(a)
Mark D. Witherell
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASKAREA & WORK UNIT NUMBERS
U.S. Army ARDEC AMCMS No. 6126.23.lBLO.0Benet Laboratories, SMCAR-CCB-TL PRON No. 1A72ZH3HNMSCWatervliet, NY 12189-4050
11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE
U.S. Army ARDEC June 1990
Close Combat Armaments Center 13. NUMBER OF PAGES
Picatinny Arsenal, NJ 07806-5000 2414. MONITORING AGENCY NAME & AODRESS(If different from Controlling Office) 15. SECURITY CLASS. (of thlil report)
UNCLASSIFIED
15a. DECLASSIFICATION/ DOWNGRADINGSCHEDULE
I6. DISTRIBUTION STATEMENT (of thls Report)
Approved for public release; distribution unlimited.
17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, It different from Report)
18. SUPPLEMENTARY NOTES
IS. KEY WORDS (Continue on reveres aide It necesemy end identify by block number)
Stress Distribution,Orthotropic.Multilayered
CylinderComposite
2G. A3rT' ACT Cthas -e w efi f N m idestitf by block mntboa)
In 1963 Lekhnitskii published the equations that describe the distribution ofstresses in a monolayered anisotropic cylinder under the influence of variousloading conditions. Recently, O'Hara simplified these equations for the caseof an orthotropic cylinder and cast them into a form that is convenient touse. This report discusses the methodology of constructing the stresssolution for a multiorthotropic-layered cylinder under plane-strain boundaryconditions, i.e., all orthotropic layers have zero axial strain. Combined
(CONT'D ON REVERSE)DI JAN 1473 EDITION OF I .oV 6 IS OSOLETE
JM 73 V UNCLASSI FIED
SECUR ITY' CL.ASSIF.ICATFOM OF "T4S PIAGE (Wh~pen Date Entered)
SECURITY CLASSIFICATION OF THIS PAGE(Wham Data Entermd)
20. ABSTRACT (CONT'D)
loadings of internal and external pressure are allowed. Solutions for aselected layup and loading conditions are compared with finite elementresults and show excellent agreement. In addition, a procedure to obtainthe stress solution when an interference exists at the interface of twoorthotropic layers is discussed. -
3. Theoretical stress and strain plots for N =10,5*(1-hoop, 1-axial), (see Table I), a(1) =1.0 in.,b(10) = 2.0 in., p(l) = 1.0 psi, q(10) = 0.0 psi........................18
4. Theoretical stress and strain plots for N - 10,5*(1-hoop, 1-axial), (see Table I), &(1) = 1.0 in.,b(10) = 2.0 in., p(1) = 0.0 psi, q(10) = 1.0 psi........................19
i
Page
5. Theoretical stress and strain plots for N = 10,5*(1-hoop, 1-axial) (see Table I), a(1) = 1.0 in.,b(10) = 2.0 in., 1.0 Win. interference at the 5 to 6layer interface........................................................20
NOMENCLATURE
a - inner radius of orthotropic layer
[A] - the material compliance matrix
b - outer radius of orthotropic layer
Co - ratio of inner to outer radius of orthotropic layer (Co = a/b)
Er,E9,Ez - engineering modulii of orthotropic layer
FZ - axial force on orthotropic layer to enforce plane-strain boundarycounditions
FZP - layer axial force contribution per unit of internal pressure
FZQ - layer axial force contribution per unit of external pressure
FZT - total axial force on multiorthotropic-layered cylinder to enforceplane-strain boundary conditions
G~j - material-geometry constant from hoop strain equivalence condition(1 , i < N-1, I - j 4, 3)
[J] - matrix of Gij values correctly positioned
[JI] - inverse of [J)
k - orthotropic material parameter
N - total number of layers in multilayered cylinder
p - internal pressure on layer
[Q] - unknown external pressure vector
PI - interference pressure
q - external pressure on layer
r - radius
[R] - result vector
RAP,TAP,ZAP - magnitude of r,9,z stresses evaluated at r=a, caused by unitinternal pressure
RAQ,TAQZAQ - magnitude of r,O,z stresses evaluated at r=a, caused by unitexternal pressure
RBP,TBP,ZBP - magnitude of r,O,z stresses evaluated at r=b, caused by unitinternal pressure
iii
RBQ,TBQ,ZBQ - magnitude of r,O,z stresses evaluated at r=b, caused by unit
external pressure
gij - compliance for cylindrical problem
6 - radial interference of two multilayered cylinders
6i - decrease in outer radius 'b' of inner cylinder per unit ofexternal pressure (interference loading case)
60 - increase in inner radius 'a' of outer cylinder per unit ofinternal pressure (interference loading case)
6u - radial interference reduction per unit pressure at interface oftwo multiorthotropic-layered cylinders
c - strainCe
Vr8,LOz,vzr - Poisson's ratio of orthotropic layer (e.g., vr@ - --
r-stress producing contraction in 0-direction) Er
o - stress
[ ]- matrix of values
(i) - pertaining to ith orthotropic layer
Subscripts
a,b - inner and outer radial evaluation points
r,O,z - radial, hoop, and axial directions
1,2,3 - directions for orthogonal coordinate system (for a cylindricalsystem 1,2,3 correspond to r,9,z directions)
iv
INTRODUCTION
The attractiveness of many composite materials is their high specific
stiffness making them an ideal choice for lightweight applications. In order to
effectively design cylinders that incorporate composites, it is important to be
able to predict the stress distribution caused by various loadings applied to
the cylinder. In the early 1960s, Lehknitskii (ref 1) published a generalized
plane-strain stress solution for a monolayered anisotropic cylinder under com-
bined loadings of internal pressure, external pressure, and axial force. More
recently, O'Hara (ref 2) simplified these equations for the special case of an
orthotropic material and cast them in a form that is convenient to use. In real
applications, however, composite cylinders are often constructed by winding
fibers or laying up fibers at various angles. Generally, the cylinder is
constructed by building up fiber windings in positive and negative wrap angle
pairs. Each of these positive and negative wrap angle pairs can be viewed as a
single orthotropic layer. The whole structure can be considered a multi-
orthotropic-layered cylinder. For these types of multilayered cylinders, it
becomes important to be able to obtain the stress distribution so that design
and analysis can be pursued. By using Lehknitskii's monolayered solution and
the proper boundary conditions, the multilayered solution can be constructed.
The equations presented herein are for a multiorthotropic-layered cylinder under
plane-strain boundary conditions with internal pressure, external pressure, and
interference loadings.
GEOMETRY
A multilayered cylinder can be viewed as an assembly of many single-layered
cylinders. It is fitting, therefore, to begin with a review of the geometry of
the monolayered cylinder problem.
For the monolayered case, the cylinder is assumed to be long with ends that
are fixed (see Figure 1). The fixed-end condition implies zero axial strain
through the radial thickness of the cylinder. The cylinder has an inner radius
'a' and an outer radius 'b'. A cylindrically orthotropic material is assumed
with its principal axes coincident with the cylinder. Orthotropic material, in
general, is characterized by nine independent material constants. These
constants consist of three engineering modulii (E1 ,E2 ,E3 ), three shear modulii
(G1 2 ,G2 3 ,G3 1), and three Poisson's ratios (V1 2 ,V2 3 ,u3 1 ). The numbers 1,2,3
indicate the principal material directions. For the above assumptions, the
1,2,3 directions correspond to the radial, hoop, and axial directions of the
cylinder (rO,z). The cylinder can be subjected to internal pressure 'p' and
external pressure 'q'. In addition, since the principal stress directions
correspond to the principal geometry directions of both the cylinder and the
applied loadings, shear effects are eliminated.
STRESS EQUATIONS FOR MONOLAYERED ORTHOTROPIC CYLINDER
Lekhnitskii's solution (ref 1) for a cylinder with one anisotropic layer can
be simplified for the case when the material is orthotropic. This simplifica-
tion was done by O'Hara (ref 2) and resulted in three stress equations that
correspond to the r,O,z directions and one equation that predicts the axial
force necessary to enforce the fixed-end constraint. These equations are given
below and are identical to those found in Reference 2 with one correction for a
typographical error in the oz equation. Also, the axial force name has been
changed from PP to FZ for clarity.
r=[PCO k+ q] rk-1 + o- [ ] ck+l( )k+l 1
2
o l---- - 1 - k F-o - 1--+ (2)L -k1 bk- L I-CO2k) ]r
-1 [PCkj q r 1- ~o-
= a L --- 2-- (Al3+kA23)() k-i --- (A13-kA23)Cok+l(k+l1
(3)
27T Lb2(AqPCk+1)(lCok+1) + kA2 3
FZ A33(I-Co 2k)L + k
+ a2(qCk-lp)(1- Ck-1) A13 - kA23 (4)
where 'r' is the radial position in the cylinder,
a (5)
and the components Aij are the elements of the compliance matrix as given in
Hooke's Law,
[e] = [A] [a] (6)
'k' is an orthotropic material constant given by
k Ali (7)A22
where, in general,
Ai3A 3 (8)Aij Aij A33
3
In developing the multilayered solution, applying the correct boundary
conditions at the interface of two orthotropic layers necessitates the use of
stress values evaluated at the inner (r=a) and outer (r=b) surfaces of each
layer. This evaluation process leads to the six stress equations given below.
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