NPS-MA-92-005
NAVAL POSTGRADUATE SCHOOLMonterey, California
AD-A248 251
0I!T!I C S,! TRA
DTIC D
,= ELECT7E• APR07 1992
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ANALYTICAL STRENGTH FORMULAS
FOR SHIP HULLS
Donald A. Danielson
February 1992
Approved for public release; distribution unlimitedPrepared for: ATLSS Engineering Research Center
Lehigh UniversityBethlehem, PA
92-08912
NAVAL-POSTGRADUATE SCHOOLMONTEREY, CA 93943
Rear Admiral R. W. West, Jr. Harrison ShullSuperintendent Provost
This report was prepared in conjunction with research conductedfor the ATLSS Engineering Research Center, Lehigh University andfunded by the ATLSS Engineering Research Center. Reproduction ofall or part of this report is authorized.
Prepared by:
DONALD A. DANIELSONProfessor
Reviewed by: Released by:
HAROLD M. FREDRICKSENChairman Dean of ResearchDepartment of Mathematics
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Naval Postgraduate School MA ATLSS Engineering Research Center
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Monterey, CA 93943-5000 Lehigh UniversityBethlehem,PA 18015-4729
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Lehigh University PROGRAM PROJECT TASK WORK UNITBethlehem, PA 18015-4729 ELEMENT NO NO NO ACCESSION NO
II TITLE (Include Security Classicatior)
ANALYTICAL STRENCTH FORMULAS FOR SHIP HULLS
12 PERSONAL AU'' SDANIELS( , Donald A.
13a TYPE OF PORI j3b TIME COVERED 14 DATE OF REPORT (Year, Month. Day) 115 PAGE COUNT
Technical 1 F OROM 9/91 TO 12/91 92 - 2 - 21 20
16 SUPPLEMENTARY NOTATiON
17 COS A I CO is " 18 SUBJECT TERMS (Continue on reverse it necessary and identfy b) block number)
FiELD GROUP SUB.GROuP ship, double hull, shell, plate, structure, solidmechanics, elasticity, strength, bending, pressure,cylinder, stress, bulkhead
19 ABSTRACT (Coninue on reverse if necessary and identi) by b/ock number)
The subject of this report is a proposed new surface ship hull concept consisting of adouble skin that wraps around the bottom, sides and main deck. The two skins areconnected by plates normal to the surfaces, forming a cellular structure similar to acardboard box. Modeling the ship hull as a circular cylindrical orthotropic shellsurrounding an elastic core, we are able to obtain analytical formulas for estimatingthe principal stresses in such a structure, subjected to end bending moments andlateral pressure. These formulas indicate that the stiffness of the proposedbulkheads in the proposed design could be reduced by a factor of 10 without incurringsignificant secondary stresses in the double hull.
20 DISTRIBUTIONIAVA L4B;L17y Or ABSTRACT 21 ABSTRACT[ S[CJRiTY C{ ,SSI IATION
5 UNCLASSIF'E[) UNOITED 0] SA %'E A S RT r" E D'Ir U ,. nS
22a NALm" OF PES ON SIOLE I;D V D)AL 22h) TEL-kf'11U)'t (Iilu~/,dp' 4rKed ode) 7'c OFFcFE syrvH3.
Donald A. Danielson (408) 646-26121 MA/Dd
DD Form 1473. JUN 86 Pc)eous edton5 are osolee VC'1
A C (ASS' I v AJ S ;'AGI
UNCLASSIFIED
ANALYTICAL STRENGTH FORMULAS
FOR SHIP HULLS
BY
Donald A. Danielson
February 1992
Accesion For
NTIS CRA&I
JustificativB y i i ., i l ...... . . ... .. ..
B y ........ ............ ........
Dit,-,b,.t o., IA vidabi~it7 y (;.
Avdit a7.6 i orDist SpccaI
Bo A-, ...,
1
1. INTRODUCTION: NEW SHIP STRUCTURE AND MODEL
The proposed new surface ship hull concept consists of a double skin that wraps
around the bottom, sides, and main deck. The two skins are connected by plates normal
to the surfaces forming a cellular structure similar to a cardboard box, as shown in Figure
1. The advantages of this type of construction have been described by Okamoto, et. al.
[1], and Beach [2].
The object of this report is to develop formulas for estimating the principal stresses
in such a structure. In order to be able to obtain simple analytical formulas, we model a
ship hull as a circular cylindrical orthotropic shell surrounding an elastic core. Under a
pure end bending moment M and a lateral pressure loading q which does not vary along
the length, the cylinder bends into a curved tube of oval cross section, as shown in Figure
2.
2. SHELL EQUATIONS AND SOLUTION
The theory governing the deformation of thin shells is well developed and available in
many forms. Here we use the semi-momentless shell theory of Axelrad [3]. ** When spe-
cialized to the linear St. Venant problem for circular cylindrical shells, Axelrad's equations
(2.122) - (2.123) reduce to
a, K2 +0 22 R 2 8 2q
84+ =9T
-4 1 9 0 (2)8i74 a 7
Here the coordinate ql denotes the circumferential angle and R denotes the radius of the
undeformed cylinder, as shown in Figure 2. The principal values of the change in curvature
tensor are denoted by ,zl(7) and , 2 (1); their geometrical definitions arc illustrated in
Figure 3. The longitudinal membrane force T1( 7 ) (force per unit length of the shell
midsurface) is related to the longitudinal strain el(v7) by the constituitive law
Ti = Elie, (3)
Here h is the thickness of the shell, and E is Young's modulus. The longitudinal bending
moment M1(r,) and transverse bending moment M 2 ( 7 ) (moments per unit length of the
shell midsurface) are related to the curvature changes ,(q7) and ,K2 (71) by the constituitive
laws
All = Dl(,c, + v, 2), M 2 = D 2 (K2 + Vtl) (4)
** Our equations may also be obtained from other shell theories, such as that of Sim-
monds [4].
Here D1 and D2 are bending stiffnesses, and Y is Poisson's ratio. The longitudinal
curvature change xCi(i) and transverse membrane force T2(i') are given in terms of the
above variables by
'vi I 227 T 2 = Rq+ I 2(5)EhR 12 , R 12
The net bending moment M acting on any cross section of the tube may be calculated
from the formula
M = R2 o T cos i dqr (6)
The strain measures are related to the displacement components by the equations
1 02wKI = -- 0- (7)
/ T 2 w (8), C2 W + W
- 2awa3 O v + 1 3o ,v0 (9 )7-W2---- = (9) a
1 Ou= 1-- (10)
e2 = (- 9 + w -0 (11)
(= U + v 0 (12)
Here R denotes the distance measured along the axis of the undeformed tube, as shown in
Figure 2, and (u, v, w) denote displacement components in the (C ,rT ,radial) directions.
We suppose that the pressure loading q(rj) acting normal to the surface of the shell
can be expressed as
= -Kw(i7 ) + w(r+7 +)] - 6(1 + cos 2 17) (13)2
at i7 = 0 is the longitudinal curvature of the cylinder axis due to the beam-like bending.
The terms in (15), (17), (19), (20) which are underlined are secondary quantities arising
from ovalization of the cross section. Proper design requires that these quantities be made
relatively small by having a large enough foundation modulus K.
3. APPLICATION TO DOUBLE HULL
As an example of the application of these formulas, let us consider the double hull
sketched in Figure 1. An average hull section contains 3 plates each of thickness t and
width b, as shown in Figure 5, so the average hull thickness is
h = 3t (21)
The bending stiffnesses corresponding to this double hull geometry are
= 7Eb2t - Eb 2t (22)=12(1-V2) D2 2(1- V2 ).
Of interest for design purposes are tiie maximum stresses arising in the double hull. The
magnitudes of the membrane stresses corresponding to the membrane forces (14) - (15)
are less than
S3(23)
6R 2 + vb 2M 4R 2
aM2 =4t 127r(1 - v 2)tR 4 + 12t(1 + a- I)
The magnitudes of the bending stresses corresponding to the bending forces obtained from
(4) and (16) - (17) are less than
bM vRa2i= + bR' (25)
aR = 77r(1 - v 2 )tR 3 7bt(1 + 2KR4
a R 3 vbM0 B2 = 6bt(1 + 2KR4 ) + 67r(1 - v 2)tR 3 (26)
From the architextural drawings of the double hull cross section sketched in Figure
1, we have computed the average values displayed in the Table of the various geometrical
and material parameters in our formulas. The value of M is the maximum value of the
design hogging bending moment. I is the moment of inertia of the hull cross section about
its centroid. If we choose the cylindrical cross section to have the same moment of inertia
as the hull, this determines the radius of the cylinder to be R = 25.75 ft. We suppose that
the foundation stiffness K is the same as the bulkhead stiffness. From experiments on
pyramidal truss cores typical of those which are proposed for the bulkhead structure, we
have determined the average value of K shown in the Table.
Now let us put some of the numbers shown in the Table into the formulas (23) - (26):
aMI = 21.6 (27)
.7aM2 = 2 .1 + + (28)
3.2
CrBi 1 -1 5 + 2KI (29)9 D2
12.4CB2 = + .4 (30)
+ 2KR41+9D2
Here all stresses are in units of ksi. Note that if there were no elastic foundation (K = 0),
the secondary stresses (those terms underlined in (28) -(30)) would be a sizeable fraction of
the primary stresses. But r- 3733, from the values in the table, so these secondary
stresses are rendered negligible by the bulkhead stiffness.
4. CONCLUSION AND FUTURE WORK
Our analytical formulas indicate that the stiffness of the proposed bulkheads could
be reduced by a factor of 10 without incurring significant secondary stresses in the double
hull. A less bulky bulkhead design would have obvious cost and weight benefits.
This analysis models only the most fundamental aspects of the complex state of stress
which could actually exist in the hull of a ship at sea. In future work the model should be
refined to make its predictions correspond more closely to reality. Features which should
be included are:
1. A more rectangular cross section corresponding to the actual hull shape. This could
perhaps be included within the framework of the present analysis by conformal map-
ping techniques.
2. A more realistic applied loading, including torsion and internal pressures.
3. Discrete treatment of the bulkheads and other stiffeners. Also, allowance for nonuni-
form stiffness properties.
4. Geometric and material nonlineaities. This is essential for an ultimate strength anal-
ysis.
ACKNOWLEDGEMENTS
The author performed this work while a visiting researcher at Lehigh University in
Bethlehem, PA, and was supported by the Fleet of the Future Program, directed by Prof.
John W. Fisher. Prof. Le-Wu Lu of the Civil Engineering Department at Lehigh proposed
the problem of determing how the secondary stresses depend upon the bulkhead stiffness.
Mr. Alan Pang computed the values of the last four parameters shown in the Table.
REFERENCES
1. T. Okamoto, et. al., "Strength Evaluation of Novel Unidirectional - Girder - System
Product Oil Carrier by Reliability Analysis," Transactions of the Society of Naval
Architexts and Marine Engineers, Vol 93, 1985.
2. J. Beach, "Advanced Surface Ship Hull Technology," Transactions of the American
Society of Naval Engineers, 1990.
3. E. Axelrad, Theory of Flexible Shells, North Holland, 1987.
4. J. Simmonds, "A Set of Simple, Accurate Equations for Circular Cylindrical Elastic
Shells," Int. J. Solids and Structures, vol 2, 1966, pp. 525-541.
Figure 1: Double hull designed by David Taylor Research Center.
91 -A069
- . - - - - -- - - - - - - - - -- -
Circ, lar Cross SectionUndeformed
q
a~ q-
Fonato Stiffness, K
q m
Section A-A
Figure 2: Idealized Model of Double Hull
1-A069
Circular
R'l xis of Revolution of Deformed Tube
- - -- -- -- - -- --
1' ' R
Figure 3: Geometrical Definitions of Curvature Changes
1 -A069
q=-IR(1+cos2Tj)2
Figure 4: Lateral Pressure Loading
31 .AO69
0 YM2
CYB2
Figure 5: Stresses Acting in Double Hull Wall
1 -A069
Table 1: Typical Parameter Values
M 665,000 kip-ft
6 0.0624 kiplft'
E 29,500 ksi
v 0.3
t 0.41 in
b 35 in
I=~TR 3h 5500 ft4
K 15 kip/in 3
91 -A069
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DEFENSE TECH. INFORMATION NAVAL POSTGRADUATE SCHOOL
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DIRECTOR OF RESEARCH ADMIN. DEPT. OF MATHEMATICSCODE 012 CODE MA
NAVAL POSTGRADUATE SCHOOL NAVAL POSTGRADUATE SCHOOL
MONTEREY, CA 93943 MONTEREY, CA 93943
CENTER FOR NAVAL ANALYSES PROF. DONALD DANIELSON (25)
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