<£.

NPS-MA-94-008

NAVALPOSTGRADUATE SCHOOLMonterey, California

STRESSES IN SHIP PLATING

by

D.A. Danielson

C.R. Steele

F. Fakhroo

A.S. Cricelli

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Stresses in Ship Plating

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Stresses in Ship Plating

19 ABSTRACT {Continue on reverse if necessary and identify by block number)

The subject of this paper is the mechanical behavior of rectangular plates subjected to

a combination of axial compression and lateral pressure. Displacements and stresses are

obtained from a Fortran code based on the von Karman plate equations. The effects of

various boundary conditions, nonlinearities, and imperfections are included.

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UNCLASSIFIED

STRESSES IN SHIP PLATING

D. A. Danielson C. R. Steele

F. Fahroo Division of Applied Mechanics

A. S. Cricelli Stanford University

Department of Mathematics Stanford, CA 94305

Naval Postgraduate School

Monterey, CA 93943

December, 1994

ABSTRACT

The subject of this paper is the mechanical behavior of rectangular plates subjected to

a combination of axial compression and lateral pressure. Displacements and stresses are

obtained from a Fortran code based on the von Karman plate equations. The effects of

various boundary conditions, nonlinearities, and imperfections are included.

INTRODUCTION

Fatigue life predictions require knowledge of the stresses in a ship under its operating

conditions [Sikora, Dinsenbacher, and Beach (1983)]. Stiffened plates are a basic structural

component of ships and submarines. The mathematical equations governing the deformation

of thin elastic plates and methods for solution of these equations are well-known [Timoshenko

and Woinowsky-Krieger (1959), Szilard (1974), and Hughes (1983)]. The objective of our

work is to use these known mathematical methods to predict the stresses in typical ship

plating.

Plates are subjected to axial tension and compression from longitudinal bending of a ship

due to wave loads. In addition, plates on the bottom are subjected to lateral water pressure.'

Deck plating boundary conditions may be taken to be simply-supported , whereas bottom

plating boundary conditions are more closely approximated as clamped. Deck plating may

be considered to have an initial geometric imperfection, whereas bottom plating is bowed in

by the water pressure.

We consider a rectangular plate with length a, width b, and thickness t. The structure

is subjected to axial force F, and possibly uniform lateral pressure p (force per unit lateral

area of the plate).

PLATE EQUATIONS

The basic differential equations of nonlinear shallow shell theory are

D V 4 w = P + $< yy (too + w) ixx +$,xx (tu + w), yy-2 $,ry (w + ^),xy ( 1

)

tt V4$ = [(u>o + u;)4

y-(u; + u>),xx (ti) + w), yy ]

- (u;j jxy -uw* u;o,w ) (2)

Here (x,y, z) are Cartesian coordinates measured from the center of the plate.

D — Et3/12(1 — i/

2) is the bending stiffness, E is Young's modulus, and v is Poisson's ratio.

wo(x,y) is the initial geometric imperfection of the plate midsurface in the z-direction, while

w(.r,y) is the normal displacement. <J>(x,y) is the Airy stress function. Commas denote

partial differentiation with respect to x or y; e.g, '.

84w d 2w 4 wV W - W,xxxx + 2w,xxyy +W,yyyy - ^ 4

+ -QX2Q 2 + Qy4

We let u(x,y) and t>(x,y) denote the displacements of the plate midsurface in the x and

y directions, respectively. These in-plane displacements can be related to w and $ by using

the strain-displacement relations

€x = u lX +wQ , x w,x +\w,2

x

txy = U,y+V,x +Wo,x W,y +^ ,y W, x +W,x W, y

and the constitutive relations

Et,Nx = Mx + VCy)

\ — V*

Et ,Ny= -{e

y + vex )

1 — v A

Etxy "

2(1 + 1/)^

The membrane stress resultants (forces per unit length of side) are related to the stress

function by

N — <J> N — $ at = — 4>"i — ^ iyy j y — '3?x i * " xy r̂ ixy

The bending stress resultants (moments per unit length of side) are related to the normal

displacement by

Mr = ~D(W,XX +VW,yy ), My = ~D(w ,yy +VW ,xx ), MXy = D (l ~ V)W,Xy (3)

The bending stresses (forces per unit area of cross section) acting on the top of sections

parallel to the yz or xz planes, respectively are

6MX 6My** = -p~, °v = -p-

We consider three different possible sets of boundary conditions:

Loosely-clamped;

b

On x = +f • w ~ ^11= Nxy = 0, u = constant, f\ Nx (+3^,y)dy = F(4)

(5)

On y — +~2

: w = w,y — Nxy =0, v = constant, f\ Ny {x, +^, )dz =

Rigidly-clamped

6

On x - +| : w = w,x — Nxy = 0, u = constant, /_% Nx(±^,y)dy = F2

On y = +| : to = u\v= Nxy = v =

Simply-supported

On x = +^ : iu = w,xx — Nxy = 0, u = constant, J26 A^. (+ f,y)(fy = F

a

On y = +| ' a' = w iyy~ ^xv

= ^' u = constant, J_2o Ny (x, + §)d;r =

The classical buckling load of a simply-supported plate with an integral aspect ratio

a- = 1,2,3,... is given by —Fcr , where6

F -1^

(6)

SIMPLIFIED EQUATIONS

For normal displacements w which are smaller than the plate thickness /, the nonlinear

terms in (l)-(2) are negligible and we can replace (l)-(2) with the linear approximation

FNx = — , Ny= constant C, Nxy = (7)

D v 4 w = p + Nx(w + w),xx +Ny(w + w),yy (8)

For the boundary conditions (4) and (6) where the lateral edges y = ±— are free to expand

or contract Ny= 0, whereas for the boundary conditions (5) where the lateral edges y = ±-

are restrained Ny= —-. If p ^ 0, equation (8) may be written in the nondimensionalized-

6

form

47T2F b

2N\y

4 W= 12(1 - V1

) + — (W + w),xx -\ ^-{Wo + W),yyPer Lf

111 f^t Twhere w = ——— , x — —

, etc. The nondimensionalized bending stresses are thenpb4 b

given by

a r t2 6MT 1

(W,XX-+l>W,yy), etC.pb2 pb2 2(1 -v 2

)

Note that in the case w = the normal displacement w increases linearly with the pressure p,

whereas in the case p = the normal displacement w increases linearly with the imperfection

amplitude.

We can investigate the importance of nonlinearities by substituting the linear solution

w to (8) into the right side of (2). The additional membrane stresses Sx ,Sy ,Sxy are then

determined by

F FNx = - + tSx = - + <S>, yy , Ny

= tSy + C = <S>,XX + C, Nxy = tSxy = -$,ry

o b

1,4V $ = [(U>0 + W)

)XJ/-(U'O + W),XX (W +W),yy]- (W

,xy -WQ ,XX WQ ,yy

If /> 7^ 0, Equation (9) may be written in the nondimensionalized form

V 4^ = [(^0 + W),l--(w + w),xx (w + U>),yy] - (^0,|y _U;0,^^0, V '/

— <&Et 5

where 4> = ——— The nondimensionalized additional membrane stresses are then given byp

2 b*

Sx t2

pb4 -

W =eF***

etc - (10)

FNote that the total longitudinal stress at the top of a plate cross section is \- ax + Sx , etc..

tb "•

SOLUTION METHOD

These equations may be solved numerically by superimposing a Fourier series particular

solution plus Levy-type homogeneous solutions with coefficients determined by the prescribed

boundary conditions. This method of solution has been recently used by Kwok, Kang, and

Steele (1991), Bird and Steele (1991, 1992), and Kang, Wu, and Steele (1993). A thoroughly

documented Fortran code was written to implement this solution.

Normal Displacement and Bending Stresses

The governing equations for an isotropic plate under axial compression and lateral pressure

load with no imperfection are from (7) and (8)

F uFNx = -, ^v= 0or — , Nxy -0

o o

D V 4 W = P + NX W,XX +Ny W,yy

with the clamped boundary conditions from (4) or (5)

On x = ±| : w — w,x =

On y = ±2 : w — w ^y=

The solution to the problem above is obtained in several stages:

1 ) I irsl ,i solution W] for constant pressure load on a clamped plate strip with Nx = Ny=

is obtained. This particular solution that satisfies

V 4^' = ^ (11)

is of the following form

"* = *(x - (t)T (12)

I he solution w-[ has zero rotation and displacement along y = ±7. From (3) the bending

moment \ly

is given by

16.4D / f1v\ 2\

My = ~ DlV,yy = —I 1 ~ 3 f y \ I tUld Mx = VMy

p b4

Coefficient A in (12) is determined by substituting (12) in (11): A = — .j s\ i v } D 3g4

I he Fourier cosine series for (12) is given by

w x — "^2 Wn cosniry

where6

4 fi niry,

pb4( -1 12 \ . nix

vVn = - / ^ cos —— dy = — + sin —

.

o 7o 6 3L> ^(nTr)'3 {rnrf J 2

The particular solution u;i has zero rotation but nonzero displacement along x = + —

2) In order to make the normal displacement zero along x = + -, we need to compute a

correction term that will achieve this goal. This new term w2 which is the solution of a

rectangular plate with inplane resultants Nx and Ny is computed in the following way: A

nny

~b~

and is determined from the condition that w2 satisfies the plate equation

olution of the form w 2 = Y^ Xn (x) cos —— is assumed, where Xn is a function of x onlv

D V 4 W = NX W,XX +Ny W,yy (13)

with zero rotation and displacement equal to —W\ along x = ±— . From the condition that

uh satisfies (13). we obtain the solution

Xn (x) = A n cosh V\X + Bn sinh U\X -f Cn cosh u2 x + D n sinh u2 x<

where /', and //•_> are solutions of

d Ht)TWt)'-^=°The- constants Bn and Dn can be taken to be zero due to the symmetry about the //-axis, and

the remaining constants .4 n and Bn can be determined by imposing the boundary conditions

along x — ±— . Then W\ + w 2 is a solution for a plate under lateral pressure p and in-plane

resultants Nx and Nywith zero displacement along all the edges and zero rotation along

•b

.;• = ± — . but nonzero rotation along y = ±—

.

3) In order to make the rotation zero along y — ±— we first compute the Fourier coefficients

of rotation on these edges and then obtain a third solution w^ = u^i + w32 with moments

distributed along the edges corresponding to the desired rotations. A solution u>3i sinusoidal

in the ./--direction is:

v^ syrnirx /cosh p,\ y cosh p 2 y\

"'31 = 2^ c im cos —

;

6- r fe-

rn „dd a V cosh ^! ^ cosh // 2 2/

where the parameters p\ and p, 2 are the roots of the polynomial

mn\ 2\ fm/K\ 2

.,

This solution is obtained in a similar manner as for solution w 2 , except that in this case

T-i is assumed to be of the form IU31 = ^ Ym (y) cos and Ym (y) is determined bymodd a

imposing the zero displacement boundary condition along y = ±-. As defined above, w3]

ahas zero displacement on all the edges and zero moment My

along x — ±— . The rotation

b.

along the edge y = - is given by

/ vi v^ ^ m7r.r / 6 6\roiy = -(u;3i,y)L= fc = V C im cos -^ tanh //i - + ^2 tanh /i 2 -

2nT^d a \ 2 2 J (15)

— 2^ lm ^llmmodd a

aThe rotation along the edge x = — is

. ^-^ m7r . m.7r /cosh /ziy cosh //2</rotx = -(u>3i,*)|x=§ = 2^ ^im— sin

2 \cosh ^! I cosh \i'in odd " " ywoii pj

The expression in the previous parentheses is an even function of y, so it can be expanded

in a Fourier cosine series:

rotx = Yl Cl*n Sin —r Yl Dl2nm cos —

—

(16)

where

4 /-2/cosh/iiy cosh/z 2y\ «7ry£>12nm = T / 7 u 7 f

COS—-dyo Jo \cosh/Ui- cosh ^2 0/ "

Anir . U7T /x? — /i~

sin62 2

(^? + (t[

)

2

)(^ + (tl

)

2

)

The moment at the edge y — — is:

My = D(-I03iw —lOWsijaw)^! = E ^1^^ (~^1 +^2) COS

m odd

= 51 ^ImCOS

niTTX

mnx

Therefore

D(-f*l+i4)

The solution sinusoidal in the y-direction which has zero displacement on the edges and zero

moment My along y = ±- is :

10

U'32 = Yl ^2n COSwiry ( cosli V\X cosh u2x

b \cosh j/j | cosh ;/•>|

w here u\ and z/2 are roots of (14

Now the rotation at the edge x — — is62

rota = -(tw32,x)U=§ = Yl Can

= E c2„

7i7ry / t^a i/2«cos —;— — V\ tanh + ^2 tanh

6 V 2 2

mrycos —— L)22n

18)

The rotation at the edge y = - is given by:

,-^ n7T n7r /cosh V\ x cosh i/2 xroty = -{w32 ,y)\ * = ^ C2n— sin—

2 ,r d̂6 2 V cosh v\ f cosh */2 f

En7r . mr ^-^ rmrx

C 2n-r-Sin— 2^ ^21nmCOS (19;

where

D 2 nAmir . m7r—— sin —

—

v\ - v\

•* + (¥)' H+(¥) i

The moment at the edge x = — is

Mr = D (-1032 ,XI -i/lU32, yy ) \x=- Y2 C2nD {-v\ + ujj COSnny

n odd

= E^» COSnny

Therefore

Con. —/•:

2n

D{-vl + vl)(20)

In terms of the Fourier coefficients of quantities rotx^ and roty, Mx , and My , we have the

following relationships: From equations (16), (17), (18) and (20) we have:

11

[rotx] = [DM12] * [My ] + [DM22] * [Mx ]21

wild r

777. IT 777 7TDM 12)nm = D 12nm *— * sin -— * -——a 2 D(f4-pi t

1

DM22)mn = D22n *D{y\ - y\

* (Identity

)

n ,

Similarly from (15), (17), (19) and (20) we have:

[roty] = [DM11] * [My ] + [DM21] * [Mx (22)

where

71 7T Tl 7T

DM21)nm = D 2 inm * — * sin— *

(DMll)nm = DUm *

2 D{ix\-AY

* (Identity)D{ti - rf

Equations (21)-(22) can be written in the matrix notation:

DM11 DM12DM21 DM22

MyM T

roty

rotx

or

[DM] * [M] = rot

Here [DM] is the flexibility matrix and the inverse [DM]~ lis the stiffness matrix. To take

advantage of the diagonal matrices DM11 and DM22, we write the following relations:

[Mx ]= [M]- 1

* {[rotx]- [DM21]* [DM11]" 1* [roty])

[My ]

= [DM11]- 1* {[roty] - [DM12] * [Mx ])

[M] = [DM22] - [DM21] * [DM11]- 1* [DM12]

(23)

12

Aftei computing the vector solution to equations (23), we have Fourier coefficients of M, and

M:/ami from (1/ ) and (20) we can compute the Fourier coefficients of W3, the displacement

corresponding to this bending moment. Then ir{ -f- ir 2

— w% is the solution which has zero

rotation on all the edges as well as zero displacement.

Additional Membrane Stresses

I he governing equation for the additional membrane stresses in a plate with no imperfection

is from (9)

T^^ = «iy - "\x*^ y (24)

with the boundary conditions

On x == ±f : f\ $ yv (±\,y)dy = 0, f\ $ ry (±-, y) dy = 0, u = constant

('25)

On y = ±| : /j« $ xx (x, ±f) cte = 0, /ja $ xy (x, ±|) dx = 0, u = constant.

The basic idea behind the solution method for calculating the in-plane Airy stress function

(f> is the use of double Fourier expansions for both $ and the out of plane displacement w.

In order to soke the governing equation (24), we need to calculate the Fourier coefficients

of w from the solution in the previous subsection. The Fourier expansions for w and have

the following form:

v^ v^ fnrx\ I jTry\w = 2^ 1^ wv cos \~^) cos

\~b~)^26

-

l even J even

—) ros ItI even J even

• = E E •«

\\ ith these expansions the boundary conditions (25) for 4> will be satisfied. The derivatives

of w are calculated by direct differentiation of (26), and the products of derivatives are

evaluated at each grid point. Then the discrete double Fourier cosine transform is used to

obtain the Fourier coefficients of the right-hand side of (24), which are stored in the Gaussian

13

curvature matrix A . The Fourier coefficients of <J> then follow from

Etl\*«= '

..2 fori = 2,4 j-.

+ (?

The in-plane displacements u and v are obtained from the strain-displacement relations.

An appropriate plane stress solution is added to satisfy the boundary conditions on u and v.

14

RESULTS

Normal Displacement and Bending Stresses

We first solved the linearized equations (7)-(8) with no imperfection and the loosely-clamped

boundary conditions (4). The shapes of the deflected midsurfaces and values of the nondi-

mensionalized longitudinal and transverse bending stresses for plates of various aspect ratios

aand axial load ratios are shown in the figures. Note that for large — the normal displacement

b

and stresses on lines y = constant are nearly uniform at a distance from the ends greater

Fthan the width of the plate. Note that as the axial load ratio — is varied from positive to

F cr

negative, the waviness of the plate near the ends increases.

Maximum normal displacements, and maximum and minimum values of the longitudinal

and transverse bending stresses are shown in separate graphs. The location of the maxi-

mum Ion gitudinal bending stresses is always on the centerline y = 0, while the minimu in

longitudinal bending stresses are always at the endpoints x = +— ,y = 0. The maximum

transverse bending stresses are always on the centerline y = 0, while the minimum transverse

b . . Fbending stresses are always on the edgelines y = + — Note that as the axial loa,d ratio —

i /' cr

is varied from positive to negative, the maximum stresses increase and the minimum stresses

usually decrease. Also, note that for most cases the minimum transverse bending stress is

the greatest stress in absolute value.

FAll values of — were greater than the buckling loads, given in the following table:

t cr

a/b \ 1.2 1 ! 1.6 2 4 oo

Buckling

Load F/Fcr

-2.52 -2.45 -2.41 -2.22 -1.99 -1.81 - 1 . ID

FAs a check on these results, the maximum normal displacements for the case —— =

' cr

(located at the origin x = y = 0) were checked against Roark and Young (1975), the

transverse bending stress distribution on the line x = was verified to be nearly parabolic

for large -, and the buckling loads were checked against Timoshenko and Gere (1961).b

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Additional Membrane Stresses

We next studied the importance of nonlinearities by solving equation (9) with no imperfection

and the clamped boundary conditions (4) or (5), The graphs show the nondimensionalized

maximum and minimum additional membrane stresses due to bending. Note that the ad-

ditional membrane stresses are at least an order of magnitude smaller than those in the

preceding section from the linear equations. Note also that the main effect of preventing

lateral displacement of the edges y = ±— is to increase the additional transverse membrane

stress, whereas the additional longitudinal membrane stress is unaffected.

For these graphs the following parameter values which are typical of ship plating were

chosen:

E = 30,000,000 psi

v = .3

6 = 36 in

I = .5 in

p = 18 psi

The additional membrane stresses corresponding to other parameter values may be obtained

from these same graphs with the vertical axis scaled in accordance with (10).

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6-5

Imperfections

Finally, we solved the linearized equations (7)-(8) with no pressure, an imperfection, and the

simply-supported boundary conditions (6). Let us consider imperfection shapes of the form

mirx n.ixyw = cos cos —— with m = 1,3,5,... and n = 1,3,5,...

a b

or

niirx . ninjWq = sin sin —— with m = 2, 4, 6, ... and n = 2,4, 6, ...

a b

where m and n are the number of half waves in the longitudinal and transverse directions,

respectively. For these shapes we obtain the analytical solution

ww = —

Fcr ( mb n2a \

~F [ ~2^' V2^6/

The bending stress resultants then follow from (3) by differentiation:

„ „ ( m 2i/n

2\

M*!Db +^ l"°

Mx = -^ —fr2 , etc.

Fa- ( inb n a \

~F V^7+2^6/

+ l

FOf course, to prevent buckling we require —— > — 1.

' cr

An alternate imperfection shape proportional to the deflection of the clamped plate under

small uniform pressure without inplane load was also allowed in the Fortran code, but for

brevity we do not report on this case here.

6^

CONCLUSIONS

We have developed a Fortran code for calculating the displacements and stresses in rect-

angular plates subjected to the effects of combined loading, various boundary conditions,

and imperfections. The next step is to extend the code to a three-dimensional assembly

of plates and beams more representative of ship grillages. Progress in the analyses of such

structures has recently been made by Danielson et al (1988, 1990, 1993, 1994).

ACKNOWLEDGEMENTS

The authors were supported by the Naval Surface Warfare Center (Carderock Division)

and by the Naval Postgraduate School Research Program. Dr. D. P. Kihl of the Survivability,

Structures, and Materials Directorate at NSWC proposed the title problem and offered

several valuable suggestions.

b7

REFERENCES

Rinl. M. I), and Steele, (..'. R. (1991). Separated solution procedure for bending of circular

plates with circular holes. ASME Applied Mechanics Reviews 44, no. 11. pari 2, S27-S35.

Bird. M. D. and Steele, C R. (1992). A solution procedure for Laplace's equation on multiply

connected circular domains. ASME Journal of Applied Mechanics 114. part 1. 398-404.

Danielson, D. A., and Hodges, D. H. (1988). A beam theory for large global rotation.

moderate local rotation, and small strain. ASME Journal of Applied Mechanics 55, 179-184.

Danielson, D. A., Kihl, D. P., and Hodges, D. H.(1990). Tripping of thin-walled plating

stiffeners in axial compression. Thin-Walled Structures 10, 121-142. •

Danielson, D. A., Cricelli, A. S., Frenzen, C. L. and Vasudevan, N. (1993). Buckling of

stiffened plates under axial compression and lateral pressure. International Journal of Solids

and Structures 30, 545-551.

Danielson. D. A. ,(1994). Analytical tripping loads for stiffened plates. International Journal

of Solids and Structures. To appear.

Hughes, (). (1983). Ship Structural Design. Wiley.

Kang, L. C, Wu, C. H., and Steele, C. R.(1993). Fourier series for polygonal plate bending:

A very large plate element. To appear.

Kwok, F. M. W., Kang, L. C, and Steele, C. R.(1991). Mode III fracture mechanics analysis

with Fourier series method. ASME Applied Mechanics Review 44, 166-170.

R.oark, R. .J., and Young, W. C.(1975). Formulas for Stress and Strain. McGraw-Hill.

Sikora, J. P., Dinsenbacher, A., and Beach, J. E.(1983). A method for estimating lifetime

loads and fatigue lives for swath and conventional monohull ships. Naval Engineers Journal,

63-85.

68

Szilard, K. (1974). Theory and Analysis of Plates. Prentice-Hall.

Timoshenko. S. P.. and Gere, J. M.(1961). Theory of Elastic Stability. McGraw-Hill.

Timoshenko, S. P., and Woinowsky-Krieger( 1959). Theory of Plates and Shells. McGraw-

Hill.

69

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Library (2)

Code 52

Naval Postgraduate School

Monterey, CA 93943 :

Professor Richard Franke, Code MA/Fe (1)

Department of Mathematics

Naval Postgraduate School

Monterey, CA 93943

D.A. Danielson, Code MA/Dd (30)

Department of Mathematics

Naval Postgraduate School

Monterey, CA 93943

Dr. Marc Lipman (1)

Mathematical, Computer and Information

Sciences Division

Office of Naval Research

800 Quincy Street

Arlington, VA 22217-5000

Dr. David Kihl (2)

Survivability, Structures, and Materials Directorate

Naval Surface Warfare Center-Carderock Division

Bethesda, MD 20084-5000

UUULtY KNOX LIBRARY

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