composites

ELSEVIER

Marine Structures 8 (1995) 1-36 1994 Elsevier Science Limited

Printed in Great Britain. All rights reserved 0951-8339/95/$9.50

Buckling and Ultimate Strength Interaction in Plates and Stiffened Panels under Combined Inplane Biaxial and

Shearing Forces

Yukio Ueda

Welding Research Institute, Osaka University, 11-1 Mihogaoka, Ibaraki, Osaka 567, Japan

S. M. H. Rashed

Technical Department, MSC Japan, Tokyo, Japan

&

J. K. Paik

Department of Naval Architecture, Pusan National University, Pusan, Korea

(Received 12 January 1992; revised version received 4 November 1993; accepted 8 December 1993)

ABSTRACT

The main portion of a ship's structure is usually composed of stiffened ph~tes. Between girders and floors, stiffeners are furnished to plates usually in the longitudinal direction. Under various loads applied to a ship, such as those due to cargo, buoyancy and waves, these stiffened plates are subjected to combined inplane and lateral loads. Imperfections due to fabrication exist mainly in the form of initial deflection and residual stresses. The be,~aviour of perfectly flat plates is, however, an important reference in design.

In this paper, buckling, ultimate and fully plastic strength interaction relationships for rectangular perfectly flat plates and uniaxially stiffened plates subjected to inplane biaxial and shearing forces are derived and expressed in explicit forms based on the results of theoretical in vestigations of the non-linear behaviour of plates and stiffened plates.

2 Yukio Ueda, S. M. H, Rashed, J. K. Paik

The accuracy of these interaction relationships is confirmed through comparison with the results of other analysis methods.

With the aid of these interaction relationships, buckling load, ultimate strength and/or fully plastic strength of such perfectly flat plates and uniaxially stiffened plates subjected to inplane loads may be predicted by hand calculation.

Key words: interactions, buckling, ultimate strength, plate, stiffened plate, compression, bending, shear.

A Ar

Ax

A~,

a I

b b' fie' ae'

beo'

b~st'

D(=Dp1) Dx, Dy, Dxy

E

m

NOTATION

Cross-sectional area of a stiffener Total cross-sectional area of a stiffener together with the associated modified effective breadth of plate Cross-sectional area of plate panel normal to x direction (= bt) Cross-sectional area of plate panel normal to y direction (= at) Plate length Half buckled wavelength Breadth of a stiffened panel Breadth of a plate between two stiffeners Post-buckling effective widths of a plate in the x and y directions, respectively Modified effective breadth of an orthotropic plate in the x direction Tangential post-buckling effective breadth of a plate panel corresponding to a stiffener Plate bending stiffness ( Et3 /12(1 - v2) ) Plate bending stiffness components for an orthotropic plate Modulus of elasticity Moduli of elasticity of an orthotropic plate in x and y directions, respectively Magnitude of eccentricity of loading Moment of inertia of a stiffener together with the associated modified effective breadth of plate Number of half-waves of buckling in x direction

Buckling and ultimate strength interaction 3

Nx, ~xcr, Ny, Nycr, Vx, Vxcr

Nxp

Gp

n

P

Pus

t

V~, ry Vrp Zm

~.rmax, O~xmin

O~ymax~, O~ymin

FB Pp Pu ? yBmi n

yUmi n

/3 x

P O" o

O'ov

O" s

Forces obtained by integrating ax, O'xcr, o'y, aycr, zx,, and "~xycr, over the respective cross- sectional area (b't or at) of a plate between stiffeners Fully plastic force of a plate in the x direction (=b' tao) Fully plastic force of a plate in the y direction (=atao) Number of stiffeners Axial force acting on a stiffener with its effective breadth. Euler's buckling strength of a stiffener with the effective breadth of a buckled plate Plate thickness Shearing forces in x and y directions, respectively Full plastic shearing force of a plate (=atZo) Section modulus corresponding to the out- most fibre of a stiffener Section modulus corresponding to the outermost fibre of the plate Stress coefficients expressing deviation of ax and O-y from o'xav and ayav, respectively Stress coefficient ~x at the locations of O'xmax and O'xmin , respectively Stress coefficient ~y at the locations of armax and O'ymin, respectively Aspect ratio of a plate between two stiffeners (=a/b') Buckling interaction function Fully plastic strength interaction function Ultimate strength interaction function Stiffness ratio of stiffener to plate (=EI/b'D) Minimum stiffness ratio of stiffener to plate for buckling Minimum stiffness ratio of stiffener to plate for ultimate strength Normal strain in the x direction Load factor Yield stress Effective yield stress (= v/(a0 2 - 3Zxy2)) Normal stress acting on the outermost fiber of a stiffener

4

fix, CTy

O'xav, O'yav

Uxcr, O'ycr, Txycr

O'xcr, O'ycr, Txycr

O'wnax, O'.vmax

O'*xmax

O'xmin, O'ymin

O'xs

Txy To

Yukio Ueda, S. M. H. Rashed, J. K. Paik

Normal stresses in x and y directions, respectively Average normal stresses in x and y directions, respectively Elastic buckling stresses of a plate under the action of each stress alone Elastic buckling stresses of an orthotropic plate under the action of each stress alone Maximum normal stresses of a plate in the x and y directions, respectively Maximum normal stresses of a stiffener in the x direction Minimum normal stresses of a plate in the x and y directions, respectively Pre-buckling normal stress of a stiffener in the x direction Shear stress Yield stress under pure shear (=fro/V/3) Poisson's ratio

1 INTRODUCTION

The hull of a ship is fundamentally regarded as a thin-walled box-girder whose major portion is usually composed of stiffened plates. Stiffeners, furnished to plates, are supported by girders and bulkheads. Under various loads applied to a ship, such as those due to cargo, buoyancy and waves, the hull is subjected to longitudinal shear, bending and torsion. Locally, each portion of the structure is subjected to lateral loads, and inplane axial forces, bending and shearing forces.

Simple methods to accurately evaluate buckling, plastic and ultimate strength of components of such complicated structures are very useful in the examination of their safety.

Plates and stiffened plates, as the main components of such structures, are considered. External loads acting on them may be divided into two groups:

(1) inplane loads, composed of axial forces (compression or tension) in two normal directions, bending and shearing forces;

(2) distributed lateral loads, caused by water pressure or pressure due to weather, liquid or bulk cargo.

In this paper, only inplane loads are considered. The influence of distributed lateral loads will be reported in another paper.


A.s rectangular plates and stiffened plate panels are small compared with the overall ship structure, inplane bending moments acting on indi- vidual panels in the deck, side or bottom plating are insignificant and may be neglected. Therefore, rectangular plates and stiffened plate panels are considered to be subjected to inplane axial forces in two normal directions and uniformly distributed inplane shearing forces.

Plates and stiffened plates in ship structures unavoidably have certain amounts of initial imperfection mainly due to fabrication. These imperfections exist primarily in the form of initial deflections and welding residual stresses. These imperfections cause reduction of the strength and stiffness of plates and stiffened plates. However, the strength of perfectly fiat plates without any residual stresses is an important reference in design. Evaluation of the ultimate strength of plates and stiffened plates with initial imperfections involves more complex procedures which are not suited for hand calculations (or a simple computer program). Numerical methods such as FEM or ISUM are more effective in such evaluations. Therefore, here, only perfect plates are dealt with.

Development of buckling and ultimate strength interaction relationships in the form of equations or graphs for plates and stiffened plates has attracted a lot of international interest for a long time. Work available in the literature may be divided into two classes. The first is a presentation of results obtained by numerical techniques taking account of geometric and material non-linearities. Example of such results may be found in Refs I, 2 and 3. The other class consists of analytical solutions, in which solutions are obtained based on suitable failure criteria. Examples may be found in Refs 4 and 5. A review of available material requires a paper to itself and it is not intended to present such a review in this paper. The available information, however, does not cover all the practical range with sufficient accuracy and confidence, in particular with regard to inplane shear effects.

In this study, the buckling strength, the ultimate strength after buckling, and the fully plastic strength of rectangular perfectly fiat plates and stiffened plates under combined inplane biaxial forces and inplane shear are investigated. Strength functions are derived and expressed in terms of applied forces. Comparisons with published results and those of analysis by the finite element method are presented.

2 NON-LINEAR BEHAVIOUR OF STIFFENED PLATES

A stiffened plate is considered as a part of a large plate structure such as a deck or side shell of a ship as shown in Fig. 1. The length, breadth and thickness of this stiffened plate are a, b and t, and its plate bending stiff-

6 Yukio Ueda, S. M. H. Rashed. J. K. Paik

I Y Oy (Ny)

"g~(V x ) ~ ~ ~ ~

O'x .~,..~ 1 (N x)

- - - ~ !

!

/ A , I ~

D

~ 0 x . 0 t ! (Nx) ""-- "*---Txy (Vx)

t t , t t t Oy (Ny)

a

t t t

Fig. I. Stiffened plate and applied loads.

X

_'2-

ness is D = Et3/12(I - 1,'2), where E is the modulus of elasticity and v is Poisson's ratio.

Assume n stiffeners of equal dimensions are attached to the plate in the x direction at regular intervals b'. The cross-sectional area and moment of inertia of each stiffener are A and I, respectively (I includes the effective breadth of the plating associated with the stiffener). It is assumed that stiffeners do not buckle prior to buckling of the plates between adjacent stiffeners (stiffeners are usually designed to satisfy this condition). Stiffened plate panels are assumed to be simply supported and edges remain straight in the plane of the panel. Inplane compressive or tensile forces are applied in the two axial directions x and y, together with uniform shear stress. Due to the assumed boundary conditions, the applied biaxial forces will produce uniform inplane compressive or tensile displacements at the plate edges.

In this paper, this loading condition is referred to as a combined load of biaxial forces and inplane shear.

When a flat stiffened plate is subjected to a combined load as mentioned above, uniform biaxial normal stresses (~rx and ~r,.) and uniform shear stress (~.~,.) are produced in the plate, while only a uniform normal stress in the x direction, axs is produced in the stiffeners. Increasing this combined load, the stiffened plate behaves in different ways according to its dimensions and the combination of the applied load components.


When the stiffened plate has a sufficiently low out-of-plane bending stiffness, it buckles in one of two buckling modes. One is overall buckling in which the plate buckles together with the stiffeners, and the other is local buckling where plates between stiffeners buckle while the stiffeners remain straight. This is controlled by the relative stiffness ratio of the stiffeners to the plate, i.e. 7(=EI/b'D). When ~: is smaller than a certain value 7amin 6, the stiffened plate buckles in the overall mode. The overall buckling strength increases together with 7.

When 7 is greater than 7Brain 6, local buckling takes place instead of overall buckling and the buckling stress reaches an upper limit regardless of higher values of 7- Thi s 7amin is given as the point of intersection between the two buckling curves representing the two buckling modes as shown in Fig. 2.

After buckling, local or overall, the stiffened plate may support further increments of the load, though with lower inplane stiffness, until it reaches its ultimate strength after plasticity prevails.

The collapse mode at ultimate strength varies according to the value of 7. Under compression acting in the x direction, one of the following modes is produced.

(a) When 7 is smaller than 7Bmin, the stiffened plate buckles in an overall mode, followed by overall collapse in the same mode of deformation as that at buckling.

(b) When 7 is slightly greater than 7amin, the stiffened plate buckles locally. As its effective stiffness decreases due to this buckling, overall collapse may.occur either due to spread of plasticity in the stiffeners or due to overall buckling of the stiffeners together with the associated effective portions of the buckled plates.

ULTIMATE STRENGTH

0~'~

LOCAL

LOCAL

BUCKLING STRENGTH

"t 0 B U 7 min Y min

Nxp : FULLY PLASTIC COMPRESSIVE BTRNGTH

Fig. 2. Reaction of buckling strength and ultimate strength of axially compressive stiffened plates to the stiffness ratio of stiffeners.

8 Yukio Ueda, S. M. H. Rashed, J. K. Paik

In these two cases, (a) and (b), the ultimate strength increases together with 7.

(c) When ~ is greater than a certain value, ~Umin6 , s t i f feners are strong enough to prevent overall collapse after local buckling. The stiffened plate reaches its ultimate strength by local collapse of the plate panels between the stiffeners followed by buckling or plastic collapse of stiffeners. In this case, ultimate strength does not increase with 7 and reaches its upper limit; Omi n is 30-50% greater than 7~mio, which is obtained by a small increase in the moment of inertia of the stiffener.

Compression in the y direction causes one of the following two modes.

(a) When 7' is less than 7"rain, the stiffened plate buckles in an overall mode, and then collapses in the same mode.

(b) When 7 is greater than 7Bmi ., the stiffened plate buckles locally, and then collapses locally. In this case, "~Bmi n ),U min"

When the plate is subjected to compression in both the x and y directions, its behaviour depends upon the ratio of ~ to a,. and follows one of the two schemes mentioned above.

The presence of shear stress reduces the compressive buckling strength in both the overall and local modes.

The values of 7Brain and 7Umio depend upon the geometry and mechan- ical properties of the stiffened plate as well as the ratio of load components. The value o f 7Brain can also vary significantly depending upon the direction of loading. Therefore, the behavior of a stiffened plate may be classified into four classes three of which are dependent on the values of 7 in reference to ~Bmi n and 7Umi. and the fourth class is the fully-plastic strength without buckling, as follows.

(1) 7 < 7Bmi, (ultimate strength condition 1). The stiffened plate buckles and collapses in an overall mode.

(2) 7Brain < ~ < 7Umin (ultimate strength condition 2). Plates between stiffeners buckle locally: ultimate strength is reached by plastification or buckling of stiffeners.

(3) 7 > ~Vmin (ultimate strength condition 3). After plates between stiffeners buckle locally, they reach their ultimate strength. Buckling or plasticity of stiffeners follows leading to collapse.

(4) Fully plastic strength (ultimate strength condition 4). Plates between stiffeners have sufficient stiffness such that buckling does not occur until the fully plastic strength is reached under the specified loading condition.


3 B U C K L I N G , U L T I M A T E S T R E N G T H A N D F U L L Y PLASTIC S T R E N G T H I N T E R A C T I O N R E L A T I O N S H I P S OF A

R E C T A N G U L A R P L A T E S U B J E C T E D TO BIAXIAL A N D U N I F O R M S H E A R F O R C E S

Buckling and post-buckling behaviour of a rectangular plate between stiffeners or girders is discussed in this section. A simply supported rectangular plate subjected to axial forces in the two principal perpendicular directions and a uniform shearing stress is considered. Buckling, ult imate and fully plastic strengths are theoretically studied and interaction relationships are expressed in explicit form.

3.1 Buckling interaction

Buckling interaction relationships for a plate under a combined load of uniform normal stresses trx and a,. and shearing stress ~.~, may be expressed by the following equat ions based on analytical solutions 7" 8.

(1) When a~ is tensile and try is compressive (trx < 0, try > 0)

. O" v "/7., O, (m2+f12) 2 a~ ~- " + = 1. ( la)

m2(1 q_ fie)2 trxc r O'ycr \ Zx,,cr /

(2) When trx is compressive and a,. is tensile (ax > 0, ay < 0)

trx (1-q-f12)2 try ( ) 2 ~- fl2)2 " t- r~3------L' = 1. ( lb) O'xcr (m 2 + trycr Zxvcr

(3) When a.,. and tr,. are compressive (ax > 0, t%. > 0)

~ W v " c c r / a,./a ,cr = 1. (lc) I - 2] + I - 2

In eqn (lc), e~ and e2 are given as follows:

for l/v/--2 ~< fl ~< x/-2, el = e2 -- 1 for fl > v/--2, el = 0.0293 f13 _ 0.3364 f12 + 1.5854 fl - 1-0596

e2 = 0.0049 f13 = 0-1183 f12 + 0.6153 fl - 0.8522.

In eqn (1), axc~, t%.c~ and zx~.cr are the buckling stresses when each stress acts alone on the plate; fl = a(n + 1)/b = a/b' is the aspect ratio of a plate between two adjacent stiffeners; and m is the number of half-waves of buckling when a plate of aspect ratio fl buckles under compression in the x direction only.

Here, when a plate is subjected to compression in both the x and y directions, the buckling interaction relationship may be expressed by


several linear interaction equations, each applicable to a certain region depending on fl and the ratio of o'x to o-.,,. As the application of several equations is inconvenient, eqn (lc) is to be preferred since it is continuous and yields a good approximation for the entire region.

Adopting these equations, a buckling interaction function I~B may be expressed as follows and as shown by the dashed curves in Fig. 3.

(1) When N~ is tensile, and N,, is compressive (N~ < 0, Ny > 0)

(m2-}-f12)2 Nx Nv ( Vv ) 2 FB--m2( l+f l2 ) 2 N , ~ + ~ + ~ - 1 . (2a)

N fiN yp

1 YIELDING AT SHORT EDGES

YIELDING AT CORNERS

, YIELDING AT LONG EDGES

' . ~ Nx/Nx p 1

BUCKLING STRENGTH

FULLY PLASTIC STRENGTH

N x ~ 11

f ULTIMATE STRENGTH

~ ~ BUCKLING STRENGTH

/ . - " /

I I t / ' l ' - - V /V . \ ,

Fig. 3. Buckling strength, ultimate strength and fully plastic strength of a rectangular plate.


(;2) When Nx is compressive, and Ny is tensile (N~ > 0, N, < 0)

N x (1 +fl2)2 Ny ( Vx ~ 2 1-'13

-- Nxc~r -~ (m 2 + fl2)2 Nycr ~ t V . - - -~c r ) - 1. (2b)

(:3) When N~ and N,' are compressive (N~ > 0, N,' < 0)

Nx/Nxc r el

. . --~-~'~xcr)2 / - 1 . (2c)

where Nx, Nxcr, Ny, Nvcr, Vx, Vxc r are obtained by integrating ax, axcr, ay, a~r, Zxy, Zxycr, respectively over the cross-sectional area of a plate between stiffeners (b't or at).

It is to be noted that when the four sides of a rectangular plate are equally subjected to the shearing stress Zxy, the shearing forces Vx and I(,, in the x and y directions are proportional to the lengths of the sides, i.e. Vx := atZxy, Vy = b' tz~v and V~lVy = alb'.

When Fa is smaller than zero, this indicates that the plate has not buckled. The buckling condition is expressed as

FB = O. (3)

FB > 0 indicates that the plate has buckled.

3.2 Ultimate strength interaction relationships and stress coefficients

3.2.1 Ultimate strength interaction relationships

When a plate buckles under axial forces in the x and y directions, the normal stress distribution along the edges of a half buckled wavelength is as shown in Fig, 4. This stress distribution is developed repeatedly along each half buckled

' t U : '' T : TENSION C : COMPRESSION

: Y IELD LOCATION

Fig. 4. Stress distribution in a buckled plate.


wavelength of the plate. Under this pattern of stress distribution, the ultimate strength of the plate is assumed to be reached when the resulting stress satis- fies the yield condition at one of the following locations, as shown in Fig. 4.

(a) The four corners. (b) Two points, one along each longitudinal edge, at the middle of each

half buckled wavelength. (c) Two points, one along each transverse edge, at the middle of the

plate breadth.

To evaluate this ultimate strength, first, denoting the uniaxial yield stress by 0-0, the Mises' yield condition may be expressed as follows when shear and normal stresses act simultaneously:

0"0 2 = O'x 2 + O'y 2 - - 0-x O'y + 3Zxy 2. (4)

Introducing an effective yield stress, a0v, 0-0v = v/O0 2 - 3rxy 2 the above yield condition may be rewritten as

O'0v 2 = O'x 2 + O-y 2 - - O" x O'y. (5 )

Here, the normal stresses at each of the above-mentioned locations may be expressed in terms of the axial forces, Nx and N,,, and shearing force Vx as follows:

0 - x ) v : b ' / 2 (l /b ') (1 + ~xmin) 0 N~ O'y)iv=0 or a : 0 (l /a) (1 + 0~vmax ) N v O'y)x=a, /2 0 ( l / a ) (1 + ~ymin) V.r Z.~y 0 0 1/b' J

(6)

where

0-X)y= 0 or b ' , O'y)x=O or a : the stresses at a corner of half a buckled wave, ax)y = h,/z, ay).~ = a,/2 : the stresses at the centre of half a buckled wave, axav, 0-yav : the average values of the stresses perpendicular to the sides, a' : half buckled wavelength.

0~xmax, 0~xmin, ~ymax and aymi n in the eqn (6) are stress coefficients which express the deviation of the stresses ax and try from the average stresses O'xav and 0-yav, respectively, due to buckling.

At the moment when a rectangular plate buckles, stresses are still uniformly distributed and the values of these stress coefficients are zero. Their values start to increase from zero as the load increases and the stress becomes non-uniform. Their values depend upon the magnitude of the load and are evaluated in the next section.

Buckling and ultimate strength #tteraction 13

Denoting the post-buckling effective widths by be' and ae' the relationship between the effective widths and the above-mentioned stress coefficients may be expressed as follows:

be' = b ' / (1 + ~Xxmax ) (7a)

ae = a/(1 + CXymax). (7b)

Dividing eqn (5) by 0"0 2

O" x O'y fix 0-), __ O'0v 2

0-0 2 0"0 2 (8)

the ultimate strength interaction relationship is obtained by substituting eqn (6) into eqn (8)

N, (1 + ax) + (1 + O~y) N, cp Nyp

= 1 __ ( V_~p)2 (9)

where Nxp = b ' tao, Nyp = at ao, Vxp = at zo, Zo = ao/x/--3. The locations of yielding vary according to the loading conditions

(Fig. 4) and associated stress coefficients should be used as follows:

(a) in the case of yielding at the corner: ~,-max, ~ymax; (b) in the case of yielding at sides parallel to the x axis: ~Xxmax , 0~ymin; (C) in the case of yielding at sides parallel to the y axis: 0~xmin , CXyma x.

An ultimate strength function Fu of the plate may then be expressed as follows:

r u - - (1-+-~x) + ( l + ~ y ) " ( l + ~ x ) ( l + ~ y ) - t. yp mxp Nyp (lo)

+ - 1

and the ultimate strength condition is expressed as

r~ = 0 . ( l l )

3.2.2 S t r e s s coe f f i c ien ts

The ,;tress coefficients ~Xxmax , 0~xmin , 0~yma x and 0~ymi n which have been defined in the preceding section are evaluated as follows.


When a rectangular plate has buckled under axial compressive forces in the x and y directions, the maximum stresses are developed at the corners of each half buckled wave, and the minimum stresses in the middle of the edges of each half wave (Fig. 4). These maximum and minimum stresses may be analytically evaluated and the stress coefficients may be expressed with superscript * as follows: 8

b ' Ny zt2 mZ D b ' 0~*xmax : T]I "-~ 7]2 a----~x a -----5~ T]3 N---~

~*xmin : --~*xmax a n y a 4 n2D a (12)

5~*vmax. : T]2 ~-~),-]-m---'~bt4 T]I - --~713 ~yy

~*vmin 2mab,ymax 2m2a2b '2 2( a2 + m2b'2) 2

rh a4 q_ m a b l 4 , r/2 a4 + m4b, 4 , 713 = a4 q_ m a b r 4

In the above equations m is the number of half buckled waves, which depends upon the aspect ratio of the plate,/3 = a/b', and the ratio of axial forces, Ny/N~; m is evaluated as the minimum integer satisfying the following equation:

fl ~< [ _ p + (p2 + 4Q)U2]l/2/2 (13)

where

P = c [ m 4 - (m + 1)4]/[2(m - 2mc - c) + 1]

Q = m2(m + 1)2(2m + 1)/[2(m - 2mc - c) + 1]

c = (Ny/NO (b'/a).

In the above equations, when N~ or Ny is zero, the stress coefficients relevant to Ny/N~ or Nx/Ny, respectively, become infinite. Actually, when the average stress in one direction is even zero, finite values of stresses in this direction are produced in the plate due to the constraint of the edges. These stresses may be evaluated by the following equations:

Nx Ny ~2 m2 D 3 ax*)y=0 = (1 + rh) + ~-7~ + T/2 -~- ~ r/

ax*)y:b,/2 = 2 Nx ) b't - aX*/y=O

O'y*)x=0 = TI2~tt-'[- 1 -q--m----~bt4r/l Ny 7c2D at b'Z t rl3

, 2Ny '~ 7 = -- Gy* x )x=a/2 at ,I x=o"


In the actual analysis, stresses are obtained as the product of the stress coefficients and average stresses. In order to avoid numerical troubles when Nx or Ny equals zero, infinitesimally small values of Nx or Nv may be assumed instead of zero so that finite stress coefficients are realised.

3.2.3 Effect of shearing force on .stress coefficients

In the following, the effect of shearing force on stress coefficients is examined numerically. It is to be noted here that the stress coefficients are functions of only Nx/N~cr, Ny/Nycr and Z/Tcr, together with plate dimensions.

A parametric study has been carried out for a rectangular plate. Load- ing histories of axial compression in one direction and shearing force are plotted in Fig. 5. At first, axial compression is applied. Then, keeping the compression at a constant value, N*, shearing force is applied. Six different levels of N* are considered and the stress distributions are calculated using the incremental Galerkin's method. 9

The stress coefficients are then calculated and results are shown in Fig. 5b. The effect of shearing force may be classified into the following two groups according to the relative value of axial compression, Nx, to the buckling compression, N~cr.

(a) When Nx < Nx~r, (curves 1 to 3 in Fig. 5(b)). When a rectangular plate is subjected to an axial compression, N~ smaller than Nx~r, it does not buckle. However, keeping the axial

3.0 -~ 6

2.0 =,L 5

== 4

BUCKLING INTERACTION CURVE

,.o : - 3

0 1.0 2.0 V x/Vxcr

Fig. 5a. Loading histories.


20

15

E

10

1 : Ox/Oxc r =0.325 2 : 0.650 3 : 0.975 4 : 1.500 5 : 2.000 6 : 3.000

-5

.c

=F -10 axmax = aymax

axmin = aymi n

1.0 1.5 2.0 "~xy / Xxycr

- - 6

4 ~ 3

-15 L -,1

Fig. 5b. Effect of shear stress on stress coefficients.

compression constant and applying an increasing shearing force, the plate buckles at a certain value of shear stress. As the shear stress continues to increase, the normal stress o-~ increases near the edges y = 0 and y = b' and decreases in the middle as shown in Fig. 6(a). The stress coefficients change gradually as shown in Fig. 5(b).

(b) When N~ > N~cr, (curves 4-6 in Fig, 5(b)). When a rectangular plate is subjected only to an increasing axial compression Nx, it buckles when N~ = N.~cr and stress coefficients change as expressed by eqn (12) (Ny = 0).

Next, keeping N.~ constant and applying an increasing shearing force, the normal stress tr x increases in the vicinities of edges y = 0 and y = b', and decreases in the middle as shown in Fig. 6. The stress coefficients change gradually with the change of shear stress as shown in Fig. 5(b).

Such changes in the stress coefficients as shown in Fig. 5(b) may be accurately expressed by the following equations:

where

Buckling and ultimate strength interaction

0~xmax ~- l '62(Nxcr/N~)v 24 + ~*xmax(f(V) + 1) "~ 0~xmin - l ' 3 ( N x c r / N x ) v 2"1 + ~*xmin(0"3 V + 1) Y

17

(15)

C(*xmax ~ 0, f (V) = 0"62 v ~*xmax > 0, V ~< 1 f (V) = 1-3 v 15 0~*xmax > 0, 1) > 1 f (V) = 1-3 v v = I VxllVx.

In the preceding case, an axial force is applied only in one direction. When the plate is subjected to compression in two directions, a shearing force is assumed to affect the stress coefficients in the x direction (0~xmax and 0~xmin ) and the y direction (~ymax and 0~ymin) in the same way as under compression in one direction. The stress coefficients under a combined load of biaxial and shearing forces, may then be expressed by the following equations:

~xmax = l '62(N,:cr/N,:)v 2"4 + ~*xmax(f(V) + 1) ) ~xmin = -1 .3 (Nvcr/NO v2'l + ~*xmin(0"3 V + 1) %,max = 1.62 (Nycr/Nv) v 2"4 + ~*ymax(g(V) + 1) (16)

~ymin = --1.3 (Nycr/Nv) v 2"1 + 0~*ymin(0"3 V --}- 1)

where

0~*xmax ~ 0, f (V) = 0.62 v ~*xmax > 0, I,' ~< 1 f (V) = 1-3 v 15 ~*xmax > 0, V > 1 f (V) = 1.3 v ~*vmax ~< 0, g(V) = 0.62 v ~*ymax > 0, V ~< 1 g(V) = 1-3 v t'5 ~*smax>0, v > 1 g ( V ) = 1-3v v = I v l/V c,

Substituting these stress coefficients into the ultimate strength interaction function (eqn (11)), the ultimate strength interaction relationship of a plate is obtained as plotted in Fig. 3.

3.3 Full plastic strength interaction

When a stiffened plate is stiff enough to prevent both local and overall buckling, it reaches its full plastic strength. Normal stresses, ax and a v, and the shear stress Zxy are considered to be uniformly distributed in the fiat plate panels. Mises's yield condition (eqn (4)) may be written as follows:

~- = 1 (17) \ ao / fro z


where

(a) Nx/Nxcr=0.65 l

Vx/Vxc r = 0.0

n

Nx/Nxc r = 1.50 I] (b)

IJ VxNxc r = 0.0

E 0.5

0.5

Fig. 6. Axial s tress d is t r ibu t ion .

1.0 1.5 2.0

1.0 1.5 2.0

~o = o0/vC-3.

Rewriting the above equation in terms of axial forces and shearing force, the full plastic strength interaction function of a plate, Fp, is obtained in the following form, and is represented in Fig. 3.

( N x ) 2 ( N y ) 2 NxNy (Vx'~2_l (18)

The full plastic strength condition is expressed as

I~p = O. (19)

4 BUCKLING, ULTIMATE STRENGTH AND FULLY PLASTIC STRENGTH INTERACTION RELATIONSHIPS FOR A

STIFFENED PLATE SUBJECTED TO BIAXIAL AND UNIFORM SHEAR FORCES

As indicated in Section 2, the behaviour of a stiffened plate is classified into four types according to the value of the relative stiffness ratio ? of the stiffeners to the plate and the value of the yield stress a0. In this section, buckling, ultimate strength and full plastic strength interaction relationships for a stiffened plate are derived for these four types of behaviour.


These relationships are illustrated in Fig. 7 which indicates how to construct them from the rectangular plate interaction equations shown by dashed line in the same figure for the rectangular plate.

4.1 Overall buckling followed by overall collapse (~ ~< ~Bmin)

Here, a stiffened plate is considered to be reinforced by many stiffeners. Thus, when a stiffened plate buckles in an overall mode, its behaviour may be approximated as that of an orthotropic plate. Therefore, in this section, a stiffened plate is dealt with as an equivalent orthotropic plate. In the analysis, the properties of the equivalent orthotropic plate may be taken as follows:

EQN (38b)

lyp

YIELDING*

o~nA ~ , ~ ON SHORT EDGES

I / .--ATCORNERS

,~>, / / / EQN(11)* - -

/-1 0 I

I

=z >" l I l / EON (19)

\ \ 2 . ~

ON LONG EDGES

EQN (34)**

Nx/Nxp

F U L L Y P L A S T I C

EQN (38a)*** EQNS (38a)**"*

EQN (38c)

ULTIMATE STRENGTH (STIFFENED PLATE)

m D ULTIMATE STRENGTH (PLATE) ~. BUCKLING STRENGTH (PLATE)" FIG. 3

* Depending on the combination of Oxrna x ,Oxrni n , o ymax, and Oymi n .

** Obtained by displacing EQN (11) a distance =o0 nA on the N x axis in the positive direction, representing fully pla~c compression of the stiffeners.

*** EQN (19) displaced by o0nA in the negative N x direction (full plastic tensile strength of the stiffeners).

EQN (19) displaced by o0nA in the positive N x direction (fully plastic compression of the stiffeners),

Fig. 7. Buckling strength, ultimate strength and fully plastic strength of a rectangular plate.


Ev = E{1 + (nA/bt)}, Ev = E "] Ax = bt, At, = at Dx (nl/b) + Dpl, Dv Dpl, 2Dx,,(Gt3/3) + v(Dx + D~,) Dpl = Et3/{12(l - v2)}

(20)

where D is flexural rigidity and I is the moment of inertia of a stiffener together with the corresponding effective breadth of plating. Subscript pl indicates the plate.

4.1.1 Buckling interaction relationship

When a stiffened plate is dealt with as an orthotropic plate, the buckling interaction relationship may be expressed by using the same equation for an isotropic plate (eqns (2)), i.e.

r B = r B (fl, Nx/Nvcr, Nv/Nlvr, Vx/~/~rcr)" (21)

However, the remaining variables present in eqn (2) should be evaluated for the orthotropic plate by

= a/b N~cr = ax~(bt + nA) I N~,cr = a~.cr(at) ? (22)

l'~cr = rx,,cr(at) J 0 -0 .C O xcr, a,,cr and xvcr are the buckling stresses of an orthotropic plate when independently subjected to normal stresses ax, ay and a shear stress, .C.~y, respectively.

4.1.2 Ultimate strength interaction relationship

In the case where a stiffened plate reaches its ultimate strength in an overall collapse mode after overall buckling, an ultimate strength condition may be described in the same form as for a flat plate (eqn (10)).

When eqn (10) is applied, stress coefficients ~xmax, ~xmin, 0~ymax and ~Ym,, are obtained by eqn (16). However, b' in eqns (10) and (16) is to be replaced by b. Nrcr, Nvcr and Vxc r a r e evaluated by eqn (22), and N~p = ao (bt + hA).

4.2 Local buckling, followed by overall or local collapse (7 > 7Bmin)

A stiffened plate with n stiffeners is regarded as being composed of n stiffeners and (n + 1) plate panels whose behaviour and strengths are dealt with separately. The behaviour of plate panels between stiffeners has already been studied in Section 3. The results of this study are used in the following.

rB = [

where

Buckling and ultimate strength Otteraction 2 I

4.2. l Buckling interaction relationship

When a stiffened plate is subjected to axial forces N~ and N,, and shearing forces Vx and E,,, and until the stiffened plate buckles locally, uniform stresses ax, a), and Zx.,, act on each plate panel. The stiffeners are subjected to uniform stress Oxs in the x direction. This stress may be evaluated from the condition of continuity that the strain e.~ along the connecting line between a stiffener and the plate is the same, i.e.

O'xs = a.~ - v o'.,.. (23)

Therefore, the relationship between the applied forces and the resulting stresses may be expressed as follows:

N~ = ax(bt + nA) - v a,. nA ]

Nv,; = a v at " I (24) __V x = Z~,, at v,_ = , ; ; b t

where A is the cross-sectional area of a stiffener. When a stiffened plate buckles locally due to normal stresses ax and a,.

and a shear stress %,, the buckling interaction relationship is expressed by eqn (1). Substituting eqn (24) into eqn (1), the buckling interaction function 1-'B can be represented in terms of Nx, N). and Vx as follows:

(1) When ax is tensile and a~, is compressive (ax < O, a,, > O)

m2(l(m2+fl2)2N*+(vnA/at)Ny+ f l 2 ) 2 N~r N,. ( VV~r)2 FB -- ~- ~ + -- 1. (25a)

(2) When ox is compressive and a,, is tensile (ax > 0, a~. < 0)

FB N~+(vnA/at)Nr (1+fl2)2 Nr ( V ~ ) 2 N ~ c r (m 2+fl2)2 = " ~ - N ) ; r ~- - 1. (25b)

(3) When ax and o 5, are compressive (ax > 0, a). > 0)

+ (v_ nM/at) Ny }/Nrcr ] el [ N,4/Nycr ]'l e2 1 ( Vxl Vxcr)2 j -t- [ l - ~ c r ) 2 J -- 1 (25c)

Nxcr = O'xcr(bt -t- nA) Nycr : O'ycr at, Vxc r : "r~vc r at

and O'xcr, O'ycr and zx~r are the independent buckling stresses of a plate between two adjacent stiffeners.


The buckling condition is expressed as

Fa = 0. (26)

4.2.2 Ultimate strength interaction relationship

When local buckling occurs in a stiffened plate, one of two ultimate strength modes may take place, depending upon the relative stiffness ratio 7 of the stiffener to the plate.

(1) 7Bmi n


to the longitudinal edges, to the buckling strength of all stiffeners with their effective breadth of plating.

N~u = nPus + Nx' (29)

where N~' can be evaluated by solving eqn (10) for Nx as follows:

ao b't 4 Vx 2 N x ' - 2 ( - i ~ 0 [ N @ p ( l + % ) + ~ - - 4 - (-E-~xp)-3I'N-~y(I+e"')]zl " l v v p

The', ultimate strength function Fu is expressed as

Nx nPus + Nx' Fu = N~p N~p (30)

(b) Bending of stiffeners (eccentric loading) In this case each stiffener, together with its associated effective portion of the plate breadth b'e, is subjected to compression and bending: The stress distribution in a stiffener and associated plate is as shown in Fig. 8. The continuity of strain of the stiffener and plate along the connection line is satisfied by eqn (23). Therefore, the stress O'xmax in the plate is related to that (a*xmax) of the stiffener as follows:

OXmax = O'*xmax + VO'y. (31)

The stress distribution in the plate and the effective breadth be' change according to the change of trxmax and can be obtained from eqns (7) and (16). The effective breadth be' = b' (Gxav)/(tTxmax) is modified to be0' so that it is expressed in terms of the stiffener stress tr*rmax-

beo' = b' axa~ (32a) tr*xmax

Since the axial force acting on the plate panel should be the same whether expressed in terms of be' or be0', be0' may be derived as shown below, using eqn (31).

beo' = be'l(1 - v O'y/O'xmax ) (32b)

The ultimate strength is obtained assuming that the stiffener with its modified effective breadth be0' acts as a beam-column. The ultimate compressive load Pus of the beam-column is determined by the condition that plasticity of the outermost fibre of the stiffener has occurred.

First, stress as in the outer fiber of the stiffener (see Fig. 8) is evaluated as follows, considering the magnifying effect of the axial force.

P P . e ( P ~ 2 ) as -- AT Z----/- secant (33a)


where P is the axial force acting on a stiffener with its effective breadth; AT is the total cross-sectional area of each stiffener with modified effective breadth beoAT'= A 4-beo't; e is the magni tude of the eccentricity of loading; and Zs is the section modulus of the beam-column corresponding to the outermost fibre of the stiffener.

The ultimate strength condit ions may be expressed as follows:

as = a0 (33b)

When eqn (33b) is satisfied, P of eqn (33a) is the ult imate strength Pus, i.e.

P = Pus (33c)

Substituting Pus into eqn (30), the ult imate strength function Fu may be derived.

The stress O'*xmax of the stiffener at the connect ion line between the plate and the stiffener (see Fig. 8) may be calculated as follows:

P.e ( P ~ 2) P + secant (33d) O'*xmax = A~ ~ - p

O x a v

i-- xmax N A -

I I I I

s s a v = P/PT

STIFFENER

t

\ Oxav xmax

PLATING ASSOCIATED WITH A STIFFENER

Fig. 8. Stress distribution in a stiffener and associated plating (eccentric loading).


where Zp is the section modulus of the beam-column corresponding to the middle plane of the plate.

Here, however, eqns (33) contain AT, Zs, Zp and I, which are functions of the modified effective breadth be0', which is expressed in terms of mean stresses. Therefore, when the axial compression P reaches the ultimate compressive load Pus, beo' should be evaluated such that it corresponds to the ultimate load. (Usually, it can be obtained by iteration.)

(2) 7 > ~Umin

In this case, the plate between stiffeners buckles and collapses locally, whiHe stiffeners do not buckle and may reach their fully plastic state. The ultimate strength of a stiffened plate can be obtained from the sum of the ultimate strengths of the plate panels and the stiffeners.

It is to be noted here that in a case of proportional loading, plate panels and stiffeners will not generally reach their ultimate strength simultaneously. When either the plate panels or stiffeners reach their ultimate strength, proportional loading cannot be significantly increased until the others reach the ultimate strength. If the loading is not proportional, some component of the load can be significantly increased until both the plate panels and stiffeners reach their ultimate strength. The stiffness of the stiffened plate will be reduced after either the plate panels or stiffeners reach their ultimate strength.

Anyway, the difference between the ultimate strength conditions representing either the plate panels or the stiffeners reaching their ultimate strength in proportional loading, and the ultimate strength to be derived here, is not significant.

This discussion is also valid in the case of the fully plastic strength which is derived in the next section.

(a) Concentric loading Here, the stiffeners are subjected only to an axial force and their ultimate strength is represented by ao hA, while the ultimate strength of plate panels is a,; shown in Section 3.

From the condition of yielding of the plate panel, which is checked either at the corners or in the middle of each half buckled wave, the ultimate strength function I'u may now be obtained as follows.

For N~ > ~o nA + Nx

- ~ ( 1 +ex) - [~--~p(1 +%.)

-- (Nxp ~yonA)Nvp (1 + O~x) (1 + ~y) -

(34) 1.


ax and 0~y in the above equation may be 0~xmax or 0~xmin and ~ymax o r 0~ymi n according to the location of yielding and they are evaluated by eqn (16).

For, Nx < go nA + f~

['u = N y / N v p - fy /Nvp (35)

where A(~ and N,. are the coordinates of the intersection point of eqns (11) and (19) as shown in Fig. 7.

(b) Eccentric Loading When eccentricity of the loading occurs after buckling, the stiffeners are subjected to bending and axial compression. By a certain magnitude of increment of y, the overall collapse mode changes into the local collapse mode which gives higher ultimate strength. The stiffness ratio y at this transition point is defined as Umin.6

Under eccentric loading, a stiffened plate collapses by yielding caused by bending of the stiffeners (with their effective breadths). From this condition, the ultimate strength is evaluated in the same manner as in Section 4.2.2 using eqns (33) and (30).

Finally, the condition of the ultimate strength is expressed by the following equation:

Fu = 0. (36)

4.3 Fully plastic strength interaction curve

When a stiffened plate does not buckle in either local or overall modes, it reaches its fully plastic strength. In this case, each plate is subjected to uniform stresses ax, ay and zx>, such t h a t Oov2(=Oo 2 - 3 z2xy) = ax 2 + ~Ty 2 - ~Yx ~Ty and each stiffener is subjected to a uniform stress equal to ao.

The fully plastic strength interaction relationship under biaxial forces N~ and N~' and shearing stress zxy is derived by the condition of plasticity of the plate panel and expressed in the following form:

( N ~ + a n A ) + ( N Y ) - (Nx+anA)Nrabfl -- aov 2. (37)

Substituting aov by v/ao a - 3 zaxy and dividing by ao 2, the fully plastic strength interaction function Fp is derived as follows:

(1) (a) N,. > 0, N, < Nx - ao nA

or Nr < O, Nx < - N~ - ao nA

p go nA + - - 1 - \ N,,p / (Nxp - ao hA) Nyp +

(38a)

o r

Buckling and ultimate strength interaction

(b) Ny > O, Nx > N.~ +tro nA

Ny < O, Nx > - f(~ +tro nA

( Nx - aonA 2 (Nx - ao nA) Nv

(2) Ny > O, ~(,: - - cr o nA


5.1.1 Rectangular plates

A flow chart of this procedure is shown in Fig. 9. First, the buckling strength load parameter PB may be obtained by using one of eqns (2) corresponding to the specified condition of loading to solve the buckling condition I~B -~ 0 of eqn (3).

Next, the fully plastic strength load parameter Pp is obtained by using eqn (18) to solve the condition I~e of eqn (19). The possibility of buckling is then examined by comparing pp with the buckling load parameter PB. If the buckling load parameter PB is smaller than Pe, the plate buckles before reaching its full plastic strength. In this case, the ultimate strength after buckling is to be calculated.

The ultimate strength after buckling is obtained by using eqn (10) to solve the condition l?u -- 0 of eqn (11). The ultimate strength interaction

CALCULATE PB CALCULATE Pp EQNS (2), (3) EQN (18)

I I

_1 ~p { --] P = P +

txmax' ~xmin ~ EQN (161 ay max ' (Xy rain J

ru=~ 1 ['u AT LONG EDGES ? E Q N (11) F u ATTRANS. EDGES.JI

I . I ,o@ Pu=P AT Fu = 0.0 ]

Fig. 9. Procedure of evaluation of the ultimate strength of a rectangular plate.


function Fu contains the stress coefficients ~.~ and ~:. which are given by eqn (16) as Ctxmax, ~xmin, ~,'max and ~ymin- These coefficients are substituted into eqn (10) according to the location of yielding after buckling. The three possible locations where the membrane stress may satisfy the yield condition and their corresponding coefficients are shown in each case as follows:

(1) the four corners of the plate (~.~max and ~,,max); (2) sides parallel to the x axis at the middle of half the buckled wave

(~xmax, ~ymin); (3) the middle of the sides parallel to the y axis (~xmi,, ~xmax).

Ultimate loads corresponding to these yielding locations are calculated and the lowest one is taken as the ultimate strength.

In the actual process of calculation, the external load is increased gradually. The stress coefficients and the ultimate strength interaction function Fu at the three locations of yielding are evaluated after each load increment. The relationship between load parameter and the value of the ultimate strength interaction function Fu at each location is plotted as shown in Fig. 10(a). The condition 1-'u = 0 is satisfied at the intersections of these relationships with the ordinate axis. The smallest load at these intersections is the ultimate strength.

5.1.2 Stiffened plates

A flow chart for this procedure is shown in Fig. 11. At first, the buckling mode and buckling load are to be obtained from the dimensions of the stiffened plate and the loading condition.

Tile local buckling load parameter PBI which causes buckling of plate

P Pc~

P~ CORNER

~ y=O,b / x:O,a

L ~ 4 , 5 " Pass o

0 F u 0 Pass

(a) (b)

Fig. !0. Itcrative procedure to calculate ultimate strength.


ASSUME SMALL O u r I

SET Oxmax = 0"~1 N,

O'xmax EQN (31)

b~ EQNS (7), (16)

b~0 EQN (32)

% AND P = .%o~, o s EQN (33a)

~SFIED NOT SATISFIED

Pus=P I

EQNS (2), (3) EQN (21) EONS (38), (39) '

YES

I

t

CALCULATE Pul USING EQNS(34),

(34)&PROCEDURE INSEC. 5.1.1

xmax EQN ~33d) EQN (31)

I

t P = PB I

P = P + & P

b~s t EQN (28)

Pus EQN (27) Pu0 EQN (29)

PLOT Pu0 " p I

p,, -- M~NI Pul. P~ I I

Fig. 11. Procedure to evaluate the ultimate strength of a stiffened plate.

panels between the stiffeners may be obtained as described in Section 5.1.1. The overall buckling load parameter PB0 may be obtained by imposing the condition FB = 0 on eqn (21). At the smaller buckling load of these two, the actual buckling occurs in the corresponding mode.

Next, the fully plastic load parameter pp is evaluated by solving eqns (38) under the fully plastic condition of eqn (39). This load is compared with the buckling load to classify the behaviour of the stiffened plate.


(1) Pa0 > Pp, PBt > PP Buckling does not occur and the stiffened plate reaches its fully plastic strength.

(2) PB0 < PBt, PB0 < PP Overall buckling occurs and the ultimate strength is evaluated using the equations of Section 4.1.2. The procedure for the analysis is as shown in Section 5.1.1.

(3) Pat < PB0, Psi < PP Local buckling occurs and the stiffened plate collapses under concentric or eccentric compressive loading, depending upon the type of loading.

(a) Concentric loading There are two cases; one is where the stiffened plate collapses in the overall mode and reaches the corresponding ultimate strength Pu0, and the other is where the plate panels collapse locally and the stiffened plate reaches the corresponding ultimate strength load Put-

The value OfPu 0 can be obtained from eqn (29) as follows. Increasing the load p gradually, the corresponding effective breadth may be obtained from eqn (28). Using this effective breadth, Pus and Pu0 may be obtained from eqn (27) and eqn (29), respectively. The effective breadth used, however, corresponds to p. It corresponds to Pu0 only when P = Pu0. To find this load, Pu0(=Pcal) is plotted against assumed values o f p (=Pass) as shown by the curve in Fig. 10(b). When Pass is equal to Pcal, P is equivalent to Pu0, which represents the ultimate strength.

On the other hand, Put can be obtained from eqn (34) or eqn (35) under the condition Fu = 0. The procedure of analysis is as shown in Section 5.1.1.

The smaller of Pu0 and PuJ is the actual ultimate strength.

(b) Eccentric loading Local buckling reduces the effectiveness of plate panels, causing a shift of the neutral axis. When the load is applied at a fixed point, an eccentricity is produced. Ultimate strength in this case may be obtained by the following incremental load scheme.

Assuming a small value of the mean stress O'sa v acting on a stiffener together with an effective breadth b~0' as defined in Fig. 7, the plate stress O'xmax and the effective breadth be' are obtained by eqn (31) with O'*xmax = O'say and eqns (7) and (16). Next, be0' is obtained by modifying be' using eqn (32).

Calculating the cross-sectional area of the stiffener, At, using this b~', the axial load P of the stiffener is obtained by the equation P ---- AT trsav.


The value of ~r~ is calculated from eqn (33) using this axial load P. These values are examined to satisfy eqn (33b). If this equat ion is not satisfied, increasing the mean stress O'sa v the analysis is to be repeated using axmax = trpl -k- v try obtained in the previous step until eqn (33b) is satisfied. This value of P is the ult imate strength Pus.

Now, the ultimate strength function Fu (eqn (30)) may be evaluated and the ultimate load is obtained as the value satisfying the condit ion Fu = 0.

5.2 Accuracy of the present interaction relationships

The accuracy of the present ultimate strength equations is examined by compar ison with results reported in the literature.

(1) Square and rectangular plates subjected to compression in one direction

The ultimate strengths of square plates as calculated by three different methods: the finite element method, l the combined elastic large deflection and plastic analysis 4 and the present method, are plotted in Fig. 12.

Results obtained by the finite element me thod with an initial deflection of Wom/t --- 0-01 in the same mode as the buckling mode (to represent a perfect plate) and the present me thod applied to a rectangular plate are plotted in Fig. 13. In the analysis by the finite element method, an initial

1.11

tO

0.8

0 . 6

0 . 4

0 . 2

i _ b ~1

i t : i i

FEM ( N O N - C O N F O R M I N G )

- ~ F U J I T A et al.

~ - ~ P R E S E N T A N A L Y S I S

I , I I I I I I 0 .5 1.0 1.5 2 .0 2 .5 3 .0 3 .5

Fig. 12. Ultimate strength of square plates by different methods.

I 4 .0

Buckl&g and ultimate strength interaction 33

1.0

0.8

0.6

0.4

0.2

E = 206,000 N/mm 2

t = 274.7 N/mm 2

t t m = l m = 2 m = 3 b = 1,000 mm

tt \ \ \ t = 12 mm

Ilmlllw ~ ~ w6111

! !

0,5 1.0

m . - - BUCKLING STRENGTH . . . . ULTIMATE STRENGTH,

FEM ( Worn/t = 0.01, NON-CONFORMING ) ULTIMATE STRENGTH; PRESENT ANALYSIS

I I I I I 1.5 2.0 2.5 3.0 3.5

a/b

Fig. 13. Ultimate strength of rectangular plates.

deflection is assumed, which reduces the ultimate strength, and the aspect ratio corresponding to the minimum ultimate strength changes. Therefore, both FEM results and present results plotted in Fig. 13 may be regarded to be in good agreement.

Considering both Figs 12 and 13, it may be seen that the present method predicts the ultimate compressive strengths of square and rectangular plates with good accuracy.

(2) Square and rectangular plates subjected to combined loads

The ultimate strength interaction relationship for square and rectangular plates subjected to compression in the x direction and shear is evaluated by the present method and compared with that obtained by the combined elastic large deflection and plastic analysis. 4 The results are plotted in Fig. 14.

The ultimate strength interaction relationship for square plates subjected to biaxial load is evaluated by the finite element method and the present method. The results are plotted in Fig. 15.

(3) Stiffened plates subjected to uniaxial compression

The ultimate strengths of stiffened plates subjected to uniaxial compression are obtained by the finite element method and the present method


o 1.0 ~o.s

0.6

0.4 0.2

I i I i

o 1.0 ~= 0.8

0.6

0.4 o.2 i

0 0.2 0.4 0.6 0.8 1.0 0 Ou/O o

.... FUJITA, REF. 4

b/t = 100~ i i

0.2 0.4 0.6 0.8 1.0 Ou/O 0

PRESENT ANALYSIS Fig. 14. Ultimate strength of square and rectangular plates subjected to compression and

shear.

oy/o 0 O FEM ~,

~ l ~ ' - ' f " ~ ~ ~ = 1 " 0 ~ ~

206,000 N/mm 2 264.9 N/mm 2 I o0 =

Fig. 15. Ultimate strength of square plates subjected to biaxial loading.

with changing relative stiffener stiffness ratio 7. The results of both methods are plotted in Figs 16 and 17. In Fig. 16, the compressive load is applied concentrically while in Fig. 17, eccentricity of the load is allowed to take place though the use of a one-sided stiffener.

In Fig. 16, non-conforming elements were used in the analysis by the


1,,0 ~o

0,,8

0 . 6

0..4

0.2

(In mm)

0 0 0 0 0 / ,p

- " '~BUCKLING STRENGTH

E = 206,000 N/trim 2 o 0 = 264.9 N/ram 2

~ ~ ~ B U 0 O CKLING STRENG'T~-I

O FEM ( NON-CONFORMING ) PRESENT ANALYSIS

I I I I I I I I 0 0,2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Nt

Fig. 16. Ultimate strength of compressed stiffened plates (concentric loading).

I 1 . 8

1.0

0.8

0.6

0.4

0.2

-='lhF~ E =206,000 N/ram 2 ~]'-- ~ ~ ~ _ ~ _ O0= 264.9 N/mm 2

3.2 (in mm)

/ / " FEM

- - - CRITERION OF Pe REF. 5 " ~ - ~ BUCKLING PRESENT ANALYSIS

STRENGTH I I I I I I

0.2 0.4 0.6 0.8 1.0 1.2 1.4 h/t

Fig. 17. Ultimate strength of compressed stiffened plates (eccentric loading).

finite element method. The inplane displacements of the edges are free, which causes the plate strength to decrease.

Also the present method assumes an orthotropic plate for overall buckling/overall collapse. This panel, however, has one stiffener only. This lead,; to collapse by stiffener yielding which is not included in the treat- ment of the stiffened plate regarded as an orthotropic plate. This explains


the deviation in the overall buckling/overall collapse range. Better agreement is expected with panels having more stiffeners. G o o d agreement may be observed in other buckling/collapse modes.

6 C O N C L U S I O N

Rectangular unstiffened and uniformly uniaxially stiffened plates are examined. The buckling strength, ult imate strength and fully plastic strength are derived theoretically under inplane biaxial and shearing forces.

Ult imate strength may be reached in one of three modes. The first is when a stiffened plate collapses in an overall mode. The second is when the plate collapses locally and the last is when the fully plastic strength is at tained without buckling. Interaction equations, as functions of biaxial and shearing forces, are presented.

Compar i son of the results obtained by these interaction equat ions with those available in the literature indicate that these equations have sufficient accuracy for practical use.

R E F E R E N C E S

!. Ueda, Y. & Yao, T. Compressive ultimate strength of rectangular plates with initial imperfections due to welding, ist and 2nd reports, SNAJ, 148 (1980), and 149 (198 l), respectively.

2. Valsgard, S. Numerical design prediction of the capacity of plates in biaxial inplane compression. Computers & Structures, 12 (1980) 729-39.

3. Harding, J. E., Hobbs, R. E. & Neal, B. G. Ultimate load behaviour of plates under combined direct and shear inplane loading. In Steel Plated Structures, Crosby Lockwood Staples, London, UK 1977, pp. 369-403.

4. Fujita, Y. et al. Ultimate strength of flat plates subjected to combined load. lst-3rd reports, SNAJ, 145 (1979), 146 (1979) and 149 (1981), respectively.

5. Rutledge, D. R. & Ostapenko, A. Ultimate strength of longitudinally stiffened plate panels. Fritz Eng. Lab. Report No. 248.24, 1968.

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