Determination of Effective Breadth and Effective Width of Stiffened Plates by Finite Strip Analyses

• i v '

ELSEVIER

Thin-Walled Structures Vol. 26, No. 4, pp. 261-286, 1996 Copyright ~'~ 1996 Elsevier Science Ltd

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Determination of Effective Breadth and Effective Width of Stiffened Plates by Finite Strip Analyses

X. Wang & F.G. Rammerstorfer

Institute of Lightweight Structures and Aerospace Engineering, Vienna Technical University, Vienna, Austria

ABSTRACT

A finite strip (FS) method is presented for the numerical investigation of two design parameters - - effective breadth and effective width - - of stir

fened plates. For the effective breadth, stiffened plates under bending are studied. Due to the transverse bending loads there is shear transmission through the plate from the stiffener which leads to a non-uniform longitudinal stress distribution across the plate width. This phenomenon, termed as shear lag, can be represented by the 'effective breadth concept', and has been extensively studied by analytical methods. A linear FS method is presented which utilizes the advantages of decoupling of Fourier terms on the one hand and, on the other hand, allows the treatment of both webs and flanges using a plate model. A definitely different situation exists for estimating the effectiveness of the plate breadth (or width) of plates in the postbuckling range. The 'concept of effect width' is based on the fact that plates with supported longitudinal edges and/or stiffeners can accept additional load after buckling under longitudinal compression, and enables the designer to evaluate the postbuckling strength of plate structures simply by using the design parameter 'effective width'. Several formulae (most of them empirically derived) exist for an approximative calculation of the load dependent value of the effective width. A nonlinear FS method is developed and applied to the investigation of the postcritical strength of locally buckled structures. An incremental successive iterative procedure is" introduced for an effective numerical analysis. Copyright © 1996 Elsevier Science Ltd.

261

262 X. Wang, F.G. Rammerstorfer

1 I N T R O D U C T I O N

The subjects of this paper are two design parameters - - effective breadth and effective width - - of stiffened plates, which have been used in structural engineering and particularly in naval and aerospace applications. Despite the common features of these two parameters ~ both of them are used to describe the effectiveness of a breadth (or width) for plate structures in which the axial, i.e. longitudinal, stress distribution across the plate is not uniform - - they are used for thoroughly different engineering purposes. Therefore, a clear definition of the problem is necessary.

In the case of effective breadth we consider beam flanges or longitudinal stiffened plates under bending. When such structures are designed to resist transverse loads, which cause the beam flange or the panel to bend out of its original plane, the distribution of the stresses across the plate (or flange) is not uniform because of the transmission of shear through the plate from the stiffener, and the axial, i.e. longitudinal, stress diminishes as the distance from the web increases. This phenomenon, termed shear lag, can be represented by the effective breadth concept. The design parameter of effective breadth enables the designers to calculate the behaviour of these structures by the use of simple beam theory.

This situation is different from the one where a thin walled beam or stiffened plate panel is designed to resist axial compressive loading. After local buckling of the beam flange or of the plate between the stiffeners, the longitudinal stress distribution is also not uniform. Designers may estimate the postbuckling strength of assemblies under compression using the concept of effective width. Schade I pointed out some historical confusion with respect to these two different concepts of effectiveness. It was suggested that the term 'effective width' should be used to denote effectiveness in the postbuckling situation, and the term 'effective breadth' should be restricted to effectiveness of the plate or flange as a component of an effective beam in bending. It is necessary to draw a distinction between them: assuming linear elastic material behaviour, under an applied bending load the effective breadth is merely a function of geometrical parameters and is independent of the load level, whereas the effective width for evaluating stresses in the postcritical range depends also on the load level.

For the numerical investigation a finite strip method, z which inter- polates the structure's behaviour in the longitudinal direction by harmonic functions and in the transverse direction by polynomial functions, is employed. Simply supported boundaries are assumed at the longitudinal ends. These conditions are satisfied a priori by the trigonometrical displacement functions in this direction. Classical plate theory is assumed in this analysis.

Effective breadth and effective width of stiffened plates

tl (y ky

: / ( 3 v y : ,! X

B

Fig. 1. Effective breadth.

263

2 EFFECTIVE BREADTH OF BEAM FLANGES U N D E R BENDING

The effective breadth bef has been intensely investigated by analytical approaches.l, 3-~0. A schematic depiction of a plate segment in bending is shown in Fig. 1.

2.1 The definition and analytical treatment of effective breadth

The Airy stress function F is related to the stresses by

OZF trxx Oy 2 '

02F tTyy : OX 2 "~ (1)

02F

x y - O x O y "

The stress function F may be approximated by a series of harmonic functions as

F = Z F. = E f . sin (to.y) (2a) /1 n


o r

F : Z Fn = Z fnCOS((J)ny)' n n

respectively, where e)~ is defined by

n g ( ' O n z - - .

l

(2b)

(2c)

f~ is a function of x. Further, it is assumed that for very thin flanges the stresses are constant over the flange thickness t. Provided that the loading is applied at the stiffeners or at the web,f~ should satisfy the Lagrange equation

04F 04F 04F ~74F -- ~ + 20x2Oy-- ~ + 0 ~ = 0. (3)

Functions fn which meet these requirements have the form

f~ = (An + Cnmnx) cosh(ognx) + (o~x) + (B, + D, conx) sinh(og,x). (4)

The relationship of four constants An, Bn, Cn, Dn is determined by boundary conditions.l ' 5

The longitudinal stress O'y p in the flange plate reaches its maximum p O'ma x

at the web intersection. The web is considered by an uniaxial beam model ignoring the lateral contraction. The maximum longitudinal stress in the web, aWax, is related to Oma x p by the compatibility condition for the strains at the intersection,

w p O ( 5 ) O'ma x = O'ma x - - V~Txx.

P is of opposite sign to o'Pnax in the case of simply supported Since a xx beams or plates, this means that there is an abrupt increase in ayy from flange to web. This leads to two definitions of the effective breadth, as follows.

The effective breadth according to the maximum longitudinal stress in the web, a~a x, is defined by

X b~n be~ - - (6a)

w w O'ma x O'ma x

where

X = aPydx.

The effective breadth according to the maximum longitudinal stress in the plate, O'ma x p , i s given by

Effective breadth and effective width qlst([lk'ned plates 265

b~f- X b~u a n - ap~. (6b)

For any single value of n, the effective breadth can be derived by

OJ;, t'

Ox o (7a) h h" ~ o:,..,. ,, dx

: ,,4.,..,, : ( ,o / j i, . , , . , O ' l n a x. n -- ~ lvl - g - "~

to.,-- + -7r--J;,) O ( o r / , )

and

bp .[~ a,.,. ,,dx Ox 0 ~ r . , , . . . . - (7b)

P "~ ' ~=O(orb)"

O'max.,,, ( ~ x

\ox-)

respectively, b~';f.,, and b p ~r.,, are the effective breadths of plates in which the longitudinal variation of the stress state can be described by a single stress function F,,. They are independent of y; i.e. they are constant over the length, can be fully determined for given boundary conditions, and are therefore referred as 'boundary functions'.

If the longitudinal variation of the bending moment M has a simple harmonic form. as for example

M = M,, sin (~o,,y) (8)

Metzer s proved that this bending moment will result in a constant effective breadth over the length. Provided that the simple beam theory is valid after introducing the concept of effective breadth, for M,, sin (,J,,y) the stress distribution can be described by a single stress function F,, =/; , sin (~,~,,y), and the effective breadth is equal to b~r.,,. Generally, if the bending moment can be represented by harmonic series

= Z M,, sin (~o,,)') M (Sa) II

the following relationship exists,

_ M - Z Z M,, sin (~sJ,, v) a,,,,. S ,, a,,,,.,, = ,, S,, " " (9)

where S and S,, are the section moduli of the assembly of the flange and the web obtained by assuming the flange as having the half-breadth h~r.,,, respectively. We define

266 X. Wang. F.G. Rammerstor/~,r

W

X = Z X,, = Z bef'"M"sin(t°"Y) (10) II I! S l l

It should be mentioned that beWf.,, is desirable in eqn (10), since eqn (9) implies that the stress, uniformly distributed across the effective flange, has the same value as the stress in the web at the intersection. Otherwise S,, would not represent the section modulus in the usual sense. If the stress in the plate will be used as the reference stress, it can be obtained by eqn (5). The final 'effective breadth' be'~ can be derived by inserting eqns (9) and (10) into eqn (6a),

bcWf.,, M,, sin (to,,),) V" w x s , - - U - be,, - - (11)

0"ma x V~ M,, sin (to,,),) /__, 3,1

The application of eqn (11) requires the computat ion of a new section modulus value S,1 for each term of the series. The section modulus S can be expressed as

where C is independent of n. There are two important cases for ft.

• f o r / - and boxgirder with lower flange identical with upper flange 3

1 h tw (13a) f l - 6 b t

where for boxgirders tw is the total thickness of webs, the sum of the web thicknesses;

• for sections with only one wide flange, and one other small flange with a r e a ,42, being so narrow that it may be regarded as 100% effective. This applies also to the case of a plate stiffened with Ts, Ls, bulbs or flat bars. Here we have j

1 h tw 4,42 + 2htw (13b) f l - 4 b t 3A2+2htw

For the special case of bars with A2 = 0, we have

1 h tw (13c) f l - 4 b t

The application ofeqn (11), as pointed out in Ref. 1, is tedious, principally for computing a new section modulus value S,, for each term of the series. Alternatively, a simplified method can be derived. From eqn (9) we have

E[/ective breadth and effective width q/stiffened plates 267

S = M

Z M. sin(to.y)" S.

n

From eqn (12),

b M be'~ = M. sin (co.y) - bfl.

+ l 3 b

(14a)

14b)

2.2 Computation of the effective breadth by the linear FS method

In the finite strip method (FSM), the plated structure is discretized into a number of strips which are joined at longitudinal (or - - in the case of shells of revolution - - circumferential) lines. The behaviour is interpolated in the longitudinal direction by harmonic functions and in the transverse direction by polynomial functions. The harmonic functions have to be chosen to satisfy support conditions at longitudinal ends. Displacement functions for different end conditions can be found in Ref. 2. Assuming the classical plate theory, the displacements in the middle plane of the plate can be described by

M

u i= ~ U ~ ( ~ ) g ~ ( 7 / ) for i = x , y , z , (15) I17 ~ [

with ~ = x/bs, ~1 = y/l, where bs is the strip width. U;,(~) and i g., (r/) are polynomial and tr igonometric functions of generalized nodal displacement parameters corresponding to the ruth harmonic series, respectively, i g.,(TI) is chosen to satisfy the end conditions. For strip elements with simply suppor ted longitudinal ends (r/ 0,1), i = g,,,(~) generally has the form:

g;~i (r/) = sin (mrtr/), ( 1 5a)

g;',i (r/) = cos (mr~r/), ( ! 5b)

g~,, (71) = sin (mrrr/). ( 1 5c)

For the two-noded strip, the polynomial functions are linear in the expressions of membrane displacements U,- and u,. and cubic in the expression of out-of-plane displacement u-:

U~,,(~) = (1 - ~)ulm + ~u~m for i : x ,y , (15d)

U;.(¢) = (1 - 3~ + 2~2)ulm q- (~ - 2~ 2 + ~3)bO,m q-

(3~2 ~ i (15e) _ 2 ~ . ) U 2 m _}_ ( ~ 3 __ ~2)b02 m for i = z.


The quantities Ulm lg2m,i ( i = x , y , z ) , Olm = [OUz/OX[¢=O]m and 02,,, = [Ouz/Ox [¢:1]m are the eight generalized nodal displacement parameters of a strip, which are connected with the displacements with respect to the polynomial variable ~ in the lateral direction.

A nonlinear FS program PIRTS is developed specifically for the computat ion of effective breadth as well as effective width of stiffened plates. In this program the stress state can be evaluated at each control node of every strip nodal line. The positions of these control points are defined by the input data. The effective breadth can be calculated for the cross section where the maximum axial stress occurs according to eqns (6a) or (6b). If the plate structure has nt equally spaced identical stiffeners connected to the plate, and one segment of the plate x E [0, b] is discretised into np strips, the effective breadth can be calculated by

bef= ~-]~TPl (ItTyy'i-~- 2ayy'i)bi (16) 2O'ma×

with jayy, i and 2ayy, i indicating the axial stresses at the first and second nodal line of the ith strip, respectively, O'ma x is the maximum axial stress across this cross section (near the stiffeners). Of course the strip discretization of the structures has an influence on the accuracy of bef. The numerator at the right-hand side of eqn (16) represents the integration of the axial stresses over the breadth b of the plate segment, amax in the denominator needs to be calculated as precisely as possible. Across the strip boundaries a discontinuity in stresses appears, and relatively large errors in stresses may be expected to occur at strip boundaries (nodal lines). For practical purposes, corresponding to a guideline for estimating the error in a finite element solution, this discontinuity can be used as a measure of the quality of the discretization. The stress difference should be restricted to not exceed a certain percentage of the maximum stress (say, less than 10%). Piece-wise continuous element stress solutiohs can be obtained in a post-processing to obtain a continuous version of the stress field, and the stresses can be estimated from the smoothed stress distribution. 12 The stresses at the node points of element boundaries can be approximated by the average values of stresses of adjacent elements. For the calculation of area x in stiffenef's at the intersection of the plate and stiffener, or the stresses at the structure's boundaries, there may be no possi- bility of solving for such average values of the stresses to obtain improved accuracy. Therefore, localized mesh refinement is to advise. This can be realized by a very narrow strip around the considered point. The breadth of such strips should be chosen to be so small that across the transverse direction the jump of the values of stresses within this strip from one nodal line to another is negligibly small. The advantage of the finite strip method

Effective breadth and effective width of stiffened plates 269

in this case is, unlike usual two-dimensional finite element models, that there is no obligation of additional transition elements around the area of interest. More accurate stress values at a certain point can be achieved by setting a localized narrow strip covering this point, so that no further modification of the stress is needed. It is also observed that this stress value is relatively insensitive to the size of neighbouring strips.

Beams with three kinds of cross sections are studied: flanges with a single web and double web or box girders with closed cross sections. For the first two cases under uniform loading, the computed results are in good agreement with analytical results, as illustrated in Figs 2 and 3. The effective breadth is calculated according to the maximum longitudinal stress in the web and in the plate, respectively. The Error o f Beam Model gives an indication of the error caused by neglecting the lateral contraction in the web due to a beam model which was evaluated by a two- dimensional plate model by

- - y O ' W x / O ' W a x .

It can be seen that more satisfactory correlation between analytical and FSM results exists for flanges with double web than with single web. In Fig. 4 the effective breadth of quadratic closed cross sections with various L/B is given. Here L denotes the structure length.

For box girders under uniform loading, Figs 5 and 6 show the influence of the geometry of the cross section on the effective breadth for LIB = 10 and 5, respectively. In this case, not only the parameter of section,/~, but also the aspect ratio LIB contributes to the variation of bef. In the web of a quadratic cross section with L/B = 2 the lateral stresses may be of the same order of magnitude as the longitudinal stresses. This can be observed

. - o

o o _

s"

FfJN ( P t . d L T I t ) " ' - - r s m ( W l m )

" ' " ' , . . . . l l I O l l t O ~ BIIUkK l l IODllg t -

0 . 0 2 . 0 4 . 0 6 . 0 8 . 0 1 0 . 0

T. /B

Fig. 2. Effective breadth (single web).

2 7 0 X. Wang, F.G. RammerstorJer

t 4 - t

o

o

o - ' " ' - - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- - JqaNIA.]L,X'W Z CAX., ] . . . . F 81m ( IPX..~'I"E ) . . . . F B M ( W R B ) . . . . . . l l ~ R O l ~ . O F Blgz~qt M O D t t L

T . 0 2 . 0 4 . 0 6 . 0 8 . 0 1 0 . 0

F i g . 3 . Effective breadth (double web).

o o

ili o o

o

l i l t

/ " - . ] - - - I B r F B C ' L ' Z V l l i l l R . E L D ~ H ( N U l l ) [ " . . . . [ . . . . . . l ~ t l a o R O F ! ! ! 1 ~ N O D B L I

o . o o 2'.oo 4 . 0 0 g . o o 8".oo xo r . / R

Fig. 4. Effective breadth (quadratic closed cross section).

particularly for large h/B values as well as for small L/B, where the structures have reasonably large resistance to bending.

The effective breadth for concentrated loading is considered to have practical applications in special circumstances and has been investigated with effort. In Ref. 1, effective breadths, be'~, related to the stress in the web are available, increasing significantly as /~ varies from 0.01 to 1.0. It is known that the concentrated load results in a stress singularity. In the finite strip stress analysis awa~ is increasing and accordingly be'~ is decreas- ing if more Fourier terms are considered. In practice really concentrated loads hardly occur, and can be only understood as the extreme case when

0

¢1o

o

. 4

o

0

o

O ,


, I , I , I , I ,

"',..

"--.,

.... EFFECTIVE BItEADTH ( PLATE )

..... EFFECTIVE BREADTH (WEE) .... ERROR OF BEAM MODEL

' I I I I 11.0 9.0 7.0 5.0 3.0 1.0 3.0 5.0 7.0 9.0

< B/h h/B

Fig. 5. Effective breadth (closed cross section, L/B = 10).

271

I

I 11.0

0

OD o-

0

0 m

0

0

0

0

, I , I ~ I , I ,

..................................................................

EFFECTIVE BREADTH(PLATE) j XFFBCTIVE ~ ( W R B ) IL~ROR OF BEAM MODEL

" " ' . L

" ' " ' - - . . . . . . . . . . . . . . . . . . . . .

r • I I I I

11.0 ,.0 s.o 3.0 1.0 3.0 s.o 7.0

Fig . 6. Effective breadth (closed cross section, LIB = 5).

9.0 11.0

loads act on a tiny patch with vanishing area. It is therefore suggested that a patch loading be introduced over a small portion o f the plate length instead o f a concentrated point loading.

In order to show that very large or very small values of fl seem to be unrealistic for beam flanges let us consider the single web case. For tw = t from eqn (13c) the ratio o f h/b should be as large as 4ft. On the other


hand, for a reasonable value of b/t satisfying the requirements of typical analyses h/t = 4flb/t. Very small values of fl would lead to a similar quantity of h and t which has obviously no practical use. For too large values of fl the structures would have relatively large bending stiffness and would mainly lead to compression of the web. Both cases will not be considered for bet.

Figure 7 shows results for a section parameter fl = 0.1, the distributed lateral load acts at the mid-length, spreading over 10% of the plate length. It can be seen that the error of the beam model of the web increases if L / B decreases. Figure 8 illustrates for L / B = 1.0 the growth of the error of the beam model if the loads are more concentrated at a narrow portion of the plate length.

In Fig. 9 the effective breadths for L I B = 10.0 but with various heights of the web h, i.e. various fl, are given. The portion of the beam at which the distributed load acts is 10% of the plate length. For increasing height of the web, h, i.e. increasing fl, the error of the beam modes becomes larger, but bef has such small changes that it can be considered as being constant with respect to ft.

It is known that a thin steel plate is very flexible when it carries loads which act in the direction of its normal, but it is extremely stiff when the loads are applied within its plane. This is the reason why stiffened plates are designed to carry lateral bending loads. If the bending stiffness of the web is very large, the behaviour of the web is not that of pure bending, but is combined with the effect of compression in its plane. Therefore, the lateral stress in the web becomes relevant, and a beam treatment of the web is not correct any more. The locality of patch loads causes localized compression of the web in addition to bending. For actual concentrated loads the peak of the lateral stress distribution in the web may lead to invalidity of the beam model of the web.

o". o

i i i I

0 . 0 2", | ,.o ,.'0 o'0 +0.o /.]B

Fig. 7. Effective breadth vs L/B, lJ = O.I.


l I I I

0 . 0 i i i J

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

Fig. 8. Effective breadth vs the length of distributed load.

m

0"

o.

I I I I

I ~ I E F F R C T X V B B R B a ~ J ~ A ~ H ( P L , / ~ T I ) m F F B C ' ~ ' r V B B R F d q D ' ~ I ( W B B ) B R . R O R O F B E A M M O D R L

0 . 0 d i • I i

. 2 0 . 4 0 ° 6 0 . 8 l . O

h/H

Fig. 9. Effective breadth vs height of the web.

. /7/

For accurate analysis, a plate model for the web is necessary. If the structure has large bending stiffness, i.e. at high h/B values or at low L/B values, and for concentrated bending loads, the complex stress distribu- tions make numerical analysis necessary.

3 EFFECTIVE WIDTH OF STIFFENED PLATES IN POSTBUCKLING ANALYSIS

Plate structures which have large width-to-thickness ratios are likely to buckle under in-plane compressive or shear loading• They may have a stable postbuckling behaviour and can carry additional loading. This postbuckling strength can be utilized in structural design. For example, if the design of cold-formed steel sections permits local buckling of their component plates, the determination of the ultimate strength and sub- ultimate stiffness is based upon the effective width concept. '3 16 For stif-


fened plate configurations which show local buckling before global buckling, this concept allows the calculation of the interaction of this local buckling with the subsequent global one or with other kinds of postbuckling failure. Several formulae (most of them empirically derived) exist for an approximate calculation of the load dependent value of the effective width. In this section, a nonlinear finite strip method is presented for the prediction of the postbuckling behaviour of stiffened plates and for the calculation of effective width.

3.1 The concept of effective width and its application

It is known that locally buckled stiffened plates under constant compressive loads in the direction of the stiffeners show stable postbuckling behaviour. After buckling the longitudinal stress distribution is no longer uniform, and an increased portion of the compressive load is carried by the material close to the stiffeners. The phenomenon that plates with supported edges can accept additional load after buckling was discovered in the late 1920s through experimental studies made in association with the structural design of airplanes. The experimental results were inter- preted by von Kfirmfin with a simple concept - - the effective width - - which allows the description of sections of stiffened plates in which, for design purposes, the non-uniform stresses over the actual width of the plate are considered as uniform over an effective width as a matter of convenience.

Let us consider one periodic section of a perfect stiffened plate subjected to a uniformly distributed in-plane compressive load P. For P ~< Pcr the axial normal stress is uniform across the plate width, with Pcr being the load at which local buckling appears.

For P > Per the stress near the stiffeners is larger than that apart from them, as illustrated in Fig. 10. For design purposes it is convenient to suppose that the load P is carried by the stiffeners and the effective width of the plate over which the stress is considered to be uniform:

\d't l

The average stress aH in the plate is determined by:

(17)

'f = a~:~, dx. (18) O'H b 0

For the postcritical range Marguerre 9 suggested that the maximum stress O'L near the stiffeners and the average stress O-H could be described by the following equation


6 Pei-iodkngth

Fig. 10. Definition of effective width.

(19)

with ~~~ being the buckling stress of the plate between the stiffeners, i.e. a plate with width b. In a modified form one linds

(20)

From eqn (19) we have

therefore,

ccr =-

GL' (19a)

Substituting eqn (17) into eqn (19a), we have the Marguerre 1 effective width formula:9

b,[( = b 3 2 for qL/ccr > 7. $

(21a)

Similarly, the Marguerre 2 formula can be derived from eqn (20):”

beCf = b(O431~+0.19) for 1 < cL/ccr < 7. (2lb)


Much effort has been devoted to giving approximate expressions for ~rn. Some well-known formulae are

von K~irmfin effective width formula: 17

ben" = b ~ k r ,°'c---£" (21 C)

Stowell formula: 9

beffZ b (0.56 ~5~ +0-44) ; (21d)

formula of Koiter: 17

beff~-b 1-2-0.65 +0"45\o'L./ / --\aL,/ " (21e)

Summary of Braunschweig University:16

beff = 0"9b~3/~c~ but (bell < b). (21f) V ~rL

The effective width beff can be derived in an iterative way by these formulae. The procedure may be demonstrated by Fig. 11.

Figure 12 shows results of these formulae for the same particular problem considered: the stiffened plate has length l = 1000mm, breadth b = 600mm and thickness t = 13mm with a stiffener placed at the midwidth of the plate with height h = 180 mm. Most of these formulae are based on experimental results. They are applicable for approximating the effective width in a large range of problems and have the advantage of simplicity. For a precise investigation of the postbuckling behaviour, the use of numerical methods - - in the present case the finite strip method - - is necessary.

3.2 The incremental successive iterative approach for FS analysis

The linearized incremental equilibrium equation of a single strip derived by the principle of virtual work can be used to form the global equilibrium equation by assembling the stiffness matrices and the external as well as internal vectors of the discretized system to obtain ~s

([K0] + [tKg] + ['Ku]) {AU} = {' ~'R} - { 'F}. (22)

Newton-Raphson methods can be applied for improving the solution for {AU} until after n iterations the out-of-balance load { '+ 1R- t+lF"}, vanishes up to an acceptable degree. Thus, the nonlinear response of the


' = i + 1

I p . = a ~ C A L + bt) ~ ~1

yes I undercritical 1 end

,~L(O) = o~. I i = 0

-I

I ond I

O" I r

Fig. 11. Flow diagram for calculation of the effective width.

b

I , I , I , I h

I ' : ~ , " . . ~ I ~ w t o m m n 2

I I I I

2 . 0 4 . 0 6 . 0 8 . 0 1 0 0

Fig. 12. Comparison of results of some classical effective width formulae.

278 X. Wang, F.G. Rammerstor]'er

structure can be determined by an incremental iterative procedure, in which the intermediate results after i iterations ' ~ u ; converge for suffi- ciently large i = n to t+ lun = t+ lu.

Because of coupling terms in ['Kg] and [tKu], for nonlinear problems, the advantage of uncoupling of Fourier terms disappears even for simply supported strips. Let [tKnl] = [tKg] + [tKu], the global incremental equation (22) can be written in detailed form as,

[K0]II "'- 0 "'" 0 )

0 '-. [Ko]jj "-. 0 +

0 . . . o " [ K o I M M

[tKnllM 1 . ." [tKnl]M j . . . [tKnl]M M { A U } M /

{'+'R}~/ { 'F I l l

where [K0]jj, ['K~,]jj, {AU}j, {~+'R}j, {tF}j are submatrices and subvec- tots, respectively, for the whole structure corresponding to the j th Fourier term. [tK,,t]jk (j :/: k) shows the coupling property of the nonlinear stiffness matrices.

[tKnt] and {tF} are functions of stresses and strains in the strips after step t, which are expressed by the vector of displacement coefficients {tU}, with

{'u}* = ({'u},*, . . . , {'u}f, . . . , {'u}*~).

After solving eqn (22a) { '+ IU} is calculated in a first-order approx- imation by adding {AU} to {'U}.

{ /+ ' v ' }T=({ ' v} , ~+{ae} ,* , . . . , { ' v I f + { A v } 7 , . . . ,

{'v}~, + {zxe}~,).

After equilibrium iterations the improved solution {'+IU} may be used for calculating ['+lK~t] and {'+ 1F}.


A modified incremental successive iterative approach is developed here: Instead of eqn (22a), an approximative decoupled equilibrium equation is derived by neglecting [~Knt]jk (j ¢ k)

([K0]~ + ['K,,lljj) {AU}j --- {t+ 'R}j - {'F}j. (22b)

We then start with the equilibrium equations corresponding to the first term of Fourier series

~[K01,, + ['K,,],,) { A U } , = {' ~' R}, - { 'F} ,

to obtain the incremental sub-displacement-vector {AU}~. An updated displacement vector {'+IU]} is used for calculating irK, t]22 and {tF}_,, with

{'+'ul}T- - {{'u}~ +{Au}, ~, ( ' u } ~ , . , {'u}[,..,{'v}~} Analogously for the calculation of [tK, t]jj and {tF}j, the updated displacement vector {t+lU)_l} is used in order to utilize the sub-incremental- displacement vectors {AUI} . . . . {AUj_ i} from foregoing j - 1 solutions at step t + 1, i.e. a step which starts at t towards t + 1

{'+'u! ,}T={{'u},~+{/xu}T, { 'u}[_,+{Au}~ ( ' U . ,{'u}~}.

J . i " ' "

{tF}i is exactly calculated. The additional error introduced by neglecting ['K,l]jk 0 ¢ k) is eliminated by the equilibrium iterations. Improved convergence can be achieved by using the updated displacement vector.

The advantage of taking into account this updated displacement vector in combination with the decoupled equation (22b) lies in the fact that results of comparable accuracy are achieved by a similar number of iterations as required for solving the fully coupled problem (eqn (22a)), however with much less computational effort.

3.3 Computation of the effective width by the nonlinear FS method

A stability analysis must be carried out to investigate the buckling loads and buckling modes. For the analysis of stable postcritical behaviour, a small load imperfection is introduced. The load imperfection is chosen to produce very small prebuckling deformations which are similar to the local buckling mode. The ratios between the maximum deflection Wc of the plate due to these initial loads and the plate thickness t are generally chosen to have the value wc/t = 0.05, if not stated otherwise. The axial stresses resulting from axial loading in the postbuckling range are computed at the midlength of the plate at each nodal line and integrated


over the plate breadth. This integral is divided by the axial stress in the plate at the plate-stiffener intersection to gain the effective width.

The purpose of calculating the effective width is twofold: the maximum axial stress at the stiffeners evaluated by the effective width gives an indication if the yield stress ov is reached, and the effective cross section, including stiffeners and plate portion with effective width, is used to calculate the stress aE at which the panel will buckle as an Euler column, whatever appears earlier. In this analysis, plates are considered which have either single stiffeners at midwidth of the plates or symmetric double stiffeners located at opposite sides of the plate at its midwidth. The loaded ends are simply supported, longitudinal edges are free. Panels with length 1-- 1000mm and width b- - -600mm are considered. The numerical solutions by FSM are compared with Stowell's effective width formula (eqn. (21d)), because the conditions for which eqn (21d) is thought to hold are much closer to the problems considered here. The critical loads required in eqn (21d) could be estimated analytically to lie between the critical loads obtained under the assumptions that the edges of the partial plates at the stiffeners are simply supported or clamped, respectively, or more precisely by a procedure described in Ref. 9. In this analysis, O'er, necessary for calculating the effective width by eqn (21d), is derived by the finite strip eigenvalue analysis using the single term approach.19

For plates with single stiffeners the neutral axis of the original cross section is shifted to the neutral axis of the effective cross section after local buckling, and the panels are no longer centrically loaded, as depicted in Fig. 13. In this case, beff is also calculated by the above mentioned procedure. In practice, for the effective width in the postbuckling range, the bending moment appears inevitably due to geometric imperfections, load imperfections or the transition of the neutral axis of the original cross section after buckling. In this respect it does not make sense to distinguish between effective width and effective breadth, regardless of which part of beff is due to effective breadth. Figure 14 shows befl for plates with a single stiffener H - - 180 mm, for which there is an additional bending moment due to the shift of the neutral axis, and beff for the case of double stiffeners of height h = HI2. H and h stand for stiffener height for the cases of single stiffeners and double symmetrical stiffeners, respectively (see Fig. 14). The calculations are carried out until the maximum stress reaches the yield stress ov, and trv/E is taken to be 5-334 x 10 -3, which is typical for steel. It is noted that even for P < Per, beff is a bit smaller than b, as a result of the introduced imperfections. Reducing the magnitude of the imperfections, the difference between b~fr and b vanishes even near P/Pcr-- 1, as illustrated in Fig. 15. These interactions of bending and compression, as illustrated in Figs 14 and 15, can be considered to be small. This is in


I ~ bb~ H ' : ~

./" ...:::i

- e ~ ---: ~-" . - - . , .x , . .o ,o , , , . . ,~- - - . " Nce~al iltis of effective cr~-scclion

A x i a l load

Fig. 13. Shift of neutral axis after local buckling.

t * h

b

• . . L 1 i I

b ~ . . [0 - S~W~T.T. - '"....~\ ] ' ~ - SINGLE STIFFRNIR

' ' I 1 I 1 0.0 1.0 2.0 3.0 4.0

P / P ,,,

Fig. 14. Effective widths vs applied loads.


I , I , 1 A I ~ I

. . . . . . • ---:-"-'---.'.-:-.:7 7." 7.'.- Z-." ~.- . . . . . . Q b

"k,

II FsM w /t=O.O01

I I I I I

0.0 0.4 0.8 1.2 1.6

~/¢P e,

Fig. 15. Ef fec t ive w i d t h s vs a p p l i e d l o a d s ( s ingle s t i f fener , H = 7 0 m m , t = 1 0 m m ) .

correlation with the conclusion made from the test results by Winter. 2° In Ref. 21 Davids & Hancock investigated the effective section of a simply supported I-beam in the post-buckling range. They found satisfactory correlation between numerical results and Winter's effective width formula. More recently, Usami 22 presented a set of new effective width formulae for locally buckled plates in combined compression and bending derived from numerical study, which are expressed as functions of the magnitudes of initial out-of-flatness and compressive residual stresses.

In Figs 15 and 16 structures are considered which collapse by global buckling, whereas in Fig. 17 a plate is represented for which the yield stress Ov is reached before global buckling appears. If the stiffness of the stiffeners is just above the minimum value required for the appearance of local buckling prior to global buckling, collapse typically appears by global buckling. If the stiffness of the stiffeners is much higher, as it is for typical metals, it is more likely that the yield stress is reached before global buckling takes place.

In the example shown in Fig. 18, plates with symmetric stiffeners are considered. The stiffener height h = 50 mm is sufficient to prevent global buckling prior to local buckling, and the postbuckling strength is very limited. For h not larger than 70 mm, collapse is indicated by global buckling in both methods (FSM and Stowell), but there is a large discrepancy between the results obtained by FSM and Stowell's approach. The FSM results appear to be more justified. The discrepancy may be partially caused by the fact that the


, I , I t I ~ I

b , a ' ~ _ '

~'.. l ~ T-6 .~.~i,, . i o" ~ ~ " [ -O- T - , F s . .J

! I o•o 1.° 2'.o Lo ,• p/per

Fig. 16. Effective widths vs applied loads (single stiffener, H = 70 ram, t = 6 mm).

283

b

, I i I ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . *

T~I3 FSM

, , ,. . 0 • 4 .8 1 2

P I P ~,

Fig. 17. Effective widths vs applied loads (single stiffener, H = 70 mm, t :- 13 mm).

global buckling load is approximated by the Euler buckling load of the effective cross section with a uniform stress distribution over the effective width• If yielding takes place before global buckling, i.e. h > 80 mm in the considered example, the Stowell formula gives reasonably good estimates.

4 CONCLUSIONS

A finite strip method has been developed for the prediction of the linear and nonlinear response of stiffened plates• The two design parameters - - effective breadth of beam flanges under bending and effective width of postcritically loaded stiffened plates --- are investigated• The validity of


o. .... ~g:

~ ~ " "

Euler Bucklin 0

"~® ~ . . . . . . . . . y , ~> Y i e l d i n g

6 0 . 0 8 0 . 0 1 0 0 . 0

h

Fig. 18. Failure loads vs heights of the stiffeners (double stiffeners, t = 13 mm).

the developed algorithm is justified by the comparison of the effective breadth parameters derived by FS formulations with analytical results and by the comparison of the numerically calculated effective width parameters with results of existing empirical formulae.

For effective breadths of beam flanges under uniform bending, the correlation with analytical results is satisfactory. However, for concentrated loading, the idealization of webs by a beam model, as is the case in the analytical analyses, results in relatively large errors, i.e. a two-dimensional plate model for the stiffeners is necessary.

For geometrically nonlinear investigations of the effective width of postbuckled stiffened plates by the FS method, the coupling of Fourier terms in stiffness matrices may lead to an extensive increase in computational effort. An incremental successive iterative approach is developed, which employs the Newton-Raphson procedure to solve the decoupled equations, with the errors from decoupling and linearization being eliminated simultaneously at each iteration step. Comparisons between the numerical results derived for different configurations of stiffened plates and classical estimates are presented with respect to the effective width as well as its use for estimating subsequent global buckling or onset of yielding.

REFERENCES

1. Schade, H. A., The effective breadth of stiffened plating under bending loads. Transactions of the Society o f Naval Architects and Marine Engineers 59 (195 l) 403-420.


2. Cheung, Y. K., Finite Strip Method in Structural Analysis. Pergamon Press, Oxford, 1976.

3. Schade, H. A., The Effective Breath Concept in Ship-structure Design, SNAME Transactions 61, pp. 410-424, 1953.

4. Schade, H. A., Thin-walled Box Girder - - Theoo' and Experiment. Schiff und Hafen, Heft 1/1965.

5. Schnadel, G., Die mittragende Breite in Kastentr~gern und im Doppelboden. Werft, Reederei, Hafen 9(5) (1928) 92-101.

6. Petershagen, H., Beitrag zur Behandlung yon Sonderproblemen bei sch(['[bau- lichen Biegetr(igern. Jahrbuch STG, 1965.

7. Petershagen, H., Datensammlung zur mittragenden Breite auf Biegung beanspruchter Plattenbalken. Schiff und Hafen/Kommandobrficke, Heft 7/ 1979.

8. Metzer, W., Die mittragenden Breite, Dissertation, TH Aachen, 1925. 9. Rammerstorfer, F. G., Leichtbau Repetitorium. Oldenbourg Verlag, Wien,

Mfinchen, 1992. 10. Kfistek, V., Shear lag in box girders. In Plated Structures Stability and

Strength (Edited by R. Narayan), Chapter 6, pp. 165 194. Applied Science Publishers, London, New York, 1983.

11. Yang, J. D., Kelly, D. W. & Isles, J. D., A posteriori pointwise upper bound error estimates in the finite element method. International Journal for Numerical Methods in Engineering 36 (1993) 1279-1298.

12. Stanley, G., Levit, I., Stehlin, B. & Hurlbut, B., Adaptive finite element strategies for shell structures. Proc. 33rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, pp. 317-330, 1992.

13. ECGS - - Technical Committee 8 - - Structural Stability, Technical Working Group 8.3 - - Plated Structures, Design of longitudinal stiffened webs and of stiffened compression flanges, 1990.

14. Rerkshanandana, N., Usami, T. & Karasudhi, P., Ultimate strength of eccentrically loaded steel plates and box sections. Computers and Structures 13 (1981) 467-481.

15. Murray, N. W., Ultimate capacity of stiffened plates in compression. In Plated Structures Stability and Strength (Edited by R. Narayanan), Chapter 5, pp. 135-164. Applied Science Publishers, London, New York, 1983.

16. Murray, N. W., Behaviour and load-capacity at collapse. In Introduction to the Theory of Thin-walled Structures, Chapter 6, pp. 301-378. Oxford University Press, New York, 1985.

17. Brush, D. O. & Almroth, B. O., Buckling of Bars, Plates and Shells, pp. 323 326. McGraw-Hill, New York, 1975.

18. Wang, X., Finite strip formulations for strength, buckling and postbuckling analysis of stiffened plates, Dissertation, Technical University of Vienna, 1994.

19. Wang, X. & Rammerstorfer, F. G., Coupling effects in linear stress and buckling analysis by the finite strip method. Z. angew. Math. Mech. 74(6) (1994) 712-714.

20. Winter, G., Discussion of the effective breadth of stiffened plating under bending loads. Transactions of the Society of Naval Architects and Marine Engineers, 59 (1951) 423-424.


21. Davids, A. J. & Hancock, G. J., Nonlinear elastic response of locally buckled thin-walled beam-columns. Thin-Walled Structures 5 (1987) 211- 226.

22. Usami, T., Effective width of locally buckled plates in compression and bending. Journal of Structural Engineering 119(5) (1993) 1358- 1373.

Determination of Effective Breadth and Effective Width of Stiffened Plates by Finite Strip Analyses

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