Schade_H_A Effective_Breadth_of 1951 TRANS

Prévia do material em texto

The Effective Breadth of 
Plating Under Bending 
Stiffened 
Loads 
BY COMMODORE H E N R Y A. SCHADE, U . S . N . , (RETIRED) , I~/IEMBER 2 
The phrases "effective width," "effective 
breadth," "mit t ragende Breite," have been used 
variously in structural engineering and particu- 
laxly in naval architecture, to describe sections of 
stiffened plating in which, for design purposes, 
stresses are reckoned as uniform as a ma t t e r of 
convenience, in situations where it is known tha t 
the stress distribution across the plate is, in fact, 
not uniform. There are two entirely distinct types 
of loading to which this concept has been applied. 
First, the plate panel 'subjected to a compressive 
load in its own plane parallel to its stiffening 
members (see'Fig. 1) is said to exhibit an "effec- 
t ive" width, or breadth. Here the inference usu- 
ally. is tha t the load produces buckling of the plate 
between stiffeners with consequent non-uniform 
stressing, b u t as a convenience in design the total 
load is thought of as uniformly distributed across 
an "effective" width which is, of course, less than 
the actual width• Thus, it might be said tha t this 
concept of "effectiveness" is a convenience to 
enable the designer to plan his complex assembly 
of plates and stiffeners as if it were a special k i n d - 
of column, or series of columns. This use of 
• [ - ' "ef fec t iveness" is limited clearly to compressive 
} l o a d i n g , since the departure from uniform stress J 
t distribution occurs only because of plate buckL_ j 
• ling. , " 
A second, and entirely different, situation 
occurs when a stiffened plate panel is designed to 
resist lateral loads, which cause the panel to bend 
out of its original plane. Under these cireum- 
stances the plate behaves as the flange of a beam, 
x Paper presented at annual meet ing of The Society of N a v a l 
Architects and Mar ine Engineers in New York, November 15, 1951. 
i Director of Engineer ing Research and Professor of Mechanical 
Engineering, Univers i ty of California, Berkeley, Cal. 
• Professor Schade was born in St. Paul , h l inn . , on December 3, 
1900. G [ a d u a t i n g from the Uni ted Sta tes N a v a l Academy, class 
of 1923, he served at sea as an Ens ign of the l ine and later became 
a N a v a l Constructor , and st i l l la ter an Engineer ing D u t y Officer. 
He progressed through the ranks, re t i r ing on February 1, 1949, from 
the N a v y ~vith the rank of Commodore. 
He received his Maste r of Science degree from the Massachuse t t s 
I n s t i t u t e of T e c h n o l o g y in June, 1928, and t h e degree of Doktor- 
Ingenieoi- from t h e Technische Hochschule in Char lo t t caburg , 
G e r m a n y . 
At present he is Director of Engineer ing Research and Professor 
of M e c h a n i c a l Engineer ing a t t he ,Un ive r s i t y of California. 
but the distribution of stress across the plate is 
again not uniform (see Fig. 1). The plate is 
loaded only by virtue of the transmission of shear 
-; ? - , 7 
~ r i ~ ~ inH (%) 
o/ 
EF?ECTIVE BRFADTH ( A ) 
FiG. i 
through the plate from the web of the stiffener, 
and therefore the direct stress diminshes as dis- 
tance from the web increases. Here, the "effec- 
t ive" par t of the plate is reckoned as tha t par t 
which, if computed as uniformly stressed, would 
be compatible with the actual flexure of the 
assembly. Then this concept of effectiveness is 
one which enables the designer to compute the 
behavior of the assembly under bending loads by 
use of simple beam theory, and clearly it is applica- 
403 
404 E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 
ble equally t o plates stressed either in compression 
or tension. 
Failure to distinguish between these two cases 
leads to design practices which are difficult to 
rationalize, and confusion has been encouraged by 
/ - - i f - 
w, = 0.85t %J----~ 
if'max 
in accordance with which for E --- 30 X 106 and 
a ~ = 30 X 103 
the use of common terminology to describe either w, ~ 27t 
situation. In this discussion, in order to avoid de- f 
scriptive repetition, the term "effective w i d t h " ~ I n contrast, for any given structure and type o f - ~ 
will be used to mean effectiveness in the first s i tua - ] load ing , effective breadth is constant so long as all 
tion; i.e., instability under compressive stressing; (,. stresses remain in the ~ r a n g e . 
n the " " " " " " " " " a d term effective breadth wdl be hmlted to The mat ter of effective width has been dealt 
effectiveness of plate as a component of a beam, 
where shear transmission, not instability, produces 
non-uniformity. This choice of terminology is 
made for no better reason than tha t the words 
"beam" and "breadth" both begin with the second 
letter of the alphabet. 
Historically, the confusion seems to begin with 
Pietzker [1] 3 who, evidently concerned with 
buckled deck and shell plating, recommended the 
use of for ty plate thicknesses (40t) as the limiting 
effective width and then, a few pages later, applies 
the same rule to plating in tension without ade- 
quate explanation. Hovgaard [2] seems to follow 
this curious pattern also. More recently, Murray 
[9] discusses the mat ter of instability (i.e., of 
effective width) but combines the discussion with 
an analytical procedure which is valid only for 
effective breadth. 
A survey of important ship design agencies 
indicates tha t in most cases an effective width 
criterion is used exclusively, even though the 
loading may be one to produce bending, not insta- 
bility. This is implied when design criteria are 
based on thickness (such as the 60t rule), since 
effective width of plate is a linear function of plate 
thickness, but effective breadth is, theoretically at 
least, entirely independent of plate thickness. 
Airframe structural designers seem to recognize the 
distinction between the two cases more clearly 
than do ship designers, even though effective 
breadth (i.e., the bending situation) is compara- 
tively less prominent in the airframe than in the 
ship structure. 
For any given structure, effective width varies 
with the load placed on the structure after buck- 
ling has begun, and effective width criteria ex- 
pressed simply as a multiple of plate thickness 
imply a certain load or a certain stress; usually, 
they give the effective width at the load which 
produces a maximum stress in the assembly equal 
to the yield stress. For example, a design equa- 
tion often used for effective width is the K~-mAn- 
Sechler formula, 
s N 'umbers in b racke t s ind ica te references l is ted a t the end of th is 
paper . 
with exhaustively, both analytically and experi- 
mentally, and the resulting design formulations 
are understood broadly and are easy to apply. 
This may be the reason that their use has been 
somewhat overextended in naval architecture. 
For example, to use 60t as the effective dimension 
of a band of plating operating with a stiffener on a 
bulkhead, which is a tank boundary, is a very 
simple design procedure; but if there are no co- 
planar compressive loads, and if the plate is on 
the tension side in bending, the procedure has no 
meaning other than that of extrapolation from 
successful past practice. 
Much less at tention has been paid to the effec- 
tive breadth question in the literature, and some 
of the work that has been done is not readily 
accessible, or involves mathematical procedures 
much too cumbersome for practical design use. 
Fur ther development in this paper will be limited 
therefore to the effective breadth question; tha t 
is, to the question of t h e s t r e s s distribution in 
plating which is acting as the flange of a beam 
being bent by a loading system normal to the 
plate. 
In Appendix 1 an analytical t rea tment of this 
situation is given, and it is shown that the effectivebreadth may be expressed by the general equa- 
tion 
)'.2 Kn 
E b 
- K + ~ 
X/b . . . . 
E K,, _ X~ 
~ + ~ 
(14b) 
The analysis (and equation 14b) shows t h a t 
effective breadth depends upon: 
(1) The boundary conditions along the sides of . 
the flange plate for which the effective breadth is 
required (see Fig. 3 for examples); this effect is 
contained in the "boundary function" 7%/b, ex- 
pressed by equation (10) and plotted in Figs. 6, 7, 
and 8. 
(2) The form of the .bending moment curve; 
this effect is contained in the "load function" Kn, 
E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 405 
values for which for conventional cases are given 
in Fig. 5. 
(3) The geometrical properties of the section 
composed of web(s) and flange(s); this effect is 
contained in the "section function" /3, values of 
which are given by equations (15a) and (15b). 
Based upon equation (14b), the ratio of effective 
breadth to actual breadth at the center of the 
stiffener (k/b) is plotted against the length-breadth 
ratio (eL~B) in Figs. 9, 10, and 11. In these 
figures L is the actual length of the assembly and 
cL is thd length between points of zero bending 
moment . Curves for symmetrical concentrated 
loading are given for three values of /3, and a 
single curve for uniform l oad i ng . .Mt hough coin- 
putations were made for uniform loading for the 
same three values of /3, the results differed from 
each other by less than I per cent, and in most 
eases were identical within slide rule accuracy. 
Consequently, only a mean value for uniform load- 
ing is plotted. 
~-~ This means tha t for uniform loading, the effective 
] breadth is independent of the geometry of the section; 
~..i .e., independent of plate thickness, web thickness, and web depth. 
The curves for concentrated loading are interest- 
ing, and may have practical application in special 
circumstances. However, an actual concentrated 
load in practice occurs very rarely, if a t all. All 
loads (including supports) are subject to some dis- 
tr ibution by connecting structure. Further, it 
must be noted tha t the effective breadths com- 
puted and plotted are at the points of max imum 
bending moment ; i.e., at the concentrated load. 
Effective breadth varies along the length of a 
member, dropping sharply a t the points of load 
application. If, as is the actual case, these loads 
cannot exist as sharp concentrations, but instead 
are, in fact, somewhat spread, or distributed, by 
structure, the reduction in effective breadth at 
such points will be lessened, and" the actual effec- 
tive breadth will approach tha t for a uniform load. 
I t therefore seems probable tha t in most circum- 
stances the design estimate of effective breadth 
should be based on a uniform load rather than 
a concentrated load. In particular, if deflection 
estimates are important , the effective breadth 
which enters into the moment of inertia computa- 
tion should be based on uniform load distribution. 
The use of Figs. 9, 10, and 11 as design curves 
for determining effective breadth in most con- 
ventional situations is very simple, and a few 
examples are given in Appendix 3. Special situa- 
tions which may require the use of equation (14b) 
can be handled most easily by using Figs. 6, 7, and 
8 for values of the boundary function. The load 
function can be determined for most cases from 
Fig. 5, but if the bending moment curve is not ex- 
pressible as an infinite series, harmonic analysis 
m a y be used to get a number of terms sufficient for 
any desired degree of accuracy. 
For very gross estimating, an algebraic approxi- 
mation to the form of the k/b curve may be con- 
venient. For example, curve (a) on Fig. 11, which 
represents uniform loading on a multiple stiffener 
(Case I I I ) configuration, m a y be represented with 
a fair degree of accuracy for values of cL/B ~ 2, 
by 
X i . i 
b i + 2/(cL/BF- 
Since a very large proportion of any ship's 
structure is made up of stiffened plating subjected 
to bending loads, this whole mat te r of the appro- 
priate value of effective breadth for use in design 
is of fundamental importance in the theory of ship 
structure. Available records of experimental 
work are not plentiful. The systematic investiga- 
tion recorded in reference [7] is very complete, bu t 
it deals essentially with flat-bar stiffening, and 
with aluminum rather than steel, so tha t the re- 
sults are not directly applicable to most ship 
situations. A systematic series of experimental 
determinations, using ship materials and ship 
configurations, would be a very valuable adjunct 
to the theoretical analysis. 
A P P E N D I X 1 
THEORIES AND ~/~ETHODS 
The general case is tha t of a rectangular I~late 
subject to bending loads, to which is at tached a 
series of webs, or stiffeners. The webs m a y be 
at tached to two parallel continuous plates, forming 
cellular construction, as in the double bot tom, or 
each web may be at tached to continuous plate on 
one edge and to an independent plate on the other, 
as in bulkhead or deck plating, with T-stiffening 
members; here the bulkhead or deck plate is the 
continuous plate, the flange of the T-bar the inde- 
406 E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 
pendent plate. Or each web may be attached to a 
plate on only one edge, as in plating with flat-bar 
stiffeners, or a T-bar loaded alone as a beam. 
In any case, consideration will be limited to 
those cases of rectangular stiffened plates where 
all stiffeners are identically loaded and equally 
spaced and are themselves identical. This in- 
cludes, of course, the case of a single stiffener. 
The edge of a plate parallel to a stiffener is called a 
"side"; the edge normal to a stiffener an "end." 
The origin of coordinates is placed at the center 
of an end, with the x-axis the axis of symmetry. 
Thus the x-axis lies on a stiffener if the number of 
stiffeners is odd, and half-way between a pair of 
// 
2,e , / 
F I G . 2 
stiffeners if the number is even. The length of 
the plate is L or 2l, and the breadth is B or 2b. 
The plate is treated as a ease of plane stress, 
loaded only by shear s.tresses imposed on it by the 
stiffeners at the lines of connection between 
stiffeners and plate, and by whatever reactions in 
the form of plane normal and shear stresses may 
be imposed on the sides and ends by the assumed 
boundary conditions. Thus the stresses in the 
plate due to its bending (that is, its deviation from 
its original plane) are ignored; they may be 
separately computed and added, but their effect 
on effective breadth, as pointed out by von 
KArm~n [3] and Winter [6], is small. 
I t is assumed further that the loading is applied 
at the stiffeners. This is not unrealistic, since 
loading applied to the plating, such as hydrostatic 
loading, will be transmitted to the stiffeners by the 
plate and can be so reckoned. 
A stress function F is employed, in accordance 
with classic plate theory, related to the stresses, 
strains, and displacements as follows: 
OaF 
°'x ~ ~ y 2 
O2F ~y = ~ - ~ 
b2F 
T 
~)xby 
bu 1 1 (b2F b*F~ 
~ bx E (~r~ - - uo'~) = ~ \ b y ~ - - l~ ~x2] 
by 1 1 (bay baF~ 
bu by 1 2(1 + #) ( i)'F~ 
3' = ~ + bx = G ~" E \ b x b y / (1) 
where 
E is the modulus =30 )< 10 s psi (for steel) 
# is Poisson's ratio ~ 0.3 (for steel) 
E G is the shear modulus 
2(1 + u) 
The harmonic form of stress function is used, 
i.e., 
F,~ = f~ sin ~x (2) 
where 
o r 
nTr 
Fn ~ fn COS CO~ 
Herefn is a function of y, of such form,as to satisfy 
the LaGrange equation 
v 4 F "= 0 
For either form of the x-function, the y-function of 
the form 
fn = (.4. + Cn~y) cosh ~y + (B. + D.t~y) sinh ~y (3) 
meets the requirements. The subscript n is used 
in the foregoing to indicate that the values differ 
for each value of n, which may be a n y integer.For simplicity, the subscript is not used in subse- 
quent text except where necessary for clarity. The 
four constants A, B, C, and D are determined in 
such a way as t ° permit the stresses, strains, or 
displacements along the sides of the plate and on 
the x-axis to satisfy the known physical boundary 
conditions, so far as possible. 
First, since the x-axis is an axis of symmetry, 
there can be no transverse displacement there; 
that is, v = 0 for any value of x. Then along the 
x-axis 
by 
- - ~ 0 bx 
and from equation (1) 
b u = _ 2 ( 1 + u ) baF 
by , E bxby 
b'u 2(1 + t~) b,e 1 / (b ,F b3F '~ 
bxby / E bxaby E \ b y 8 -- p ~--x-~y] 
or 
bSF + (2 + z) b~F by--- 3 ~ = 0 
For the form of F given by either of equation (2) 
this means 
E F F E C T I V E B R E A D T H O F S T I F F E N E D P L A T I N G 407 
CASE I. Single web, flange 
with free sides. 
c=A (wb +sinh wb cosh 
w 2 b 2 + j sinh 2 mb 
D = 
A (~ + c o s h 2 r o b ) 
2 b 2 + J sinh 2 wb 
~b) 
O'y : O, r : O 
v : O 
S 7: 
1 ~ ~web 
flange 
CASE II. 
C=O 
Double web, flange 
bounded by webs. 
A 
wb tanh wb 
_L" flange 
Cry:O / 7 
._y:o__ z, c__0_. "__:" 
J-- X ~ w e b s 
Case III. Multiple webs. 
C = O 
D - - - N 
Atanh w b 
wb- j tanh wb 
63f 
~ (2 + u) J ~y~ -- ~y = 0 Y = 0 
Subs t i t u t i on from equa t ion (3) gives 
, B 1 - - p -- = 0.5382 C j , where j 1 -t-/~ 
Fro. 3 
and this e l iminates one cons t an t from equa t ion 
(3), so t ha t 
f = (A -t- Ca,y) cash ~y -.}- (Cj -t- Dc,,y) sinh ~y (4) 
and the der ivat ives which are necessary in the 
subsequen t deve lopment are 
408 E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 
iv--z--. 
)t 
tw--~ 
= b - - - - - -~ 
*- 2 h 
Fro. 4 
~f =~[(A -b D -t- C~y) sinh ~y -b 
(C(j % 1) -t- D~y)cosh ~y] 
~ ' f _ ~y--i -- ~ 2 [ f _}. 2 ( D cosh ~y -t- C sinh wy)] 
T h e physical conditions which can be satisfied at 
the ends of the plate by the assumed form of stress 
function are next examined. 
If F = f sin wx 
then O-~ = ~s in~ox = 0 
and O-~ = --co~f sin cox = 0 
~f but r = o ~ c o s o ~ x ~ 0 
$ - 0 
and if F =fcoscox 
then O-~ ---- ~ y 2 e ° s ~ x ~ 0 
and O-~ = _ ~ 2 f cos o~x ~ 0 
but r = --~.~---fsino~x = 0 
(5) z ~ 0 
Thus the stress function F cannot satisfy com- 
pletely the condition of a plate with free ends; that 
is, with no stresses there. For this case, the sine 
form gives the closer approach to reality, though 
i t does require shear stresses at the ends. In 
practice, these usually would be supplied by ad- 
jacent structure, since a completely free-edged 
p l a t e is rare in ship structure. The cosine form is 
c l e a r l y compatible with a continuous plate, with 
symmetry about the ends, since direct stresses, but 
not shear stresses, would in fact exist at the ends 
of the span under such conditions. 
With respect to boundary conditions at the 
sides and along the x-axis, three cases of practical 
interest are shown in Fig. 3 and attention will be 
limited to these three. Case I represents a free- 
edged flange (or flanges) with a single web, such as 
is represented by beams, whether rolled or built- 
up, of I, H, or T forms. Case I I represents the 
double web beam with a flange bounded by the 
webs, such as a box beam (open or closed ). Case 
I I I represents multiple web combinations, repre- 
sented by plating with repetitive stiffening, such 
as bulkheads, decks, sides, or by cellular structure 
such as the double bottom. For each of these 
cases the side boundary conditions are shown. By 
writing two condition equations for each case, 
corresponding to the two conditions given (the 
condition that v = 0 when y = 0 has already been 
used to eliminate B) and solving these equations 
simultaneously for C and D, the values for C and 
D shown on the figure result. 
Fig. 4 shows a representative distribution curve 
for longitudinal stress in the flange plate #,. This 
stress reaches its maximum at the web intersec- 
tion, designat~ed a*. The stress in the web is n o t 
necessarily equal to the flange stress, however, be- 
cause lateral contraction is trivial and therefore 
ignored in the web, but not in the flange. The 
condition which must be met is that the strains in 
web and flange must be equal at their intersection. 
If # is used to indicate the inaximum longitudinal 
stress in the web and ~* in the flange, this con- 
dition is expressed as follows: 
b2F b~F-] (6) 
= # * - - ~'o-~* = ~ y 2 - - ~ bx21~=o Cot ~) 
Since the ~* is of opposite sign to 'u* with the 
boundary conditions assumed, this means that 
there is, according to theory, an abrupt increase in 
stress at the web, indicated by the jump in the 
stress curve in Fig. 4. In reality, this jump cannot 
exist as a discontinuity, but a more gradual change 
must take place. This situation, however, has 
given rise to two different definitions of "effective 
breadth," indicated by X and X*, respectively, on 
Fig. 4. 
In both forms, the force in the half-flange per 
uriit flange thickness, represented obviously by 
the area under the ~, curve, is divided by an 
optimum stress to give an "effective breadth"; 
i.e., if effective breadth is designated as X, and 
the force b)/X, 
X fo bO-~dy (7) 
O ' m a x O - m a x 
E F F E C T I V E B R E A D T H O F S T I F F E N E D P L A T I N G 409 
i) Concentrated load P at center, free ends. 
n-1 
"2- 
2 ( - l ) l ~ nIVx 
M----~P L ~, n2 sin---~ 
I~ k _I 
V 7 [P 
t l 
n = 1 , 3, 5 , 7 
1 
At center, K n = 
2) Uniform load p, free ends. 
i sin n~x 
n = I, 3, 5, 7 
n-1 
2 1 
At center, K n = (- i) n3 
P 
t t 
3) Triangular load, free ends. 
n-I 
2 L 2 
n = I, 2, 3, 4 
i sin n~ x 
Maximum moment L X " - - 
At .x = 
n÷l 
L 1 n~ 
~9 K n = (- i) n-- ~ sin--~_~ 
FIG. 5 . - -FORMS OF LOAD F.UNCTION K. 
~ P 
t ? 
o 
If the web s t ress is cons idered to be t he o p t i m u m 
stress, t hen for a n y single va lue of n used in t he 
s tress func t ion 
fob~ay bf7 b 
~. ~yJ o (8) 
6y2 -'.J v=o (or b) 
while if the pZate s t ress is used• as t he o p t i m u m 
ff~,~y ~ l ~ 
X,,* = = by" o 
¢* a~f l 
~y~l z,-o (or b) 
(9) 
These va lues Of effective b r e a d t h for a s ingle 
va lue of n can be cal led " b o u n d a r y func t ions , " 
a n d i t is to be n o t e d t h a t t h e y ar~ i n d e p e n d e n t of 
x; t h a t is, a re c o n s t a n t ove r t he span . T h e y can 
410 E F F E C T I V E B R E A D T H O F S T I F F E N E D P L A T I N G 
4) Concentrated load anywhere, free ends. 
~--~ Z I nS d n'~x M = P L ~-~ sin --K-- sin --K-- 
n = i, 2, 3, 4 
i n~d 
At load, K n = n- ~ sin 2 L 
F L 
t 
5) Cor~entrated loads at quarter points, 
free ends. 
~2 ~ 1 sin~ M= PL V 
n = I, 3 , 5, 7 
sin n ~ x 
L 
1 
At load, ~ = n2 
/ V 
6) Equal moments at both ends. 
M=4 M Z l_.sin nSx 
n & 
n=l, 3 , 5 , 7 - - - 
At center, K n 
n - 1 
- (. l) n 
FIG. 5.--FORMS OF LOAD FUNCTION Kn ( C o n t i n u e d ) 
be evaluated by inserting the value o f f f rom equa- 
t ion (4) with the appropr ia te values of the con- 
s tants C and D. T h e y are then funct ions of 
2o~b only, and if we write, for simplicity 
2n~b B 
a = 2~b = - " - ~ = n~ E 
o 
they become, for the three cases shown ih Fig. 3: 
/ M 
Case I. 
~a 4 
b 
sinh ~ + 
~2 
(3 -- a) (1 "4- ~) cosh a -4- ( 1+ /z) 2 ~ + (5 -- 2~ "4- ~z) 
X* 4 sinh a d- 
(3 + ~ cosh ~ -t- (1 + tt)-~ + (5 -- g) 
E F F E C T I V E B R E A D T H O F S T I F F E N E D P L A T I N G 
7) Central concentrated load, fixed ends. 
l Z M=----~PL 
n = I, 3, 5, 7--- 
l co s n ~ x 
n 2 
1 At center, or ends, K n = n-~ 
I 
P 
411 
Pt 
8) Uniform load, fixed ends. 
1 L Z 1 n]~x P T cos !IIiIIt t! 
n = I, ,2, 3, 4 
(n+l) 1 
At center ~= (- i) n- ~ 
At ends 1 
n 
pU 
°-2'L--1 7.f--J 24 
! / Nl- 
L t J 
i - ~1 
FIG. 5.--FORMS OF LOAD FUNCTIONK. (Continued) 
pL 
12 
Case I I . 
~, X* 1 sinh ~ -t- a 
b b a cosh a + 1 
Case I I I . 
~. 4 cosh a -- 1 
b- = ~ (3 - ~)(1 + u ) s inha -- (1 + ~ ) ~ a 
X.* 4 cosh a -- 1 
-b- = ~ (3 + #) sinh a -- (1 + #)a (10) 
T h e l imi t ing va lues of these b o u n d a r y func- 
t ions are t a b u l a t e d on page 412. 
These func t ions are p lo t t ed in Figs . 6, 7, and 8. 
I t will be no ted t h a t the two forms given for Case I 
and Case I I I do n o t differ g r ea t l y f rom each other . 
T h e fac t t h a t X,/b for Case I I I has a l imi t ing va lue 
g rea te r t h a n u n i t y is in te res t ing ; th is i s d u e , of 
course, to t he effect of the t r ansve r se r e s t r a in t ; 
i.e., Po isson ' s effect. T h e fac t t h a t t he two forms 
are iden t ica l for Case I I resul ts f rom lack of 
t r ansve r se s tress ~ in the p la te a t the web in te r - 
section. 
T h e whole purpose of the effective b r e a d t h con- 
cep t is to enable the des igner to use s imple b e a m 
theory . Accord ing ly , if the app l i ed be nd ing 
m o m e n t M per r epea t i ng sect ion should have a 
s imple ha rmon ic form, as 
M = M . sin eox (11) 
where M,~ is a func t ion of n, t hen s imple b e a m . 
t h e o r y ind ica tes ~ ~ 4 f ~ : 
M 1~, sin ~x 
. . . . s . S . (12) 
where S . is the sect ion modu lus of the a s sembly of 
flange(s) and web(s) o b t a i n e d b y reckon ing the . 
- "412 ' E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 
LIMITING VALUES 
CASE I 
A 
n 
CASE II 
A 
~t 
n 
CASE III 
L 
n 
1.0 b 
1.0 b 
1.0 b 
b 
_ jr~.d -- LO98b 
I 
h b _ 2 L _ 0 . 1 8 x h L 
( 3 - ~ ) ( X * / ~ ) a 7 r ( 3 - , ~ ) ( 1 , , ~ ) n n 
4 b _ 2 L - 0 .1932 L 
3 ÷ /~ o< 7t (3 . /-c) n n 
b _ 1 L - 0.1592--L 
oC 2~" n n 
(3 -/4)(1 ÷~) ~ 71"(3 -~(i -~) n n 
1.0 b 4 b 2 L L 
- - 0.19.32 
3 +~ O( 71"(3 + ~) n n 
flange(s) as having the half-breadth k.. Accord- 
ing to equation (7), the force in the half-flange per 
unit thickness is then 
X, , = X,.M,, sin wx (13) 
S. 
I t is desirable here to use ~. in equation (13), 
and not ~.*, since the use of equation (12) implies 
tha t 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 of the term; i.e., the momen t of inertia of 
the section divided b y the distance from the neu- 
tral axis. The difference between the two is not 
great, and from the design point of view unim- 
E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 413 
por tan t , but in analyzing stress measurements in 
experimental work it needs to be understood. 
Since it is as easy to develop design criteria based 
on the form X~ as on X~*, the form X~ Will be used 
in the subsequent development. When the value 
of effective breadth so obtained is used to com- 
pute an effective section modulus, which is then 
used in the =simple beam formula, the result will 
be the op t imum stress in the web at the flange 
junction. If, for some special reason, the stress in 
the flange, which will be slightly less, is desired, it 
can be obtained by the use of equation (6). 
The actual bending moment , of course, nor- 
mal ly will not have a simple harmonic form, but 
whatever its form it can be represented by a 
Fourier series of terms like equation (11), which 
can be writ ten 
M = y~ ]fin sin wx (lla) 
which is the equivalent of superposing a number of 
harmonic bending moments. The forces and 
T:~ stresses are linear with respect to M, so tha t they where / _ d b-~ . . . . . . . . . . . ~ . . . . . . . =-_. 
2J/Y" may be regarded as superposed also, and equation 
(12) and equation (13) become 
• ~ Mn sin cox (1-2a ) 
Crmax ~ Sn 
X = ~ sin cox (13a) 
The final "effective breadth" for the actual 
applied bending moment represented by equation 
(1 la) i s then 
2 },,,M~ X ~ sin cox 
x (14) cr=ax ~ M,, sin cox 
& 
This is the general equation for effective breadth. 
The foregoing analysis is exact, within the limits 
of plane stress theory, the validity of the assumed 
boundary conditions, and the convergence of the 
series used. I t is essentially a generalization of tha t 
given by von Kfirm/m [3], Timoshenko [5], and 
Schnadel [4] for specific cases. Application for de- 
sign purposes in the form of equation (14) is 
tedious, principally because of the necessity of 
computing a new section modulus value S~ for 
each term of the series. 
A further simplification can be made. The 
value of M~ sin ~0x is K~ times a constant, where 
K~ is a function of n and x, and the value of the 
constant depends on dimensions and loading. 
The constant cancels from equation (14), which 
may b e w r i t t e n 
Z (}~,,/b )K,, 
X &, (14a.) ~, = X-" 
2_.. (K~/&) 
Values of K~, called the "load function," for 
various 10adings are given in Fig. 5. 
The value of section modulus S, applicable to 
the upper (A 1) flange of the general double-flange 
case, is 
h 12A,A2 4- 4A,~(A, 4- A2) q- A,~ 2 (15) 
S = ~ 2A~ 4-A,~ 
Here Ax and A2 are the total section areas of the 
@rective upper and lower flanges respectively, A~ 
is the total area of the web, and h is one-half the 
depth of the web. (Note: In a box girder, A~ is 
the area of both webs.) 
There are two impor t an t spec ia l forms of S: 
(a) Lower flange identical with upper flange 
(i.e., same thickness t and same boundary con- 
ditions) 
S = 4hbt [b 4- ~ (15a) 
1 h t,o 
6 b t 
(b) Lower flange, with area of A2, so narrow 
tha t it may be regarded as 100 per cent effective; 
this is applicable generally to plating stiffened 
with T's , L's, bulbs, or flat bars (in the lat ter case, . ~ 
A2 = 0) -" C ~ 
\ S = 8hbt (3A' 4-2ht"'~ [~ . ,~15b) 
] 
where . . . . 
1 h & 4A2 4- 2htw 
4 b t 3A~. 4- 2ht,. 
I f S., expressed in the form of (15a) or (15b), is 
inserted in equation (14a), the factor of the t e rm 
in square brackets, being independent of n, cancels 
from numerator and denominator, and equation 
(14a) becomes 
(),n/b)K,~ 
x (}~/b) ~ (14b) 
2 K. (X./b) + 
The effect of/3 will be small, in any case, and for 
design purposes consideration of its limiting 
effects will normally suffice. Note tha t it is con- 
s tant (i.e., the same value for any value of n) while 
~ / b diminishes rapidly with successive values of n. 
As the first limit, consider tha t /3 is so large in 
comparison with X,/b tha t the value of X,/b -I- 
is virtually constant for all values of n. 
Then equation (14b) would become (for /~ -~ ~ ) 
414 E F F E C T I V E B R E A D T H OF S T I F F E N E D PLATING 
X ~-~flk./b)K. 
(14c) 
Equa t ion (14c) would be applicable, then, to an 
assembly with ve ry heavy, deep, closely spaced 
webs. This form, incidentally, results when it is 
assumed t h a t the form of the stress curve, ra ther 
t han t h a t of the m o m e n t curve, is known and is 
expressed as a Four ier series, as was done b y 
Winte r [6] and M u r r a y [9], t hough in this case 
),.* should be used ra ther t han k,. 
Secondly, consider t h a t fl is so small" in com- 
parison w i t h 7~,,/b t h a t it m a y be considered zero. 
Then equat ion (1413) would become (for B --~ 0) 
k ~ K, 
(14d) b ~ K. 
(X,/b) 
Equa t ion (14d) would be applicable to shallow, 
thin, widely spaced webs. 
A P P E N D I X 2 
COMPUTATIONS AND CURVES 
The " b o u n d a r y funct ions" (viz., X,/b and 
k,*/b , for the three cases t reated) have been com- 
puted, and the results given in Tab le 1 and Figs. 6, 
7, and 8. Bo th forms are given in order t h a t com- 
parisons m a y be made with o ther work, if desired, 
a l though only the X,/b form is used here in the 
effective b read th computa tions . 
TABLE 1.--BOUNDARY FUNCTIONS ~ AND ~n* VS. 
Case I Case II Case I I I 
0.2 0.990 0.992 0.993 1.091 0.990 
0.4 0.963 0.968 0.962 1.063 0.969 
0.6 0.920 0.932 0.936 1.016 0.933 
0.8 0.868 0.886 0.903 0.955 0.890 
1.0 0.810 0.833 0.855 0.892 0.844 
1.2 0.748 0.777 0.803 0.827 0.790 
1.4 0.689 0.722 0.748 0.766 0.737~ 
1.6 0.631 0.668 0.697 0.699 0.684 
1.8 0.581 0.618 0.641 0.643 0.635 
2.0 0.534 0.571 0.591 0.591 0.590\ 
2.5 0.437 0.472 0.479 0.488 0.491 
3.0 0.366 0.390 0.392 0.402 0.415 
3.5 0.314 0.340 0.326 0.341 0.356 
4.0 0.275 0.297 0.276 0.296 0.310 
4.5 0.245 0.264 0.239 0.261 0.274 
5.0 0.221 0.238 0.211 0.232 0.246 
6.0 0.186 0.200 0.171 0.195 0.203 
7.0 0.161 0.172 0.145 0.164 0.174 
8.0 0.j42 ~'" 0.151 0.126 0.142 0.152 
9:0 "0. '126 0.134 0.111 0.127 0.135 
10.0 0.114 0.121 0.100 0.114 0.121 
In comput ing effective breadth, eleven terms of 
the series were used in every case, in order to have 
the same basis for all. Computa t ions were made 
for Case I, Case I I , and Case I I I , with three dif- 
ferent values of/3 (i.e., B -~ co, fl = l/e , and ~ --~ 0) 
for each of the following forms of load funct ion: 
( ~ K. = I (n = 1 , 3 , 5 , 7 . . . ) 
K, = ( - - 1 ) ("-1)/21 (n = 1 ,3 ,5 ,7 . . ) n3 * 
K. = ~ (n = 1 , 2 , 3 , 4 . . . I 
( - 1 ) . + ~ ( .=1,2,3,4 ..) 
n n2 • 
This made a total of 36 combinations. 
Within pract ical limits of accuracy for design 
purposes, it was found t h a t the two lat ter forms of 
K , in the foregoing gave results for d is t r ibuted 
loading, fixed ends, wh ich could be d u p l i q ~ _ 
use of the f i r s t two form~ with. the length betwew_~nen 
points of zero bending moments subst i tu ted for ' 
the actual length L : Consequent ly , 
values of ~/b corresponding to the first two forms 
of K , are plot ted. 
Fur ther , it was found tha t with the second form 
of K , , values of ~/b for the three different values 
of /3 were pract ical ly identical, Consequent ly , 
on ly the values corresponding to /~ --- t/6 are 
t abu la ted and plot ted for this form, which repre- 
sents uni form load. 
E F F E C T I V E B R E A D T H OF . S T I F F E N E D P L A T I N G 415 
1.0 
0.8 
0 .6 
0.4 
0.2 
0 
0 
" xx. 
\ 
h_J 
b 
2 ' 3 
FIG. 6 . - - C A S E I. 
4 
S I N G L ~ W B B . 
FOR oc > IO 
"Xrl = 4 I 4 I 1.140 - - = = 
b ( 5 - j k ) ( i , ) o( (2 .7 ) 1.3) o< o< 
.11 
X...gn = 4 4 1.212 
- - I = l 
b 3 + F ¢¢ 3 . 3 o¢ ¢< 
I 
5 6 7 8 
<x = n l l B - - " 
L 
BOUNDARY FUNCTIONS ~,n/b AND hn*/b 
. 
._l 
I0 
1.0 
0 . 8 
0 .6 
0 . 4 
0 . 2 
\ 
~n Xn* 
/ - - - -6- ='-~- 
\ 
\ 
FOR o~ > I 0 
"Xn _ Xn ~ 1.0 
b b o¢ 
0 
0 I 2 5 
F i o . 7 . - - C A S E I I . 
4 
D O U B L E WEBS. 
= B 6 7 8 
o(. n l l - C 
BOUNDARY FUNCTIONS ~,n/b AND hn*/b 
9 I0 
416 E F F E C T I V E B R E A D T H OF S T I F F E N E D PLATING 
1.0 
0 . 8 
0 . 6 
0 . 4 
0 . 2 
1-" 
\ FOR o~ > 
~._~n = 4 I = 
b (3-p,) ( I+/~) o~ 
b 3 + / ~ o¢ 
\ 
~ , Xn" 
I0 
4 I _ = 1.140, 
(2.7)(I.3) o~. <X 
4 -- = 1.21_.__.2_2, 
3..3 O< oc 
/__ X n ~ ~ " ~ ~ ~ 
b ~ ' 
I 2 3 4 5 6 7 8 9 
~ . = n'n" .-.~-B - - . ~ . 
L 
FIG. 8 . - - C A S E I I I . MULTIPLE WEBS. BOUNDARY FUNCTIONS ~n/b AND Xn*/b 
I0 
1.0 
0.8 
0,6 
b 
0.4 
0.2 
o o 
V 
(o ~ ~ : : : : : 1 1 1 ---- 
J 
,~ f / ' . ~ 1 I ' ' ' - 1 1 i ~ 
/ y / / f (a) UNIFORM LOAD 
# / (b) SINE LOAD 
~ / + ' - (c) CENTRAL LOAD,,8----oo 
, ,,.,- (d) . . . . ,8 = 0,167 
, . , . . . . , , - . o 
NOTE: cL IS DISTANCE BETWEEN 
POINTS OF ZERO BENDING 
MOMENT. 
I I 
I 2 3 4 5 6 7 8 9 
c L 
B 
Fio, 9.--CASE I. SXNGLB WEB. EFFECTIVE BREADTH RATIO ),/b FOE TYPICAL LOADS 
I0 
EFFECTIVE BREADTH OF STIFFENED PLATING 4 t 7 
I.O . . . . . . . . . . . . . 
o~ ........ , ~ % ~ i i ~ _ . . i 
0 . 6 , , , . 1 _ ..,.~ ' J~7 ~ - to) UNWORM LOAO 
. . . . ( rb) S i N E L O A D 
b / / ~ t ~ ~ , ~ (C) CENTRAL LOAD, B - - ~ o 
• , , ...... {d) . . . . . .B= 0 ,167 
• iT~ , / " , . , , . ~ - o 
o ........ .,.,I ..... ~-= I 1 1~-1 i 1 ..... I . - - - ~ 
0 I 2 3 4 5 6 7 8 9 I0 
cL 
B:" 
Fi~. I 0 . - - C a s E II . [)OUBL/~ XV/~EB, EFFECT/V'/~ BREADTH l~,.AT[O k/b FOR TYPICAL LOADS 
0 .4 
1 . 0 . . . . . . . . . - - . . . . 
t o ) . ~ , 
"/ 
O . E / 
0 . 2 
I 
0 I 2 3 
Fz~. l l . ~ C A s g i1I. 
/ Q ..... 1.1 / I 1 
1 , ~o, UN,FORM LO, D . . . . { 
( b ) S INE LOAD " / . 
t i ( ~ OENTRAL ~0AD, a-*DO 
- - - ' J ~ " ~ ( d ) . . . . ,8 = 0 . 1 6 7 
{ N O T E : e L IS D I S T A N C E B E T W E E N " 
POINTS OF ZERO BENDING 
MOMENT, 
4 5 6 7 B 9 
cL 
B 
~V~ULTIPLB WEBS. EFFeCTiVE BRI~,~I)TH ]~ATIO A/3 FOR TYPXCAL Lo.~)S 
iO 
418 E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 
< 
< 
(3 
N 
N I 
e4 
,.a 
< 
i 
. . . . . . . ~ 
S £ N d 4 ~ d d ~ 
¢5 
t 
o 
¢ J t . 
¢ $ Q 
~--. 
m2 
A P P E N D I X 3 
E X A M P L E S 
]~XAMPLE 1 
A box-girder, length 10 feet, breadth 2 feet, 
depth 1 foot, web thickness 1~ inch, flange thick- 
ness 1/6 inch, is supported at the ends without 
fixety and subject to a uniform load of q pounds 
per running foot. What is the effective breadth of 
the flanges and the maximum stress? 
L = 10' 
B = 2' b = 1' 
h = 1/6' 
t~ = 1 ~ ( twice ac tua l th ickness , s ince there are t w o 
webs) 
t = ~ " 
L / B = 5 c = 1 c L / B = 5 
From Fig. 10, X/b = 0.94, or the effective 
breadth (total) is 2 X 0.94 = 1.88 feet. 
From equation (15a) 
S = 4 h b t ( ~ + ~ ) 
where 
1 h tw 1 1 2 
. . . . . . . . . . . 0 .167 
6 b t 6 2 1 
( 1 ) ( 1 ) 1 . 1 1 . . 
S = 4 - (1) (0 .94 + 0 .17) = - ~ - tt ~ 
M 12.5q (12 .5 ) (12) q lb per sq ft 
* S S ~ 1.11 
(12.5)q 
(1.11)(12) = 0.937 q psi 
If the same load were concentrated at the center, 
X/b would be approximately 0.76 instead of 0.94, 
for f5 = 0.17. 
0.93 
S = (0.76 + 0 .17) = ~ - ft 3 
so that the section modulus decreases about 16 per 
cent because of the load concentration. 
Application of the conventional 60t rule would 
mean that X/b = 1, since actual flange breadth is 
less than 60 thicknesses; i.e., 
1.17 
S = - ~ - ft 3 
so that conventional design would underestimate 
maximum stress by about 5 per cent for uniform 
load, and by about 21 per cent for concentrated 
load. 
E X A M P L E 2 
Vertical bulkhead stiffeners spaced 3 feet apart 
E F F E C T I V E B R E A D T H O F S T I F F E N E D P L A T I N G 419 
are 10 fee t long be tween suppor t s . B u l k h e a d 
p l a t i ng is 0.38 inch th ick . W h a t is effect ive 
b r e a d th ? 
(a) I f s t iffeners are b raeke t l e s s (i.e., zero 
f ixety) 
(b) I f s t iffeners are b r a c k e t e d top and b o t t o m 
(100 per cen t f ixety) . Assume f l anged-p la te 
s t i ffeners a re 7 't X 4" X 7 ~ 6 P ' for (a), and 6" X 
31/~ " X 5~6" for (b). 
(a) F o r the b racke t less case, the size of t he 
s t i ffener is u n i m p o r t a n t in d e t e r m i n i n g effective 
b r e a d t h . F r o m Fig. 11, us ing the un i fo rm load 
curve as a p p r o p r i a t e for h y d r o s t a t i c loading, and 
c L 10 
en te r ing the curve wi th B - 3 ' the va lue of 
effect ive b r e a d t h is o b t a i n e d a t once as 
b ----" 0.94 
X = 0.94 X ~ = 16.9" 
$ 
(b) F o r the b r a c k e t e d case, . i t is necessary to 
c o m p u t e /5 to ge t t he effect ive b r e a d t h a t t he 
suppor t s , which r ep re sen t concen t r a t ed loads. 
F r o m equa t ion (15b) -. 
1 h t~ 4A2 + 2ht,~ 
4 b t 3A~ +2h t~ 
Here 
h = 3 v 
b = 18" 
t~ = 0.312" 
t = 0.380" 
A*. = 31/~ X 0.312 = 1.094 
1 3 0.312 4(1.094) + 2(3) (0.312) 
0 4 18 0.380 3(1.094) + 2(3) (0.312) 
= 0.0414 
(1) 
n 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
(2) (3) (4) (5) (6) 
kt n X 180 ° 
b x / 2 s in (4) (--1)n + t X 
0.7854 0.965 127.3 +0.7955 +1.0000 
1.571 0.710254.6 --0.9641 .0 .1205 
2.356 0.503 21.9 +0.3730 +0.0370 
3.142 0.380 149.2 +0.5120 --0.0156 
3.927 0.302 276.5 --0.9936 +0.0080 
4.712 0.248 43.8 +0.6921 --0.0046 
5.498 0.215 171.1 +0.1547 +0.0029 
6.283 0.188 298.4 --0.8796 --0.0019 
7.069 0.165 65.7 +0.9114 +0.0013 
7.854 0.146 193.0 --0.2250 --0.0010 
8.639 0.132 320.3 --0.6388 +0.0007 
F o r the cen te r of the stiffener, a p p r o x i m a t e l y 
- - ~ (0.58) ,~ 1.93 
B 
and a t the suppor t s 
1 
= (0.42) -= 1.40 
Then f rom Fig. 11, us ing curve (a) 
at center/~ = 0.72 
and b y in t e rpo la t i on be tween curves (d) a n d (e) 
for 3 = 0.0414 
x at ends ~ = 0.31 
Th i s second va lue m a y be r ega rded as a p p r o p r i a t e 
for use in c o m p u t i n g a m a x i m u m stress, unde r t he 
a s sumpt ion t h a t the s u p p o r t force is a c t u a l l y con- 
c e n t r a t e d . I f (as is p r o b a b l y more real is t ic) i t is 
a ssumed t h a t the s u p p o r t force is d i s t r i bu ted , 
curve (a) m a y be used, resu l t ing in 
at ends /~ = 0.58 
If deflect ion is t he des ign cr i ter ion, the l a t t e r 
va lue would be governing. 
EXAMPLE 3 
Using the genera l formula , c o m p u t e effect ive 
b r e a d t h for a f ree-ended st iffener wi th a t r i a n g u l a r 
load d i s t r ibu t ion , where 
cL 1 
- - = 4.0 and fl = - B 4 
T h e resu l t is o b t a i n e d b y us ing e q u a t i o n (14b), 
w i th t he va lues of k,~/b o b t a i n e d f rom Fig. 8, a n d 
wi th 
1 m r 
K, = (--1) n + l n 3 sin ~ forn = 1 , 2 , 3 , 4 . . . 
(see Fig . 5, i t em 3). The c o m p u t a t i o n in t a b u l a r 
form is g iven on this page. 
x 0.7204 
0.9330 0.7721 
(7) (8) (9) (10) (11) 
(5) X(6) (3) X (7) (3) + 1/4 (8)/(9) (7)/(9) 
+0.7955 +0.7676 1.2150 +0.6317 +0.6547 
+0.1250 +0.0855 0.9600 -~0.0890 +0.1255 
+0.0138 +0.0069 0.7530 + 0 . 0 0 9 1 +0.0183 
--0.0080 --0.0030 0.6300 --0.0047 --0.0126 
--0.0079 --0.0023 0.5520 --0.0041 --0.0143 
--0.0032 --0.0008 0.4980 --0.0016 --0.0064 
+0.0004 +0.0001 0.4650 +0.0002 +0.0008 
+0.0017 +0 . 0003 0. 4380 +0.0006 +0 . 0038 
+0 . 0012 +0 . 0002 0.4150 +0 . 0004 +0.0028 
+0.0002 +0 . 0000 0. 3960 +0.0000 +0.0005 
--0.0004 --0.0001 0.3820 --0.0002 --0.0010 
Totals 0.7204 0.7721 
T h e resu l t is 
Th i s is s o m e w h a t less t h a n the a p p r o x i m a t e 
effect ive b r e a d t h which would be o b t a i n e d d i r ec t l y 
f rom Fig. 11, unde r t he a s s u m p t i o n t h a t t he t r i - 
angu la r load ing is e qu iva l e n t to un i fo rm loading. 
Th i s a s sumpt ion was used in E x a m p l e 2, and 
420 E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 
clearly leads to results which are somewhat too 
favorable. 
The type of computation made in the foregoing 
is representative of the procedure used in com- 
puting each of the points used in constructing the 
curves of Figs. 9, 10, and 11. 
R E F E R E N C E S 
[1] Pietzker, Felix, "Festigkeit der Schiffe," 
Berlin, 1914. .- 
[2] Hovgaard, W., "Structural Design of 
Warships," 1940. 
[3] K~rm~n, Th. v., "Die Mittragende Breite," 
Springer, Berlin, 1924. 
[4] Schnadel, G., "Die Mittragende Breite in 
Kastentr/igeru und im Doppelboden," Werft, 
Reederei, Hafen, 1928. 
[5] Timoshenko, S., "Theory of Elasticity," 
First Edition, 1934, pages 156-161, inclusive. 
[6] Winter, George, "Stress Distribution in 
and Equivalent Width of Flanges of Wide, Thin- 
Wall Steel Beams," NACA 784. 
[7] Hartman, E. C., and Moore, R. L., "Bend- 
ing Tests on Panels of Stiffened Flat Plating," 
Aluminum Research Laboratories, Technical 
Paper No. 4, Aluminum Company of America, 
1941. 
[8] Boyd, G. Murray, "Effective Flange 
Width of Stiffened Plating in Longitudinal Bend- 
ing," Engineering, December, 1946, pages 603- 
604. 
[9] Murray, J. M., "Pietzker 's Effective 
Breadth of Flange Re-examined," Engineering, 
Volume 161, page 364, April 19, 1946. 
[10] Raithel, Wilhelm, "The Determination of 
the Effective Width of Wide-Flanged Beams," 
Technical Report No. 61, Ordnance Research and 
Development Division Suboffice (Rocket), For t 
Bliss, Tex. 
DISCUSSION 
PROFESSOR HENRY C. ADAMS, II, Member: 
Dr. Schade's paper clears the fog tha t has sur- 
rounded the problems of effective breadth and 
effective width, and has presented, for the use of 
the profession, the means of determining the 
former quickly without requiring calculations 
• whose length frightens the usual designer. While 
it is recognized tha t most analyses have to pass 
from the stage of complicated and involved cal- 
culations to simplified ones, it is hoped tha t future 
authors of such papers will t ry and simplify their 
work to some such form as Dr. Schade has done in 
this pap er. 
The validi ty of the results of a paper of this 
kind is strengthened when compared to practice. 
For example, the American Bureau of Shipping 
requires tha t where there are no centefline open- 
ings the required 'area of the strength deck shall 
be increased 5 per cent; i.e., the effective.breadth 
is 95 per cerft. Assuming a 400-foot by 55-foot 
vessel, cL/B = 7.27. Using Fig. 1O (Case II, 
Double Webs), the effective breadth is 97 per 
cent for a uniform load and 96.5 per cent for a 
sine load. The various full-sized tests corroborate 
this. 
An application of the methods of this paper 
has been made to one of the specimens of deck 
girders tested by J. Bruhn and reported in the 
Transactions of the Insti tution of Naval Archi- 
tects in 1905. For this application Specimen 
No. 64, as shown in Fig. 12, was used. This 
specimen failed by the shear of rivets in the 
upper intercostal angle a t a load of 49 tons, 
applied at the center by a hydraulic press. This 
load imposed a bending moment of 1,470 inch- 
tons. 
I t is well known tha t when, in a double flange 
girder, the areas of the flanges are equal, increases 
in the area of one of the flanges cause a slowly 
decreasing increase in the inertia of the section, 
bu t tha t due to the shift in the position of the 
neutral axis the minimum section modulus in- 
creases a t a relatively slower rate. Experiments 
dealing with effective breadths, therefore, should 
E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 421 
TABLE 3 
~ / B = 0.312; X = 15"-~ ~ X / B = 0.50; X = 24"-.~ ~ X / B = 0.82; X = 39.4"-~ 
Load, tons 15 23.47 36.72 t5 23.47 36.72 15 23.47 36.72 
Observed deflection 0. 090" 0.195" 0. 555" 0. 090" 0.195" 0. 555" 0. 090" 0.195" 0. 555" 
Calculated deflection 0.084" 0.131" 0.205" 0.069" 0. 108" 0.169" 0.057" 0. 088" 0. 139" 
Observed deflection 
Calculated deflection } 1.07 1.49 2.70 1.30 1.80 3.29 1.57 2.21 3.99 
Ratio of ratios 1.00 1.39 2.52 1.00 1.38 2.53 1.00 1.41 2.54 
Shear rivets, T/in. 5.71 8.93 13.97 6.18 9.67 15.13 6.37 9.97 15.61 
Tensile stress, T/in. 6.84 1 0 . 7 0 16.74 6.34 9.93 15.53 5.94 9.29 1.4.54 
Yj 
[ 
~3/, I ! ° 
S- ~. Rivers Each FI~J~\ 
I 
FIG. 12 
-,, Io. 3~x3Ex ~ Lugs 4t~x~ I. t" S"l 
,,r,_-7-1 
be based on deflections rather than stresses. 
Three values of X/B were used: 
X/B = 0.50, corresponding to concentrated central load. 
X/B = 0.82, corresponding to uniform load as suggested 
by the author when dealing with deflections. 
X/B = 0.312, corresponding to the time honored 30t. 
The deflections were calculated using the 
foregoing breadths of plating for the inertia of the 
section with the results shown in Table 3. 
From the foregoing it would appear that the 
old 30t is more nearly correct. However, if tha t 
were true, the ratio of the observed deflection for 
the value of X/B should remain constant irrespec- 
tive of the loads, provided of course that the 
stresses were within the proportional limit. As 
is indicated, it is slightly over in the case of the 
36.72 ton load. However the shear stress in the 
rivets in theintercostal angle is of such a value 
that slippage probably has occurred, so that the 
observed deflection is due to some extent to this 
cause. Any projected experiments based on 
deflections should use welded specimens. 
The use of the methods provided in this paper 
will assist designers in complying with the Amer- 
ican Bureau requirements relative to the area 
of deck plating needed to balance properly the 
face area of deck girders (Section II , paragraph 3). 
In this connection, the discusser presumes that 
either Case I or Case I I could be used but would 
like to inquire whether the greater stiffness of 
the vessel's side in comparison with tha t of the 
girder would make the application dangerous 
Dr. Schade's paper has advanced the knowledge 
of the profession on a hitherto disp.uted point, 
and its use should be of great benefit to all. 
Both he and the Society should be congratulated 
upon its presentation. 
MR. JOHN VASTA, Member: Dr. Schade has 
presented a clearly organized discussion of the 
problem concerned with the effective breadth of 
stiffened plating. The two concepts which have 
confused the designer in the past (namely, the 
effective width associated with coplanar forces 
as distinguished from the effective breadth asso- 
ciated with bending loads) have been clearly 
separated. The design charts presented in Figs. 9 
through 11 are simple to follow, and no doubt 
will find useful application in design problems. 
The theory presented in this paper is a distinct 
improvement over the classical one. By intro- 
ducing the "section function" /9 the author 
recognizes the influence that t h e stiffener plays 
in the determination of the effective breadth, and 
for the case of concentrated loading he gives 
various values of ft. However, for the case of 
the uniformly distributed loading the author 
concludes that "the effective breadth is inde- 
pendent of the geometry of the section." This 
is a difficult conclusion to accept. I t would 
appear that, i f ' the geometry of the section is a 
factor to be considered in the case of concentrated 
type of loading, it should be a factor also for the 
distributed loading since the latter may be as- 
sumed as a large number of concentrated loads. 
During the past year the Bureau of Ships has 
pursued an independent theoretical study of 
this problem which will be published shortly. 
In the main, the Bureau's s tudy follows the 
approach of the classical theory. There were 
introduced, however, some significant departures 
which were dictated largely by the recognition 
of the fact that the characteristics of the stiffener 
play an important role in the determination of 
the effective breadth. The problem suggests 
that the effective breadth of stiffened plating 
422. E F F E C T I V E B R E A D T H O F S T I F F E N E D P L A T I N G 
is inf luenced n o t on ly b y the th ickness of t he 
pla t ing, t he spac ing a n d span of the stiffeners, 
b u t also b y the shape f ac to r of the stiffener, as 
well as b y the genera l s t ress level. T h e equa t ion 
which re la tes these t e r m s is : 
t h e o r y are no t only cons i s t en t wi th the t e s t resu l t s 
b u t show the closest a g r e e m e n t wi th them. 
T h e compar i son m a d e w i th t he t e s t d a t a of 
Tab l e s 4 -6 m a y no t cover the b r o a d spec t rum 
2X 
F 
.IT 1 [ 
¢~,, Cs et l 
Sinh ~ (1 
Sinh ~- 
p z) ~rI, { ( 3 - - u ) ( l + u ) c ° s h ~ - - ( l + v ) 2 ~ c°sh 
+ ~ +e-~/L 4 
where 
t, ~, b, and X have the same meaning as in this paper. 
2L = span. 
q = ultimate compressive strength of panel 
evp = yield point of material. 
I r = total moment of inertia of plate and stiffener. 
Cc = distance from neutral axis of combined section of top 
fiber of plate. 
e = distance from neutral axis of stiffener alone to center 
of gravity of plate. 
o = radius of gyration of stiffener alone. 
Is = moment of inertia of stiffener alone. 
T h e t rue va lue of a n y theory , however , depends 
on how closely i t can p red ic t the phys ica l be- 
hav io r of t he s t ruc ture . I t is comfor t ing t h a t 
some expe r imen ta l d a t a a re ava i l ab le which can 
be c o m p a r e d wi th exis t ing theories . Th is com- 
par i son is m a d e in Tab l e s 4, 5, and 6 where the 
bu lk of the d a t a was t aken f rom some recen t 
Br i t i sh t ests~.. These tes t s were c o n d u c t e d on 
~-fh-H-size s t r uc tu r a l m e m b e r s of such p ropo r t i ons 
as are c o m m o n l y found in ship s t ruc tures . 
Refe rence to Tab l e s 4 a n d 5 shows the effect 
of p l a t e th ickness a n d st i ffener size on the effec- 
t ive b r e a d t h for t he case where the spec imens were 
sub j ec t ed to un i fo rm loading. I n T a b l e 4, the 
spacing, span a n d st i ffener p ropo r t i ons r ema in 
cons t an t . 
I n T a b l e 5 the va r i ab le is the s t i ffener cross- 
sec t ion a rea ; p l a t e th ickness , spac ing and span 
r ema in ing cons tan t . T h e t a b u l a t e d d a t a show 
t h a t t he theore t i ca l p red ic t ions m a d e f rom the 
Bureau of Ships ' s t u d y are in excep t iona l ly good 
a g r e e m e n t wi th the t e s t resul ts . T h e cor re la t ion 
of t he t e s t resu l t s w i th t he class ical t h e o r y is in 
eve ry case poor , while the cor re la t ion wi th t he 
a u t h o r ' s t h e o r y is i n t e r m e d i a t e be tween the two. 
T a b l e 6 shows some expe r imen t a l resu l t s 5 for 
t he case where the spec imens were loaded wi th 
c o n c e n t r a t e d loads. T e s t resu l t s are aga in com- 
p a r e d wi th theory . I t will be n o t e d here t h a t 
the a u t h o r p red ic t s an increase in effect ive 
b r e a d t h as t he th ickness is decreased. T h e 
e x p e r i m e n t a l d a t a show the oppos i te . Once 
more , t h e p r e d i c t e d va lues f rom the B u r e a u ' s 
4 N o r t h E a s t Coas t I n s t i t u t i o n of Eng inee r s and Sh ipbu i lde rs , 
Volume 61 (1944-1945) . 
Amer i can Socie ty of Civi l Eng inee r s , Vo lume 64 ( J a n u a r y 1938). 
T A B L E 4 
Plate 
thickness, ~ E f f e c t i v e breadth, X / b - - ~ 
in. Tests Timoshenko Schade BuShips 
0.32 0.60 1.45 1.03 0.62 
0.44 0.88 1.45 1.03 0.89 
0.63 0.89 1.45 1.03 0.87 
Span = 16 ft. Spac ing = 24 in. St i f fener a rea = 6.22 sq. in. 
L o a d i n g - - u n i f o r m l y d i s t r i bu t ed . 
T A B L E 5 
Area 
stiffener, ~ E f f e c t i v e breadth, ) , [ b ~ 
sq. in. Tests Timoshenko Schade BuShips 
6.22 0.88 1.45 1.03 0.89 
7.72 0.82 1.45 1.03 0.82 
8.61 0.82 1.45 • 1.03 0.87 
10.72 0.79 1.45 1.03 0.81 
P l a t e th ickness c o n s t a n t = 0.44 in. Span = 16 ft. 
24 in. L o a d i n g - - u n i f o r m l y d i s t r ibu ted . 
Averages of 4 to 8 Tes t Loads. 
Spacing = 
T A B L E 6 
Thick- - Spac- -~------Effective breadth, X / b ~ 
ness, ing, Timo- Bu- 
in. in. Tests shcnko Sehade Ships 
0.73 24 1.00 1.23 0.81 '0.98 
0.68 26 0.67 1.14 0.89 0.75 
Span = 16 ft. C o n c e n t r a t e d loads. 
of t he p rob lem, y e t i t shows some h igh ly signifi- 
c a n t t rends . T h e suggest ion m a d e b y the a u t h o r 
to pe r fo rm a c c u r a t e l y con t ro l l ed expe r imen t s is 
therefore a w o r t h y one. Th i s should be done wi th 
full-size mode l s and wi th i m p r o v e d e x p e r i m e n t a l 
techniques , m a k i n g sure t h a t all t he p a r a m e t e r s 
bea r ing on the case are e v a l u a t e d p rope r ly . 
N o t w i t h s t a n d i n g the l ack of s a t i s f ac to ry agree- 
m e n t be tween the theo ry p re sen ted b y the a u t h o r 
a n d the t e s t resul ts , i t is cons ideredt h a t th is 
p a p e r has se rved a useful pu rpose in b r ing ing to 
l igh t some v e r y p e r t i n e n t concepts of the p rob - 
lem, a n d in s t i m u l a t i n g in t e re s t in fu r the r exper i - 
m e n t a l work. 
E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 423 . 
P R O F E S S O R G E O R G E W I N T E R , 6 Visitor: C o m - 
m o d o r e H. A. Schade's distinction between 
"effective width" and "effective breadth" is a 
welcome clarification of terminology which may 
help the designer to distinguish between these 
two very different cases. The reason for the 
non-uniform stress distribution and the con- 
sequent "effective width" in stiffened compression 
~01ate elements is, of course, due to buckling and 
the consequent stress-redistribution in the post- 
buckling state; i.e., in the slightly buckled 
plate. On the other hand, in a stable plate, 
such as one in tension, the non-uniform dis- 
tribution of stress and the consequent "effec- 
tive breadth" is due to the phenomenon which, 
in recent years, has become known as "shear lag" 
in aeronautical literature. By this is meant 
the fact that the longitudinal stress is transmitted 
to the flange not at the ends and uniformly but 
through shear at the junctions with webs. In 
spreading from the webs, the consequent normal 
stresses show a lag with increasing distance from 
the web, which results in the non-uniform distribu- 
tion of such stresses, and the consequent effective 
breadth. 
I t should be recognized, however, that even 
in beam flanges the design according to the lat ter 
"effective breadth" (i.e., on the basis of shear 
lag) is correct only if the flanges are stable. To 
be sure, tension flanges are stable under any 
conditions and, therefore, should be designed 
accordingly. However, wide and thin compres- 
sion flanges buckle in precisely the same manner, 
no matter whether they are components of com- 
pression members or of beams. I t is well es- 
tablished by many tests by the writer 7 and others, 
that in this case, for compression flanges of beams, 
as for compression members, the "effective width' ' - \-- 
~ r a t h e r than the "effective breadth" governs the 
performance. Thus, for example, in a wide, 
thin-flanged box-beam it often may be necessary 
to use the "effective breadth" for the tension 
flange, but the "effective width" for the compres- 
sion flange in order to determine effective cross- 
sectional properties, stresses, and strength. 
There is a possibility that the transverse 
curvature which forms in very wide beam flanges 
as a consequence of the beam deflection [6] s may 
contribute some stabilizing influence for the 
compression flange, in which case the "effective 
6 Head of Department of Structural Engineering, Cornell Uni- 
versity, Ithaca, N. Y. 
7 Geo. Winter, "S t rength of Thin Steel Compression Flanges," 
Transactions. American Society of Civil Engineers, Volume 112, 
page 527 (1947); also Cornell University Engineering Experiment 
Station, Reprint No. 32. 
s Geo. Winter, "Performance of Thin Steel Compression Flanges," 
Preliminary Publication, 8rd Congress, International Association of 
Bridge and Structural Engineering, page 137 (1948); also Cornell 
University Engineering Experiment Station, Reprint No. 33. 
width" of beam compression flanges may be 
conceivably somewhat larger than if the same 
flange were a component of a straight compression 
member. On the other hand, there is also the 
possibility that in beam compression flanges the 
individual reductions in effective width due to 
buckling and due to shear lag may, to some extent, 
be additive, in which case the effective width of 
beam compression flanges might be expected 
to be somewhat smaller than if the same flange 
were a component of a straight compression 
member. The writer's admittedly limited test 
evidence seems to indicate that these interaction 
effects, if at all present, are quite small. He 
recommends, therefore, that the same effective 
width expressions which have been developed 
(by many authors, the writer among them) be 
applied to all compression flanges, no matter 
whether they are parts of beams or columns. 
Conunodore Schade's interesting determina- 
tions of effective breadth are more inclusive than 
the writer's in two respects: (a) he introduces a 
parameter characteristic of the main cross- 
sectional dimensions of the beam, and (b) he 
investigates the case of multiple webs, in addition 
to the single and double webs which also were 
analyzed by the writer. Exactly as the latter 
the author finds that ahnost exac t ly the same 
effective width ratios hold for single and for double 
webs. For multiple webs he finds values which, 
in some cases, are significantly higher; i.e., more 
favorable. This is easily understood since, in the 
first two cases, the transverse stresses are zero 
at both longitudinal edges, whereas they are not 
zero in continuous plating over multiple webs. 
These tranverse stresses contribute to an equaliza- 
tion of the stress distribution. 
Regarding the influence of the cross-section 
parameter the following may be noted. For 
uniform load the writer's curve is located but 
I slightly above the author's, both curves being 
,valid for any value of ft. For concentrated load 
ithe writer's curve falls reasonably close to the 
middle of the band given by the author 's three 
curves c, d, and e (for different fl's). The param- 
eter fl essentially determines the shape of the 
distribution of the shear transmitted from the 
web to the flange. This distribution does not 
produce a jump even at the point of application 
of a "point" load, in view of the distributing 
action of the web. In this connection it may be 
well to remember that in reality "point" loads 
do not exist, since all loads are distributed over 
some length of bearing, a fact to which the author 
draws attention. Hence, the a c t u a l effective 
breadth will depend not only on fl but also on 
the physical width over which the "concentrated 
. 424 E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 
load" actually is distributed. This influence is 
p robably of the same order of magni tude as tha t 
of t , and is not accounted for explicitly in either 
the author ' s or the writer 's t reatment . In fact, 
the ideal case of "point" load is appt:oximated, the 
closer the more terms are considered in the 
Fourier expansion. Yet , the author ' s t r ea tment 
represents an improvement in accuracy in tha t 
he considers a t least one of these two influences 
purposely neglected by the writer for the sake 
of simplicity. 
Finally, a t tent ion is drawn to the author ' s 
s t a tement tha t for concentrated loads the reduc- 
tion in effective breadth is a localized phenomenon 
restricted mainly to the immediate neighborhood 
of the loads. For this reason it m a y be less 
impor tan t practically, than is sometimes be- 
lieved, since overstressing in this restricted region 
under static loading will lead simply to some plas- 
tic stress redistribution without other adverse 
effect. In cases of fatigue, however, this local 
effect, which has the character of a stress concen- 
trat.ion, needs careful consideration. 
MR. E. E. JOHNSON 9, Visitor: The author 
is to be commended for having brought clearly 
to the a t tent ion of the naval architects and en- 
* David Taylor Model Basin, N a v y Department, Washington, 
D . C . 
gineers the confusion tha t has existed with regard 
to effective width of plating. He has pointed 
out once again the difference in plate effectiveness 
of stiffened plating in direct compression and in 
bending, as well as the complete lack of justifica- 
tion for determining the effective breadth of 
plating for a stiffened panel in bending in terms 
of a multiple of the plate thickness. 
In reference [10] a comparison is made oftheoretical results similar to those presented in 
the author ' s paper with experimental da ta 
obtained from tests of two wide-flanged T-beams. 
The results, while indicating general agreement 
with theory, show appreciable scatter. While 
no investigation of effective breadth has been 
undertaken a t the Model Basin, in connection 
with tests of stiffened panels in compression 
one five-stiffener panel was loaded in bending. 
The panel was loaded by a concentrated load 
a t the center and supported on rollers spaced 
a t three different distances apar t so as to obtain, 
in effect, three different L/b ratios. The effec- 
t ive width was determined a t the center by mea- 
suring the strain on the plate side and on the 
flange of the T-stiffener. Assuming linear varia- 
tion of strain through the depth of the stiffener, 
the location of the neutral axis was obtained. 
The effective breadth was then determined by 
balancing moments of area on each side of the 
I.Z 
1.0 
0.8 
,Zl..~ 0.6 
0.4 / 
From F{g. II of Commodore Schade's 
Paper j3 is approaching 7_era 
I 
I X . / 
I 
Exl,er~menfal resulfs w{fh pla~ing o 
~n compression 
x Ex )er~men#al resul%s w~th pla%{ng 
m %enslon 
0 
0 1 2 3 ~ 5 6 7 8 9 
ch 
B 
FIG. 13.--EFFECTIVE-BREADTH' RATIO VERSUS LENOTH-V%/IDTH RATIO AS ~ APPROACHES ZERO 
E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 425 
neutral axis. The results of this minor investiga- 
tion are compared with Case I I I of the author 's 
paper in Fig. 13. The results from the tests 
with the plating in compression indicate a trend 
in line with the theory. However, when the 
panel was reversed and loaded with the plating 
in tensidn no reasonable trend was discernible. 
Effective breadth determined experimentally in 
this manner is extremely sensitive to errors in 
strain measurements however, and the maximum 
observed deviation from the theoretical curve 
can result from an error of only 20 mieroinehes 
per inch in the strain measurements. Within 
the .possible experimental error, the experimental 
results thus appear to confirm the theory. 
ZS0 I O0 
260 90 
I Pla~qng/~ 
z 80 
Flange I ~ ?.0 
I0 
8O 0 
Z 6 I0 I~ 18 XZ Z6 30 
Bread'~h of PIafing Assumed Effecflve 
FIG. 14.--VARIATION IN'MoIv~NT OF INERTIA AND SI~CTION 
MODULUS WITH EFFECTIVE BREADTH 
~Z20 
-~ zoo 
C 
~80 
E 
o 140 
In connection with plating stiffened by T's , 
L's, bulbs, or flat bars, it should be noted tha t 
the effective breadth has but little effect 'on the 
maximum design stress. Fig. 14 shows the varia- 
tion of momen t of inertia and section modulus 
with effective breadth for the usual proportions 
of stiffeners and plating encountered in normal 
practice. The governing stress occurs in the 
stiffener flange, and it is observed from the illus- 
t rat ion tha t , for effective breadths from 10 to 40 
times the plate thickness, the section modulus 
referred to the flange and consequent max imum 
stress is altered less than 10 per cent. I t would 
appear, therefore, that , from a stress standpoint , 
effective breadth is of minor importance for this 
case. 
For symmetrical sections such as box girders, 
however, the effective breadth is of greater ira= 
portance. In this instance, unlike the case of 
plating stiffened by T ' s or bulbs, the stresses 
in the plating become appreciable and it would 
seem reasonable to expect a reduction in s trength 
resulting from buckling of the plating in compres- 
sion. I t appears, therefore, tha t effective breadth 
and effective width cannot be dissociated com- 
pletely, and the author ' s comments in this re- 
gard would be appreciated. 
Finally, it appears tha t the assumptions neces- 
sa W to obtain the simplified formula (15) for 
section modulus m a y introduce appreciable error. 
In checking the section modulus applicable t o 
the upper and lower flanges, the calculated values 
based on the author ' s formula (15) were from 3 
to 50 per cent larger than those calculated in the 
usual manner. The greater difference occurs 
where A1 is large compared to A2. The extent to 
which this may alter the actual values of k/b has 
not been determined and perhaps the author 
would like to comment on this point. 
The opinions expressed are those of the discusser 
only. 
PROFESSOR J. H. EVANS, Member: In the 
ideal case, design data may be visualized as 
consisting of three elements; viz., the theoretical 
analysis, the experimental analysis, and the 
correlation of the two presented in a digested form 
most suitable for design purposes. In his paper 
Commodore Schade has at tained the first and 
third of these objectives in an admirable way, 
a n d there is lacking only the experimental verifica- 
tion (as he points out) to complete the t r iumvi- 
rate. Certainly n o p r e s e n t a t i o n can be more 
simple to use than Figs. 9, 10, and 11. I should 
like to raise the question if, in actuality, the curves 
of Figs. 9 and 10 should not be considered iden- 
tical despite the difference in boundary conditions 
arising from the difference in origin of the two 
eases. 
The extent of computat ion necessary as ex- 
emplified by Example 3 might be a little frighten- 
ing to anyone setting about the determination 
of effective breadth of plating, which calculation 
m a y have to be performed m a n y times in the 
design of a ship's structure. However, the 
convergence of the solution generally appears 
to be so rapid tha t only the first two or three 
values of n need be used. Better still, the curve 
for triangular and non-central concentrated 
426 E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 
loadings might be added to those already given 
(expressing several values of d in terms of L for 
the concentrated load). 
Judging from his book "Structural Design of 
Warships," Hovgaard did in fact recognize the 
difference between the cases of "effective width" 
and "effective breadth" of plating (see page 37 
of the 1940 edition) and in several examples 
throughout the book distinguishes between them. 
His proposal is to use a constant 50t as the "effec- 
tive width" and a constant 80t as the "effective 
breadth." 
I should like to express my personal thanks to 
Commodore Schade for the time and effort spent 
in exercising his skill on this problem and for 
the masterful result. 
VICE ADMIRAL EDWARD L. COCHRANE, U.S.N., 
(ret.), Past President: I rise to discuss this paper 
orally with a great deal of diffidence, for a number 
of reasons. First is the fact that I was personally 
demoted from any association with technical 
things some years ago, and I recognize, likewise, 
the fact that an old man who rises to the platform 
and starts to reminisce can quickly become a 
nuisance. 
I do want to say, however, in this connection, 
first, that I believe very strongly, and, as Chair- 
man of the Technical Research Committee of The 
Society, welcome and urge this sort of approach. 
Out of my experience, personal experience, I do 
want to soflnd a note of warning, and that is that 
while one can analyze quite neatly the question of 
effective width or effective breadth, as we have 
talked about it here one isn' t always clear that 
the plate which may be the bot tom plate or the 
bulkhead plate will be well informed as to which 
one of these situations it is working in, and I 
think there ought to be some way to clarify that 
and to make sure tha t tha t part which is width 
knows it is width, and that part which is breadth 
knows it is breadth, and works so, too. 
Almost every structure in the ship is loaded in a 
complex system. I t is impossible, so far as my 
own experience is concerned, to have any plating 
structure in which the stiffeners are loaded purely 
in bending, and correspondingly, it is impossible 
to have one in which stiffeners are loaded purely 
in compression; as a mat ter of fact,on top of that 
• come secondary effects of bending in the right 
angle direction, such as one experiences, of course, 
in the bot tom of a big t anker - - the easiest example 
I can think of--where there is a system of trans- 
verse bulkheads which carry part of the load, and 
longitudinal bulkheads, so there is bending in a 
transverse direction as well. Added to those 
complications, are the complications of shear 
which come in through the effectiveness of these 
various elements in carrying the over-all stresses 
in the ship, which almost invariably involve 
some shear problems. 
I recognize quite clearly and want to second 
what Professor Adams pointed out, and that is 
tha t deflection has a very serious influence in all 
of this, and I likewise want to urge that when the 
comments of lX~r. Vasta are presented, some dia- 
gram or inforanation be given as to the exact form 
in which the tests to which he referred were set 
up. Manifestly, a table rests on tests. We do 
not have a background of how the test was con- 
ducted, so that those figures are not too effective 
in one's own adaptation to his own experience 
and problems. 
Of course, what I said first is an a t tempt to 
emphasize what Professor Troost read of Pro- 
fessor Evans ' comment, namely, tha t all of these 
tests and studies need to be rationalized into our 
ship experience. 
To go back again, I think that Professor Schade 
has pointed out tha t this system is interesting as a 
basic approach, and I am not quite sure whether 
his comment on the right-hand column of page 
404, bot tom of the first full paragraph, was said 
with sympathy or not, but I am sure that both of 
the analyses that have been made are valuable to 
us as a basis of extrapolation from experience in 
ships to new designs. 
Now I recognize that that may be the archaic 
way to do things in this modern age of science 
and research; on the other hand, it is very com- 
forting for any designer to check back and see 
whether his analytical approach is rationalized 
with previous experience. 
This is a field, gentlemen, in which, I am sure, 
we, as the older profession, should associate our- 
selves with our younger and very vigorous pro- 
fession of aeronautical engineering. Over the 
years such engineers grew up with a form of 
structure consisting basically of a frame only. 
As you know, they originally had only fabric 
panel coverings. The fabric was untreated at 
first, and later treated fabric was used, and still 
later metal was adopted. The wing surfaces are 
beginning now to be of sufficient strength to 
take part of the compressive loading• Over the 
years the aeronautical industry wisely omitted 
considering any compression strength in the plat- 
ing and assumed that the plating worked only in 
tension. Many of you may have noticed the com- 
pression wrinkling on the surface of an airplane 
wing, which is disconcerting to a ship designer, 
but in shear you will see the tension diagonal come 
into play very effectively. 
E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 427 
This is a very interesting subject, but I want to 
make this comment of warning, that while anal- 
yses are essential, the realities of life should not 
be overlooked. 
M. MARCEL JOURDAIN, 1° Visitor: We must 
be grateful to Commodore Schade for distin- 
guishing so clearly the two concepts of "effective 
width" and "effective breadth" and pointing 
out the nonsense of using the formulae of the 
kt- type when bending is involved, as the de- 
signers do so widely. 
At first reading of Appendix 1, the writer had 
been perplexed about the basic hypothesis con- 
cerning the loading of plate and stiffeners, be- 
cause, in the very common case of loading applied 
to the plating, the reckoning of the loading trans- 
mitted to the stiffeners depends upon their flexi- 
bility, which is just unknown; a glance at table 
29 (§43) of Timoshenko's Theory of Plates and 
Shells will show how great may be the error 
when flexibility is not considered. 
But the doubts were removed by the survey 
of the curves given in Appendix 2. As a matter 
of fact, one must say that, at the approximation 
needed by the designer, all the curves related t o 
distributed loads are fairly identical, whatever 
be the so-called boundary, load and section 
functions. In other words, for practical use, 
provided that no concentrated loads are present, 
the effective breadth ratio k/b is function of the 
ratio cL/B only. 
For designers who prefer formulae to graphics, 
it seems that : 
x 1 
b 1 + 2/(cL/B) 2 
would offer a good combination of simplicity 
and accuracy. The writer thinks this simplicity 
will be an efficient support to substitute this 
formula for the kt-type ones, since designers 
are always reluctant to use intricate formulae. 
So, the flexibility of the stiffeners may readily 
be known and taken into account for the stresses 
in the plane" of the plate to be computed. In 
this regard, we must bear in mind that the ex- 
ternal loading of the plate and the normal or 
shear loading due to its contribution to the rigidity 
of l~he stiffener have not a similar nature, with 
the consequence that the addition of their effects 
must be performed according to a hyperbolic 
law and not a linear one. 
Another remark may be of interest; the stress 
computed by the effective breadth theory may 
be seriously raised up, if the plating is subjected 
1o Ing6nieur en Chef du O6nie M a r i t i m e (H .C . ) , Ing6n ieu r a P i n - 
s t i t u t de Recherehes de l a Cons t ruc t i on N a v a l e . 
to a local bending and this increase has to be 
estimated prior to fixing the optimum stress. 
The last two remarks relating to the plating 
itself, the study of which would constitute a 
second step in the question, but as far as the stiff- 
eners only are concerned, the paper under dis- 
cussion is, in the writer's opinion, an outstanding 
progress from the designer's point of view. 
PROFESSOR DR.-ING. GEORG SCHNADEL, Asso- 
ciate Member: The paper presented by Professor 
Schade comprises a very interesting and important 
subject of shipbuilding. I am interested in this 
subject, as I have r e a d ' a paper in 1925 about 
effective breadth (STG 1926, 27.Jahrg.) and later, 
(1927) as mentioned by the author. Two further 
important publications may be cited: 
Metzer, Dr.-Ing.: Die mittragende Breite Luft- 
fahrtforschung 1929 Band IV, S.1 
Miller: 0 b e t die mittragende Brei te 'Luf t fahr t - 
forsehung 1929 Band IV, Cont. 
These publications contain most of the essential 
formulae important for the subject. Only some 
short remarks may be added: 
The calculation with a restricted number of 
terms is not always sufficient to get the necessary 
accuracy for the effective breadth, if concentrated 
loads or fixed ends are to be considered. I t is 
possible to sum up the infinite series and to deter- 
mine the limit of error of the calculation. For 
some spread or distribution of the concentrated 
load an exact calculation may be obtained in a 
similar manner, but not by using the results for 
uniform load. 
The term/5 in Sehade's paper or the moment of 
inertia of the web has a very great influence on 
the effective breadth in the case of concentrated 
loads or fixed ends. 
The author has calculated the case of central 
load in Table 2. Really in case e, (/~ -* 0), the 
series is not convergent. For fixed ends, uniform 
load, we get 
1 L ~ I 1 1 
L 
If/5 --+ 0 and a > 10 we get Xn = ~ n and 
00 
I ~ E , 1 _~ o~ and x_~o. 
~ = ~ 1 ~ 
Indeed the effective breadth may be very small, if 
fl is small. Therefore, Figs. 9, 10 and 11 and 
Table 2 in Schade's paper cannot be used for in- 
terpolation in this case. 
428 E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 
In every case of concentrated load it is necessary 
to calculate the limit of error. I t is possible in 
every example mentioned by ProfessorSchade to 
determine the upper and the lower limit of the 
series and to get the right value with the limit of 
error. 
Mathemat ica l ly the effective breadth is only 
independent of the geometry of the girder, the 
-relations of web and plate thickness, if t h e m o m e n t 
has .the harmonic form. In all other cases the 
effective breadth is different along the length of 
the girder. The difference is only small in the 
case of uniform load with free ends, but it m a y be 
great ifl other cases. 
The series of the deflections have a very good 
convergence, as they are calculated with the 
momen t series by double integration. 
PROFI~SSOR GEORG VEDELER, Member: Com- 
modore Schade's presentation of the p r o b l e m o f 
effective breadth mus t be very welcome to naval 
architects, who for m a n y years have been led 
somewhat as t ray by Pietzker 's a n d Hovgaard ' s 
reference to thickness, but who gradually realize 
tha t span is more important . I f I should make 
some remarks regarding Commodore Schade's 
clear exposition, I would express first the wish tha t 
in addition to Case I, which includes beams of 11 
H, or T forms, he would also have said a few 
words about beams of U or Z forms, for which the 
effective breadth is very much less. 
Secondly, I believe tha t a presentation of effec- 
t ive breadth a t the point of max imum bending 
momen t only is not always sufficiently enlighten- 
ing, tha t is, in connection with deflection. The 
effective breadth of an infinitely wide flange is a 
certain fraction k0 of the span and the variat ion of 
k0 along the span is very illustrating. This is for 
Case I and a uniformly distributed load k0 = 0.38 
a t mid-span, reducing gradual ly to half this value 
a t the supports, assuming no fixity. For a con- 
centrated load a t midspan, however, k0 = 0.19 a t 
the load and increases to 0.55 a t the supports (for 
tL/Aw = 10, flat bar stiffener). The values of 
ko = f(x) can be computed once and for all for the 
most impor tant types of load and then curves 
similar to Schade's Figs. 9, 10, and 11 could be 
given, but with k0 as a parameter , thereby enabling 
a ready calculation of flange efficiency anywhere 
along the span, also for finite flange widths and 
without the tedious calculation of sometimes 
badly-converging Fourier series. To make the 
effective breadth more a t t ract ive for daily use, an 
a t t empt in this direction was made in m y paper 
"Calculations of Beams" (Transactions of the 
Inst i tut ion of Naval Architects, 1950). I t un- 
doubtedly can be improved upon. Especially the 
curves of flange efficiency as a function of L / B with 
k0 as parameter should be given a correction for 
different values of ft. I have given some examples 
only of the variat ion with ft. As mentioned, only 
the curve for k0 = 0.36, which is the value for a 
sine load, is entirely independent of ft. 
From m y curves of ko = f(x) it can be seen tha t 
it is correct to use distance between points of zero 
bending moment as span for a concentrated loadl 
but for a distributed load the character of the k0 
curve changes, especially near the points of zero 
moment, when end fixity is introduced. Only at 
mid-span the value of k0 varies roughly in the in- 
verse proportion of the lengths between zero 
moments. All cases with fixed ends are, however, 
only approximate, because the bending moment 
curves used have been obtained on the assumptiofl 
of a constant moment of inertia over the span. 
Actually the bending moment curves should have 
been corrected according to the varying moment 
of inertia due to the variat ion in k0, whereby one 
would get a slightly different k0. I t might be 
interesting to ask a s tudent to carry out such an 
iteration process to see what the result will be. 
MR. J. M. MURRAY, 11 Visitor: This paper is 
valuable in that , in addition to reducing to simple 
terms for design purposes a rather complex mat ter , 
it clarifies the di§tinction between what the au- 
thor has so happily named "effective width" and 
"effective breadth ." On this point, the writer 
would like to acknowledge the reference which 
the author has made to his note on the subject in 
Engineering where tha t distinction also was ob- 
served. The conclusion tha t for uniform loading 
the effective breadth is independent of the geom- 
e t ry of the section is of interest, as is also the 
rather unexpected conclusion tha t with uniform 
and sine loading the effective breadth may ex- 
ceed unity as a result of the "Poisson" effect. 
The experimental determination of effective 
breadth may present difficulty on occasion since 
the operation hinges on the determination of the 
precise position of the neutral axis which is n.ot 
always easy; such information as exists, however, 
tends to confirm the val idi ty of the author ' s de- 
sign curves. 
11 Lloyd's Register of Shipping, London, England. 
E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 429 
COMMODORE H. A. SCHADE: Since Professor 
Adams is the man who originally proposed a 
paper on this subject for our 1951 meeting, his 
remarks are very welcome. I t seems apparent 
tha t in the 1905 Bruhn tests cited by him, either 
r ivet slippage or buckling occurred, since de- 
flections are not proportional to loads, even at low 
stresses. The boundary conditions applicable to 
the deck girder question raised by Professor 
Adams produce effective breadth curves slightly 
higher than those for Case III , since one may 
assume a rigid convection of the deck plating at 
.ship side. Case I I l may therefore be used for 
these conditions, noting that the actual breadth 
B is here the distance from the girder to the side. 
With reference to Mr. Vasta's comment to the 
effect tha t section geometry is theoretically a 
factor with distributed loading as well as with 
concentrated loading, the author does not disagree 
with Mr. Vasta's point of view; the analysis 
shows only that the section geometry significance 
is so small with distributed loading that the author 
recommends it be disregarded. 
Publication of the 13ureau of Ships study, cited 
by Mr. Vasta, will be awaited with much interest. 
Detailed coniment on the formula given would 
not be appropriate here, since its derivation is 
omitted. The appearance of the material yield 
i~oint value in it suggests tha t it is valid only 
when yield point is reached somewhere in the 
structure. This implies a definition of effective 
breadth quite different from that adopted by the 
author, which is simply tha t of an artificial 
dimension to be used in place of the real width 
in computing section modulus for use in the 
flexure formula, valid for any maximum stress 
less than yield, and limited to situations where_ 
buckling_ does not o q~ccur.. T h e later req_uirement 
limits application to the tension side_, or the 
~ ~ d e when the average stress is below 
b-fi-dPdih--~-IiNit-~-. The close agreement between 
t ~ t - ~ d - ~ - y Mr. Vasta and the prediction 
based on the Eureau formula is certainly inter- 
esting, but further comment must be reserved until 
the promised publication, which presumably will 
give a description of the test conditions and stress 
levels. 
Professor Winter 's comments are very welcome 
in view of his great experience with this question, 
particularly in the field of air-frames, where the 
term "shear lag" is usually used. The author 
is in complete agreement with the points raised 
by Professor Winter. 
I t is quite true, as Mr. Johnson points out, for 
the case of flat-bar or similar stiffening on a single 
plate, where the effective breadth of only a single 
flange is in question, that the maximum stress 
in either web or flange is not very sensitive to 
variations in effective breadth. Consequently, 
experimental verification by strain measurement 
is difficult,and perhaps futile, from the design 
point of view. Deflection measurements are not 
only easier to make, bu t the deflection itself 
is more sensitive to variation in effective breadth 
since it is a function of moment of inertia rather 
than section modulus. The theoretical effect 
upon moment of inertia and stiffener section 
modulus of variation of effective bread th from 
zero to infinity is shown by the following: 
When X ~ 0 When X ~ 
h 2 4 h 2 . 
I =_-flAw I = --3- Aw 
S = h gAw S = Aw 
This is based on the approximation of Equa- 
tion 15, and shows tha t while moment of inertia 
increases by a factor of four, section modulus 
increases by only a factor of two, when effective 
breadth goes from zero to infinity. 
However, when two parallel flange plates 
are stiffened by webs between them, both 
moment of inertia and section modulus are very 
sensitive to variation in effective breadth, and 
increase from the same lower limits to infinity 
in both cases when effective breadth goes from 
zero to infinity. 
Equation 15, is, as pointed out by Mr. Johnson, 
an approximation. I t is based upon the assump- 
tion of plane stress in the flange, which is com- 
patible with the analysis. This comes to the 
same thing as imagining tha t the stress in the 
flange middle surface exists unchanged through 
the flange plate thickness, and measuring web 
depth f: o n that middle surface, the same assump- 
tion usually used in computing the properties of 
the ship girder section in the conventional strength 
calculation. The discrepancies noted by Mr. 
Johnson are due to these simplifications; their 
magnitude depends on the ratio of flange plate 
thickness to web depth. For example, in the 
single flange plate configuration, stiffener section 
modulus may be written 
hAw 4A + A,o 
S = -3-- 2 A - + Aw ' (from Equat ion 15) 
hAw 
S - X 
3 
421 + A w + ~ \ ~ ] J I~(A + 4 A w ) + 3 A 
2 A + A w + 2 /~ A 
(exact) 
430 E F F E C T I V E B R E A D T H OF S T I F F E N E D P L A T I N G 
The terms in t/h represent the effect of the omis- 
sions noted above, and become insignificant when 
t/h is small. Since the omissions were made in 
order to use the paramete r /3, which the analysis 
shows to have only a second-order effect on effec- 
t ive breadth, the omissions are clearly justified. 
In computing section properties for stress or 
deflection deterrninators, however, the appropriate 
conventional method m a y be used for more exact 
results. 
Certainly, as Professor Evans points out, the 
curves of Figs. 9 and 10 are very similar and for 
practical design purposes they could be used 
interchangeably, ignoring the differences between 
them. They were presented by the author to 
bring out their similarities, ra ther than their 
differences, l~ would be interesting also to have 
the curves for tr iangular load (i.e., the hydro- 
static load on bulkhead plating) but the labor 
involved in computat ion is quite large, and it was 
only by vir tue of a research grant from the Univer- 
sity of California tha t the author was able to 
get the computing done which was applicable 
to the curves presented in the paper. 
In response to Admiral Cochrane 's remarks 
concerning the inabili ty of the plate to know 
which par t of its is width and which breadth, the 
author should emphasize (in the same vein) tha t 
these concepts of width and breadth are merely 
convenience to the designer, not to the plate. 
I t is imPor tant tha t the difference between them 
be recognized by the designer so tha t he can bet ter 
predict the performance of the plate. In the 
long run, be t te r understanding of analytical pro- 
cedures, combined with knowledge of successful 
practice, will produce bet ter distribution of ma- 
terial in a ship structure. 
I t is not clear whether M. Jourdain intended 
to omi t the 10 per cent increase in the denomina- 
tor of the empirical formula or not, bu t the author 
believes it should be retained. Perhaps it could 
be regarded as a hidden factor of safety. 
Dr. Schnadel 's comments are very penetrating, 
and are especially interesting to the author, since 
his own interest in this subject was first s t imulated 
by Dr. Schnadel m a n y years ago. Dr. Sehnadel 
points out tha t the series does not converge for 
the case of a concentrated load with /3 = 0. 
This is true. The curves for /3 = 0 represent 
a physically impossible condition in any event, 
since they represent a vanishing web. In the 
opinion of the writer, however, they are usable 
for interpolation purposes to save computat ion 
for physically practical web configurations, and, 
when so used, will give the value of effective 
breadth for a bending moment curve represented by 
the first eleven terms of the series. The degree 
of failure of the first eleven terms to represent 
the actual bending momen t curve for a concen- 
t ra ted load can, as pointed out by Schnadel, be 
assessed by establishing upper and lower limits 
for the series with an infinite number of terms. 
This can be done by correcting a value of X/b 
obtained by interpolation from the curves in the 
paper for any specific cases. Fo'r example, 
if cL/B = 7r and /3 -= l/e, the value of X/b for 
Case I I f rom the curve is about 0.64; application 
of the theory of limits for an infinite number of 
terms in the series indicates tha t the most probable 
value for a mathemat ica l point load is abou t 0.535. 
Curves for low values of /3 could be constructed 
to show the most probable values for a mathe- 
matical point load rather than values for the 
eleven-term series; this would require a great 
deal more computat ion, and has not been done 
in view of the author ' s impression tha t only 
limited interest a t taches to the mathemat ica l 
point load. A desirable extension of this work 
might well be the computat ion of curves for 
the probable values for the infinite series, rep- 
resenting a true point load. For uniform load 
and for a point load with high values of r , this 
question is of no significance, since the correction 
for the neglected terms is vanishingly small. 
Professor Vedeler's comments concerning U 
and Z forms are certainly cogent. These un- 
symmetrical forms, unless restrained against 
torsional displacements, exhibit drastically re- 
duced effective breadths; for example, a single 
web with a single unsymmetr ical flange shows a 
maximum X/b value of 0.5, so tha t less than one 
quarter of the flange material is effective. The 
reference cited by Professor Vedeler is not avail- 
able to the author, bu t he is in agreement with 
the points made. The variat ion of X/b as a 
function of x is t reated extensively also by Raithel, 
in Reference [10] of the paper. 
PRESIDENT KING: Thank you, Dr. Schade. 
There certainly has been a need for some clear-cut 
thinking on this subject and your paper is a valu- 
able contribution to our Transactions. I want to 
thank you on behalf of The Society and also thank 
all those who contributed to the discussion.