REPORT NO. IS S SEPTEMBER 195/e
STUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE
AFDELING ,,SCHEEPSBOUW" - NIEUWE LAAN 76 - DELFT
(NETHERLANDS RESEARCH CENTRE T.N.O. FOR SHIPBUILDING AND NAVIGATION)
(SHIPBUILDING DEPARTMENT - NIEUWE LAAN 76 - DELFT)
INVESTIGATION OF THE STRESS DISTRIBUTION
IN CORRUGATED BULKHEADS WITH VERTICAL
TROUGHS
BY
PROF. IR H. E. JAEGER
IR B. BURGHGRAEF
I. VAN DER HAM
Page
Summary 3
CHAPTER I Introduction
Stresses resulting from the hydrostatic pressure 3. Local instability
§ 4. Stress distribution in the welds of the plating 5. Some notes on bulkheads with horizontal troughs
s 6. Comparison
bulkhead 14
Appendix i - Three-moment equations 16
Appendix 2 - Geometrical properties of half a corrugation 18
Appendix 3 - Calculation of bending moments M 20
Appendix 4 - Application of the three-moment equations to an
expanded strip 21
References 21
CHAPTER II
§ 7. Introduction 22
5 8. Strain measurements 24
The calculation of the stresses at the tested points 27
Comparison between the measured and the calculated stresses 27
Conclusion . 28
between a corrugated bulkhead and a common flat
3 3 7
12
INVESTIGATION OF THE STRESS DISTRIBUTION IN
CORRUGATED BULKHEADS WITH VERTICAL TROUGHS
by
PROF. IR H. E. JAEGER, IR B. BURGHGRAEF, I. VAN DER HAM
Summary
This report deals with the stress distribution in a transverse corrugated bulkhead with vertical troughs. Its aim is to arrive at a method of calculation suited for use in practice. A method, based on elementary principles, and the necessary assumptions are discussed and an example is given. The problem of local instability is discussed and the stresses in the welds of the plating are considered. The effect of the adoption of corrugated bulkheads on weight
is discussed briefly.
In Chapter II some tests on a corrugated bulkhead are described and a comparison is made between the calculated and measured stresses. It is shown that there is a reasonable agreement, which, in the opinion of the authors, justi-fies the method of calculation discussed in Chapter I.
The desirability of more extensive stress measurements on a bulkhead of this typeis mentioned.
CHAPTER I § 1. Introduction
For a long time it has been the practice in
ship-building to construct plain watertight or oiltight
bulkheads, consisting of a relatively light plating strengthened by rather heavy stiffeners riveted to
the plating.
The development of electrical welding brought
along a small improvement, as it made possible the use of toe-welded angles or T-sections replacing the riveted
L-
or 1=-bars.However, this method of stiffening bulkheads is far from ideal, because of the unsatisfactory
distri-bution of stresses, as has been known for a long
time.
An improvement of the stress distribution may
be obtained, however, by the use of corrugated
bulkheads, as was suggested some 35 years ago.
The development of welding was required to
overcome the practical difficulties of construction.
At present, corrugated bulkheads have become
fairly common. The increase of tanker tonnage of recent years no doubt has contributed to the wider
adoption of this kind of bulkheads.
The profile of the corrugations, in use at present,
may be rectangular, trapezoidal or triangular. Of these, the trapezoidal profile is probably the
most frequently applied and therefore oniy this
profile will be considered in the following invest-igation. Moreover, the troughs formed by the
cor-rugations may be horizontal or vertical, and it is
obvious that horizontal troughs are the most suitable ones for longitudinal bulkheads, as vertical troughs
would make the bulkhead more or less useless as a component of the longitudinal strength of the ship. To avoid difficulties at the crossing of
longitu-dinal and transverse bulkheads, vertical troughs
are nearly always adopted for the transverse
bulk-heads.
Indeed, only a vertical plane strip of sufficient
breadth, to which the longitudinal bulkhead can be welded, is required in the transverse bulkhead.
Even if for some reason horizontal troughs are
wanted in the transverse bulkheads, there are
methods to overcome the difficulties. Some of these
methods are patented (Isherwood, Tom Petersen). In this investigation, only transverse bulkheads with vertical troughs will be considered.
The investigation itself does not claim to be an
exact one.
Its object is only to show a method of calculation
of the stresses in a corrugated bulkhead, based on
elementary principles [1].
If such a method leads to results which are in
reasonably close agreement with the actual stresses,
it will have its value for use in practice.
The agreement between the calculated stresses
and the actual stresses can only be shown by stress
measurements.
In Chapter II the results of some stress
measure-ments at a corrugated bulkhead, which were made
available to the authors, are compared with the
stresses calculated according to the method discussed in Chapter I.
§ 2. Stresses resulting from the hydrostatic pressure
In this investigation of the stress distribution in a
corrugated bulkhead, it will be assumed that the
bulkhead is to be considered to consist of a number of similar bars, the cross section of each being half a corrugation.
Each bar is laterally loaded by the hydrostatic
pressure.
For a bulkhead supported only at its top and
base, the calculation of the bending moments would be a simple matter.
Generally, there will be stringers to strengthen the
bulkhead, which are considered as supports. This
means that the bars mentioned above are to be
4
11
ALL DIMENSIONS IN cm.
t = 1.05 (1.25)
Fig. I
considered as continuous beams and the calculation of the bending moments can be carried out by using
the well-known three-moment equations of
con-tinuous beams.
It is necessary, however, that the profile of the
corrugation is uniform throughout the height of the bulkhead, even if the plate thickness is not constant.
In fig. i a bulkhead of a tanker, strengthened by
two stringers, is shown. There are two longitudinal
bulkheads, and moreover there is
a web at the
centre line, by which the bulkhead is divided into four panels, each consisting of a number of con-tinuous corrugations, supported at the ends and at
two intermediate points.
Each panel may be considered to be built in along the vertical edges, so there will be bending moments
along these edges. As the flexural rigidity of the
panel along a horizontal section is small, the
in-fluence of these bending moments is supposed to vanish at a short distance from the edge.
If a bar consisting of half a corrugation at the
middle of the panel is considered, it may be assumed to be free of the influence of the clampins moments,
X
Fig. 2
that is to say, the bar is assumed to be part of a
panel of infinite breadth.
In Appendix 1, an example is shown of the
cal-culation of the bending moments by making use of the three-moment equations.
The stress distribution over the cross section is
given by:
M . z
=
I where: = bending stress M = bending momentz = distance from neutral axis of section
I = inertia moment of area of section.
The coordinate axes are taken as shown in fig. 2.
This means that the X- and Y-axis are in the plane
of the bulkhead.
In fig. 3 the cross section of half a corrugation is shown, with the stress distribution according to
the ordinary bending theory of beams.
The shearing stress on a cross section can be
cal-culated in the usual way by means of the formula:
D.S
= t
. I
Fit. 3 1035 465 570 A t =1.05 B 505 t =1.05 C 310 = 1.25 s t 290 DJ-- -
45where:
= shearing stress
D = shearing force acting at the section considered
t = breadth of the section = plate thickness I = inertia moment of area of section
S = first moment of area, above or to the right of
the intersecting line considered. (I and S referred to the neutral axis)
In Appendix 2 the geometrical properties of the
section of half a corrugation are shown.
The shearing stress at a point fo at the distance x
from the right-hand end of the profile will be:
D.h. x.t
t.I
for z = h, at d0 therefore: D .h. b1. t
D . h. b1 . t So at e0, 'r = O and at d0, 'r=
t.I
the shearing stress is increasing in a linear way
from e0 to d0.
At a point go at the distance z from the neutral
axis:
h+z
zD{h.b1.t+
2(b2
sin t}'r =
(see fig. 3)tI
as before and in c0, for z = O:
D (h.b1.t+.h.b2.t)
'r-
t.I
The shearing stress is increasing from d0 to co
according to a parabolical law (see fig. 3).
On two planes, perpendicular to the Y-direction
and at a distance dy from one another, the shearing stresses will differ by an amount
dt
.dy=.
S dDQ.S.dy
.dy=
dy
t.I
dyt.!
where Q = the component in the Z-direction of the load per unit length of the hydrostatic pressure q
at the point considered.
q
Fig. 5
Q = 2q (b1 + b2 cos a)
If the shearing stresses on planes at a distance of unit length from each other are considered, the
dif-ference may be written:
At
Q.S
t.!
From the bar an element of unit length is cut out
by two parallel planes perpendicular to the
Y-direction and its equations of equilibrium are
con-sidered.
In fig. 4 the hydrostatic load on the bar element
is shown.
Only the difference of the shearing stresses on the
parallel planes need be taken into account. At a0 and e0:
A
'r = o
At e0 and do: A 'r
Q . h . b1
Atco:A'rQh
(b1+b2)
The resultant shearing forces are:
Q..t
on the parts doeo and aob0: S11
=
2!on the part bodo: S1
2Q.h.t.b2
(b +
b2)It is clear that the component in the Z-direction of S must be equal to Q but of opposite direction.
Q = Sz . sin a
This is easy to see:
2Q.h.t.b2 (b1 +
sinSz . Sin
=
I
2 h2.t. (b1 +)
(as b2.sin a=h).
In fig. 5 the system of forces necessary to pro-duce equilibrium is shown. In order to satisfy the
condition of equilibrium, a force = 2R11 in the
X-direction must be applied.
Though the bulkhead is considered to consist of a number of similar bars, this does not mean they
are separate beams.
Q
'r =
(C)
Fig. 7
-
(C) = (b) ROTATED OVER I80 THE RESULT MUSI 5E() VEUVE M. = M. R. = R.
-Fig. 6
So there must be an interaction of forces and
bending moments between adjacent bars.
The forces R will act at a0 and e0 and must be the result of this interactiOn. For the same reason
there will be bending moments acting along the
edges of the bar. On the bar element there will be
bending moments at a0 and èo. It is easy to see that
the moments at a0 and e0 must be equal as well as
the forces at a0 and e0 in the X-direction (see fig. 6).
cos
Ra=Re=Rx=Sx+Sz.
2
+b2.q.sina
The loading to which the bar element is subjected
is known now with the exception of the bending
moments at a0 and e0. These external moments
can-not be obtained from the equations of equilibrium. If Ma = Me = Mo, a simple calculation shows
the bending moments at b0 and d0 to be:
Mb=Md=Mo
b1.q
and at c0 : M0 = Mo - (b12 - b22)
Moreover it is clear that the curve of bending
moments from a0 to b0 as well as from b0 to c0 will follow a parabolica! law.
I'IIII'I!!
n
n
il
n
--
n
n
In fig. 7 the curve of moments set off on the
expanded bar element is shown, assuming Mo to
be known.
From the symmetry of two adjacent bars it may be concluded that the slope of the deflection curve
must be zero at a0 and e0, or:
JMx.dx = O
Hence Mo = (b12
+
2h1 b2 - 2b22)(See Appendix 3)
So Mo does not depend on the angle a of the
pro-file of the corrugation. This circumstance leads to the supposition that it must be possible to consider
an expanded elemental strip, cut out from a panel of infinite breadth by two parallel planes perpendicular
to the Y-direction and a distance of unit length apart, as a continuous beam, the corners of the
corrugations acting as supports and carrying a uni-formly distributed load (see fig. 8).
The three-moment equations of a continuous beam would then be applicable, and indeed this
assumption will hold when there is no deformation
of the various parts of the corrugation in their own
planes.
This means that the distances MN, NR, RS, etc., must remain constant, and this has been
as-sumed throughout this discussion, though, of
course, it is not quite correct.
x=2(bi + b,)
Fig. $
The deformations mentioned, will be very small, however, and have been neglected in this
investiga-tion, which does not claim to be an exact one. In Appendix 4 it is shown that by applying the three-moment equations the same results are obtained as
before, with the exception of the distribution of the
shearing stresses, which has to be calculated in the
usual way.
It follows from the foregoing discussions that the stresses as., produced by the bending moments in the
continuous beams of which the panel Consists and
the stresses tx in a bar element, caused by bending
moments and forces R, as well as the shearing
stresses can be computed.
IIIIII!IIIIIIIIIIIIUJIIII!I!IIIIIIII!IIIIIIIIIIII!IIIflhIflI M 2b, N 2b5 R 2b S 2b5 T 2bT 2b5 R 2b S 2b5 2b M N T
As each flat part of a corrugation may be con-sidered to be in a plane-stress condition, it is
pos-sible to determine the "effective" stress rEFF, i.e. an
imaginary normal stress which is judged to be
ex-actly as dangerous to the material as the actual
composition of normal and shearing stresses, as computed.
According to the theory by Von Mises, Huber
and Henc-ky, which is based on -Maxwell's theory, the «effective" stress is expressed by:
= /r,1 +
- x0v + 3
From this formula rII'F may be computed, and
hence it is possible to determine where the severest conditions will occur.
§ 3. Local instability
When a bar consisting of halfa corrugation, of which the cross section is shown in fig. 3, is sub-jected to -a bending moment, one of the flanges is in compression. In the case under discussion, the.
flange d0e0 is in compression, the stress being uni-formly distributed over its widthb1.
The web of the profile, b0d0, is subjected to a stress varying linearly from a maximum tension at
b« to a maximum compression at d0.
In the compression flange, as well as in the web, instability may occur at a certain critical stress.
-If the bulkhead of fig. i is again taken as an
example and when it is considered as simply
sup-ported at top and bottom, the flange d0e0 will be ih
compression over a height,..DE in the span between .211 stringer and bottom, as-may be seen in fig. 19.
- If, for the moment, the webs are left out, it is
clear that the compressiòn flanges of two adjacent half corrugation profiles form a rectangular plate of width b = 2b1, and of length a = DE, which is
STRINGER C D I I I '1 - -BOTTOM E a==n.X Rd D Fig. 9
loaded in its own plane by the compressive stress a, where a will be -assumed to have its maximum value over the whole length, a, of the plate.
MOreover, it is loaded in its own plane by the
forces R along its edges (see figs 4 and 5), which also will be assumed to be uniformly distributed.
Now the buckling- problem of the compression
flanges is reduced to that of a rectangular plate,
loaded as shown in fig. 9.
For such a plate, when simply supported along
its four edges, Ti-,n.oshenk.o [2] gives the following relation between and a:
xm2+ay.n2=ni(m2+n2)2.. (I)
where ai = buckling stress according to Euler, and
in this case given by:
rE =
12(1 - vb)
(t\2
E Young's modulus Poisson's ratio V =b = width of plate and
=
a = length of plate and
= a
t = plate thickness
rn = number of half-waves in the X-direction n number of half-waves in the Y-direction
in which the plate will buckle.
For any given value of ay, depends on m and
n. If m is assumed to be constant, then a may be
d a
dedúced from
.. = 0.
c1n
-2(m2 + n2 ) OEl: - = 0, from which n2 may
be solved.
By substitution in (I), the minimum value of
cr is found to be:
a
=a(l
)4m- rE
In this solution n may have any value, but ob
viously the plate can buckle only in a whole number of half-waves, so only these solutions are important.
Assuming m = i and taking n = 1, 2, 3,
equation (I) represents a family of straight lines,
b
for any given value of the ratio
=
Taking as an example the bulkhead of fig. 1,
it is to be seen in fig. 19 that a DE = 240 cm
and b = 2b1 = 48 cm, so a = 5 or = 0.2.
The set of straight lines represented by equation
(I) for = 0.2 is shown in fig. 10.
All those straight lines are tangents to the
para-b X
/
Ybola: r = ...(1
ay ), independent of the value of .'5v
From fig.
lo it follows that when rx = O,
= 4 r and that the plate wilibuckle in 5
half-waves and in the case under consideration a = 5.
The plate will buckle in the number of
half-waves, nearest to the ratio
=
With increasing o, r. will decrease and the
critical value of r. can be deduced from formula
(I) or, for the case taken as example, from fig. 10,
where the shaded line indicates the critical values of
c and av. It can be seen that with increasing rx, the
plate will buckle in 5, 4, 3, 2 and i half-waves.
From formula (I) it follows that in this example, where Ç = 0.2, n would become < I for any value of a < 2.04 r1, which of course is impossible.
This means that in such a case the plate will
buckle in one haifwave, the formula (I) cannot be
used; the critical combination of r and r may be taken from the shaded line for m = 1 and n = i
in fig. 10.
For practical purposes the critical stresses may be
approximated in a simpler way for any value of
0.25 or a 4.
If in fig. 10 the parabola x = Y
(1 - 4) is
drawn in, it is readily to be seen that for all values
Fig. IO
AIL THE STRAIGHT LINES ARE REPRESENTED BY THE EQUATION = .(m'+ fl'.
WHERE m = i AND = b/a O T
1-5 MEANS m = i (X.DIRECTION)
n = 5 (Y-DIRECTION)
THE STRAIGHT LINES ARE TANGENTS TO THE PARABOLA
. . (i - .WHICH HAS ITS APEX IN
= v. v = 2
AND INTERSECTS THE AXES IN (0.0) AND (0.4 G)
of > 2 rE the parabola does not materially differ
from the shaded line indicating the critical stresses. As for all values of ,the straight lines represented
by formula (I) are tangents to the same parabola it is proposed to replace the broken line which
indi-cates the critical stresses by this parabola for all
values of 0.25. The critical stresses determined from this parabola will be slightly smaller than those
deduced from formula (I) and therefore on the
safe side.
It is clear that for values of a < 2 rx this
para-bola is not to be used, but to be replaced by a
straight line a = rE. If in this way, for a < 2 rE,
a is taken as = or, a slightly smaller value will be
found than from formula (I).
A graph representing the approximated critical values of r and r for all values of 0.25, is
shown in fig. il.
In the foregoing discussion m has been assumed
to be = 1, and indeed this is the only possible
vaitLe of m.
By taking m = 2,
a simple calculation will reveal that a larger value of is found than form1.
The plate will buckle in only one half-wave in the X-direction.
i 2
The assumption of m
=
i corresponds to themost unfavourable condition, and formula (I) might
be written: r +
ry n2=E
(1 +
2For a plate under compression in the Y-direction only, Tinioshenko [2] expresses the buckling stress in the following way:
2.E
(\2
12(1v2)
where k =plate factor, depending on the ratio
,
the edge conditions and the number of
half-waves, as shown in fig. 12.
In the case of the simply. supported plate, the
minimum value of k is 4 for all values of n, and for
r,-CT = (II)
long plates k remains practically constant at a value of 4. Only a small error is made if, for all values of
1, the plate factor is taken to be 4.
By substituting a variable k' for the plate factor
it would be possible to take account of.r in
for-mula (II). The new factor k would depend on
the ratio - and could be taken from fig. 11.
In the foregoing discussion a simply supported plate in compression has been considered, but the
influence of the hydrostatic pressure in the Z-direc-tion, to which the compression is due, has not been taken into account. This waterpressure tends to
pro-duce membrane stresses in the plate, and these, in
FOR ALL VALUES OF i 0.25 OR 2 4, THE CRITICAL
STRESSES AND , (OR THE VARIABLE PLATEFACTOR k')
ARE GIVEN BY THE FULL DRAWN CURVE, FOR = '/ OR = 4, BY A BROKEN LINE
?='/
OR 2=2,,,,, = i OR 2 = 1, BY A STRAIGHT LINE Fig. 11UNIT
E 3 11k 12 13 4 05 10 15
=a/b_
Fig. 12 2.0 25 3.0turn, tend to increase the stress at which instability
will occur.
This phenomenon has been described by one of
the authors, Prof. ir H. E. Jaeger, in a recent number
of "Bulletin Technique du Bureau Véritas" [3].
Levy [4] has shown that this increase is very
small for plates as used in shipbuilding, as the ratio
f/t is small, f denoting the total deflection of the
plate and t being the plate thickness.
Hence, this increase of the buckling stress may
be neglected.
The web of the corrugation profile, subjected to
a linearly varying stress, might be treated in a
simi-lar way. A critical stress would be found for the
special way in which it is loaded in its own plane.
Thus, a plate factor kwEn might be deduced, con-sidering the web as a rectangular plate, simply sup-ported along its edges.
Returning to the actual bulkhead it is obvious
that the buckling problem is not as simple as that
of a simply supported flat plate with free edges.
Actually the plate is part of a trough of the
cor-rugated bulkhead, and the webs adjoining the plate
in compression will contribute a considerable stif-fening against buckling. The interaction between flanges and webs, in this respect, has been
invest-igated by Caldwell [5], who introduces plate factors
for flange and web kb and kb. which differ
some-what from those used by Tinjoshenko, but are
related in the following manner, as shown by Miles
[6].
=t v'kn
or kFL andk2
= z
y' kwr. or kwEB b,The linear distribution of ry across the web is
replaced by Caidwell by a uniform compression, but a reduced breadth of the web is calculated so that in
the equivalent section the critical stress is equal to that at the edge of the actual web.
If K = plate factor of the original web and
kb plate factor of the reduced web
then the critical stress in Caldwell's
no-tation is expressed by:
K2D
kb2.D
= t
.(2b,)2 - t. (2. X. b2)2
where:2b2 = breadth of the actual web
E'
D
= 12 (1
- y2)
2 X b2 = reduced breadth of the web
Hence:
=
Theoretically, however, an exact solution cannot
be obtained in this way as no account is taken of the different number of half-waves in the buckled form of the actual web and the reduced web.
CaIdweil determined the most suitable reducing
factor X by experiment and found it to be X = 0.45. From his experiments he also concluded that after
buckling, the common edge of web and flange remains straight and that the angle between web
and flange remains constant.
An approximate solution of the problem may now
be obtained by treating a trough of the corrugation
as consisting of a plate of width b = /2b1 joined by 2 webs of reduced breadth 2 Xb2, all being subjected to the same uniform compression. The loaded edges
are assumed to be simply supported, the common edges to remain straight.
Both web and flange are assumed to buckle in
one half-wave in the X-direction and in n
half-waves in the Y-direction. This last assumption is not correct, but will give a good approximation.
The critical stress may be expressed in the form:
kb2 . D kb,2 D
t .
(2b1) - t
. (2 X b2)12b1 a
where kb1 depends on the ratio
2 X b. and on a
=
In most practical cases the ratios a are such that( k a
b
12.)
A -2 697 4.00-1277
The relation is shown in fig. lo kb1 6 kb 8 2 4 2 o between k11 and kb, 13. 2 b1 and kb1 kb2 2 3 2b1
-r-
u2 Fig. 13=
13k1 is virtually independent of a and is governed
only by
2 X b;
2 b1As mentioned, Caidwell found a value of = O.4
for the ratio X. In the case of the bulkhead used as an example in this investigation, the width of the
plate in compression is b = 2b1 (see fig. 1) and the breadth of the web is 2b2.
For t}e sake of simplicity, X is taken here as being
= O. and the expression for aycr may be written:
kb2 . E ( t \2 kb 2 E
(t\2
12 (1
_v2)
2b1) - 12 (1 vI)
b,1=
2b1
where kb varies with - and a.
In these formulae, no account has been taken of a in the flanges.
For a plate simply supported along all 4 edges,
this might be done by making use of a variable
plate factor k'FL in the formula:
Ycr = k'FL .
-12 (1 - y2) \2 b1)
k'FL.'2.E
( twhere k'FL is taken from fig. 11.
Obviously the form of buckling of a plate
stif-fend by the webs cannot be the same as that of
the simply supported plate, and hence the critical
stresses will be different. However, it is proposed to
make use of the same way of taking account of a by introducing the variable factor k'FL again.
For
a, = O,
in the simply supported plate,k'FL = k = 4, and if the assumption is made that
the influence of a is proportionally the same as for the simply supported plate, then the critical stresses may be expressed by:
for the web (actual breadth = 2 b2):
12 (1 - y2) b,
E
(t)I
for the flange (actual breadth = b = 2 b1):
a,.
=
. .1cr
4 12 (1 2) \2b1
k'FL
kb2.E
¡ t )2where the plate factors kb1 and kb are taken from fig. 13, and the variable factor k'FL, depending on
the ratio -, is taken from fig. 11.
rE
It must be repeated here that this solution of the buckling problem of a corrugated bulkhead does not pretend to be a rigoroustheoretical solution.
It is to be considered an approximation which
may be used to estimate the critical bending stress
cc
In the foregoing discussion it has been assumed
that nowhere in the plating is the proportional limit of the material exceeded.
As a tank bulkhead is designed to withstand the pressure of the contained fluid without leakage or
permanent deflection, it is obvious that the
pro-portional limit must not be exceeded, and therefore it would appear to be superfluous to investigate the behaviour of such a bulkhead in the plastic range.
However, as pointed out by Caidwell, such an investigation may be of importance with regard to subdivision bulkheads, for which the principal re-quirement is that the bulkhead will not fail when
an adjacent compartment is flooded. Such a loading
condition will rarely, if ever, occur during the
life-time of a ship, and in these circumstances the
bulk-head may be expected to suffer permanent
defor-mation.
It is interesting to note that in the plastic range
the critical stress may be found by replacing rcr by
aer E
, where r is the ratio
= - < i and E =
v'tv E
tangent modulus, as indicated by Bleich [7]. Even in the plastic range the bulkhead can
with-stand buckling, and the critical stresses in the plastic range may be expressed as follows:
for the web (actual breadth = 2 b2):
Ycr kb 2 E
(t\2
- 12 (1
y2)for the flange (actual breadth: b = 2b1):
a2.k'FL
kb2.E
(t
- 4
12 (1 - v2)where kb. and k1 are to be taken from fig. 13 and
k'FL from fig. 11.
2000
/
1760/
/
/
/
/
2390After calculating from the above formulae,
ry may be taken from fig. 14, where graphs of y
as functions of are shown for steel St. 41 and
St. 60.
The proportional limit rs. and the yield stress r
are also given for both steels.
A calculation of the critical stresses in this way will make it clear that local instability will hardly
ever occur in a corrugated bulkhead, where the
corrugations are dimensioned as is usual in
ship-building.
4. Stress distribution in the welds of the plating
Corrugated bulkheads are usually built up from plates which are bent to the required form of half
a corrugation.
The half-corrugations are then welded together,
and in this way the weld will be situated in the
cross section at a0 and e0 in fig. 3.
Stresses parallel and perpendicular to the direction of the weld will be acting in the weld. As indicated
by Roi, such a combination of stresses suggests an
effective stress which determines the strength of
the weld.
According to Roi, this effective stress can be
cal-culated from the following formula, based on
experiments:
rv = 0.046
+
+ V 0.91 r2 + 0.83 r112 - 0:69 r11 + 2.28
As this formula is meant for dynamic loading,
the effective stress v is called by Roi the limit
of fatigue.
In the notation of this report r = r of Roi and = r. The component stresses o and r1 can also
be taken from the ellipse shown in fig. 15, where
Oi, however, must be taken as r + .
Another method has been proposed by Vreeden-burgh, who calculates the resultant of the stresses in the two directions and gives an admissible maximum
value for this resultant, depending on the angle
between its direction and the direction of the
mi-nimum depth of the weld.
Both methods have been described by one of the
authors, Prof. ir H. E. Jaeger, in the "Revue
Uni-verselle des Mines" [8]. I: ST 41 i. = 1760, 2320 kg.cm' Il: ST. 60 = 2390, : = 3160 ka.cm' I 3000
/
I/
/
I / f 2500 1500 2000 2500 3000 3500 4000 5000 10,000 15 000 Y (kg.cm 2) Fig. 14According to Vreedenburgh there is a linear
re-lation between the admissible effective stresses due to static and to dynamic loading.
If this may be assumed, the ellipse of Ro (fig.
15) may be used as being applicable to staticloading,
as the unit of stress in this graph is the admissible r in the case there is a tensile stress, perpendicular to
the direction of the weld, only. From his
experi-ments Rof concludes that for dynamic loading the admissible r1 -tension may be taken at r
=
1500kg/cm2.
Fig. 15
In using the ellipse in fig. 15 for static loading,
a higher value might be allowed and accordingly,
the unit of stress would have this higher value too.
In any case, in using the value indicated by
Ro for dynamic loading, the calculation would beon the safe side.
From an examination of fig. 15 it is evident that
the most suitable place for the weld joining the
component plates of a corrugated bulkhead would
be where
is small, as r, =
the web of the profile.
r is constant across 15
liii
ji
ill
L
llUU!IIiI
IiIi1I!;IIII
su
hululAI
l,pp
liii
Ijo
::
t--r--te
u
uuuuuiuuuuuuruu
HiiflIViHII
fl
U
!'
I
2 ¡ I. J JADMISSIBLE STRESSES. AND IN . AND K -WELDS = ADMISSIBLE TENSION PARALLEL TO THE WELD
61 =
PERPENDICULAR TO THE WELDThe point indicated by the character I-I in fig. 18 complies with this requirement as x O and r is
oniy small.
Should the weld be placed in the neutral axis, in the web, then r O and a would be rather small,
but ' would be large. Since shearing is the most
dangerous stress in a weld it is evident that such a
proceeding would not be advisable, not to mention the practical difficulties it would present.
§ 5. Some not es on bulkheads with horizontal
troughs
In the Introduction, it has already been mentioned
that horizontal troughs are adopted usually for
longitudinal bulkheads in oil tankers, and, as stated, this investigation refers to vertical troughs only.
However, some remarks on the calculation of the stress distribution in this type of bulkheads may be made here.
The principal stresses, c. and rz, according to the
notation in this report (see fig. 2), as well as the shearing stress, can be found in a way similar to
that in the case of vertical troughs.
But as the hydrostatic pressure, q, varies
accord-ing to depth, it is different for every
half-corru-gation profile.
This brings about some complications in the cal-culation, as the reaction forces and bending moments
resulting from the interaction between adjacent
profiles are no longer symmetrical as is the case for vertical troughs (see fig. 6).
Moreover, the stresses in a longitudinal bulkhead
due to hogging or sagging of the ship are to be superimposed on the stresses resulting from the
hydrostatic pressure on the bulkhead, when
con-sidering the most unfavourable conditions.
These stresses can be calculated in the usual way
for standard conditions, and the stress distribution over the height of the longitudinal bulkhead is the
well-known linear distribution.
In calculating the effective stress, it is evident
that in general the highest values will be found
where a due to hogging is largest, that is to say, in the half-corrugation profile nearest to the bottom.
The compression r is combined there with rather
high values of r,., which is a tension, and hence a
high value of the effective stress will result.
On the other hand it must not be overlooked that in oil tankers the largest compression may occur in the deck in the sagging condition.
For this reason it will be necessary to calculate the effective stress in the longitudinal bulkhead near the
bottom when the ship is in hogging condition and
near the deck when the ship is in sagging condition.
It is not possible to say beforehand which of the two conditions will be the most unfavourable one.
The buckling problem in the case of the
hori-zontal troughs will probably be more pressing than
it is for vertical troughs.
Summarizing, it may be said that the problem of
the corrugated bulkhead with horizontal troughs, though corresponding in many respects to the one
discussed in this report, presents some entirely
dif-ferent questions and therefore requires an inves-tigation of its own. The authors intend to discuss
these problems in a future report.
§ 6. Coni parison between a corrugated bulkhead and a common flat bulkhead
One of the authors, I. van der Ham [9],
cal-culated the stresses in a corrugated bulkhead and in
a common flat bulkhead, loaded in the same way, basing on the following assumptions:
the thickness of the plate is the same in both cases;
the distance between the
stiffeners on the
standard form of bulkhead is
equal to the
'pitch" of the corrugations;
the section modulus per unit of width, s, is
equal in both cases;
the inertia moment per unit, s, of the corrugated
plate is 85 % of that of a common flat
bulk-head;
both bulkheads are supported on 2 stringers at
the same levels.
These assumptions coincide with the requirements of the classification societies.
The stiffeners on the common type of bulkhead
are toe-welded angles.
Cross sections over a unit of width, s, are shown in fig. 16. FLANGE Y s t NEUTRAL AXIS CORNER WELDS b, NEUTRALAXIS' Fig. 16
Van der Hain [9] calculated the effective stresses at the points a-e of the half-corrugation profile and found them to attain the highest values at d1 and eu.
In the flat bulkhead he found the highest values of
H
course, in the flange of the stiffener (see fig. 16).
The stresses attain a maximum value in the span betweensecond stringer and bottom somewhat below the point where the bending moment M, due to the hydrostatic pressure, is maximum.
The calculations were carried out for two kinds of corrugations, both of trapezoidal shape, one of
the regular type, where b1 = b2, or nearly = b2,
and the other of the shallow type, where b1 )) b2. Though it was to be expected that the assumption
of simple support at top and bottom would result in the highest stresses, a calculation for clamped
ends was also included.
As there is a tendency in the construction of
cor-rugated bulkheads to avoid brackets, no brackets
on the bottom were adopted in this case. The results
of these calculations are to be found in Table I.
In general, the stresses in the common flat
bulk-head are higher than in the corrugated bulkbulk-head, and as might be expected, the difference is largest for the regular type of corrugation with clamped
ends.
But even in the case of simply supported ends the
stress remains small and it is quite possible when
designing a corrugated bulkhead to keep the stresses within the elastic range.
It is
rather remarkable that the rather high
effective stressesin the common flat bulkhead
differ so little in the two cases of end supports. This may be partly explained by the local stresses in the stiffener giving rise to considerable stress concen-trations.
The stresses in a corrugated bulkhead are more
evenly distributed (see Tables II and III in
Ap-pendix 1).
The advantages of a more efficient stress distri-bution are evident.
With regard to the weight of a corrugated bulk-head, it does not matter if a small number of larger
corrugations of the regular type are taken for a
certain width,, or a larger number of small
corru-gations; the expanded width is the same in both cases.
By adopting a shallow form of corrugation, a
slight gain in weight may be obtained, as compared
to the regular form.
An appreciable saving of weight, in comparison
to an equivalent flat bulkhead, can be achieved only by decreasing the plate thickness, but then, in order to ensure adequate strength, the troughs should not be too shallow. Moreover, a minimum thickness is
imposed by the danger of buckling.
As the plate thickness of a bulkhead in an oil tanker is chiefly governed by requirements as to
corrosion, there is no advantage in the adoption of shallow corrugations. Consequently, the saving of
weight does not play an important part in the choice
of the type of bulkhead to be employed in an oi1
tanker. 17 C .2 -'C V O C .2 C o
-i
B o - N.. ,, O 'qi
I
-
N.. C' 1._ 'o'
N.. C' C.. LCC'
N.. -C. o = ..0 o 'C oI.
.-...-
t'... c'i l0lic
1.C) F'.. . I. I I -C .,. o0 2 o 1. :: -o 0. ':' 0. 8 .-'12 '.V'Q o -'C t"."o
LC.,.C'4 I, " C'C '.0 I I o --j CC' oC' 'C-
-t'..-
C c_4 -'7 -. 'C -j C -j V U c_C 'C " '.0 '.0 '7 II IC.0 IC' O o-.
I N..N. --j -N..'.0 0 '.0 C_4N.. c_4-
'7-
C_4 -N.. 'C iiOn the contrary, the gain in weight may be
con-siderable for corrugated bulkheads with shallow
troughs in ordinary cargo-ships, where, as pointed out already, even stresses in the plastic range need
not be considered beforehand as being inadmissible.
By keeping the height of the corrugation profile
equal to the height of the stiffener of the equivalent
flat bulkhead, loss of volume in the hold can be
avoided. But as in this case the component plates
may be thinner, the weight of the corrugated bulk-head is reduced.
For a bulkhead in a cargo ship of 10,000 tons deadweight, Van der Hain calculated a gain in
weight of 37 %.
For this particular case, he also examined the
pos-sibility of local instability of the bulkhead plating and arrived at the conclusion that the corrugated
bulkhead is superior in this respect as well, as might be expected.
APPENDIX i
Three-moment equations
For a continuous beam simply supported at the ends and at one intermediate point, if the supports
are in line, the three-moment equation is in general:
in / ill ill +I ill +i
M11-1 .+ 2 M.,(+
J + M,±In \In 111+1/ 111+1
6L11 6R11
= l . h
- L
+ . L +where:
M , Mn and M11 ± are the moments at the
supports
L1 and h + are lengths of span In and In + are inertia moments L11 = first moment, referred to the left-hand
sup-port, of the area of bending moments on
span ln, as caused by the load on this span, considered as a separate, simply supported
beam
R11 = as above, but on span in + and referred to the right-hand support.
In the case of a continuous beam on m supports,
and simply supported at the ends, there are m - 2 equations for the m 2 unknown moments over
the supports.
In the case of the same beam but with clamped
ends, there are of course, m unknown moments and therefore two more equations are needed. These are provided by the end spans, for which the equations in general are:
6R0
2M0.11+M1.11=
- _l1
bulkhead shown in fig. i is taken as an example. The profile and dimensions of the corrugation are
also shown in fig. 1.
(z = 45°; h = 14.75 cm; b1 = 24cm;
b2 = 20.85 cm)
In this example, half a corrugation, at about
midway between centre line and longitudinal
bulk-head, is considered as a continuous beam, both
simply supported and clamped at top and bottom of the bulkhead.
In this case the beam is loaded by hydrostatic
pressure, the distribution of which is represented by a trapezoid.
The bending moment for a span l, considered as
a separate, simply supported beam, is:
My=611.{_3111.y2.Qn_i+yl.(Qn_i
Qn) + ln. y (2 Q + Qn))
where:
Qn
- =
load per unit of length over support
n-1
Qn = load per unit of length over support n
When q = hydrostatic pressure per unit of area, thç load per unit of length Q = 2q(b1 + b2 . cos a) (see fig. 4). It should be noted that only the
com-ponent of the
total load perpendicular to themiddle plane is considered. For the span In:
L11
= f y.Ms.dy =
(7Q + 8Qn)
and for the same span:
=
+ 7Qn)For the beam under consideration, the geometrical properties are:
505 cm; 1 = 14140 cm4; t = 1.05 cm
l9=3lOcm; 12=14140cm4; t=1.OScm
l=290cm; 13=16830cm4; t1.2Scm
Taking a head of water of 245 cm, the hydrostatic
pressure and the load Q, over the support, will be:
q0 = 0.245 kg/cm2; Q = 18.99
kg/cmq1 = 0.750 kg/cm2;
Q, = 58.125 kg/cm
q2 = 1.060 kg/cm2;
Ql = 82.15
kg/cmq, = 1.3 50 kg/cm2; Q = 104.62 5 kg/cm
Substituting these values in the equations, the
bending moments can be calculated and then the reaction forces at the supports can be computed as
well.
For the continuous beam with simply supported
ends is found:
atA: M0=
OR0= 6140kg
at B: M, = 984320 kg.cm; R1 = 24730 kgat C: M2 = 630160 kg.cm;
R2 = 25520kgatD: M3=
O R0 = 11910 kg and 2 1Vl,,,.i1I-1Vl1_1 .l,,= Im 6 Lrn= RENDING MOMENTS
D = SHEARING FORCES q ' HYDROSTATIC PRESSURE.
SCALE
F
For both cases the curves of bending moments and shearing forces are shown in fig. 17.
In table II, bending moments, shearing forces,
bending stresses and shearing stresses, for the con-tinuous beam with simply supported ends, are given
at intervals of .50 cm, both for the outer and the
inner side of the flange of the profile, in relation to
the neutral axis. In addition, the strésses in the
X-direction are. calculated and collected in Table II.
The distribution of r, over the cross section of a
-half-corrugation profile is shown in fig. 18. The notations at, all, b1 and b11 in table II refer
to fig. 18.
It may be noted that a. in the flange under
consideration, is a tensile stress, caused in b1 and b11
by R = Sx +
. Sz.. cos a + b2. q . sin a, and inai and ail by R - S.
SIMPLY SUPPORTED ENDS
rIFI = / r2 +
-
r + 3
= 1.000,000 kg r,
20,000 kg -= 2 kg 1fl
Fig. 17
In table III a is the combined stress
+
from table IL
It is clear that the normal stresses in aI and cii are of equal value but of opposite sign. This is also the case for air and er, b1 and.dii, b11 and d1 (see figs 3
and 18).
Of course, the shearing stresses in d1, d11, b1 and b11 are equal.
Hence the "effective" stresses in ai and, eu are
equal, as is the case in ail and er, b1 and du, b11 and d1.
In fig. 19 graphs of the various stresses and
"effective" stresses are shown.
From. table III and fig. 19 it is to be seen that the severest "effective" stresses will occur at the corners d1 and b11 of the corrugation.
19
and for the continuous beam with clamped ends: is again a bending stress, compressive'ìn aI and
atA: M1=-806100kg.cm;
R0 8170kg b11, bu.t tensile in ari and b1.at B: M1 = 762910 kg.cm; R1 = 22290 kg In table III ay, a and r in ai, au, b1 and b11 are
atC: M-533010kg.cm; R2=23090kg
given as well as the "effective" stress according toFig. 18
APPENDIX 2
Geometrical properties of the cross section of half
a corrugation
See fig. 3.
Area of the section: A = 2 t (b1 + b1)
TABLE II
First moment of upper or lower half of area:
S = h.t.b1 + .h.t.b2 =h.t (b1 +
Inertia moment of area:
I =
.b2.t+ 2.h2.b1.t=2.h2.t(b1+
(First moment and inertia moment referred to
neutral axis).
The formula for I is a very good approximation, 2
as oniy the term .b . t. has been neglected.
As h
b2 sin , I
2 . b22 . t (b1 + ) . sin2Fora=45°:
I=b2.t(b1+)
Forb1=b2b:
I=b.t.sina
andforb1=b2=band=45°:
I=b.t.
t i for=
this is y in cm bending moment shearing focc D hydrostatic pressure q load per cm Q 3y ¡n kg.cm2 Ifl Dy in Ig.' m 2 IO t in kg.cm2 IO Dx2 in kg.cm2 111 2 kg.cm2 I Dx ifl kg.cm 1 rn kg.cm in kg.em ja kg in kg.cin2 in kg. m' all and b11 aj and b1 b1 and b11 b1 and b11 d1 and d11 b In auA o O + 6,140 0.245 18.99 0 0 154 +12
+18 227 +157
50 +281,610 + 5,090 0.29522.86 + 304 +283
128 +15+22 274 +190
100 +506,070 + 3,850 0.34526.74 +
+510 97 +18+26 320 +222
150 +663,700 + 2,420 0.395 30.61 + 717 -l---668 61 +21+30 366 +254
200 +744,770+
790 0.44534.49 + 805 +79
20 +22+32 413 +286
250 +739,630- 1,030
0.49538.36 + 800
+745 26 +25+37 l-59
+318 300 +638,590- 3,040
0.54542.24 + 690
+643 76 +28+41 505 +350
350 +431,950- 5,250
0.59546.11 + 467 +43k
132 +30+44 552 +382
400+110,040 - 7,650
0.645 49.99 + 119 +111 192 -1--33+48 598 +414
450 --336,85010,250
0.69553.86 364 339
257 +36+52 644 +447
B 505984,320
13,330
0.750 58.121063 991
334 +39+56 695 +482
B 505984,320
+11,400 0.750 58.121063 991
285 +39+56 695 +482
555488,940
+ 8,400 0.80062.00 - 528 492
210 +41+60 741 +514
605148,550
-F 5,200 0.85065.88 - 160 150
130 +43+63 788 +546
655 + 27,150 + 1,810 0.900 69.75+ 29 + 27
45 +46+67 834 +578
705 + 28,470- 1,780
0.950 73.63+ 31 + 29
44 +48+70 880 +610
755154,270 - 5,550
1.00077.50 - 167 155
139 +51+74 926 +642
C 815630,160
10,340
1.06082.15 - 682 634
259 +54+79 981 +680
C 815630,160
+15,180 1.06082.15 - 577 530
320 +45+66 693 +480
865 + 23,990 +10,980 1.11086.03 + 22 + 20
231 +47+69 725 +502
915 +462,080 + 6,580 1.160 89.90 + 423 +389 139 +50+73 758 +525
965 -t--677,410 + 1,980 1.210 93.78 + 619 +569 42 +52+76 791 +548
1015 +657,310- 2,800
1.260 97.65 + 601 +552 59 +54+79 824 +570
1065 +393,080- 7,790
1.310 101.53 + 359 +330 166 +56+82 856 +593
DuOs 0 --11,910 1.350 104.63 o 0 251 +58+84 883 +611
b. t3 t-In the last case the error is only:b, E FE
i
'r
GEFE / I //
/
/
a, SCALE Fig. 19 b,, EF F a , b 1000 kgcrn' 21 Ir EFE a, b, a,, b,, -GEF a, IN , b, e,, d,, e, d,TABLE HI Stresses in kg.cm2
TABLE III (continued)
APPENDIX 3
Calculation of bending moments M
In order to satisfy the condition that the slope of
the deflection curve at a0 and e0 mist be zero,
x=2(b+bz)
dx must be zero (see fig. 7).
As the curve of bending moments follows a
para-bolica! law, this integral can be computed in a
simple way. Y b11 b1 a10 -
+ 29 + 27 + 29 + 27
6 55 .788
45 +880 45 +645 0511
0 EFF 807 870 631 525+31 ± 29
+ 31 -i-- 29832 ±928 +680 540
705 44 44 0 0 rEFF 851 917 665 555.16? 155 167 155
-875 +977 +716 568
r 139 139 0 0 840 1090 812 .509682 634 682
C,815577 530 577 -30
±22 ±20 ±22 ±20
678 +772
+571 433
865 'r 231 231 0 0 rEFF 797 861 560 443 +423 ±389 +423 +389 o708 +808
±598 452
0 915 'r 139 139 0 EFF 1019 740 533 729 ±619 +569 ±619 +569739
+84:3±624 472
965 'r 42 42. 0 0 JEFF 1180 748 622 903±601 +52 '
±601 +552'770
±878 +649491
Ô 1015 59 59 Ø rEFF 1195 775 626 904 ±359+330 ±39 ' +33°
-r--800 ±912' +675 511
1065 166 166 0 0EF'
1067 850 585 ' 734 D,1105 a . O O O O Y b11 b0 a11 a0 A,0 00'
0 0 +304 ±283+'3°' +283
259 ±289 +212 168
50 'r 128 128 0 0 JEFF 536 '362 270 395 ±547 +510 ±547 +510 100 'r302 .
. 97 +338. ±248 97 0196
0 752 ' 480. 484 631 +717 . +668 ±717 ±668 150 r 61 +3876.1 +284 '0224
O EFF 944 590 626 804±805 ±79 ±805
+749 200 r391 .
20 +43520 +318 '0254
0 rEFF 1057 652 702 903 ' ±8004
+7 +800 +355 +745281
250 r 26 +48426 0 0 1085 656 694 918 +690477
±643 ±690+391 +643309
300 r 76 +53376 ' O ' O EFF 1025 610 599 841±467 +434
+467 ±4345,22
+582+426 338
350 'r 132 132 0 0 887 572 448 670 ±119+1'll
+119 ±111-565
±631 +462-366
400 192 192 0 0 EFF 715 672 416 432364 339 364 339
608 ±680
4:499 395 450 , EFF 257 692 25.7 0 1003 750 0 37.0 B, 505 r1063 991 1063 991
'528
700
492 '-528
±782 +5492
452
'r 210 210 0 0 rEFF 729 1171 955 474160
- 745
150 160
+831 +609150
483
605 1301 130 0 0 rEFF 716 943 703 428X2(bi+b,)
fMx.dx = 2.
( b13 +b3) 2 (b1+ b2)
q - Mo) =
(b13 + b23) - qb12 (b1 + b2) + + 2 (b1 + b3) Mo. .: t
+ qb12 (3 b12 2 b12 + +2 b1 b3 2 b32) = (b12 + 2 b1 b2 2 b22) Then: M2 = M0 = Mi) (b13 + 2 b1 b2 - 2 b32)MI,=MI=M
bq
=_ (b12b1b2+b22)
M0M -
(b12b32) = ( 2 b12 +
+ 2 b1 b2 + b22) qb2 For b1 = b3 = b: M2 = Me = -APPENDIX 4Application of the three-moment equations to an
expanded strip
In applying the three-moment equations to an expanded strip, cut from a panel of a corrugated
bulkhead, provided the panel is assumed to be of
infinite breadth, it is clear, from symmetry, that
the, moments M21 and MN are equal.
Moreover, for the same reason, MN = M11, and M11 = M5, hence M31 = MN = M11 = M5 = M.
'(See fig. 8.)
The three-moment eqúation reduces to:
2 M2b1 + 2 MN(2b1 + 2b2) + 2 M11b2 +
+(8b,q+8b33q)=0
or 3 M.(b, + b2) + q (b1: ± b3) and M = - (b12 - b1 b3 + b21), as before.The reaction at every support is q (b3 + b2) for
the expanded strip.'
For the actual corrugated strip it should be kept in mind that the reaction must be split up into two
parts. For instance at R, a corner of the actual strip:
qb2 perpendicular to NR and' qb1 perpendicular
to RS.
These partial reactions are to be resolved in
com-ponents in the direction of NR and RS (fig. 20).
qb1 qb. q
InNR:.
sin a+
tg a sin_Q
Sz2sina
2 and in RS: qb1+
qb2 tg a sin a cos a= q.
.(b1+b2.
a=
q(b1 + b2 cos a) cos
+ q. b2 sin
sin
=
.cosa+q.b2sina=RxSx.
This again is the same result as before. Only Sx, the shearing force in RS, is not found in this way,
but must be deduced in the usual way, as indicated
in § 1..
References
H. E. Jaeger: La détermination des dimensions des cloisons trans-versales construites en tôles ondulées plis verticaux. As-sociation Technique Maritime et Aéronautique, 1954, Paris.
S. Timoshen/ro: Theory of elastic stability. MacGraw Hill,
New, York, 1936.
[31 H. E. Jaeger: La détermination des dimensions des tôles soumises à uric pression d'eau ou comprimées sur la tranche et sou-mises latéralement à une pression d'eau. Bulletin Technique du Bureau Véritas, Février 1954, Paris.
S. Levy: Bending of rectangular plates with large deflections. N.A.C.A. Techn. Note 846 and N.A.C.A. Techn: Report
737, Washington D.C., 1942.
J. B. Ca/dwell: Bending strength of corrugated plate, Engineering, November 7, 1952, London.
A. J. Mi/el: Stability of rectangular plates elastically supported
at the edges. Journal of Applied Mechanics, Volume 3,
page A 47, 1936, and Thesis at the University of Michigan,
Ann Arbor (Mich.), 1935.
F. Bleich: Buckling Strength of Metal Structures. MacGraw
Hill, New York, 1952.
H. E. Jaeger: La détermination des dimensions des soudures en construction navale. Revue Universelle des Mineé, Mai 1953,
Liège.
I. van der Ham: Vergclijkendc spanningsberekeningen van door waterdruk belaste vlakke schotten en .vouwschotten. Schip en Werf, February 26 and March 12, 1954, Rotterdam.
q sin a R (b1 cos a + b2) = q.b, q.b, tg Sif101. s
23
cosy a -- sin2 a cos)=
M b = Md = qb2 M0 qb2= M2 = M0.
a (b1 + b9 cos a7. Introduction
By courtesy of the Netherlañds Dock and
Ship-building Company at Amsterdam, the results of
measurements carried out on a corrugated bulkhead
of a 16,000-ton tanker, were made available for
comparison of the stressesdetermined by experiment with the stresses calculated according to the method developed in Chapter I of this report.
These tests were made some years ago and were
only meant to provide some insight into the values of the stresses to be expected in a corrugated
bulk-head loaded by hydrostatic pressure. For this reason,
- 505 30 290 260 155 145 FLANGE 20 20 13 C» 10 11 14 BOTTOM CHAPTER II 96
only a limited number of resistance straingauges
were used, and no- attempt was made to determine
-the shearing stress. The bulkhead on which the
measurements were made, is shown in fig. 1. It is the
bulkhead taken as example for the application of
the three-moment eqúations and for the
calcula-tion of- the stresses in X- and- Y-direccalcula-tion.
Particu-lars of the bulkhead are to be found in fig. i and
in Appendix i.
The location- of the resistance strain gauges is
shown in fig. 2 1 . They were fitted only on the dry
side of the bulkhead, as- the technique of
water-proofing straingauges at the time was not considered to be -reliable, and as the bulkhead had to be loaded
by water pressure, the strain gauges on one side
would have been exposed to the water.
This is to be regretted, as a much more complete
comparison would- have been possible if straingauges
had been attached on opposite sides at the same
points. Moreover, strain gauges for the X- and the
Y-direction shouldhave been- used at every point to
be tested, as the stresses, in a case of plane stress, have to be calculated from- t-he well-known
for-mulae: - -
-y = Poisson's ratio, taken as = 0.3.
- As can- be seen from fig. 21, o. and were
measured at 6 locations only (ños 2-6, 4-7, 9-13,
11-14, 16-20 and 18-21); at
all the remaininglocations only t, was measured.
- The reason for this omission is to be found in the
assumption that a- is constant over t-he width of the
flange of the half-corrugation profile, and
there-fore t, as well, and that the distribution of r. and thereföre of e. across the height of the web is linear
according to the elementary beâm theory. If this
assumption should be correct-, r = O and y = O at the neutral axis of the profile (locations 3, 10 and
17 in fig. 21).
-It would then be sufficient to meásure on the
flange only. -
-Obviously, the influence of the lateral contraction must be taken into account if such an assumption is
to be justified.
Considering the contraction in the X-direction,
caused by a, it
isclear, from the shape of a
section ii-i the X-direction, that it may be assumed to have its way and thus will cause no additional stress -in the X-direction.
On the other hand, it may -be assumed that the
contraction in the Y-direction, caused by ,will be
prevented and só will bring about extra stresses in
the Y-direction.
- An elongation or shortening of a vertical strip 24 - x + ' y
+
-=
E . E and rv , .=
- 1v-where e and1v-are the measured strains and 660 SCALE - 100cm Fig. 21 M N 0 EXPANOEDPART M-N R S OF BULKHEAD ALI DIMENSIONS IN cc - I - 50 cm', 232.5 FROM CL 155 FROMC.L DECK A WEB 10 Al FLANGE 21 19 24 20
(in the Y-direction) of the bulkhead might be
considered possible, depending on the rigidity of deck, bottom and stringers, but a is principally abending stress and therefore linearly distributed over the thickness of the plate, so that it will be
tension on one side of the plate,but compression on the opposite side. Moreover, r' changes its sign over
the width of the flange, or the web, as shown in
fig.
18, where the distribution of the bending
stresses over a section in the X-direction is shown.
As a does not change its sign over the height
of the bulkhead, lateral contraction would bring
about elongations in the Y-direction, where r is a
compression (-), and shortenings where a is a
tension (+).
Deformations, such as these, in the plane of the
plate, might be produced by a bending of the plate, but in this case it would mean curvatures in
oppos-ite directions of the long edges of the plate and its
centre line, which is impossible.
In conclusion it may be said, that the assumption
of the prevention of the lateral contraction in the
Y-direction is a reasonable one.
Then it is sufficient to measure at some point of the flange only. As there will be no additional
caused by lateral contraction, - will be constant
over the width of the flange, and
,. = O at the
neutral axis of the profile. In comparing calculated
and measured stresses,
it must be kept in mind,
that additional stresses in the Y-direction will be set
up. These have to be superimposed on the bending
stresses r., computed from the three-moment equat-ions. The additional stresses have the sign as
indicat-ed in fig. 18, where + = tension and - =
com-pression, and are to be computed from - V rx. As mentioned already, these assumptions had to be made because of the limited number of straingauges used. Nevertheless it is thought worth while
r-145 145 1SSi--l55 M, = - 557630 q, 0,955 q, =0755 M.., 305520 q, = 1 060 q,=0665 M, 365430 755 q, =0465 M.., = 40100 M.., 666670 M,., 544050 35280
°7a4:
0245 630160 = 1155 M, = 984320 q, = 0865 LOADING CONDITION II q,= 0.555 M,., = 428190 M...66910 - 630530 245LOADING CONDITION III
q = 0.355 M,., = 135340 LOADING CONDITION IV 155 M, 69500 660 M,., 727450 350 960 260
25
ALL DIMENSIONS IN cc,BENDING MOMENTS M IN kg en,, HYDROSTATIC PRESSURE IN kg cc, Fig. 22. Corrugated bulkhead viith electric strain gauges fitted. Fig. 23
q = 350 LOADING CONDITION I
M,., 11070 q,=0.155
to compare the results of the experiments, obtained
by basing the calculation of the stresses from the
recorded strains on those assumptions, with the
results of the calculations based on the method dis-cussed in Chapter 1.
8. Strain in-casurements
For the tests 2 1 resistaice sraingauges were used,
the location of which is shown in fig. 21 and
illustrated in fig. 22. The strain gauges in theX-direction were fitted at three different distances
from the top of the bulkhead, the numbers 1-5 at
Y = 260 cm, about midway between the top and
the first stringer, the numbers 8-1-2 at Y = 660 cm,
midway between the stringers, and the numbers
15-19 at Y = 960 cm, midway between the second
stringer and the bottom:.
It will be noted that the remainder of the strain
gauges, those in the Y-direction, are attached at Y = 250 cm, Y = 650cm and Y = 950 cm.
For the calculation of the stresses it is necessary to
know both .. and at the same level on the
bulk-head. The recorded values of have to be corrected for this difference.
TABLE IV
Assuming the strain measurements to be in ac-cordance with the calculated stresses, a correction
of t can be computed from the bending moments M at- the locations of the strain gauges,
As has been shown already, it is reasonable to
assume that depends only on the value of ry,
calculated from the three-moment equations.
If
is measured at Y = 260 cm and
t atY = 250 cn, then
M.
21111 =
M =2r,0 .Y=25Ø
Readings were taken for 4 different- loading
con-ditiòns, which are defined in fig. 2-3.
The recording apparatus is pictured in fig. 24. In Table IV the results of the measurèments for
the, 4 loading conditions are given in "microstrains",
where 1 "microstrin" = 10 X
= io X E.
E X "microstrains"
= 2.12 >< «micro
Hence: r lo;
strains".
0) (Two sets of resistance strain gauges were used, the- correction factors, being different, as indicated in tie table)
direction and number Readings * ) correction E'acor
Recorded strain in "micro-strains"
-II 1V - -i ii -ni iv X, 1 +200 +160 +128 + 93 0.966 +193 +155 +124
: + 90
X, 2- 32
255
206
156
0.987321
252
203
-154
X, 3 - +123± 98
+ 79
+ 57
0.966 +119± 95
+ 76
+ 5
X, 4357
281
227
171
0.987352
277
224
--169 X, 5 +260 +204- ±163 +121 0.966 +251 +197 +157 - +117 Y, 6 +317 +254 +216 +176 0.966 +306 +245 ±209 +170 Y, 7235
187
143
102
0.966227
181
138
- 99
X, 8 - +286 +204 +138 -+ 72
0.966 +276 ±197 +133 -- 70 X, 9270
200
149
94
0.987266
197
147
93
X,1O ±100-+ 80
- 55± 29
0.966± 97
± 77
± 53
+ 28
X,11391
-273
193
117
0.987386
269
190
115
X,12 ±218 +168 +122.+ 72
0.966- - ±211 - +162 -±118+ 70
Y,13 ±107+ 96
+102± 82
0.966 - ±103 - + 93 -+ 99
+ 79
Y,14- 38
- 60
- 59
- 32
0.966 -- 37- 58
-- 57
- -31 X,15 + 93 ± 61± 32
+ 17- 0.966+ 90
+ 59
- + 31± 16
X,16194
- lOi
33
±
5 0.987191
100
33
+
5 X,17 +107+ 62
± 27
+ 11 0.-966 ±103+ 60
- +
26+ 11
X,18156
- 88
- 40
- 7
0.987154
- 87
- 39.
- 7
X,19 - +253 ±133± 42
+ 2
0.966 +244 +128+ 41
+ 2
Y,20 +47g +240 +105± 36
0.966 +459 +232 -+101 + 35 Y,21402
198
51
+ 27
0.966388
191
49
± 26
loading conditión I = hydrostatic pressure at bottom 1.35 kg/cm2 (fig. 23),, - ,, II ,, 1.15 5 ,, (fig. 23)
,, ,, III ,, 0.955 ,, (fig, 23)
Calculated stresses:
TABLE V. Loading condition I
-Loading condition II
= stress computed from bending moment M
bending stress
normal stress (tensile or compressive)
= combined stress =
+
= additional stress, caused by prevented cont'raction resulting total stress
=
-f--see fig. 18 see fig. 2 1 27. from * location on cross * * -number of strain recorded strains in mkro-strains" measured stresses in kg/cm2 calculated stresses - in kg/cm2 -. -
-0P section gauges . lx y -C1 x1 x2 x Cyc
ail
