The tendency toward economy of material and lightness of structure has long since led to the increased application of beams having large ratios of moment of area W to cross-sectional area F. This paper tries to provide an answer to how thin the beams can be

Huber, Maximilian T

English

NASA Technical Reports Server

No. 793
.—— —. —..
BENDING OF Z?3!Al,fS03? THIN SECTIONS
By ~.[axtl~ilianT . Hllber
Instytut Bada6 Technieznych Lotnictwa
~,wartalne ~~r”
S~ramozdanie J. 3
Warsaw, 1930
.,
. Washington
April 1936
https://ntrs.nasa.gov/search.jsp?R=19930094624 2020-06-17T00:44:54+00:00Z
. . ... .,. .
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3117601441 1681
NAT IOifAL ADVISORY COMMITTEE
——-——————
!FECHNICAL MEMORANDUM
,, ,.. . . . . .
.
-— —.-—-—
TOR AX R0NA4UTIcs
No. .793
BENDING OF BEAMS OF THIN SECTIONS*
By Maximilian T.’ Huber
.,
1 .“ INTRODUCT1:ON
,..,
The tendency toward economy of ‘material and lightness
of structure has long since “led to t“he increased applica-
tion of betirnshaving large ratios of moment of area W.to
cross-sectional area F. According to the elementary theo-
ry of bending, within t-ne limits of application of HookeIs
law, the critical value of the ’bending moment M crit’ also
called briefly, though incorrectly, the breaking moment,
is proportional to W (for the same “beam material). Since
the weight of each unit of length of the beam is propor-
tional to F, the ratio I’:W is a measure of the light-
ness of a beam subject to any giv~~ bending moment. As
this ratio has the dimension CL] it c“hangqs with change
in unit of length L. This inconvenience is evidently not
possessed hy the ratio ,.
F3/2
:W=p
which is nondimensional and can therefore be used for the
comparison of different forms of cross-sectional areas
with respect to lightness. For a rectangle of base b
and bcight h, for example,
I’=bh,
that is, a beam of rectangular cross section is lighter
the smaller the ratio b:h.
Similarly, we find for an ellipse with semiaxes b
and 21 the ratio
p=4@~= ~7.09
—-————————_——————___..———________._.———___———— -————————————————
*“Zginanic belekprostych. o przbkrojach wiotkich.’t Insty-
tut ‘Badan Technicznych Lotnictwa, Sprawozdanie l~wart-
alne, NO. 3, pp. 5-13. Warsaw, 1930.
2 I?.A. C.A. Technical Memorandum No. ’793
...
and therefore for the same ratio of I) to h an ellipti-
cal cross section gives a considerably heavier beam.
These and. similar sections,belong to the class of
, close, or compact, cross sections. If the ratio b:h is
too small, there may occur a deformation of the beam in
the plane of the h axis: ??or tilis reason alone we cann-
ot go’ too far in decreasing the value of this ratio. The
magnitude of the critical bending moment Merit would then
be decided not by the value of If but by other cross-sec-
tional magnitudes alongside with the elasticity of the ma-
terial and the distribution of external loads. (See sec-
tion 15 of !!Study of I Beams~” Warsaw, 1923, by the auth-
or.) In this case we classify the beams as thin or slen-
der learns.
Now the most typical forms of cross section as well
as the most important from a practical viewpoint are the
double T, the ‘1’boxl!form, and the tubular (figs. 1 to 3) .
Let us investigate the lightness of beams of these
cross-sectional areas, that is; compute the ratio ~, as-
suming for simplicity that the thickness of the web is
very small in comparison with the other dimensions.
a) Double T section of width b and height h meas-
ured between centers of flanges of thickness 8. The web
is of thickness 8,
from which it is easily found that
The same expression’ evidently holds for the box cross sec-
tion with the exception that al is replaced by 262. It
.
clearly shows the advantagesof these sections compared
with the recta-ngular. It is clear that these cross sec-
N.A. C.A. Technical Memorandum No. 793 3
tions, too, in the case of too large a decrease in &l and
,,. b, and therefore ~.;undergo -deformation ,and.-the.,expres-
sions given by the usual bending theory lose their signifi-
cance unless deformation is avoided by the use of streilgth-
eaing ribs.
b) For the tubular cross section of radius r meas-
ured to the center of the wall of thickness 8,
I=mr38, 17=7_rr2fj,
$ =mT/g=
“ ‘JF=‘*”’RF
I?or this section the lightness ratio is, to be sure, less
than that obtained for the others, but on the other hand,
the stiffness of the curved tube wall is undoubtedly great-
er than that of t-he flat plates and therefore the tubular
section can be made much thinner.
To the question, How much thinner? we shall attempt
to provide an answer in our present paper.
2. THEORy OF llT~~~SVERSEll BEYDI~?G
It is well known that the exact solution of the equa-
tioils of the mathematical theory of elasticity is not yet
sufficiently developed to explain the behavior of deflect-
ed beams for the case where the displacew.ent of its points
is of the same order of magnitude as t’he least of its di-
mensions. More accurate results are given by the classical
theory of the bending of ‘Infinitely thinll elastic beams
leadiilg to the well-known equation:
1 M
–=.-r .
P
Here EI is a measure of t-he stiffness of the beam in the
principal plane, M the bending’moment at the section, p
the radius of curvature of the neutral axis. (A more de-
tailed consideration of this subject the reader will fiild
in the author!s work, ‘tCriterion for the Steadiness of
Equilibrium, IIpublished by the Academy of Technical ”Sci-
ences, 1926, especially in paragraphs 14, 17, and 19.)
But it is not difficult to show that this theory, too, is
4 N. A. C,.A*’ Techilical Menor’andum Ho . “’793
only approximate in all cases where p is not very large
in comparison With tb.e cross-sectional dimensions.
Let us imagine a 11fiberlt element of length ds paral-
lel to the neutral axis cut out from the beam, and choose
the Z axis as shown in figure 4 in the principal lending
plane. (The Y axis on the figure is Qirected toward the
olserver.) Then, the u~lit stress ~=l~z of the classi-
1
cal theory actiilg on each of the filer elements of area
dT = dy dz gives rise io two forces o dF inclined to
each other at an ailgle da and therefore not fulfilling
the coilditioils of equilibrium for this element. This
leads the ileighboring elements to exert on each other
transverse forces who,se resultant o d? da is perpendicul-
ar to the axis of the fiber element’ and- lies in the plaile
of bending.
This resultant is evideiltly determiiled by the trans-
verse stresses of which are fu-nctions of z and Y
and which the classical theory neglects.
The stresses al satisfy the equation of equilibrium,
Or substituting
Since
ds – P+ ‘= 1+ ~
.——— _
dso P P
d.al
——
dz (
l+”?-)=%
p/ p
Siilce t“~e presence of the transverse stress 01 modifies
tile.loil~itudixlal stress ~: we see that we may not, in
ge:eral, simply put 1.= AL as given by the classical bend-
P EI
ing theory. However, we may consider the a%ove as a first
z/p to be small compared withapproximation. ar.d assuming
1, obtain” the following expression for. the transverse
stress:
IN.A, C.A. Technical Hemorandurn i?o. 793 5
da 1 (s1/1 2——. = .— =
–% z,.—, .
.dz. EI EI - .,.___
As a result of the presence of the stress 0’ the fiber
element of longitudinal area dy dx is subjected to a
pressure which is always directed toward the neutral layer.
‘The stress on each unit of length of fiber is
The resultant of the stresses along the chord DD’ ‘(fig.”
5) exerted ly the area ??l on the remainder has the value
Here S denotes the static moment of the area indicated-
in the figure. The same dependence on S is shown by the
shearing stress per unit length of the beam arising from
the shearinE stress T, namely , ~! = ;5. Hence the
s-hearing stress PI on a deflected beam subject to a bend-
ing moment U has its largest value at the neutral layer
similar to the shearing stress T!.
The important difference in the two kinds of stresses
consists in this, that T1 has the same direction over
the whole area depending on the direction of T, whereas
the stresses P! are oppositely directed on both sides of
the ne~~tral axis, always pointing toward this axis.
It is clear that both these types of stresses play a
subordinate part in the case of compact beams and may be
neglected. In the case of slender beams, however, these
stresses may become of first importance and must be taken
into consideration to obtain a more accurate evaluation o.f
the strength of such learns.
3. DOUBLE T AND BOX SECTIONS
(Figs. 6 and 7)
If we consider the beam divided into ,sections by
planes at unit distance apart, then each section will act
as a uniformly loaded frame under the stresses PI (if we
—. . --.—
6 N.A. C..4. Technical Memorandum No. 793
neglect for the present the effect of any transverse
force) . The flanges will act as horizontal beams under a
uniform loading of magnitude,
aild the web through connection with the flanges will be
exposed to longitudinal stresses and bending. The web
will be acted upon first %y the flange stresses:
secondly, by the stresses:
distributed continuously along half the depth hl , and
finally, by the constant and statically undetermined mo-
ment 11!.
‘We compute the latter, using the principle of minimum
strain energy for the whole frame, considering only the
energy of bending, and introduce the following notation:
30 in kg cm transverse bending modulus of each sec–
tion of the flange of width, 1 cm.
Bl the same for the section of the web with a view
toward eventual strengthening by stiffening ribs.
1:11 in kg bending moment of a section of the flange
or wet of width, 1 cm.
Since the bending moments of the projecting parts of the
flanges do not depend oil the magnitude of the unknown mo-
ment If1;.1, it is sufficient to express the energy for the
middle parts of the flanges and the web. We therefore
have for the flange (fig. 8) :
in wl.ich
I
.N.A. C.A. Technical Memorandum No. 793
=.,(PJ
],t2& ho
P .—. — ——2. = ho .= ~~2,..z .
z .—2
/
. ..~””-“
.
/7 -,.-”
/“
.2
~~y,. t]~ew~~ Mt is simpl’y equal “to Ml ~:~ = 1 for both
parts and differentiating the strain energy with respect
to Ml, the condition is obtained:
or
Substituting the values and solving for Ml , we find:
M2 a ho 80 2a2 - ~%~
Ml = “————_ -—.-____.
6 E12 B.2a -t-–– ho
BI
It is seen that the most
web are obtained when
in which case the moment
under compression only.
ends has the value
advantageous conditions for the
a= approximately 0.8 a
M ~ disappears and the web is
The longitudinal force at the
or about equal to Q!. “’
Under these conditions the web,must be insured against
buckling and the critical values of If1 and Q’ must be
computed. We have to deal l~ere with the general problem
of Yasinsky, introducing some very laborious calculations
8 17.A. C.A. Technical Mcmorandurn No. 793
in order to o%tain an exact solution. In view o
proximate nature of the whole theory, however, w
sumc the beam to be loaded at the end sections o
plify the problem.
The value of the loads can be taken simply:
f the ap-
e may as-
nly to sim-
~1 +
——..
2
-Q. jyl
or
.-.
~1
1/12ho
_————
2X12 [( a a + ho-1-
I?o
mcnt
given critical val UC of the principal beriding
mo
and
where J?I de
of the web of
strengthening
?20 t
1
by
es
cm
“-v
the
mid
erti
ar
th,
cal
lea of cross secti
taking into acco
ribs.
on
un
of a section
t eventual
In the general case where Ml * o,
consider the web as a. rod on w~ose ends
pression forces of magnitude IT1 and v-o
case the coefficient for the cross sect
takei~ for a.width of 1 cm (taking stiff
sideration) can be computed to a suffic
proximation from the equation:
it
are
ment
ion
enin
ient
is necessary to
acting com-
S Ml. In th~t
of the web WI
.g ribs into con-
degree of ap-
——..——
‘crit
.—__——-.
g:.
‘)Nl~/
TT2
BI
-.—
ho Z
N’ E
—
The
tly
ing,
ocri
hy?
shearing
does not
but only
t in the
othesis.
for
affe
req
a b’o
IJet
ce 92 on the ~i
ct the cb.aracter
wires a decrease
ve expressions a
the normal stre
ven
of
in
Cco
Ss
cross section ev
the transverse
the assumed valu
rding to some sui
for an n-point
.e
t-
ide
ben
of
abl
il
d
e
N.A.C.A. Techilical Memorandum No. ’793 9
loading computed from the expressions given above, be equal
to, (s1 and-the continuous, str.ess..d,etermined by the trans-
verse force T, be equal to T1. If the material is of
elastic metal, then according to the most accurate present-
day energy hypothesis, the strain of the material is meas-
ured by the effective stress:
b
A certain complication in the computation is introduced by
the circumstance that CSl is”proportional to the square
of the load and not to the first power as T1 is. In
this case the effective stress will not be proportional &i-
ther to the load or to its square.
In those cases where the” load is carried in a given
mailn.eron one of the flanges, additional normal stresses
arise in the longitudinal cross sections which for the
case of thin sections must also be taken into account. On
this matter we must refer the reader to the work already
mentioned, IIStudy of I Beams,” pointing out, however,
that this work still follows the old hypothesis of maximum
coiltinuous stress, which in recent years has given way to
the theory of maximum strain energy used above.
4. TUBULAR CROSS SECTION
(Figs. 9 and 10)
The ring cut out by two planes separated by 1 cm dis-
tance, is subject to transverse bending by the forces per-
pendicular to the neutral axis and directed toward it.
The magnitude of these forces on a fiber element of area
dF = ~ r d~ is given by the expression found a’hove,
namely ,
If2
—–z Z dl?
EI
Dividing this’by r d~, we evidently obtain the force per
unit area of the cross section of the cylinder of radius
r, or’
.,
10 N. A. C.A. Technical- Memorandum No. 793
if we put C = j$ 5 (kg/cm3). Considering half of the
ring on one side of the neutral axis and replacing the
combined forces by Q arid the statically undetermined mo-
monts by 110, we easily find the expression for the bend-
ing moment for the section defined by the angle cp to be
v
1$’1= MO - Qr (l- cos Cp) + J C z r d~ r (cos ~- cos cp)
o
The force Q is obtained from the projections
whence Q=Cr2
?utting this value into the expression for M and performi-
ng the integration, we obtain:
M= Ii.- C r3 (1 - cos T) + C r3
(
3
~+*cos2q- Cos Cp)
Since 3$_= 1, the condition for the determination of MO
o
is of the form
n/2
fMav=o
o
whence lio = ~ C r3 and Ml = (M) = - $ C r3
cp=~
are the limiting values for the bending moment.
The dangerous section is evidently that which is act-
ed on by MO , since the largest longitudinal force Q
appears here. Computation shows that for a sufficiently
small value of &/r, it is possible for the stress result-
ing from this transverse bending to exceed considerably
the principal bending stress. In this case the tube is re-
inforced by strengthening members fixed to the tule wall
with distances b letween them (fig. 11). .The presence
of these stiffening members causes an increase in the trans-
verse bending’ stiffness of the tube sections of width b.
Under the most advantageous condition wher’e the stiffening
members are only weakly joined to the tube wall, the stiff-
ness of this section Bb will at least be equal to the
sum of the stiffness of the wall itself; that is,
N.A.C.A. Technical Memorandum No. 793 11
E
-———— Q--$% (where p, is Poisson’s ratio)
“’~’’’-’v~ “ ““ -“
and the stiffness of the member Bw = EIW
or
Iil this case the bending moment MO b divides itself be-
tween the wall and the stiffening member in simple propor-
tion to their stiffness, and therefore the mo’ment on the
wall will be
~lo = MO ——.--——————————..—.-.—
12 31 (1 - ~’)
1 + ––––~–––~-–--––
Ebb
As far as the compression force Q is concerned, a cer-
tain part, to be sure, is supported by the strengthening
mei~ber, but on the assumption of very loose Joining this
pai’t will be very small. We shall therefore assume in
what follows that the entire force Q is taken up by the
cross section of t’he tube wall. Under these conditions,
we compute the limiting compressive stress in the material
of the wall from the expression:
Considering that I = n ra~ and denoting the area and u.o-
ment of inertia of a member by J?w and IT, respectively,
we find on substitution of the proper values the following
expression for the limiting stress in the tube wall result-
ing from the transverse deflection caused by the principal
bending moment M:
or
M2
(
3rlO* = “–5––––Z–= 1 + – - —————
Tr Er& 261+(X )
If there are no stiffening members, then obviously, a
must be put equal to zero in the above expression.
I
12 3T.A. C.A. !lechnical Memorandum No. 793
Let us yet consider the case where the stiffening mem-
bers are ,perfectly joined to the tube wall. In this case
the transverse stiffness may be measured exactly as for a
plate of thickness a with ribs separated as distance b.
This stiffness is equal to that of a cross section formed
of tk.e plate together with the rib, having a so-called ef-
fective decreased width bl. This width bl differs from
b the more, the smaller ~ is compared to r, aild rz
compared to b. Oil the %asis of certain theoretical con-
siderations, bl may be assumed equal approximately to
With these assumptions, we obtain another expression
for O* , namely,
Q ‘b l:!Ob
CT*= –F– + –––– e2
2 12
Here Y2 denotes the section made up of the annular sec-
tion of the tube of width b and the section of one stiff-
ening ~nember; 12 is the moment of inertia of the comlined
sectioils of effective width. ‘bl: firially e2 is the dis-
tailce of the remotest filer in the section corresponding
to 12. Substituting the values of Q and 110, we ob-
tain:
or
312 b ( lF2erO* = ‘-–––––—–––– 1 -1-– ——————
,TT2I1 F2 r4 8 4 12 ),.
Translat~on ‘by-S. Reiss,
National Advisory Committee
for Aeronautics.
N.A.C..&.
8
Technical Memorandum lTo.793 Figs. 1,2,3,4
F—b— ‘i
l?igures1- 3.
BI ul+~xdz
z
(fx@
Crxdl’X&
ox xl?
Fi~re 4.
;_
(u
m
l
w
w.
-1m0 —>
N
Ln
.
u I
k--is ----=
w :— .
Cn
-3
.
<
.
m
—N.A.C.A.
.. . ,,,.,.,___ ,.
Technical Memorandum.No. 793
,.,
Figure 9.
“if
M.
T
Q
+-”k M.
Q
Figure 11.
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Bending of beams of thin sections

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