This report examines the stipulation contained in the 1918 BLV, that a girder subjected to longitudinal compression under the influence of half of the specified breaking load, along with Euler bending safety with an initial deflection of 1/200 of the length of the girder can, at most, be subjected to half the stipulated breaking strength

Hoff, Wilhelm

English

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y. /’2
/3 -/
CCM!UT.4?’IYEFXkWTION OF’RENDING STRICNGTHOF GIRDERS
ORIGINALL3?’OUR’VEDAND SUBJECY#T.)TO LONGITW)INAL CX)MPRXSSION.
By Wilhelm Hoff
From ‘Zeitsclmift f& Flugtechriikur..dI[o$oxluftschiffahrt,”
April 13, 1922.
.
October, 192&
.
https://ntrs.nasa.gov/search.jsp?R=19930083156 2020-06-17T19:44:36+00:00Z
, ,, ,, , ,...,.! .m.-— ------ . . ... . . ... . . .--———-——
,,..,,.,. .... . ..... . . . . . ,., ,., . . –,, ., --
The 1S18 military specifications (BLV) c~ntaj,ned%li&i!ijlJ.owA
ing paragraphs:
“It must not only be demonstrated that the longitudinal
Stress generated i-na girder is bel.omthe Euler compression force,
but also that, with an initial deflection equal to 1/200 of the
length of the girder, the stress produced in it by half the pre-
scribed breaking load, does not exceed half the breaking strength.
“Short compression struts, whose cross-sectionmay have a
radius of gyration i, must in so far as G/i<lo5 for cast iron
and steel and s/i6110 for wood, not be computed according to
the Eulex, but according to the ‘1’e~ajer Compression formula.
Th$s stipulation is based on an article by MfillerBreslau.**
It is accordingly requested that at least double the actual load
on the levea arm, 1/200 of the length of the girder, should at mo:
create a stress near the limit of proportionality.
Since it is diffioult to determine the actual load in air-
plane construction and since the introduction of the breaking loa$
into the strength computation was customary, the MfillezBreslau
requirements, of double the actual load and the limit of propor-
tionality, were conve=ted into the stipulations of half the break-
ing.load and ~alf,the breaking strength. . .
The following consideration is based on the designations in
* From “Zeitschriftf~ Flugteohn2k und Motorluftschiffahrt,“ Apri
i3, 1922, pp. 92-95. (54th Repoxt of the German Experimental In-
stitute for Aviation, Adlershof.)
**MfiUer Breslau$ “~er exzentrisch gedrtiehtSt&be und tibezKniok-
festigkeit, 1, Des elnteilige Stab,tiin “Der Eisenbau,” September,
Fig. 1. The
indi~te,i,by
.,,..—..
girder s(cm)
the,expression
f =+%r
~g.~
long nas an incipient p.arab~~ic~~-rve
.$,..,..,
.,... .,., “’”’
(1)
The curve is the greatest in the middle and has the value
fmax = r s.
Under the influence of the longitudinal forces S(kg), the
girder is bent to the position of the dash line. At the point P,
with the coordinates x and y I-f, there is a bending m~~lent
M’ S(y+f) (2)
This moment is held in equilibrium by the elastic forces of
the girder.
M=- EI$# (3)
in which E (kg/cmz ) is the modulus of elasticity and I (cmA)
the moment of inertia of the girder.
Expressions 1, 2 and 3, after the introduction of
k =
r
EI (4‘,
T
are combined in the differential equation
(5)
This is integrated in the usual manner, as also, after in-
troduction of the integration constants for x = o and x = s
and the+simpllfioation
m,, , . ,,,. -.,,,,,.. ..-. .-.-—----
-3-
r 2s (6);.,,,=,,,SE..I.,.,= !,..... . .,,,,.,,.,.’.,,,.,,.,.,,....
The solution:
The bending moment from equation (z) is transformed to
v=srss[co~g+(l -Cosa)!.. Q2 I k sin~-1sin a k 1
and consequently increasea proportionally
deflection, y s.
This bending moment attains, with x
r
Mmax J=8 r S -+-Cos +2
On the other hand
?!lmax= C3w
with the maximum ini+:~
= 1/2, its maximum vzl~-f
(9)
(10)
in which a(kg/cma ) represents the bending stress in the middle
of the girder and ‘W (cm3) the resistance moment of the girder.
According to the already cited stipulation of the 1918 milita-
ry specifications> the bending stress under the action of the nt3-
st:ess. FrOm this requirement
part of the bendicg load as longitudinal load, must “notexceed a
,,.,,.
certain nth part of the breaking
follows
K-b Sk
o=~-—
mF
(11)
-4-
in which the designa~iofis Kb (kg/ems ), for the breaking be?idin
stress; Sk (kg)0 for the Euler bending loaa and F (cm=), for the
.
,,. . .. $,., .,, ,,. ,.,, ,,, . -..,,”. ‘,.,,,.. 1,.,.,
,,, ,.,, ,,, .
cross-bectionalarea of the girder, aye newly introduced.
.
By empioying the Euler equation
SJ n2E I
=—
sinequation (6),wehave
By coia’bininge~~ations (9) and (10) ma introducing
(12) and (13), we have
‘1?g&- TT2EI 1
-1 r 1=i3z’s -1n ---3--7 ‘l-rCos —— EI—.S2
(13)
equation=
(14)
whence we obtain, for the length s of the girder, the quadra-:ic
equation
If D (cm)’designates the distance between the outexmost fib-
ers of the cross-section of the bent girder under considerations
we may substitute
(16)
If we make the radius of gyration i, through the equatior!.
i= e D dependent on D, we may also substitute (17)
i
—,. —— . . ..- ..
-5-
$=f=~ 2 D2
~n which are introduced
A ,= 2rn~
Kb
(18)
(16) and (18), we fin-
(1Q)
[
1
1
-1 (~c?
-l-r
Cos2./=- J
and
E
/
nE
=ne — (21) ‘
% m Kb
The quantity A is independent of the cross-sectionof the
girder and increases in direct proportion with the quantities r
and n and decreases with increasing values of m. The quantity
B increases -~ithincreasing n and decreasing m. .
This important result means that, for a given cross-section
of a girder, designated by the factor e, for a fixed ratio of
E to Kb, as well as for chosen quantities, r, m, and n, there i
z single ratio s/1), which fulfills the requirement that, with ~.z
initial maximum deflection r s of the s-long girder fcr the ~.’
part of the Euler bending load, also the nth part of the breaki:.
stress will be attaineCL
Since the value s/D, thus computed for any cross-sectiGn.
applies to all cross-ssctionspossessing gsonetrioally similar s2::
like e values, it constitutes a great simplification, as wiii
equation
-6- ,
demonstrated in the exmple foz cylindrical piFes,
producedby the bending load Sk is given by
(22)
Accordingly, for a fixed ratio” s/D, also the stress producs~
‘:)Ythe beit(i~ng load ~em~i.ns the s~,ne fox all simila~ c~oss-s~ctio~.
Computation exam~le.
section of a Cyiindriml
Cylindrical tubes,- For the cros+
tube the equations
and
F ; (D’-<).=---
a~ply, when D (cm)
side diaweter of the
designates the outside and d (cm) the in-
tube, hence:
In ?ig. -.39 e’ am d/n are plotted against each ctber.
After choosing the ratio E/Kb = 550, which corresponds ap-
proximately to the malleable, seamless ,steeltubes employed in ai.
plane construction, the ratio s/D is plotted against the quantity
r (Fig. 3) for the quantities m = n = 1.8, ‘m= n = 2.0 and
m =n= 2*2, as well for solid rods as fG~ tubes with infinitely
thin walls
The curves of Fig. 3 indicate that the influence of the tk.ic
-...
— “,< -
ness of ths walls vaniches for high valu~s of 2, i~e~ la~ge initi~
deflections of the girder. For a vesy small.initial deflection,
hence mall values of r, the influence of the thickness ‘ofthe
wall cannot be disregarded. lf the initial deflection is taken *OO
mall> me obtain s/D ratios, which lie beyond the region of appli-
.~.’.lonof Vke Euler “oendingfoxuula for which the abOVf3 cOTlSide~~-
,,tonalone holds good, The Euler-Tetmajez-limitsare in%zoduced irl
TO Fig. 3, in accordance with the ratio’vhich applies to cast iron
steel (See pazag~aph 2 of the quotation from the 1918 BL”Jat
beginning of this treatise).
g @fy’-~-
Solid cylindrioaalrod
.
Tube with infinitely thin malls
g
D
In the 1918 BLV,
~ivzs values of s/13
~ 38*9
.
the stipulated quantity r = 1/200 = 0.005
whioh lie bsxely above the Tetmajer limits.
If ~e examine the struts of air~lanes which have given satis-
factory service, we find no simple illustration of the distributio~.
of the s/D values”. ?Janyof them lie in the Tetmajer and othezs
far within the Euler region.
Since with slender wing stzuts, bucklings are not infreque
-.—
a.3 a cor..sequwwe of some injury an.a sitice.such struts have greats
e/D values than would correspondto an initial buckling of l/20G
or2= 0.005, a ~eparture fzom the 1918 BLV is justified,
The ordinarily employed seamle~s stee~ tube$ hatiea diame$er
ratio of d/D = (),94
r =0.02andm=n=
‘i’liestxess
‘fk
For this value, as aiso for the qu~fi+iti~s
2$ we have
: = 65’
for a cylindrical tube is found to be
For E = 2150000 kg/cma, for d/D = 0,9 and fos a/D = 65,
we have
Ok
If, as is usually
r =0.01 andm=n=
must be ascertained at
‘.
= WC) kg/cm*
the case, s/D is not computed for
2, but is differently emp3.eyed,then it
what share of the bending 3oad the ~esired
nth part of the breaking stress occurs. For this purpose, for
fixed values r and n with equations (19), (20}, and (21), the
m value must be found at which equation (19) is fulfilled,
In Fig, 4 m and s/D are plotted against each other for the
fixed values E/K-~= 550, d/D = 0.9, r = 0.01, and n = 2:
For s/D = 65, Correspondingto the previous computation,
m= 2. For smaller values of s/D, a larger value of m must h
used, for instance, the value m = 2.9 corresponds to s/D = 40.
lesser utilization
sm311 gix~sz.sthan
mmstance is based
of the resistance to axf~l eotipfessiohin
in large ones, In the computation, this di2-
on the bending strength.
The results explairi~din tie example for cylindrical tubes
nay also be found in a simi,la2way for other materia3.sand otk:
xmss-sections, Here it should be remembered tt-atthe value n
is specially aikaptedto the mate~ial. For girders with a high
elasticity limit, n<2. For wood, however, n = 2 is recom-
mended.
Goncl.usion.-?Yeexamined the stipulation contained in the
1918 BLV, that a girdersubjected to ~ong~tudinal compression un-
der the influence of half of the specified breaking-load, along
with the Euler bending safety’,withan initial deflection of
1/200 of the length of the girder, can, at the most$ be subjected
to half the stipulated breaking stress, We found a generally ap-
Miicable ratio fo’rthe relation’of the length s.(cm) of the
girder to the distance D (cm),between the outermost fibers of
the czoss-seotionsubjected to,bendtng. This vatio gives, for
all geometrically simi~ar girder cross-sections (also e = i/D
,.
constant), a le-ngthlimit fulfilling these conditions.
By means of an &a.mple, we showed that the stipulation of
the initial d-eflectionof 1/200 of the girder lerigthis not ful-
filled by compression struts i’nuse, but that, on the contzarY,
the assumption of a greater.initial deflection, e.g. 1/100 iS
advisable.
.
-m”-
Fig. 1
,“
0.353
c)
4
3
m
~
1
‘0 20 4i) 60 ~ 80 100 120 140 lfxl
T
JJFig. 4
I
H
I
~
I
I
~..– —._____
llllllillllllllmMBilllllililllll=
.
.
31176014373295 ‘-——


Computative examination of bending strength of girders originally curved and subjected to longitudinal compression

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