The author presents a comparison and combination of the numerical and geometrical determinations of the maximum M, when calculating the maximum bay moment

Muller-Breslau, H

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NAT IONAL ADVISORY CZOIdNUIITTEE FCR AEROiVsZlTTICS . 
*Nt ’ i  .- . - *-rc.r.r . * _. -./ 
_GP,.LCULJLT I O N  OF WING SPARS . 
3Y 
H. M6ller-Bresla;u. 
Translated froin 
!!Zeitschrift far Flugtechnik und Slotorluftschif fahrt 
Ajjril 30, 1920. 
... . , 
. 
https://ntrs.nasa.gov/search.jsp?R=20050241729 2019-08-29T21:07:49+00:00Z
1) ~ 3 1176014399944 CALCULA~ION OF WING SPAES. __- ____. - -- - 
I 
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I 
M h e  r-Br e s  1 a. 
2)  
e-=%*---, - e'- . . 
In t h e  same i s s u e  of t h i s  magazine, IMr. Ratzersdorfer re- 
f e r s  t o  my a r t i c l e  i n  t h e  October, 3919,. number and leads me f ; ~  
cmpare and combiiie t h e  numerical and geometrical  determinations 
of the rm,ximuni M. 
The object  of my 191s article,  viihich ;-;as i n sg i r ed  by Mr. 
11 
P r o l l ,  was the s im2l i f ica t ion ,  not only of t h e  aL3proximate, but 
a lso  of the exact ca l cu la t ion  of  t h e  maxinum bz;y mornent, whereby 
espec ia l  imgortance is  attached (even with a very low Euler  s a fe ty  
fac tor )  t o  the rapid and accurate c d c u l a t i o n  of  $' =n2 E J/S s2 . 3 )  
The geometric representat ion o f  t h e  bay moments cons is t s  i n  
mul t ip l ica t ions ,  p rescr ibed  by Tormulas,  of l i n e s  n i t h  c i r c u l a r  
functions.  M r .  Ratzersdorfer starts :3ith t h e  formulas i n  my 
%raphische S t a t i k a n a  ;iJ.hicb have o f t e n  been given i n  t h i s  magazine. 
1 w i l l  now use formulas develoi2ed i n  my 1919 a r t i c l e ,  p. 200. 
4) 
From 
Apr i l  30, 1920, 3 2 .  102-105. 
See N . A . C . A ,  f i l e  1116.247. 
The case of lower vtzlues of I' i s  e spec ia l ly  ixportant ,  s ince  
i t  i s  customary i n  airplane c o n s t w c t i o n  t o  ca lcu la te  -.-:ith 
loads near the  breaking s t r eng th .  
1 w i l l  soon give a report  on the experiments xhich sup- 
por t  t h i s  r;ethod. They were made, according t o  a plan worked 
out by myself, i n  t h e  Prussian xateridl-testirLg o f f i ce ,  f o ?  
both bending and breaking, rt i th t zo  bay ;;rooden spars furn- 
i shed  by the former "Flugzeugmeissereiti (Aircraft Inspect ion 
and Engineering Dspartment ) : 
See also  nix t r e a t i s e  "Zur T'rreorie der  Biegungsspannungen i n  
Fachwerktragern (On the Theory of Bending St resses  in T r u s s  
Girders) ,  Allgemeine Bauzeitung, 1885. 
Zeits chr i f  t fGr Flu@ echnik und Not orluf t s chif f akrt , 
From the formula 
x. 
&: t a M = C, cos 2 -t- Ca s i n  - - g k 
K 
I .  1 % .  
- - I  v .. ._" -
for the  moment at the  distance f r o m  the middle of the ba37 
(Fig- l), it d i r ec t ly  fo l lovs  t h a t  M + g k2 (Fig- 2) can be rep- 
resented by the projection . a d  of the' r ight  angle a b c, 
formed by C, and C,, on a straight l i n e  which, w i t h  Ci, in- 
closes the  angle 
It follows from the formulas 
U 
C, = D sec - 2 
t h a t  C, and C,, through t h e i r  project ions 
are determined on a st raight  line which, w i t h  C,, forms the 
'b 1) angle; - . 2 
. Almost without excep2;ion w i l P  now be required. It 
then suffices t o  draw the ri@t angle t r iangle  a b  c whose hy- 
potenuse a c gives the value 
- 
I 
g k2 + 
1) M and MB he sam8,direotion as g. In my 1919 a r t i c l e  i n  agreement 
w i t h  P r  1, I represented. MA and MB as positive re l ieving 
manents 
are positive &en, according to Fig. 1, they act in  
Y , 
.. 
.I *'* . . 
(1 If the  absolute v d u e  ox" tks- mzle  c a b 
then Ma,=. equals the gyeate;:. o f  %he end moments MA and MB. 
is grea te r  than 
Formulas (16). and (17) on p-200 (1919) can be combined into 
the following f o r n u l a  
2 2 %naX. = / C 1  + C2 - g . k 2  . . . . . . (1) 
which may also be obtained from Fig4 2. Usually Mmax, l i e s  so 
near the  middle of the bay, that w e  nay write 
C 2  
- 2% 
= C, + A  - g  k2 . . . . . . . ( 2 )  
i n  agreement w i t h  formulas (ll), (18) and ( 2 0 )  of my 1919 art- 
ic le .  If cc l i e s  i n  the neighSorhood of  n, then C 1  must be 
calculated. The formula 
on p.198 
factors ,  
of .  I" we 
(1919) and the table  on p.200, for v e r y  small safety 
lead quickly and accurately t o  the goal. For a low value 
may write : 1) 
1) Point d (Fig. 2 )  l i e s  on a c i rc le ,  whose diameter i s  a c. 
M r .  Ratzersdorfer determines t h i s  c i r c l e  b y  means of the condi- 
t ions M = MA f o r  x = 0 and M = MB for x = s. If a varies 
only a l i t t l e  from TI, then the three points determining the cis- 
cle l i e  nearly in  a st raight  l ine .  I give the calculation the  
preference and cal l  the at tent ion of readers, who likewise prefer 
t o  calculate, to t w o  tables:  
J. H r a b k ,  "Prakt ische Hilfstabellen" (Pract ical  Tables) (Pub- 
l ished by Teubner in  Leipzig). Table V, "Trigonometrische, Linien 
von Minute zu Minute,11' a l s o  contains the often used secants and 
o w e  cants - 
C. Burrau, IITafeln der Functionen Cosinus und sinus m i t  den 
natdrlichen Zahlen aLs Argument" (G. Reiner, Berlin) - 
The secant t ab le  with the argument l' (1919, p-200) a l s o  en- 
ables the calculation of C z 2 ,  since - 
cosec' = secZ 
sec2 - 1 
In  the numeric21 ~ x z ~ $ L P , E  cf xy 3.91.9 ar-bide, I made th9 c d -  
culations w i t h  excessive z,CO.Ij:zCVa becaiae 1 wished to obtain s 
----. ascurate a ccmparison -as p s s i b l e  of the results obtained by the 
different formulas.. I will therefore give still another exmplc 
f o r  the accurate determination of %=. together with the  round 
numbers t o  be recommended,. i n  pract ice  
Let J = 314 R = 64 ern.=, F = 28 an.*, s = 305 cm., 
S = 3840 kg., g = 2kg/cm-, E = 120,000 kg/cm.2, MA= -- 44.82kg.m. 
MB 2..- 324.4:kg.m.; ';%<A +.$I,) = - 184.6 kg.m. (M A -. MB> = 140 
kg.m. 
2 2- 
We obtain: 
g k 2 = g r -  E - 9812.5 g = 196.3 kg m. 
t 
a 1.041 +G.272 = 32.0 r = rr2k? = 1.041, sec  - = 
S2 2 0.041 
C, = (196L3 - 184.6)32.0 = 374 kg a. C, = 140 kg m. 
20300 + 3840 = 454 kg/cm,a 
64 28 . 
Q =  
For the determination of the  flexure, according to the oalcu- 
l a t ion  of 
Mo = g $- = 232.6 kg E. 
(Capare 1919, 3.198, formulas ('7) and. (4). 1 'The following formu- 
. e -  las serve: 
= 0.040 m. 203 - 232.5 + 184.6 3840 - v m a x  < 
- 5 -  
If t h e  safety f ac to r  is very large and a is consequently 
s m a l l ,  then the  moment g k2 = g sa /a2 becomes inconveniently 
*. 
l a r g e  for georfietrical-r2presentation. 2 )  
* 
For example, l e t  
_. 'A = - 0.628, - = - 0.096, MA MB = - 0-362,  
MO No - Po 
Ve obta in  f o r  Ido = 1 
C = (20-264 0 - 3 6 2 )  1.0515 = 20,927, C, = - 0.266 x 3-236 = 
- 0.861, Mma = 20*927 -I- 2 x 20.927 
1) For s m a l l  values of a 
supporting moments of s;?ars -zith severa l  bays. 
1 ,  
- 20.264 = 0.681 0 861" 
-the vzlues used f o r  determining the  
are cdcula-bed n i t h  t h e  forwiulas cb'iiaj-nea frowi a s e r i e s  develog- 
ment 
f o r  t-ihose usefulness t h z  statement vouches, that for& = 20' 
( a  = 0-34907) they s t i l l  give t h e  values 0.33604, 0.16904 and 
0-04217, which d i f f e r  only i n  t h e  fifth decimal glace from t h e  
exact values 0.33607, 0.16907 and 0,04218. 
by Itonig, i n  th i s  rzgazine,  1919, No. 2 1  I remarked that t h e  case 
of very small values o f a  ( i n c l u d i n g a =  0s i s  t r e a t e d  i n  ny 
" G r ~ ~ h ~ s c h e , S t a t i k "  Vol. 11, Chzg- 2 ,  3-  289. 
In t h e  in t roduct ion  t o  'che ar-cicle I'Zur Knickungsbiepng,ll 
. . 
0 
\ 
If we disregard thoJ infldence of I? on t h e  moments, we ~ K - F  
0 266 
2 1 - 0-362 + (-) = 0.656 
The calculat ion gives t he  r e s u l t  quickly and exactly for all 
safety factors, but presupposes the  use of tables of ' funot ions.  
11. 
In the l i m i t  case u = n, sec a = cr. z 
main f in i t e ,  D must equal 0. This gives 
If the moments t o  re- 
and C, = 0 x a. At le& one of. the moments MA and IdB must be 
a function of a .  1) 
A genera3 investigation of t he  value 
c, = (D S W % )  ' = - 2 (q 2 a=n dU UEfl 
w i l l  be given i n  a special article. Here I will confine myself 
t o  a simple example. I will t a k e  a beam (Fig- 3) centrally load- 
ed and r e s t i n g  on three rigid sug~ports, and t o  whose ends are ap- 
p l i e d  the moments MA and NB" If > then may 0; > IT . 
This may happen, without exceeding the p ropor t iona l i t y  l i m i t  
when, f o r  abbreviation, we w r i t e  
The in t roduct ion  o f  *hi:; vd.--ie i c t o  the  expression 
, .. ,.., , 1 
gives, after a simple intermediate ca lcu la t ion ,  
2 c ct a (sin o, - a, cos a, + y,'-w: s i n  = 
= + g ,  s.2 
1 ' 1  
[ s i n  3 (4 + a,2 + 2 Y % ' )  - 2 COS 9 I 2 
From t h i s  follows f o r  U, = n: 
v2 " 2)  9 2 + Y " z  
-+ 'A n - Mc r( 
Numerical. example.- 6, = 2.0 kg/cm. sT = 330 an., 
MA = -25,000 kg/crn., hence, g, s1 = 217,800 kg/m. and f o r  U = TI 
1) See my articles i n  t h i s  magazine, 1918, NOS. 1 7  and 18, and i n  
'"Zentrabblatt  der Baxuremaltung, n 1919, No-84. 
t he  influence of a permanent uneven l o a d  is  considered. 
2 )  If s > = % D  and J, = 4, then a, =a, may exoeed 
the l i m %  n only f o r  symmetrical loading and MA = Me. 
We then obtain f o r  a = TT 
In the la t ter ,  
g, = g2 
- p fla ( 4  + nq , +MA ~ 2 
% - 2 7f= 
V 
The values MB and C, are independent o f  &, s2, S, I M c ~  
The only condition is as, > C$ 
If now 
in order t h a t  we may have a, > f i  b 
% 2 ------ 
J2 = Jl # % = Z '  23- = 1.44, then .'& = - ,/1*44 = 0.8, 
81 3' s, 7 3  
S S  25 
s s 2 .  . 24' 
y = - L A = -  % = 0.8 TI = 2.5132741 (144*), 
Vb ' 0.7245714 
and we obtain 
C, = 0.52328 gls17 - 0,24140 gz + 2-11518 M A  - 1-09139 MC 
= 61090 - 11684 ga - 1.0.9139 Mc- 
The fo l loy ing  table shows the  great influence o f  g2 and MC on 
the makimutn  bay moment s 1  . 
_ _ _ ~ _  _ _  
%lm. : %a;c. 
.: MC : C, : Bay 8 , :  Bw s2 
(kg/cm) ; (kg/m) : (kg/m) : (kg/m) : (kg/m) 
3.0 : -80 : 347.7 : 128.2 : 80.8 
3.0 : -40 : 304.0 : 84.8 : 148.1 
2;o : 0 : 377.2 : 157.6 : 58.6 
3.0 : 0 260.4 : 41.4 : 215.9 
We can a l s o  determine the t r u e  value of C, = 0 x 00 by cal- 
culating the value of C, for the value of 6 ly ing  the nearest 
poss ib le  to that  of 
- 9 -  
.*-I . . 
1 
I 
We t ake  a* = leOo 5 C 1 ,  dt = 0.8 ul= 144' 40' and obtain 
v i  = - 215.9847, Y ' '  - -   218.0077, u~"' = - 44.0683 
= f 0.7435 I f 1  V Z ' s  + -4.5617, UZ" = .t 3.3659, u, 
a 
2 
= 21864.83 k g / c m .  see 3 = - 137,511. 
For the loading case g& = 3 kg/cm. and Idc = - 80,000 kg/cm., 
there follows 
MB = - 19236.95 kg/cm., D = g,klz  M A  + 2 = - 253.64 kg/cm. f 
C, = 2.5364 x 137.511 = 348.8 kg/m., 4 = = 28.8 kg/m., 
2 
This result v a r i e s  but s l i g h t l y  from t h e  value 128.2 kg/rn. 
obtained f o r  ut = TI. In t h i s  comparison, no account i s  taken 
of the f a c t  t h a t ,  w i t h  increasing values f o r  cx and S,  t h e  
loads * .  g a.nd the end moments also increase somewhat. 
Whither leads, therefore  t h e  valuat ion o f  the  mori,ents i n  the  
l i m i t  case 6 = n, as given by Mr. Ratzersdorfer i n  h i s  second 
figure? e 
The three po in t s  a, , 0, by which determine the  c i r c l e ,  
whose rays o c represent t he  moments g k2 -t Id, l i e ,  when 
a = Tf i n  a s t r a i g h t  l i n e .  The center  of t h e  c i r c l e  t hen  only 
remains finite, when the poin ts  U, and b, coincide- This 06- 
curs, when a$rees with the condition 
D = 0. Opposite the i n d e f i n i t e  form C, = 0 x cp stands the in- 
d e f i n i t e  t&. of;,' dmsing a ciz&e .ata which o3L;Ly t w o -  poinie - .O: a d  
g k2 + M A  = - (g k2 + MB) 
' .  
5. - 1-0 - 
b4 are given. In his 1919 tr-atis  
. .  
i n  the "Osterr. Flugzeit- 
schr i f t "  Mr- RatzersdcSr.fer takes the distance 
as the diameter of the c i r c l e  and obtains the  demonstration re- 
peated here in  Fig. 4- This gives, f o r  the bay s, . of our exam- 
ple, moments which are independent of the ldading of %he bay 
and of the moment M,. 
% 
The foregoing investigation presu2poses a Anif ormly loaded 
bay with a c o n s t a t  end load 'SI Ir, rea l i ty ,  the air pressure is 
solved5by the ribs i n to  a number of individual loads. krrther- 
more, the inner 3ension'betKeen the two wingzspars resu l t s  in the 
application of oblique s t resses  n i th in  the bay of a spar,. . The 
bending s t resses  generated i n  the spars tyere, t o  the bes t  of my 
knonledge, first considered i n  pr in t  i n  my t r e a t i s e  on the 
s t rength calculation of airplane spars i n  the  August, 1918, num- 
ber  of the Technische Berichte (Technical Bulletin) of the  ''Flug- 
zeugmeisterei!' 
f w a s  principally concerned, through the application of gen- 
e r a  formulas, practica33e for every c2se of loading, i n  invest- 
igaeing, with a few, examsles, as t o  iyhether the customary a- 
sumption, i n  practice,  of .a c o n s t k t  pressure and uniform lo&- 
ing, instead of indivi&d loads, is sufficien3ly accura;Lue, but 
(Aircraft Inspect ion a d  Engineering Department. ) 
\ 
i 
!.. ': 
- 11 - 
also i n  demonstrating, cn an important specimen, the  great influ- 
ence o f  the direction c% the  stresses resulting from the inner 
--rp--7ytension. 1 will go .more thoroughly i n t o  these  . t  questions i n  a 
special. treatise. 
(Translated by t h e  National Advisory Committee f o r  Aeronautics.) 
. . .  c 
-. 
I . .  


Calculation of wing spars

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