Roy, J Albert

English

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- I 
• 
ARB No. L4F24 
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
I I~ 
OItIGINAlLY ISUD 
June 1944 as 
Advance Restricted Report L4F24 
THE IlETERMINATION OF EFFECTIVE COLUMN LENGTH 
FROM STRAIN ME.ASUREMENTS 
By Evan H. Schuette and J. Albert Roy 
Langley Memorial Aeronautical Laboratory 
Langley Field, Va. 
WASHINGTON 
,.A NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of 
advance research results to an authorized group requiring them for the war effort. They were pre-
~ viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution. 
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NACA ARR No . L4F24 
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
ADVANCE RESTRICTED REPORT 
THE DETERMINATION OF EFFECTIVE COLUMN LENGTH 
FROM STRAIN MEASUREMENTS 
By Evan H. Schuette and J . Al bert Roy 
SUMMARY 
A method is presented for the experimental deter -
mination of the effective length of a column for which 
the end conditIons are unknown by establishing the points 
of zero curvature from readings of strain gages dis -
tributed along the length of the column . Tests of four 
colQ~ns of different cross sections indicated that the 
proposed method gives satisfactory results even when 
there is considerable scatter in the strain- gage readings . 
INTRODUCTION 
A suitable method is needed for experimentally 
evaluating the effectiv e length from column tests in 
which the end conditions are unknown . One such method 
consists in establishing the deflection curve from meas -
urements of lateral defle c tions and estimating from this 
curve the locations of the inflection points . The prin-
cipal objection to this method is that the determ i nation 
of inflection points from an experimentally established 
curve is an inherently inaccurate procedure . A more 
accurate method of es tab lishing th e inflection points 
consists in measuring the curvature rather than the 
deflection and establishing from such measurements the 
points of zero curvature, which define the inflection 
points. The present report shows how strain measurements 
can be used to indicate curvature and thus to establish 
the effective column length. Four columns of different 
cross sections we~e tested to provide an experimental 
check of the method. 
2 NACA ARR No . L4F24 
SYMBOLS 
L actual length of column , inches 
Le effective length of column , inches 
(1 ' ( 2 strains at two points in cro ss sect i on 
d distance between points in c r oss se c tion for 
which strains ( 1 and ( 2 are taken, meas -
ured perpendicular to neutral axis , inches 
r radius of curvature, inches 
METHOD OF MEASURING CURVATUHF. AND DETERMINING 
EFFEC'J.'IV5: COLUMN LENGTH 
Ii' it is assumed that sections r emain plane a.f t e r 
bending, the curvr:.ture l/r of a column at a give n c ro ss 
s e ction is related to thB difference in s t ra In a t t wo 
points on the particular cross section according t o t he 
equation 
This equation indicates a convenient exper i men tal me t hod 
for' determinln€:, curvatur e from strain measureme nts t aken 
along the length of the colwnn . The method consist s in 
attaching a number o f strain gages on opposite s i des 
of the colwnn along its length and recording the dif-
ferences in strain on the t\O sides . It is the n ne c essary 
only to plot the curvature - or , if d is cons t ant, t he 
strain diffe rence (1 - (2 - against distance a l ong the 
length of the column and to determine the points of z ero 
curvature . 
SPECII'v1ENS AND TEST ·EQ,UIPMENT 
I n order to check experiment ally the prac ti cab il ity • 
of the strain- measurement method for determining effec t i ve 
• 
NACA ARR No . L4F24 3 
len.gth, cohu:ms o!' fou:::, different types of cross section 
were t e sted . A rostan ,,';'..l l Dr OEer ,\,8.8 t ,3 sted wi th strain 
gages p:i..Rc'3d opl:'·,)::-,it~ (, .v,;0. ,·;.1-181" ,')~'l tI'J") wld":lr s:UJ.'3 s. The 
cross :>s('t~orls aJ.)(.~ the ':".""'cDt: '):11': ()f t"'.8 re sis tance -·type 
wIre sL>J. ~n ('-',[88 fo r 'U'18 z- -j ·~ (,t: ,)~~ ccLmin , the s!dn-and-
s ';:;1 1'f'3110r Co l. ur.n , and thE hi:.. t-st : .' .:'<' :.sJ pa.ne l column are 
shown in fi gure s 1, 2, and ), r c . ..:-resGJ.vely . 
The wire strain ~ages were at t ached along the length 
of each column at S::r.8=.:i.. :i.nt'3 c"..-al3 . T1l8 a pl'r'oxima te 
dis tanc e d betwe8n strain [?gP~ at a part2nular cros s 
section , measure d perpend icula r to th e neutral axis, was : 
The over- a ll accuracy of the strain measurements was 
within 2 percent. 
The specimens were tes ted flRt - ended in hydraulic 
testing machi~es having accuracies within three - quartars 
of 1 percent for the range of lo ad used . ?igure 4 shows 
the Z- sectio!l column under 109.d . 
RESULTS AND DISCUSSION 
Th8 results of the t ests are p resented in two sets 
of figures . The first se t (figs . 5 t o 8) shows the 
vur v6.-cure of' tl:e coluJll::1s , a3 gi ven by the differenc es in 
s t!·81.)1 ..l or VlE' g'lC;8 S a;~ each eros s sec t:l.CYl, >~ >~ [',ed 
aCb.i~J.:lt r;is:::anee i-.lol,:S th.e lcngtn of tb 8 c.O-'Il':-:- _:'0 1" 
s everal l c? 1::- rlectr t'1? r.l::!.,X.tl1.Ull'! load . T1'8 '--;' 1._-rLlta l 
lin'3s r'er.2.',,;cct i.Cl'O ,c.I.'05 ... 1 d jffere. nce . 'L -~ ' ... .'J1d set 
(l'igs . S' L.u ~ 2) 3~YH\;:' C~,0 l.~ E1. tio of effe ct:L \'~ i , ·.h to 
ac tual :en~"'~: J ob J.:;ained from the first Get of i ig:.1.res , 
plotted against load . 
Except for the Z- section column , the values plotted 
in figures 5 to 8 re present the increase of curvature 
4 NACA ARR No . L4F24 
caused by loads in excess of a particul ar initial load 
near the maximum . The use of a fairly high init i al load 
in the calculation of strain differences had the effe c t 
of eliminating a large part of the scatter in the t es t 
data . It was believed that this ~rocedure would not 
affect the accuracy of the result~ , because the stra i n 
differences at the particular initial load were qu ite 
small in comparison with the differences that were 
recorded at loads near the maximum . For the Z- section 
(fig . 6) , however, a somewhat more satisfactory p l ot was 
obtained when the strain increments were taken from zero 
load , because large strain differences were recorded as 
soon as loading was started . 
A part of the scatter in the plot for the skin- and-
st:i.ffener column (fig . 7) results from tte fact t hat 
slightly different curvatures were measured on the right 
and left sides of the column. Near the bottom of t he 
co lurn..l1. , especiall::,' on the left side , some of the sca t ter 
is thought to be due also to local buckling of the sk in. 
All strain gages were placed on the stiffener , but i t i s 
probable that the effect of the buckles in the sk i n 
carried over to some e~tent into the stiffener and inf l u -
enced ~he strain readings . 
In the plot for the hat - stiffened panel (fig . 8 ) 
the effects of any possible differences in curva t ure 
between the two stiffeners to which stra i n gages we r e 
attached were eliminated by using averages of the co r re -
sponding gage readings 9n the two stiffeners . 
Figu res 5 to 8 establish the effect:i.ve lengths for 
several loads below the maximum . In order to extrapo l ate 
from these values to the maximum load, the ratio of 
effective length to actual length is plotted against load 
in figures 9 to 12 . In the extrapolation , greate r weight 
was given to the values obtained at the higher loads . 
This procedure seemed justified because the absolut e 
scatter in the curvature plots was approximately the 
same for all loads and thus relatively less important 
in comparison wi th the grea ter cur'va bU'es that existed 
at higher loads . Theoretical considerat i ons , mo r e over, 
lead to the conclusion that the shape of the elastic 
curve for maximum load is more and more accurate l y 
approximated as the maximum load is approached , and t he 
effective length thus approaches a defini te value . 
• 
NA CA ARR No . L!.jl<12h 5 
It will be noted that, for all except the Z- se c tion 
column ; a reasonable extrapolation to maximum load was 
accomplishec1. by drawing a horizontal line through the 
test points . This fact is of particular interest in the 
cas::; of the skin- and- stiffener column (figs . 7 and 11) , 
for which quite consistent results were obtained e ven 
though con siderabl e scatter was evident in the curvatur e 
plot . For the Z·- section column (fig . 10) , the extr a -
polation was accomplished by drawing a curve t h r ough 
the points for high loads in such a manner that the 
tangent to the curve was horizontal at the maximum l oad . 
Be c ause of the presence of ineffe c tive widths of 
skin , it was impossib l e to check the loads obtai ned from 
the skin- and- stiffener c81umn and the hat - stiffened pane l 
against the values gIven by the Euler column formula . 
Such a check was made for the other two col~~ns , howeve r, 
and the results are given in the following table : 
Column 
Experimental 
maximum load 
(lb) 
Euler l oad 
based on experf -
mental Le 
( lb) 
Bar I 50 , 000 47 , 800 
Z - sec t ion I 2 , 800 2 , 930 
~_________ L ______________ ~ ________________ ~ 
The calculated values of ~uler load , based on the experi -
mentally determined effective length , are within 5 per c en t 
of the test values . 
LIMITATIONS OF THE METHOD 
The method pres e nt0d for the determin&tion of effec -
tive column length from strain measurements is not 
applicable in all cases . If there is local buckling of 
the column , the strain measurements will be adversely 
affected . Jhether the results are completely invalidated 
will depend on the extent to which the local buckling 
takes plQce at the points where the strain gages are 
located . In the tests of the skin- and- stiffener column 
reported herein the local buckling did not carryover 
into the sides of the stiffener , where the strain gages 
were l ocated , in a sufficient degree to invalidate 
6 NACA ARR No . L4F24 
completely the results obtained but the effects of the 
buckling 1JIlere evident in the strain measurements . 
Another limitation is imposed by the size of the 
column being tested. It is evident that the accuracy of 
the results depends to some extent on the distance d 
between strain gages . If the column cross section is 
such that this distance must be small, the accuracy of 
the results will be impaired, especially if ther~ is a 
variation in strain across the width of the strain gages . 
CONCLUDING REMARKS 
The t es t data presented indicate that the proposed 
strain-measurement method for the experimental deter -
mination of the effective length of a column for which 
th e end conditions are unknown gives satisfactory results 
even when thdre is considerable scatter in the strain-
gage readings . 
Lang ley Memorial Aeronautical Laboratory 
National Advisory CommIttee for Aeronautics 
Lang ley Field , va . 
NACA ARR No . L4F24 
Figure 1.-
Strain gage 
NATIONAL ADV~SORY 
COMMITTEE FOR AERONAUTICS 
Cross sect/on of Z- secflon 
column. 
Fig . 1 
NACA ARR No. L4F24 
Strain qage 
NATIONAL ADVISORY 
COMMITTEE FOR AERONAUTICS 
F ig. 2 
Figure 2. - Cross section of' skin-arxi-st/ffener 
column. 
NACA ARR No. L4 F24 
.---- Strain gages 
Fig. 3 
NATIONAL ADVISORY 
COMMITTEE FOR AERONAUTICS 
Cross sectior; of hat - stiffened panel 
column. 
NACA ARR No. L4F24 Fig. 4 
Figure 4.- Z-section 
column under load. 
NACA ARR No. L4F24 
Top 
47.5 
1.0002 
~ 
Dislonce along column, In. 
Bollom 
NAnoNAL ADVISORY 
COMMITTEE FOR AEROMAllTICS 
Curvature, 
l/in. 
figure 5. - Curvalure of'rectangular 
bar. 
Fig. 5a 
NACA ARR No. L4F24 
Top 
Load=4B5 kips 
~ 
Disfance along column, in. 
Figure" - Confinued. 
Bottom 
NATIONAl ADVISORY 
CCMMITIEf FOR AERONAUTICS 
Curvature, 
l/in. 
Fig. 5b 
NACA ARR No. L4F24 
Top 
Load= 4.9.5 kips 
7i 
Disfance along column, in. 
Fi'gure.5.- Concluded. 
Bo/lom 
NAllONAL ADVISORY 
CONMlnEE fOR AERONAUTICS 
Curvature, 
l/in. 
Fig. 5c 
NACA ARR No. L4F24 
Top Bollom 
Load = 2.40 kips 
2.45 
2.50 
~ I·0002 
Dlstance olon9 column, in. 
Curvature, 
0 17. 
NATIONAL ADVISORY 
r.OMMITTEE FOR AERONAUTICS 
Figure 6. - Curvafure of' Z-section 
column. 
Fig. 6a 
NACA ARR No. L4F24 
Top Bottom 
~ 
L oad= 2.55 kips 
fsi 
Distance 
2.60 
2.65 
10002 
along column, in. 
Figure 6. - Continued 
I 
NATIONAL ADVISORY 
COMMITTEE FOR AERONAUTICS 
Curvature, 
Yin. 
Fig. 6b 
NACA ARR No. L4F24 
Top Boffom 
Load = 2 .70 kips 
2.75 
~ 10002 
Distance along column, in. 
Figure 6. - Concluded. 
NATIONAL ADVISORY 
COMMlITEE FOR AERONAUTICS 
Curvature, 
1jtn. 
Fi g . 6 c 
NACA ARR No. L4F24 
Top ~0 8ottom ~ ~ood= 27.0 kiPS~~1 Q:l~ ~ 
00 --__ ..--/ 0 
-~-O---
~~ f----~ 28.0 ~ 
o 
o 
o 
o 
0° 
9 
o 
29.0 
20.5 
" / 
" / 
...... / 
...... -0----
30.0 
/ 
1 
NmO~AL ADVI, ORY 
COllMITTEE FOR AERONAUTICS 
\, /1-11 <' ~ "'--B- ..... .....- .00005 Curvature, 
/~ . l/in. DI.stancg along column, m. 
<:) Right side 
o Left sIde 
Fig. 7 
Figure 7. - Curvaturt? of skin-ond-stirrengr column. 
NACA ARR No. L4F24 
Top Bottom 
<> o 
Load = 30.5 kips 
/ 
\ / 
\ / 
\. /1 
" <> /' ....... _-
o 
3/.0 
\ / 
\ / NAnONAL ADVISORY 
\ / COMMlnEE FOR AERONAU TI CS 
\\ //100005 Curvature, 
Ijol ,,_~ / l/in. 
o 
Dis tance olong column, in. 
o 
Figurg 7. - Concluded. 
Right side 
Left side 
Fig. 7 b 
NACA ARR No. L4F24 
Top 
~ 
Bottom 
NATIONAL ADVISORY 
COMMITTEE FOR AERONAUTICS 
I Curvature, .0002 l/in. 
Dis/once along column, in. 
Figure 8. - Curvafure of' haf-.sfiffened 
panel column. 
Fig. 8a 
NACA ARR No . L4F24 
Load =/22 kjps 
\ / 
\ / 
\ / 
\ I 
" / 
''-0-/ 
/25 
/ 
I 
\ / 
\ / 
\ I 
\ I 
BoHom 
\ I 
\ NATIONAL ADVISORY \ I COMMlnEE FOR AERONAUTICS 
~ 
\ I 
\ / 
\ / 
\ / 
\...0-/ 10002 
Disfonce along column, in . 
Fi'gure8.-Concluded. 
Curvatllr@, 
l/in. 
Fig. 8b 
NACA ARR No. L4F24 
.6 
.5 
.4 
. .3 
.2 
.I 
o 
45 
. 
. 
46 
" 
"" !J 
47 48 
Load, kips 
Fig. 9 
v ~ 
Maximum load --.. 
I 
I 
t 
T~ 
L Le 
L 
~ 
NATIONAL ADVISORY 
COMMITTEE FOF AERONAUTICS 
49 50 
Figure 9. - Rat/o of' efFective /Bng th to actual 
length for rectang"lar bar. 
NACA ARR No. L4F24 
. 5 
.S 
.4 
. .3 
Le 
L 
.2 
./ 
. 
• 
0 <;> 
Fig. 1 0 
~ I ",100.... 
V'1axlinum load-+-
Irl~ 
L Le 
l:=1 
NATIONAL AD ISORY 
C DMMIITEE FOR AE RONAUTICS 
o 
2.3 2.4 2.5 2.6 2.7 
Load, kips 
Figure 10. - Ratio of' effec five length to actual 
length Tor Z -secfion column. 
2.9 
NACA ARR No. L4F24 Fig. 11 
.6 
.5 
.4 
.3 
Le 
L 
.2 
.I 
o 
I> 
26 21 
D ~ 
rr= 
L Le 
~ 
28 29 
Load, kips 
(:) L\ 
Maximum load---. 
NATIONAL AD ISORY 
CPMMITIEE FOR AI RONAUTICS 
30 31 
. 
Fftgure II.-Ratio if effecfive length to ClCfuol 
lengfh for s/(in - and- .stiffener co/ulnn. 
32 
NACA ARR No. L4F24 
.6 
.4 
. 3 
Le 
L 
.2 
.I 
o 
110 1/4 
Fig . 1 2 
/"1aximu m load ~ 
• 
NATIONAL A[ ~ISORY 
OMMITIEE FOR A~RONAUTICS 
1/8 122 126 
Load, kips 
130 
riqtlre 12. - Rafio of' effective lengfh to actuol 
lengfh for hof - .slirfened panel 
co/ulnn. 


The Determination of Effective Column Length from Strain Measurements

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