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DIVIS ION REPORT, S ERIAL No. 2712
AIR CORPS INFORkMATION CIRCULAR
(AVIATION )
PUBLISHED BY THE CHIEF OF AIR CORPS, WASHINGTON, D. C.
Vol. VI March I , 1927 No. 584
PROGRESS REPORT ON THE STUDY OF
TORSION ON WING FRAMEWORK
(AIRPLANE SECTION REPORT )
Prepared by N. R. Bailey
Materiel Division, Air Corps
McCook Field, Dayton, Ohio
September 24, 1926
UNITED STATES
GOVERNMENT PRINTING OFFICE
WASHINGTON
1927
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CERTIFICATE: By direction of the Secretary of War the matter contained
herein is published as administrative information and is required for t he proper
transaction of the public business.
(n)
< PROGRESS REPORT ON THE STUDY OF TORSION O~
WING FRAMEWORK
INTRODUCTION
Because much difficulty has been experienced with
wing flutter or unstable torsional oscillations on large
monoplane wings, the question of torsional r igidity
has come to be one of great concern to designers of
monoplanes. There are several reports in circulation
which treat the subject from the standpoint of aerodynamics
and kinetics, but there has been very little
work done on the structural problems connected with
obtaining the desired torsional rigidity. This project
is limited to a study of ways and means of furnishing
the torsional rigidity demanded by aerodynamic considerations,
while the problems of determining the
amount of torsional rigidity required will be left for
solution by engineers specializing in the study of
aerodynamics.
The problem is being attacked both experimentally
and analytically. In order to prevent confusion, the
detailed discussions of the report are divided into two
parts. Part I deals with the experimental work thus
far carried out. In Part II is given an outline of the
progress which has been made to date on the theoretical
or analytical study.
OBJECTS
The principal objects of this study are as follows:
1. To determine the type of wing structure which,
for a given weight, has the greatest torsional rigidity.
2. To develop a method by which any type or size
of wing structure can be analyzed to determine its
actual torsional rigidity more accurately and quickly
than is possible by the best methods known at the
present time.
3. To determine which members of a metal truss
type wing structure are the most important in regard
to their contribution to the torsional stiffne s of the
structure.
4. To discover some definite laws, according to which
a wing frame deflects under torsional loads, that can
be relied upon generally for use in the aerodynamic
solution of the cause of wing flutter, and which can be
used for designing a wing to give stable deflections.
SUMMARY OF RESULTS
The most important results thus far obtained tuward
accomplishing the principal objects of the study are as
follows:
1. It has been found that the torsional rigidity of a
cantilever wing structure depends mainly on three
characteristics:
(a) The stiffness of the spars.
(b) The location of the spars a long the chord.
(c) The type and design of the drag structure.
Of these three, the last characteristic has proved to be
the most critical and the most difficult to handle.
However, all are dependent for their determination
upon the airfoil used, the movement of the center of
pressure, the degree of torsional stiffness required, and
the maximum weight to be permitted.
2. A method has been discovered for computing
approximately the stresses in a tru s type wing frame
subj ected to simple torsional loads, and which has
several redundant members, by the use of static
equations. The stresses are conservative for members
of the drag structure and can be used for comparing
different types of drag structures in regard to their
contri bution to the torsional stiffness of the frame.
This method of comparison is much more rapid than
any previous methods available.
3. A criterion for determining which members of a
given metaltruss type wing frame are most important
in regard to the stiffness of the whole frame, under
any loading, has been worked out. For torsional
loads, certain members of the drag structure are generally
the most important.
4. Tests made with different sizes of tie rods for
drag bracing proved the importance of using the proper
design of drag structure in order to obtain torsional
stiffness. An increase in the size of rods, which increased
the total weight of the frame tested by only
13.3 per cent, decreased the torsional deflections by 56
per cent.
5. It was found that, in general, the torsional deflections
vary uniformly in direct proportion to the torsional
couple, or to the torsional moment of a ll the
loads about an elastic axis.
6. Deflections obtained by computation compared
closely with those measured in the experiments and
were found to be on the conservative side.
7. The tests indicated that there must be an elastic
axis, parallel to the spars of a cantilever wing frame,
about which the outer members rota e in circular arcs
when the frame is subjected to t sional moments.
The tests did not give, directly, the location of this
axis. However, a method was worked out, in the
analytical study, for using the test data to check the
theoretical location. A very close check was obtained,
by the use of this method and data from the te ts made,
on the theoretical location determined by analytical
computations.
8. The theoretical location of the elastic centrum
can be located quite accurately by means of the Method
of Least Work for computing stresses and deflections.
However, this process is too long and tedious for practical
design work; therefore a more rapid method was
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sought. A method was found which gives the approximate
hori zontal location very readily.
9. T he effect of combined bending and torsion loadings
upon the torsional deflections was determined by
the tests and the analytical study. A formula was
derived which may be used for Rredicting the torsional
deflections, under any case of combined loadings, from
ex perimental data for simple torsion loadings.
10. By means of the Least ·work computations, a
definite relation was found for the di tr ibution of
stresses in a cantilever wing frame acted upon by
simple tor sional loads. These computations showed
particularly that the stresses in member. of the drag
stru cture are the greatest at the outer d rag bay and
decrease toward t he supports. The stre es in the
spars a re the smallest at the outer end and increase
toward the supports.
Rear Elevation
3
The object of making these deviations from the normal
wing in both loading and relative sizes of members
was to obtain simplicity of computation. It was also
believed that the complex problem of the monoplane
wing of normal type can be solved more easily if the
first step consists in the study .of a simplified type of
framework such as used for obtaining the data given
in this report. The actual sizes of members used were
determined chiefly by the following considerations:
(a) The structure should be sufficiently flexible to
give easily measured deflections under r elatively small
loads.
(b) The individual members should be large enough
and heavy enough to allow for the use of screws to
attach a duralumin heet covering for a series of test ·
to be made for determining the effect of such a ·covering.
(c) The sizes used in the two spars should be such
End Elevofion
FIG. 2.Diagram of testing apparatus
OUTLINE OF PRECEDURE
A small welded steel tubular wing frame was constructed
for t he first tests and to serve as a typical
structure upon which to base t he general study. This
frame, as shown in Figure 1, is 144 inches long and 30
inches wide. It has two spars of Warren truss type,
and there are five bulkhead of Warren truss construction,
which divide the frame into four identical bays.
The front spar is 9 inches and the rear spar 6 inches
deep. Welded lugs are provided for tierods in both
upper and lower surfaces. The frame is supported at
one end by four horizontal pins passing through clevis
type fittings at the four corners of the frame.
Since the primary object of the project was to study
the effect of torsion loads, the same sizes of tu bes and
wires were used for the homologous members of the
different bays. In this respect the test framework
varies fundamentally from the framework of an int
<!rnally braced monoplane in which the members
would increase in size as the inner (fuselage) end is
approached. The loading also differs greatly from
that on a wing in flight, as it consists of concentrated
loads at the outer end of the structure instead of a
di stributed load along the entire length.
that tubes of twothirds their sectional area would be
available. This criterion was due to the plan to construct
a threespar framework, in which it was desired
to have the same amount of material in the three spars
a in the two spars of the first framework.
The first tes_ts •ere made on the twospar framework
just described, the only variations being in the sizes
of tierods for drag bracing, the number of pins in place
at the supports, and the applied loads. The loading
always consisted, however, of a vertical load acting
down'.\~rd at the outer end of the front spar, and another
vertical load acting upward at the outer end of
the rear spar. These tests were planned with the
following objects in view:
(a) To obtain data on the deflections of the frame
to use for compari on with the analytical computations.
(b) To determine the effect of size of drag bracing
upon the torsional rigidity of the structure.
(c) To determine how the tor ional deflections vary
with uniform variations of a torsional loading in which
the two loads are equal in magnitude.
(d) To determine the effect on the torsional deflections
of other combinations of loads in which the two
loads are unequal in magnitude.
(e) To determine whether or not there is any fixed
center, or elastic centrum as it is called, about which
points on the outer bulkhead members rotate in circular
paths when the frame is subjected to torsion loads.
(J) To determine the effect of supporting conditions
upon the torsional rigil!.ity.
It was also planned to make tests on the twospar
framework with different types of drag structures, such
as (a) using steel tubes for bracing instead of wires,
(b) use of diagonal trusses, and (c) with a corrugated
duralumin covering m place of drag bracing, in order
to determine how much each system contributed to the
torsional rigidity of the complete structure in proportion
to its weight. A threespar framework is under
construction , and tests will be made on it to find out
whether or not there would be any worthwhile advantages
in the threespar construction, particularly with
regard to torsional rigidity.
After enough tests were made to obtain sufficient
data for use in studying the phases of the problem for
which the first tests were planned, it was decided to
suspend testing operations for a time. The accumu
4
lated data are as accurate as it was possible to obtain
with the best experimental apparatus that was immediately
available. It was, therefore, impracticable to
do any more testing until it could be determined what
further tests, such as would warrant the design and
construction of more efficient testing apparatus, are
most needed. Consequently, it became necessary that
the next st ep should consist in a careful study of the
data obtained. Also, it was .found advisable at t his
point to take up the analytical and theoretical study
of the problem, since the decision as to what kind of
tests are most desirable would be determined very
largely by the character and res ults of the computations.
In Part I , following, is given a description of the
testing apparatus, an outline of the tests made, and a
discussion of the experimental results. Part II is a
general outline of the various teps in the theoretical
or analytical study as far as it has progressed at the
time of writing. All tabulated data together with the
results of special computations are included in the
Appendix, and these are explained in the text of the
report.
•
PART I
DETAILED REPORT ON EXPERIMENT AL WORIK
DESCRIPTION OF TESTING APPARATUS
The wing frame was supported horizontally, as shown
in Figure 2, at about 5 feet from the floor, by its
female clevis type fittings (F), these in turn being connected
through horizontal pins to corresponding male
fittings which were attached to the side of a large rigid
supporting structure (A).
The down load on the outer end of the front spar
consisted of a platform (Pf) suspended by wires directly
underneath the spar, on which the front spar load was
placed in the form of shot bags. For the upload on
the rear spar, a cable attached to the outer end of the
spar passed vertically upward over a small ballbearing
pulley directly above, then horizontally to another
pulley, over and down again to a second platform (Pr)
carrying the load. For the initial loads the necessary
number of shot bags were placed on the platforms so
that after the weights of the platforms, cables, and
the wing frame itself were properly taken into account
the true load at the front spar was 100 pounds acting
vertically downward, and the true load at the rear
spar was 100 pounds acting vertically upward.
For measuring the deflections, long indicators (I)
consisting of wooden rods were fastened along the top
chords of each bulkhead so that they extended about
21 inches in front of the frame and 21 inches to the
rear. Small nail points were inserted in the ends of
each rod to serve as sharp tips for the indicators. A
wooden structu re (B) was built around the wing frame
so that there would be 10 by 1 inch vertical uprights
in line with each bulkhead, both in front of and to the
rear of the frame, where the indicator points could
move up and down against their surfaces. Horizontal
pencil lines from which to measure the movement of
the indicators were drawn across the surface of each
one of these uprights.
OUTLINE OF TESTS MADE
For the purpose of determining the effect of size of
drag bracing upon the torsional rigidity, various ests
were made using the following sizes of t ie rods:
No. 1032 swaged round rods.
7428 unswaged round rods.
fi24 unswaged round rods.
In order to determine the effect of suppo ting conditions
in addition to running tests wit . the frame
supported by four pins, tests were made 'th just two
pins in place, and others with three pms. However,
for the twopin support condition, orily torsion loadings
could be applied, since it was necessary to keep
the upward and downward loads eq al in order to maintain
a balance.
All of the data obtained from the tests are given in
Tables 1 to 12. A detailed outline of the tests and
explanation of the data in the order tna the tests were
made is given below.
PRELIMINARY TESTS
The first few experiments consisted of trials to
mine the most desirable ways of conducting the tests,
especially with respect to type of loadings to use and
their order of application. Another function of these
trials was to find out what refinements would be necessary
in the testing apparatus. Tests were made both
with No. 1032 rods and ~28 rods as bracing, and
both for a fourpin support and a twopin support.
The loads were chosen somewhat arbitrarily, ranging
in value from 100 to 500 pounds and changing in increments
of 100 pounds. I nHI st ca~es the loads on front
and rear were kept equal, but :t few c ci inations were
applied in which they were unequal, so give
various amounts of net bending load. Samples o t
data obtained in these preliminary tests are given in
Tables 1 and 2 of the Appendix. These data were
plotted in sever::t ways to show the variation of torsional
deflection, ,,)th load. It was revealed that not
enough loadings were used to give very definite information,
particularly in regard to the effect of direct
bending and torsion loads combined. Corrections in
the testin apparatus were al o found necessary, such
as stiffenfng the indicator rods, and replacing the
cable supporting the rear platform with a more flexible
cable as to make for more uniform operation and to
reduce the fri ction as much as possible. Due to the
unre· iable character of these data, ,they will not be
referred to again in this discussion.
TESTS WITH NO. 10 32 RODS
For the next tests, which were made with the No.
1032 rods, a definite load schedule was worked out
and followed through. Combinations were used in
which the loads varied from 100 to 350 pounds in
increments of 50 pounds. There were six readings for
the case of simple torsionthat is, with equal loads on
front and rear. Also several readings were made for
cases in which the loads differed by 50, 100, 150, and
200 pounds, with varying amounts for the small load,
and both with the small load on the front and on the
rear. Readings were made of the deflections of the
indicators at the four outer bulkheads. ·Table 3 gives
the deflections for the case of a fourpin support and
Table 7 gives those for a twopin support. The positions
of the indicators with the weight of the frame
acting alone were taken as the zero lines from which
to make the measurements. In the tables, A. signifies
(5)
6
til e verqt ical deflection of the rear indicator point at
b lkhea i A, Ar the vertical deflection of the front
1. nu d_1 cat or point, and similar symbol the pointer deflections
at t l~e other bulkhead . A positive s_ign den_otes
1 ct •ng upward or an upward deflection, while a
..... a~g~0a·: '1tive st"gn denotes a load acting downward or a
downward deflection . The last columns give the
algebraic di erence between the deflection of the rear
indicator and the deflection of the front indicator and
the angular eflection in radians. These angles as
given in radian a re actually the sines of the angles of
deflection obtained by dividing the difference A, A 1
by 72.5 inches, ti1edistance between the front and rear
indicator poiJ1ts. For the small angles involved, the
difference etween the value of t he angle i radians
and it · e is so small that there is no error in assuming
equal, within the limits of accuracy of these tests.
TESTS WITH %28 RODS
For the tests with 7:1:28 rods used as drag bracing,
the load schedule was revised somewhat, a few ore
combinations being added to the list of loads. The ·e
ranged from 100 to 400 pounds and were arrangec in
such an order so that the smallest and least criti al
loadings which should be the least liable to cause a
permanent set in the structure would be applied to
the frame fi rst. The 'Mll!:ed deflections for a fourpi
n support are · in Table 4, and those for a twopin
SU p • Taole 8.
.1.c:....,...."~'1 supported by the two upper pins alone, it was
ound that, due to the downward deflection of the
front spar, the front lower fitting found a bearing on
the supporting structure, there r ; being enough
clearance between them. This c ~ion was really
that of a threepin support, and the deflections were
somewhat smaller than would be obtained with a true
twopin support. Further tests were made with three
pins in place in order to illustrate this effect, and the
readings for 7:1:28 rods are given in Table 9.
TESTS WITH 5/16 24 RODS
When testing the frame with ,\24 rods, another
still more extensi_ve load schedule was used, practically
twice the number of readings as in previous tests for a
fourpin support being taken. Loadings were added
which were thought unnecessary when testing the
frame with 7:1:28 rods, but which were later found
desirable in order to give sufficient data for plotting
complete sets of curves. The range of loads was
also extended to a maximum of 450 pounds. Table 5
gives the values of deflections obtained with a fourpin
support for the complete set of loadings. Table 6
gives the data obtained for a second series of readings
with a fourpin support in which only torsion loadings
were applied . These additional readings were made
for the purpose of checking up on the previous test to
see if there were any permanent set in the fr~me or
give in the supporting structure which would account
for the fact that the indicators did not come all the
way back to zero after the loads were removed at the
end of the test. This point will be discussed in more
detail in connection with the plotted curve . Readings
were also made with a threepin support to compare
with those for 7:1:28 rods, the data being•given
in Table 10. Table 11 gives data for tests with a twopin
support like that used for the smaller rods, while
Table 12 gives the deflections obtained with the lower
front and upper rear pins supporting the frame, instead
of the two upper pins.
DISCUSSION OF EXPERIMENT AL
RESULTS
EFFECT OF SIZE OF DRAG BRACING
In Table 13 a comparison is made of the angular
deflections obtained for all the applied loadings with
different sized tie rods for drag bracing and a fourpin
upport. Table 14 makes a similar comparison for the
deflections obtained with a twopin support and a
threepin support. Columns 1 and 2 give the loads
which were applied vertically at the outer ends of the
two spars. Columns 3, 5, and 7 give the algebraic
differences in inches between the deflections at the
front and rear indicators at the outer bulkhead A.
Columns 4, 6, and 8 give the corre ponding sines of
the angular deflections, these being equal to the angle
in radians within five decimal places.
Under columns 9 and 11 are the ratios of the deflections
obtained with the larger rods to those with the
No. 1032 rods. In columns 10 and 12 are the
amounts of the differences between the deflections
with the larger rods and those with the No. 1032
ods .
It is to be noted that the use of 7:1:inch rods decreases
th deflections by an average of 55.8 per cent for all
th loadings, while the total weight of the frame is
increased only 13.3 per cent of that with No. 1032
rods. With the /,iinch rods an average of 63.2 per
cent decrease was obtained for the deflections by an
increa e of 27.5 per cent in the total weight of the
frame.
When the angular deflections for each loading are
plotted against the total weight of the rods, as shown
in Figure 3, the effect of comparat ive strength of drag
bracing upon the torsional rigidity is most clearly indi cated.
Tt~ese curves how that a certain amount of
increase in "ze of tie rods decreases the angular deflections
by a comparatively large amount, but beyond a
certain point any more increase in their size does not
decrease the deflections enough to compensate for the
further increase in weight. The analytical study of
the problem as discussed later ha shown that this
relation can be assumed to apply to all types of drag
structure.
VARIATION OF TORSIONAL DEFLECTIONS
WITH SIMPLE TORSION LOADS
In Figures 4 to 8 curves have been plotted to show
t he variation of torsional deflections with load for the
case of equal loads on front and rear. The points as
plotted are the actual mea ured torsional deflections
that were obtained with the different sized rods and
different supporting conditions without any correct
ions being made for imperfections in the testing
apparatus. .
The curves are all very clo e to being straight lines
and therefore show that for p ractical applications it can
7

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be considered that the tor ional deflections increase of this test is seen to be practically traight over the
uniformly in direct proportion to the torsional moment. largest portion of its length. Therefore the bend in
It is to be noted that the curve are plotted on fairly curve ABC would mo t likely have been caused by
large scales in comparison with the amount of preci the effect of tho net bending loads on the supporting
sion obtainable in the reacli ngs, so that small offsets structure, since the individual load applied in the secfrom
the straight line can be expected. The location ond test were as large as those applied in tho previous
of the .points in Figure 4 for the No. 1032 rods is I test. A permanent set in certain members would have
an illustration of this effect. HO\rnver, in the case of this same effect, but computations showed that none·
the tests with ;!42 rods and fo24 rods the curve could have been stressed beyond 60 per cent of their
for the fourpin upport in Figure 5 and curve ABC I allowable loads.
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FIG, 5
in Figure 6 are seen to be concave upward. Thi tendency
might be attributed either to the taking up of
lack in wire as the design loads are applied or to a
certain amount of give in the supporting structure
caused by the larger net bending loads. The latter
explanation is shown to be the most reasonable by the
resu lt of the test on December 4. Th.is was made
for the purpose of checking up on the previous test to
find an explanation for the fact that the indicators did
not come back to the zero marks when tho loads were
removed after the last readings. Only equal loads on
front and rear were applied, and the same reference
lines from which to measure the deflections were used
as before. Curve EFG of Figure 6 for the first part
In Figure 6 the difference between curve EFG for
increasing torsion loads and curve GH for decreasing
loads is attributed to the effect of friction in the
pulleys over which the cable to the rear load platform
operated. This effect was brought out by not raising
the platforms between readings, as '\\as done in previous
te ts, and illustrates the importance of using the right
kind of testing apparatus, in which friction is a
minimum.
None of the curves as plotted in F igures 4 to 8 show a
zero deflection for a true zero load . Thi is due to the
fact that it was necessary to locate the horizontal
reference lines from which to measure the deflections
arbitrarily, since there was no direct way tv determine
10
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~ 5 /u ?~ <;>,,h<  ./,..;; "',,,,, r,,, ~A~I ! I
~ / Ir.. · · . ~ ;; ~ v. b/ :,,, 1w~ Ir , ~,',, I r }
? i.,., i: I /;r,,, lJ ,,,,, ~a " rf.
,,..,,,,,, ·~ I ,,.,.. "'" ~    _, ".JA ,,;
.I r~ "P. ,,,,. Z r  / 1,,, , ,) vo '.5
I? . ...,,., ...:
~ r I
\.  ~ .J
,, " I f) .,../
I _v
~ _v
< _v ,, ,,.., ;..... ~/
,, /.,.,,,,
~ .,./
~ L1 v
f /) Q/) ..,<;; h 1....."
"l L.1
.....
0 ;
/ v
 / ,.., l...rl'
~Tr T 1 ~ / IA l.M ~ r. ,_, '· T
, _
''rin ,,_/ 1 ~, )~ o
HJ, ,, ~ <>. iv> I .? r,., 2/YI ;; .r/J A.,,... '19
F IG. 7
_? OA I 1 I ' I I I I
I I I ~ I
.5/1l ?A =  I I \
I
? 1<n
; ' s;:.:,.... . ,_ / l _,../_  ,. l.LA ,.,
c ..
'
? ')/)
l I
~ "'"
'
I t/l I
!
"" I ''"'
,
J
_( .., '
112 ""'
FIG. 8
12
t he posit.ious of t he indicato rs for a t rue zero load.
Therefore, fo r the te ts made, the r eference lines were
located at t he original posit ions of the indicators with
just t he weight of the wing frame itself acting. In
order to p icture more defin itely the effect of d ifferent
. ized rods a nd d ifferent suppor t ing conditions upon the
torsiona l defl ections, the best average slopes were chosen
from the cur ves in Figures 4 to , and straight lines
having these slopes \Yere d rawn, as shown in F igure 9,
t hrough the intersection of the coordinate axe . The c
curves give the best approximation. to the correct
variations of to rsional deflections with p ure torsion
load, since t he errors involved i n t he location of reference
lines arc all practically elimi nated. From F igure 9
Q.°(170
~
m . I T"> •
" l . >m h
ltJ. '>L , D ln ' <'L · ~ l+ fr'/, r1,..,,.1'.,
IF' 1,,r,; In rh ""' '°
II=?. ,, 11 .r~u
, .
IF' ,/ Ir~ h,Y '~ F'r h~ ~r ..,,., ~ """,,...
1"1 """h ~ ·~· · ~bl ·r _, . · hr In ~
'" fil) '/' · T In/'. l,,t
.__ ~ ., v
In hA "l ( v
'
_( ff'
oV ·~
,.,~ ,p v
In h:>h , IY /
' _, o/ 6)
~ ' li7 ~~
I t :Y # Y
'"' y,.,;, ~ v v
t ings when the larger load were applied with the t wopin
support the front lower fitt ing pressed aga inst the
supporting structure. For thi reason, with only t he
two up per pins in place, the wing frame was supported
in almost the same manner as with the two upper pins
and also the front lower p in in place. I n F igure 9 it
can be seen t hat with 1
5  24 rods the deflections wit l1 a
th reepin support are nearly as large as t hose wit h a
twopin suppor t.
T he question concerning the behavior of an actua l
monoplane wing with only a twop in support is naturally
not of very much practical interest. However ,
an application has been found for thi suppor t ing co nd
it ion in developing a convenient method for comparing
/
v /
/ v / .,,
v v
v v
v v
v /v
v / v I/
v v
v
/v
~ i..
............
_...... v v'""
~V v 
L.."' ..... _...... v
v ........ . _v b:::= j,..oo"".
 L." brfi ~ ..... _ i::: t;;:;i"" L,... ....
ln12 v ..... u:'v  ~ ~ t!..! μ:;.....
·~ v ~ v _, :;!I""< Bli ~ _t::::; pr,;~ ? ~~
"'l v / v 1 ~1J ~ ...... i;t.B 111.'T k::: l:;:::::: ""'.:. k;;'i 5"~
__ ,
I/ I/ 1/ A [!.? 1:: L."
~ " 1'. Id. ~ F::: f:;;s ~ f'P'
I/ ]/ i.:,i: / A.1 ,,.., av ~ ~  Z ·~ h .... h v v _i.:: t: ......  "',4'1
~=
/,. v
1::::: t;:::::. '"" ~ ~ ...
,b ,...., ;::::;:: ~ ..,.... •7;, .T hr ~ (Jr, F .or, '/" · ~h' '}, 'A/' T"" ?ri ,..,, ,.., / "}>, hn ~ .//,
V2
,_
" ~ ~ n ~l:vl ' 4 ~17 P,1¥) 2 1'17 .,,ty) ~ fi'J .A.rll'J 4 ro
FIG. 9
a better comparison of the deflections with different
sized rods is also obtained. F or a fourp in support the
deflections with 7;;128 rods are 44.2 per cent of those
with No. 1032 rods, or are 55.8 per cent less. T hose
with ft24 rods a re seen to be 30. per cent of those
with No. 1032 rods, or 69.2 per cent less.
EFFECT OF SUPPORTING CONDITIONS
Figure 9 sh ows very clearly the effect which the
number of p ins in the supports has upon t he torsional
defl ections. It i seen that with o. 1032 rods the
deflections with only two pins in place are 22 per cent
greater than those with four pins; with 7;;128 rods
they a re 12 per cent greater, and with ;:\24 rods
t here is a d ifference of 10 per cent. Due to the fact
that t here was not enough clearance between the fitdifferent
types of wing structure by analytical comp utations.
This method will be explained later in connection
with the analytical phase of the problem.
EFFECT OF NET BENDING LOADS UPON
TORSIONAL DEFLECTIONS
When the loads on the front and rear a re unequa I
the loading consi ts of torsional and net bending load s
combined. The net bending load is equal to the difference
between the two loads. If the up load on t he rear
is the large t, the net bending load is acting upward ,
while if the down load on the front is the la rgest t he
net bending load is acting downward. T he torsiona l
deflection is brought about not only by t he coupl e,
the loads of which equal the small load, but also by an
additional torsional moment caused by the net bending
13
In rn,
., · ,/ r · . .; Ir fV ' \h
Ill f? . 17? ,y, · ;W '°' ....,,
1/1 O/.h lo ' ,, hi 14
 . ·~ ,;.
~ I ~:,,,
 ~ ·'?":. I~  Ai'. ,f . ·o;o:sllJ
t>., /
v v )
v / / / /
/ / l/ / _,b'
llJ ,,.,,, .  / v .: /
,! 7 / .,,,. I/ .<'l v
& v ol y v /. ~ 'f
~ 6 ;.' J'/ v. ~ ,,, n> • f.'/ < ~./ < 6,.. ·~
I .r !/ 1.i:: v ! ~~ v ~ [2 L~ V
I ~ ,.. ~.v ~'· i.: ~
' / v i,. o'? , '{ ~ <>' ~
/'J. " I V / .. .... 6 ,,,. \;~ ,...
"'l ..,...v v ~ ~
/ Iv " ~ v v 1r
n ~ ,, .Lv IV
v /
llJ V>
'!.,_,, 1,,, r ln t",., ln1  lh·
0 I? .uin LtO 21Jn P lrlJ . .?h?
FIG.JO
·~ ,_ ' lh
OR ,, ,,, . p.,,.,
" I'\"
In. ··· I< <" " I T~  " ·
,,, · ~" Ir> o.!. ,., "'" "·'I )\ ,,, '' /cl?·
'
"' .,, f i
'
n , ...,.
'
'
' ......
"'"'' ':  J 111 ~
_,...
'1 .. ,,,t <" t,;. ....... ,,.., d E'..v ......  ~
VJ IV ,![::.'  1 f\.[ ~S·  ~+r ,_ rro 11' :.. .......
J.,t>l' ....... . ,,, 1w ............ ~
l~ l,. J]fC 1'11 . ,_. .....
,,, '° ....... Q JU ..
,.J•   v
..... 
n ''"'
117""' } Ir,, jl. fTnlATo .)~ in .If lif
~ V> ' ,,., , k:,, " tvi
FIG.11
14
'
In 1.rl/l 7'.  ·   . ~..~ ...
/~ ?A "' _,_
' ,,., ., ,_ J fi'
I ,, .,_ .,,, " ,.._
l'l •?•'/'> ~ . '·~ I Al,.,/
,_ "'
,,... "' v_ A/, '2 ,a .,,
< v ,, ,,
r::: ,, ,, ,,,"" v /v
In h ? on ! "' .)"' v v /
...; ..J I'./ v I/ if' /
~ ' ~' v 1 .... 1 P/" v L.J ~
<:t i'IV: "'~ (; V' () i;v,...." ~v
In h o ln ~ ·'_... ~~ v~ ~(; ;:;. . ..: ,; I .. ~ v
lV> v .,.o I/ ·~~ ~ v (; ;:;.' ~ .,,
i,,v. rp">· / l~~ :/ _.. II.~ ,,, <D ;
/ ·~/ < ::>:,, v IS'. v ,'iV
In n 111n I/ ~v l<,01 ~ ~ p~
./ .~ v
b~ v
,() ~';:: v
in hn ICn 1'} ,,,.
In h/1h
n ,1, h/l '"" 7n.l'J 71>/l ~.M
I '"" ,,.,, " l ,., L1 ,., " ...... h1 /f l~
I
FIG. 12
I ~ In .# ~· le
/I  t>~  le>, , ,.,,~
b LI kl " ,,..
""'""
'"' ,,., I = '" ,,.,, ,,¥ I~ Ll: .. b~
f ;!,vi i " :lr.1 ,, h .. JJ. .i n. IA
,.,, : ;> , '/) "" r ..... .i. i o / ,,,A
'! ' · 011 '" ,,._, ,, ... iJ. I .. A
d '°'"'lhs M <'.'t IE• 'ill. 'in ,, ~
5 .. b . I "" , ~ n i · 1  lrioi .... II'> ..... ~ I T'I '"
/ v"
' v v
r / v ,,,,... ~ .... v ,, v,,,, '}""
~ / ,, /i/ v
( / /v /V
... v /0 '/ ... / i/ '//."'
/ /V .f/'
11/ I/" ...;.V
, v VI/
Mn<o ,; ,) v
3 15 ·
""' h " '"" '" '"" , ,,, ..,,.,,,, "' l!nnfl ,_, ,~ '" I
FIG. 13
15
111.•., I I I
0 I
I ~   L 1. .. _' ~ L I
~ ,, ,,, L ·· l..J.
~ • •v ii.
___ [,
•A
" In .. ''" I ,. ' A /'  I/ I
I/ L/ ,,, __
· ~
r _ _, •nl.c r,. l,, ·A ··· 1,. '
...
I 
\n,1• ' I '
T ... _.... ..  
I ~N " :;..... ...... __..i ;;;;;
l/'I hu " < hJJ ~ !. _L.~
I S  .. • \5~ .....
.........  "' iio i1J:   b
11>"'" • =~ A 
_ .....
In 1, _ t.. ' · Inf'/ ~ ..... 
~ .o IAD I V~ , i..  ...,. _t.

1 ~ i.....

" ' "
n 1>n1
I
;Ir i , __ ..  . T  .  111'
!. ~,.,  _,M ~"' 
F IO . 14
I
.
Ju, _ ~ , . _ l/ !
·n ,., ' / 11' L  _I
I nn~ I
' i L/ , _
l
n/ I ' ".
 _, L bO
. ' • · = JO
In,,.,,
blr,t.< .., , r ,_  .... _ '  "· ' ,, ...... I _,,.._ .,jc,.,
FIG. 15
3340427 3
16
load being located away from the elastic axis. Curves constant. It is to be noted that most of these curves
have been plotted in Figures 10 to 19 to show how the are practically straight lines and are generally parallel.
torsional deflections vary with load for different com These two facts tend to prove that the torsional
binations of loads and with the different sized t ie rod . deflection due to the torsional moment of the constant
~
 ~
I 'Tl., 'In o/'J !or rin1 '5 tJ'j 'fh
l1R.h IM ;;>. 11 J:: lriri .,_J ;n, ,..z. PJ'rJ c;, ,...,.. ,,.,,..,
' }~ "' ' ,, ,,). "
~ fr., ~ro '""" ni {_ ~ I Th \lA) inr'7'
i...: h L ' nni1
I / /Y, n 1,,,1 .. ,.,, • 'Tl bc.1  In, ~I" i:? .! .:? ~q ~5
'
'i
" ' ~
11 .... 1/J ' /
v,.,
' .)P'>l'l v
( v ..... v / 49
Q.. ...... v ./ v
I I <1"1 ~ I v ..... v ~ /""'
ti Y) r, _........v v
;;; Ail" ~ ~ 0. v / ~B
~ Id< ~v j(\ 1 l>::.r v / 1720
I/ ,1n ·' .. 5) v , o'' [./. J9 A /v v '
~ 1'70 lJ _) .,,, ~ :J:.. v •uOl v ..... v. ~'!::/ v
/
' V' ,i,.)',J v Ai( ~ ~n v"' v ~ I' .
"I v
v / 11b ~;.v .,e' V' "' ~o. .... v 12 ~/ '1'i ..... v.~1  ~ ie:., v .1<1C\ t:/ v h~ v In lln ""O
·~ I/" 1 n Y\v lo.t r1:; v /l!J pfl v
v l ~VJ '(:{ ~e v .. . l(l0 .>/V
/
v l"" "' I
D"" ...... v .• <'; V I ~
In :n 8 v""' k"1. 1': ty'
v v ... I~ t/v _/ .
~o I/
v Ai! ~
.,
~
_,..v  ~ ~
•/'I lfn .:J ie /
..
/ z
In P.n
c ~"" "" T.I" nr: , ,., .... ~,., 1,,,1 ,.... /J ~"
/~ / ~ . ~ lll1 ~rn :?.ll".l
FIG . 16
A fourpin support wa used for all tests in which there
were net bending loads.
In Figures 10, 12, 13, 16, and 17 curves are given
showing the variation of angular deflections with the
variation of the small load for a series of loading in
which the net upward or downward bending loads are
net bending load is constant, while that due to the
increasing couple increases uniformly in direct proportion
to the small load. This relation is illustrated
more clearly in Figure 20.
The same data used in the cu rves just mentioned is
plotted in another way in Figure 11, 14, 15, 1
1
8, and
17
19. These curves show the variation of the angular
deflections with the variation of t he net bending load
for a series of loadings in which the small loads on the
front or rear a re constant. They are also seen to be
A furt her proof for these two relations is the fact
that the tests with equal loads on front and rear
show that t he torsional deflections increase uniformly
in direct proportion to t ho loads. Since with unequal
I
~R/
·~, ~· "';, I r) M · '<;° ll\] ,'ff,
~ .f,{L  "1 I Pm {<; Pr. lflr P1: ") ( + ft
R, ,/k ,,..,, "'Jr! ,,.,) "
.......
II LI I ('.'"' ·iJ. '.;,' 6f1 ..,,., h<:'. 'Jn 1T ln1. l/l"l AH". r"1
z::  _J T ,....,, 'A J
r:i,.., ~ n1 7;, 1.,+  Ja. "' ~ I ,f i~' Q; '~
:
IA~ I
lo.. ,) )54
~
... / ff 5
,. ./ [/'j'w
il'll ,) 65 /v / ! I
t:: I/" ~ //
I / v / ,. I
"f~
I
// v ~ / Zo
~ k1' \>9 ~ ZI / lllll / 'i7 ell.
~ V '~ rJ'J c. ~ !V J7 / _,.
/" ty.: .J.'¢ ,.G,,v ,.« ~
~ 1 .... / ,,,.1.1 ~ ~ .v ft
nRJ Ii::: ~/ ~~ v v,,,. ~ .._v_; :v
. ~ • .I 'P )J#q, PY ~· ~ !Y ~ ~"
~~) v ,,,.6 ~ ~JI r ~
.. ,,.
Ir>  ,f]J' / ,,i1([,. v .iG...v
/lL h '<:I r / v / "/"'! ... t. ~
~ v _,.V / t~
16" v v / / ./~ < v ,,,.v
111.A.h ~ v
/"

1bn, 'nf< A ,..e 71r. ~ in ,,, .,{,,, ,.. I ;,.,f
D ,,.,,, AT"' "" ifnJ .,,,,,.
J
I,,~ "I
/11
l'i'n ,.,/, ,,,,,., r1 ~,.., p,,, ,,,. /h<;
/,')/) I· "'7 ?. ']() ::on q, '}<?
FIG. 17
practically straigh t line and generally parallel. T here loads t he t orsional deflections increase uni formly when
fore, the torsional def1ection due to the con tant coup le plotted against t he net bending load, t he net torsional
is constant, while that due to the increasing net bending deflection mu t be d irectly proport iona l to the amount
load increases uniforml.v in d irect proportion to t he of net bending load.
net bending load. Figure 21 illustrates t his relation.
18
RESULTS OF TESTS IN SHOWING THE LOCA horizontal location of this center. A few samples of t he
TION OF THE ELASTIC CENTRUM drawings made for the purpose are given in Figures
The fact that the torsional detlections due to a net 22 to 29. Figures 22 to 26 show how the indicators
bending load are proportional to that load indicates on top of the bulkhead defl ected, with the different
that the moment arm of the net bending load is con sized tie rods, for the case of equal loads on front and
Tr: re: ,,,, ,,,.,, l) A ·~ '"' 'A ,j 'h
I
(/' SA ~ Jl >rv 'i Pn. ;r Din C::; rnr ,..,,...,
l:  . ~, . I
. ~ {}, ,,., IP• r ,/'I .,.,, >C:1 hn I < ·,,..,, .,,,
I J; ...... ,, , ; >n Fr on'f v '
Tl; '"""" n1 ~.,;. ,.,j l _ in .. ~,. ::> .d." 1a :::>'f
l/JJ
::.; : I . 
...... 
II?/ ! I !
I ~ v
.! I ,........~
· :"i. i... · 11n1 I
JI ... ~ oc. ;:;. ,........
"'1~ L. r1 f (C )I I
~
,........ ·v
~
1 oO Cl. ..... ~,.. __..... v
'ir ri..O..... . :... .....
/\ ~~ · ~
...........~
_J ...... ..... I ~t <
, j. : v .......
""' h
_,
~ _v ....
• /'l f J1 v i
( ........... ...........
~ ....... ,, ,_,o ~ ~ ,........v
1..
j n~ ........... 1nG J...i,_• ........... [...."
~ .I ~v; 1t ~ ...........
/IL .~,..o   _.. ......... .......
~A or'I" v '<:i v
y _ .... t. JI ~ v
.J v ~
ln.11. l ........ ...........
.......... v .....

/) ?'1
lfl/ I/I
IN of 11rl\A/1 ~rri T:: .1 •
" 1 , . r .r: nr. he:
1) '"1 l1tln I· ~/) ?.?;')
FIG , 18
stant. In other word , t here must be a fixed point in
the plane of the bulkhead a bout which all points in
the outer bulkhead rot a te in circular a rcs.
The measured deflections of the outer bulkhead were
plotted in such a way as to show the approximate
rear. Figure 27 a nd 2 show how the bulkhead
deflect ed wi th serie of loadings having constant upward
and downwa rd bending loads, and Figure 29 is
t he same for t wo serie of loadings in which the small
load was constantly equal to 150 pounds .
19
Figure 22 for the t ests with No. 10 32 rods tends to Neither did t he tests of December 2 and 3 with h show
a center 6.75 inches behind the front spar. inch rods and equal loads on front and rear give the
This does not agree with the t heoretical horizontal locat ion of a ny p oint which could be depended on as
locat ion of the center, 9.48 in ches behind the front being the elastic centrum, according to Figure 24.
spar, as well as might be desired . Also, if the hori This could have been caused by the same thing which
Tr tr:<; t/ir nl n ,j'/ · ~ ·..l..r,•'n
IML ~?.4 p ,.,,rl } r,, Ir J:. vn r:;,',;..,,, lrf
t . D If ~ ~"
:•
1 lr.1 ~n P<>i m r. in. 'F>fdnf <:;, ,,,..,,t,;
I r,..J,.,n /]/ ,;::. IPn,,..
n, .f,p ntl~ ~c:f n Pr ?, .~ 10 t::'"l :
/A.i {
'
~
l/?.i
~
(
//}'()
(~ I  i ~ '7(}/ O/bS ·  I ~ ' I 0 17 JC. c;u• i II T ,..17( ~  .:> JIU' .... ·/)~ /) ~ ~  I
I ~ '"" ~ ,.._ ...
L.... '""
 _...., ~ ,_ "" f :Jl
f..'~0  '__. .... al ]., 70t v
~ '""  ~,,
111 :/J I .....,
....~ i I  ....__.  ,, 'o,,, '<> 1.C 'U ~ .  ar __. I
'I J Joe u ~· '1ma __. ~
In •n • ~   L  I : ,__
I
11 ~I]
I
/JI 1/J
M ~f} ),,,   _'_,/ , /:=?. •n l1'r'. ,/.J I ,/7/ ~d /h~
'~ "IVl . 1 7/J ~· 'ri/J ,;:>, ']/]
F IG. 19
zontal movement of the indicat ors had been measured I made the curves in Figures 5 and 6 for variation of
and t aken into account, the results of this test would torsional deflect ion with equal loads deviate from
probably have shown t hat t he center is even 1 or 2 st raight lines. The fact that the st iffer counter wires
inches less t han 6.75 inches from the front spar. ca rried some load in compression and did not defl ect
No not iceable center of rotation at all was obtained uniformly in proportion to the load might be another
with the 7,1inch rods, as may be seen by Figure 23. possible reason for this peculiar behavior of the frame.
20
The friction in the pulleys over which the cable to the
rear platform operated could have had ome effect.
Figures 2.S and 26 for t he tests of December 4 with
fir inch rods show a general center of rotation at a
· di tance of approximately 19 inches from the front
spar or nearer the rear par. Theoretically, it is not
to be expected that th is center should be located nearer
the rear spar than t he front spar , even when taking
into consid eration the effect of the lope in the upper
27 does have a crossing approximately at the same
horizontal location as the ones in Figures 2.5 and 26,
wh ile the other three groups do not give any noticeable
general crossing.
The deflections plotted in Figu1·e 29 do not have any
application for locating the elastic centrum, but
p icture how the frame acts under series of loadings with
varying amounts of net bending load, and in which
the small load on the front or rear i kept constant.
Fiqure.s to Show Effect of Net Bendinq Loads
Upon Torsional Deflections
Deflection Due to Couple The
Loads of which Equal The Small
Load
Deflection Due to Torsional
Moment of the Net Bending
Load
Deflection Due to Couple the Loads
of which Equal the Small L oad
Net Bendinq Load lbs. ~/6' . 2/.
FIGS. 20 and 21
plane of t he fram e. Also, t he points are too high
above the datum line to be considered as centers of
rotation for equal loads. The only reason that can
be found for t his action at the present stage is t he
effect of the coun ter wires in taking compression loads.
Figure 27 a nd 28 show how the outer bulkhead
deflect with series of loadings which have a constant
upward or downward bend ing load. According to
t heory all of these readings should show the same
horizontal location of the center of rotation as the
series of readings for the case of equal loads on front and
rear. I t is seen t hat t he upper group of lines in Figure
The fact that the lines a re not parallel shows the effect
of the torsional moment of the net bending load .
In Figures 22 and 23 dotted lines were drawn
t hrough the theoretical centers of rotation and parallel
to the full lines. It is seen that there is a much more
even spacing between the ends of the dotted lines.
Thi fact justifies a theory that s ince the differences
of deflection vary uniformly with load, the theoretical
centers of rotation are more nearly correct than the
apparent centers shown by the full lines.
These tests tend to prove that the wing frame must
have an, elastic centrum or at least an approximately
21
definite point about which all points on the members movement of t he indicator points should be measured,
rotate in circular a rcs. Therefore it would be practical as well as the vertical movement, in order to make
to make further tes ts on this frame or one of similar possible the determination of the vertical location of
construction to see if it is not possible to locate a the elastic centrum and to obtain the horizontal
4.r I ~f
. . i I . ·n ~ ' .,;1.,
. #,,b i;:o; b,.,, Ve lr,,,,r b;,, <'. · · ,..,.. ~ ... ,.,
'\ F'. ~'" "" ,...,... ffc Inn ,..,., hni I ~ o,., nr
I'\ I><., 1,q, . " _, " ~, I\ "
" ~ P.,.~ n hf, ,.,. i.7 boof  IN: bv '(4 l/C. t?
'· K:,. '/V ·, [_,<.'
I"'\ ' ~ ~A
' '"" x I\.
' ~ ~" 'b_, ' § ' ~"' ~ "\
' [_,<. ' .:;n .......... , '"' ' ~ "\ .....
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)<  " r ..... ~ ~ ' ' '\
........ "' "' ;::9 )~ i'..: " '\ ' l+I ?O
""'2 :/.1 ')" ........ ' ~ I'\ (.
I'..._ ~ ,._:/: ..... '1'. '\ >n r.... '
7 t?i "'"" .......... '..
r   " ~ '\ I C.,:1_(. 'OJ
.....
' ,_ I". ~ ~ .~ _,_,, ,..,..  ~ r r ~ "' ~ ~r'\.
  . ...; ....._ ........... ~ ~ . ' t. ['...._ t ~ ·""'I "" ...
~~t. ',.".. . ~',__ '1.1'.
~ .:,:,:_:.. ..~. !'..'\ ' I'.,. rr  ~ ' 1,"' N ' f'.... ..... l':" r..... rr ......_
I i ,, I ~ 'f'.... I". .....
" f".. i:,._ " "" 'I". 1'. k?i
..... 9. 'r48' .... I'. "'r..... ~ f'"....
~ ~
~ ~  I.... ~
er .~ 'I'.' '.,. Otl
~ I 
..... 'I\." ~'
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)//" I
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 '4 ~ . ~ ',
Fm. 22
definite center of rotation experimentally . When I location more accurately . No more indicators will
making t ests for t his purpose more efficient apparatus be required than have been used before. For equal
should be used for applying the loads than heretofore I loads on front and rear, when a line is drawn between
in order to eliminate friction as much as possible. . The two positions of an indicator point, the perpendicular
counter wires should be left out. Also the horizontal bisector of that line will pass through the true elastic
22
centrum. The point where a bisector for the front
indicator crosses one for the rear indicator gives the
correct location. Two bisectors for the same indicator
will also locate the point correctly.
~. I I !
 '  J_
I I I
LL I If .J n~ fl~
I ·
I ~ '?8 t:>n. If., ..+rt)
,..n.,,.,, r wi.LJ.
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I
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I' ..... ~ ~ ' .• 1....;:,.r..o..
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;.. ;::;.?: !,la.  I' '
..... 1' r.. _J ....... '  .1..c..._ . ........ ~
 . ... . ..... I' ....... lh ' r~ ' ~,....._ ..,.. ti:::::~ l"Ci....:. .... I
+~....' I .: ~ctz ... r.;; ~ .~  1 . +j=" ;::. i=l _ r..,, ~~   r" ~
   L ~   ~ ,._,,,,.
I I I
I i
.·
' I 
~· ~I
·   i ~ I
I ( I  i v
l
~
lo.!

errors caused by taking up t he slack in ires, and the
readings with different types of drag structure could be
compared more easily.
A method for determining the position of the elastic
! : I 4f
i J
>rh '>n l lfifh
1r Dir s. 11n1 •n~ !,< ? l<"n.
ri r ~,.,, J. ~ .. i>, Pnr
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. I+' ""'"'
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t. ~
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i " 1':' ~ ~ r::::
' Hg· ~4: ~  ,.._ ~
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'
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~ ' r,
I~  ' t?t?
~
e1 it .., 1 · ~
l~~ t I~ .., f "
~ . .,   
 "'/)
I
FIG.23
In order to simplify the work of compiling the data
an.d p utting it in a form suitable for practical use,
it would be better to make the zero readings of t he
indicators come at the positions for equal loads of say 50
pounds on front and rear. This would eliminate the
centrum analytically is explained in the second part
of this report. T his will make more apparent the
value of being able to check the theoretical location
by experiment.
23
I
i
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r..~~~
']) ~"'i,' "''/, n ,., '"' °""'Iii~ iliiii;
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3340 27 4 FIG. 24
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FIG . 26
26
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~  :::::: F=::::: ::::::::, ::::=:: E:::::: 'i::+ •nr.
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'r ....... ...... ,_  .... :'::: ::::  la\ I
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i:::::R ............ ........;  :::::::   ~ /) FJ/) r !'... ...  ,r__: :: p:::::,,
""'  ,.__ c:"' E=:::: f:::::: ....... ~ .......
l (f)~Q . + £1111 '.) ,/V) .. ...;:  IGn '"" ·+l m,, ... ,... .. =:!!!! i;;;;;::: ~11 'ii')
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FIG. 27
27
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w; h.a "'~ '"·•·r I n< INW h
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r.r In'< ifnr If I
 r . .. .L n I. llnr~
~ ..... Er. 111/J I//.; 1717 n ~ r: '' .L '  ~, ~ '' 1:1;: 1171'.J ....... r.. ........
.. I'. 'TJr ~f,o Inf IT"° c:: f ~ r DPr 2 ".3 IC, I? ' ...... r::: r::::. ::::: '...
~ r ~ ~ i:::...... t:'rir l ~+n nf 1n rn' '~ r orn 1   r ~ f':::o,
,
r """' l.,t I l'>t:J
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1(2) "''' ;:,....
+ d/l/ b  D/Vllr ...;;:: r l® j,,.o
..... r:::::: i::: rI"""= + ii.cn b  , ... ,, ...... i::::::,..
(4) !"~ ? + onn .., _ rnnt> r  r... r ..... ~
(5\ ;t.,.,  r. r..... .......
+ ,,..,., .,_ i..nr
(6) ~ ., r  + >Ni b llN.~ r...
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rr ~ r... 
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FIG. 28
PART II
OUTLINE OF PROGRESS MADE ON THEORETICAL OR ANA~
L YTICAL STUDY OF A WING FRAME UNDER TORSIONAL
LOADING
In ~arall~l with the tests . de~crib_ed in Part I , . a I stresses in all members and the deflections are directly
study is bemg made of the d1stribut10n of stresses m proportional to the loads when only one load or a
a wing frame under torsional loads, and of various single couple is applied to the frame. By combining
methods for computing these stresses and deflections. the stresses or deflections due to a couple with those
This is done with the object of finding the most prac due to a single load on the front or rear, the stresses
tical method of analyzing a wing structure to determine and deflections can be obtained for cases in which two
its actual degree of torsional rigidity. The same two unequal loads are applied.
spar wing frames (fig. 1) on which tests have been In Figures 30 to 34 are given line diagrams of the
made is used as the basis of this study. wing frame with all the stresses and reactions that were
For t he purpose of investigating the distribution of computed by the Method of Least Work for five of the
stresses and the deflections under different types of cases listed above. These stresses were checked at the
loadings, different supporting conditions, and with joints and all were found to be correct except for small
d ifferent sizes of wires, the stresses and deflections errors in the last two digits, which can be neglected for
have been computed by the Method of Least Work for practical purposes.
the following cases: The stresses were not computed by the Method of
Case Support Load at front Load at rear Bracing
Ai   Fourpin_ l,OOOpounds l,OOOpounds No. l (}32 rods.
down . up.
~;~~:~~~~~~~~~ ::JJ~~:~r~J~~~~~:~ ~~~Jr~~~~~ g~~~::::: c, _______ _______ do __ ____ ___ __ do_        ~28 rods.
~:::::::::::: :::::::::::!:::::::::::::: _'.'~d:~~~~~ :~::::~~ods
In addition to the study of the distribution of stresses
the results of these computations are used for the
following purposes: First, for determining which
members of the metal t r uss type structure are the
most efficient with respect to their contribution to the
torsional rigidity of the structure. Second, for
determining . the position of the elastic centrum or
center of rotation of the outer bulkhead and the effect
of different sizes of drag truss wires upon this position.
Third, since the method of least work for computation
of the stresses is too long and tedious a process for
practical design work, the stresses computed have
been used as a basis for comparison in seeking to find
a shorter method which will give the same results or
approximations which are close enough for practical
purposes.
EXPLANATION OF COMPUTATIONS
The 1,000pound loads were used to take the place
of unit loads, so that the stresses and deflections could
be easily computed for any magnitude of loads by
simple proportion without the inconveniences of
having to work with large decimals. Analytically, the
Least Work for case B2, but it was found later, in
another way, that the stresses for this case a re the
same as those for Bi. However, special computations
were necessary for the deflections in case B2. The
stresses were not computed at all for cases D, and D2,
b ut the deflections were determined from those for
cases Ai, C,, A2, and C2 by making use of Maxwell's
Law of Reciprocal Deflections. This law states that
the deflection at any point (a) due to a load P applied
at any other point (b) of t he frame is equal to the
deflection at (b) when the load P is removed from (b)
and applied at point (a).
I t would be desirable that the computations made for
obtaining the stresses and the deflections be in.eluded
with this report, but they are so voluminous that they
would make the report too bulky for convenience.
Therefore, it was considered impracticable to include
them. However, all the stresses obtained by the
Method of Least Work are given in the line diagrams
of F igures 30 to 34, and the deflections that were
computed by the Method of Work are given in the
Appendix. Therefore, the results obtained can be
checked by anyone who so desires, as the basic data
are given, and these are standard methods given in all
complete textbooks on the "Theory of Structures."
Both of these methods of computation are described
in Chapter II of "Airplane Design."
COMPARISON OF COMPUTED DEFLECTIONS
WITH MEASURED DEFLECTIONS
\Vith a fourpin upport, as in cases A1 and A., the
computed torsional deflections turned out to be 17.5
per cent larger than the actual measured deflections,
as plotted in Figure 9. This is a very good agreement,
(28)
29
considering the n umber of members in the frame and
the rigidity of the welded joints. Another encouraging
fact is that the computed deflections a re on t he conser
vative side.
Ar
'T . __, ......
"
I<" / ,., IA r_ ,,...~ £>
V',., n<'l .,,,. s 'nn
A ii .t _, .. ". !JS> In, >f~ /")] 7 !><" r:.:: ~ "L 0..1' r.  r. ':!!' >'...;: r. ;_
.... r:::: ~t> rr.
r,.._
r ~ ~ :50 lo 
1 r,...._
r l t r
r~ ~<:?i:;: le  ......., ,_
r t r  r..:J t !50 r.> 
· r r ; ,_
II<.,,_  t
r · ~ I?~  r r ~
1 r,...._
rF ~ f5"o,
,_ r
 I>
I rt 1' .?.:i  rr r  ~! ?,... · w;,..,. r r,__
!~ ~ ~ t1"o~ I r r
r,__ ,.._ r r
I
r
f
v
~
Q:
"lj .I "' ~ )
~
In compa ring the computed deflections with different
sized rods for a fourpin suppor t , almost an exact
agreement is obtained with the ratio of the measured
deflections. The ratio of t he deflect ions with 7,1'.
I f'l.1
i I
IT.ATI ~h
iv,, ,,..., p;,.., s ·
~/!
,  /l:i<.: U N 
"IA ~ J. ~/)/
1 !)..., ,,... ?. I~ ~J· ~; rr ~
~
·~ 1 ''>11
r .... r.,_
r r. r ; . r ....... + l/lr.
 r r r  t r r rr
t  r
 r,...._ 1'"" r r  r r rr .....
r r r r  r r7 IN
rr r  rr,...._
rr:_
rr r ..... _, ~"11 ,..._ r r  rr,..._
rrr
rt  r;;; r  § rr
fl)
~  "ill
;;::
, .. ~? ., J'
~. 14 1 ~ 1F
~ '/Y.
FIG . 29
For a t wopin suppo rt, t he comp uted deflections a re
61 per cent la rger than t he measured deflections with
No. 10 32 rods and 33 per cent la rger with t he U
2 rods. T hese greater diffe rences, compa red to t he
fourpi n support, can be attributed to t he fact t hat a
real twopin support was not obtained in the tests.
2 rods to those with No. 10 32 rods is 0.446 for t hose
computed and an average of 0.442 for the measured
deflections. With a. twopin suppor t , not such a close
agreement is obtained , but t his is to be expected from
consideration of the result given in t he previous pa ra graph.
The ratios for this condit ion are 0.336 for
.1000
Srre.sses ,,;,, 11"')/?_p Fro"7'?t!? Co.r~ Az Fovr p,,;,7 Sv,..o,Por;t'
/()00 /b.J' . Oown on Fron~/000 /~.J: Upon Rear# /(J 3p RoQ"'J"
bbc9·,~;...;;;...;.;;....:..~77~~~'7'J~.:...:::..::;...;....:._:;~r:.._:I_::J~l~7:...:~...:..:..:~~.,,..::...!:...:.~~..,.::...:....::::..:::..~,.r~~~.
0 !>•':>
"' ·~ ,o 0
t ~z9 .................... ~~"'~"':l"='~::..L.~~¥.........:..::::i.~~..l.L'~~e~o~~:..::...~_.::~~~~~~.J..::!:.!!:::.~L~~::.___::~...£._..11
D
r ~ Tens/on 19 9
Co177_press /on
.Front Spar
c
Plan
.Qear Spar
FIG. 30
1000
B .Front A
Stress /n W/np rrarne case A 2, roe/r P/'n c5<./_pporr
1000 IOI}() !Os. Oown on Fro/Jr/ /tJtJO lbs. v;0 1v1 Rear ~Z~ RoO"s
J,e3~+_2_7_~_o_ __ _,~+~1_14_0_ ___~ _+_6_0_0_ _~ ~~7~~~0........,Pr".'~l~8~io'~~14_.:;.;65__~~~5~6~5~~1~7~7S:;..___,,
E D
+  Tens/on 24
 '"' Co/77pr@..s.s/on
Fm. 31
c B A
front
..51re.ss~.s /n W/n_J:I rre::nne CPJ"e L!JJ, T11Vo,P/n c5vppor/'
.tooo 1000 //J.s. Oovvn on Fron~ /000 /bS. v,,40/'? ..T"ea..... #/0.32Rods
E D
T  'T'en.s/on  14
Corn,,oress /on
"1 7. 0
FIG. 32
/792
c B A
.Fronf
JO 0.r
E
Srresses /n J.'V/np /='"r&une Co.re C.z, ro~r .P/n c.5'v,Ppor1"
/000 //Js. 001Vn on Fron!' On/y.:#: /032 Rod's
/000
D c B A
Front
Rear
4Joa._·~..,..;i~£2::f...._,..,...;~~2....,.,.,..Z~~~~~~~~"90:~~":r.~~1.«.c::.~~!'l'::""~~~:""?'~~.t.:~
'fJOIJ'i.~'~~~~..::;:::~;::_...::i,~~u:.<~~~.::.::.~'~~~~~::...:;;~:;.:c.:::....._.;:~~~~~..:;_..:;.::~'~~~
~ ,.. rsn.s ... G..n ~$
 ... Con?/'re.s.r ... oN
FIG. 33
1000
SJ';4!'.sses /n W/A_R rrO""e CPJ"e c 2 .,..Covr ,P/~ vv,.o~or/
/000 /bs. Oon o,, Fron~ On/y 74 ..c'B Ho<YJ
~/~~d'llr"'.r..:..:.~....... ..L'~;..._'~~'=::..::.:::;:;_...,...._..~~"~~~....Z~~:l~.Z
I
r '"' Te,n.s/on 76
  Co/npre.s.s/on
FIG. 34
35
the computed deflections and 0.369 for the measured
deflections.
DISTRIBUTION OF STRESSES
On the question of the distribution of stresses the
facts discovered which thus far have been found to be
of the greatest importance are as follows:
For case B1 , where the frame is supported by the two
upper hinge pins only or by just the vertical reactions,
t he stresses in the homologous wi res and members of
the front and rear trusses are the same, respectively,
for all four drag bays. Similarly, t he stresses in
members of the three inner bulkheadsB, C, and Dare
all the same, while tho e in the end bulkheadsA
and Eare dependent upon t he position of the applied
loads and reactions. It is seen by Figure 32 that the
stre es actually obtained in this case for the homologous
members of the four bays differ by a few pounds.
This is due to the fact t hat, although t he most cri t ical
comp.utations were made with a calculating machine,
certain parts which did not require such precision were
carried out with a slide rule in order to cut down
considerably on t he tedious labor without impairing
the accuracy of t he computations too much for practical
applications.
For case A1, with a fourpin upport, it was assumed
that t he frame was supported by four horizontal
reactions in addition to two vertical reactions. The
large differences between the stresses in cases A 1 and
B1 show the effect of these additional reactions. The
particular point to notice is that the differences between
the stresses in the wires for the two cases decrease
rapidly the farther out the drag bay is from the
supports.
In an actual wing there would also be drag reactions,
and there could be two more vertical reactions in addition
to t he horizontal and vertical reactions assumed
for case A1. However, the only effect of the drag reactions
for torsional loadings is to relieve the vertical
reactions by a mall amount and change the tresses
in members of the inner end b ulkhead E without
a ffecting the stresses in any of t he other members.
Al o, the additiona.I vertical reactions would only affect
t he members of bulkhead E. Therefore, in order to
have fewer redundancies and to cut down on the computations
required, only two vert ical reactions were
used, and the drag reactions were assumed to be equal
to zero. The effect of t he drag reactions was verified
by later computations.
CRITICAL MEMBERS
Those members which are most critical with respect
to their contribution to the stiffness of the frame as a
whole i.n proport ion to their weight for any given type
of loadmg are the ones which have the largest value of
p .
AE where P 1s the load in the member, A is the area,
and Eis the modulus of elasticity. The reason for t his
criterion can be readily understood from t he formula for
computing deflections of a truss by t he Method of Work.
When all members a re made of materials having the
same modulus of elasticity, as in the twospar frame
under consideration, the value of ~. or the unit stress,
serves as the criterion for determining t he most critical
members. For cases A1 and A2 with No. 1032 wires
and ~  28 wires, the wires were found to be the most
critical members for the case of equal loads on front
and rear. The lower chord member of the rear spar
nearest the support came next in line as being the most
critical. However, with the larger wires the ratio
between t he ~ for the wires and t hat for the chord
member was only 1.93 as against 3.55 with No. 1032
wires. This indicates that as the sizes of wires are
increased this ratio decreases, and that when a cer tain
size of wire is reached the chord members mentioned
would become the most critical and the ratio would be
less than 1. It would appear that this relation would
just about be obtained with the ft wires, by consider.
ing the rate at which t he ratio has dropped as the area
of wires was increased. The stresses with ft wires
have not been computed, so no definite figures for the
ratio of~ with these wires can be obtained immediately.
The effect of the size of wires on the ratio of unit
stresses in the wires and spar members is illustrated by
the curves of Figure 3, which have been previously
referred to in Part I. The way in which the slopes of
these deflection curves decrease as the size of wires is
increased compares with the rate at which the above
given ratio of unit stresses decreases with increase in
size of wires.
ELASTIC CENTRUM
Theoret ically, there is a line called the elastic ax is,
parallel to t he spars and within t he boundaries of the
frame, about which all points in the frame rotate in
circular arcs when the frame is subjected to torsional
moments acting in planes parallel to the plane of
symmetry of the airplane. The point at which the
elastic axis pierces the plane of any bulkhead is called
the elastic centrum of that bulkhead. In the frame
under consideration, in which all bulkheads are of the
same size and shape, the elastic centrum has the same
location for all bulkheads.
An important property of the elastic centrum is that
when a single load is o applied that its line of action
passes through t hat point t he front and rear spars will
both d eflect vertically by the same amount. Also,
when a single horizontal load is so applied that its
line of action passes through t he elastic centrum, the
top and bottom chord members of both spars will
deflect horizontally by the same amount. These
properties justify the theory that the location of the
elastic centrum is dependent upon the relative stiffness
of four longitudinal t russes. The trusses considered
are the two vertical t russes or the pars and t he two
horizontal trusses which a re composed of the wires, the
chord members of the spars, and the chord members of
36
the bulkheads. The deflections of each of the four
t russes are computed separately, all for a unit load
acting in the same plane as the truss. Then the
perpendicular distance from each truss to the elastic
centrum is directly proportional to its defl ection at the
outer end, where the unit load is applied. If instead of
t russes making up the frame there were four beams of
uniform sections, all having the same modulus of
elasticity , the distance of the elastic centrum from
each beam would be inversely proportional to the
moment of inertia of that beam.
In trying out the abovegiven theory, the deflection
of each of the four trusses were computed for a unit
load by the method of work. TJte horizontal location
of the elastic centrum at the outer bulkhead was determined
by taking into account the relative stiffness Of
just the two spars in the following manner :
Dr= Deflection of the front spar under a unit load .
D,= Deflection of the rear spar under a unit load.
a= Distance between spars.
ac= Distance from front spar to elastic centrum.
(See fig. 35.)
FIG. 35
a,= Distance from rear spar to elastic centrum.
Then
ac De
a;= D,
ac+a,=a
D,
ac+acD
1
=a
(l+Dr) Dc+ D,
ar De = ac~=a
aDc
ac= De+ D,                                (1)
De= 0.005792 iuch. D,= 0.012875 inch. a= 30
inches.
30X 0.005792 .
ac 0.018667 9.32 rnches.
If the upper and lower trusses were both horizontal
or if they had equal and opposite slopes together with
same sized members, this value of a1 would be theoretically
correct. Also the vertical position of the
elastic centrum would be midway between the upper
and lower trusses. However, no satisfactory way has
yet been devised for taking into account the effect of
the difference in slopes between the upper and lower
trusses upon both the horizontal and vertical position
s. In order to study this effect, the deflections at
the outer ends of the two spars, with t he whole frame
acting together, were computed for both cases, A 1 and
Az, by the method of work. This was clone by, first,
computing the deflections at the front spar for cases
C, and C2. Then by combining these with the differential
deflections which had previously been computed
directly for cases A 1 and .112 and by making use of
Maxwell's Law of Reciprocal Deflections, the deflections
at both front and rear were obtained for the cases
A, and A2 with equal load on front and rear.
From these deflections, obtain ed for all pa rts of the
frame acting together, t he values of a1 were computed
by the same formula u ed above, except that Dr eq uals
the downward deflection of the fron t spar and D, the
upward deflection of the rear spar, for the two loads of
1,000 pounds acting as a couple. A value of 9.48
inche was obtained for ar with the No. 1032 rods,
and 9.94 inches with ;!428 rods. A comparison of
these values with the value 9.32 obtained above by
considering t he two spars aloue show the effect of the
slope in the upper truss. The fact that the greater
value was obtained for the >i 28 wire corresponds
with the fact that la rger loads a re obtained in the wires
for this case than for the case with No. 1032 wires.
This indicate a tendency of the upper tru s to add
to. the stiffness of the rear spar or to carry more of t he
load to t he front spar.
Due to the extraordina ry amount of tedious labor
involved in computing the defl ections at front and rear
by the method of work for all parts of the frame acting
together, a shorter method of determining the exact
theoretical position of the elastic centrum is very much
desired . This should be one of the problems to be
taken up in more detail for further study. Being able
to definitely locate the elastic cen trum will greatly
aid the aerodynamic solut ion of the problem of torsional
vibrations. If it can be checked more closely by experiment,
the above approximate method for determining
the horizontal location will be found of considerable
value until a more exact method is discovered.
Theorectically, the elastic centrum has the same
location in the bulkhead, no matter what t ype of load
is applied, whether the load on front and rear are
the same, or whether one is greater than the other so as
to give a net bending load.
The to rsional defl ection of any loacling, regardles.
of the points of application in the plane of the bulkhead,
is equal to t he torsional deflection of t hat cou ple
whose moment is equal to the combined moment of all
the applied loads about the elastic centrum . This
relation provides a special way of checking the theoretical
position of the elastic centrum from the test
results. When We is the down load at the front spar
and W, the up load at the rear spar, the equivalent
couple load is
W= ·W1ac+ W.(aac)_ (Wc  W,) ~r + W, _______ (2)
a a
This formula was used to ee how the values of
a1, which were computed by the method of least work
for cases A1 and A2, would check up when applied to
the experimental data thus far obtained. In Table 15
the eq uivalent couples are given which were computed
by the formula for the unequal loads used with t he
smaller rods. Since no computations were made for
the deflections with ..ft rods, a valu e of ac equal to 10
inches was assumed in order to obtain an approximate
check on t he data. Columns 1 and 2 of the table
give t he unequal loads. In columns 3, 6, 9 are the
loads of the equivalent couples which should give the
same torsional moment as the unequal loads. Columns
4, 7, and 10 give the angular deflections in radian
/
37
for these couples sca' ed from the curves for fourpin
support in Figures 4 and 5, and curve ABC in Figure 6.
In columns 5, 8, and 11 are the ratios of the deflections
for the couples to those for the unequal loads, as given
in Table 13. It is seen that a very close check was
obtained for the above formula. Also, since the ratios
show t hat there was not more than 10.5 per cent
difference between the deflections for the unequal
loads and those for the equivalent couples, the computed
values of a1 must be very close to the correct
experimental values.
When the above formula is altered for use with
distributed loads, it will be found very useful for
aerodynamic studies. If experimental data are at
hand giving the variation of angular d eflections with
equal loads and the value of a,, for any given >v ing,
the formula can readily be used for determining the
angular deflection for any case of unequal loads.
FOKKER METHOD OF COMPUTING STRESSES
The first method investigated in the search for a
way of figuring stresses more quickly than by the
Method of Least Work was one used by Fokker for
taking into account the effect of plywood wing covering
in resisting torsional moments. This was used in the
stress analysis of the Fokker monoplane transport
FVII.
The method consists of first locating the position
of the elastic centrum by means of formula (1) in the
previous article. The formula used by Fokker differs
from this slightly ; in place of the deflections for a unit
load, the r eciprocals of the average moments of inertia
of each of four beams were used. The two spars
formed t he vertical beams. The upper beam consisted
of the upper flanges of the two spars and the top plywood
covering, while t he lower beam consisted of the
lower flanges of the spars and the bottom plywood
covering. Because of the difficulty in thinking of the
truss type spars having a moment of inertia, formula
(1) was specially derived to use with trusses. An
imaginary moment of inertia could be computed from
the deflection formula for a simple cantilever beam,
but this was found to be unnecessary labor.
The next step is to divide the total beam component
of the air force acting along the center of pressure into
an equal beam load applied at the elastic centrum and
a couple which will give a twisting moment equal tc
the origina l moment of the beam load about the
elastic centrum. When the center of pressure is at
the rear of the elastic centrum, this couple would
consist of a beam load acting upward at the rear spar,
and an equal beam load acting downward at the front
spar, and vice versa iJ the center of pressure is forward
of the elastic centrum.
For computing the stresses it is assumed that the
total twisting moment of t he couple is divided between
the horizontal and the vertical trusses, so that there
is an individual couple acting on each pair. Each
one of the four trusses takes a load acting in its own
plane which is proportional to its relative stiffness
and inversely proportional to its perpendicular distance
from tbe elastic centrum. For the h ·izontal trusses
the effect of the slope in the top of the frame was
neglected , and it was assumed thai both t he upper
and lower trusses had t he same stiffness in a horizontal
p lane. Their deflection under a unit load was computed
in the same manner as for the spars, using the
distance between the spars for their depth. The
average depth of the frame7.5 incheswas chosen
as the distance between the horizontal trusses, this
being the moment arm of their resisting couple. For
computing the values of the individu a l couples another
assumption found nece sary is that the bulkhead
remains perfectly stiff under the action of the twi ting
moment and that all points in its plane revolve about
the elastic centrum t hrough the same a ngl e.
Fokker derived formulas for determining the values
of the individual couples, in accordance with the above
assumptions, which involved the u e of the moments of
inertia of the four beams. In order to use t hese formulas
on t he wing frame, t he imagina ry moments of
inertia of each of the four t russes were computed from
the deflections under a unit load by use of the formula
for deflection of a simple cantilever beam. However ,
formulas can be easily derived for using the defl ections
under a unit load direct, which would be much simpler
than the F okker formulas, and t his will be done if any
furth er use is found for t he method.
The individual couples were computed for the two
sets of trusses for a 1,000pound couple acting on the
frame as in case A,. The stresses in each t russ were
then computed fQr its individual load by pure statics.
Since the ch ord members of the spars are common to
both t he horizontal and vertical t russes, their stresses
were computed by taking the algebraic sum of t he
stresses contributed by the two a dj oining t russes.
The stresses computed by this method do not compare
very closely with those computed for case A1 by
t he Method of Least Work. The stresses obtained
for the wires a re the same for all wires, which is in correct
for a fourpin support. However, it was interesting
to find that they a re eq ual to 90 per cent of the
average of t he correct stresses for a ll the panel wires.
The stresses obtained for tbe chord members of the
two spars are anywhere from 10 per cen t to 150 per
cent conservative, and a few near the outer end are of
opposite sign to the correct stresses. The web stresses
in the front spar, which a re the same for a ll bays, are
equal to t he average of the correct web stresses in that
spar, which are not the same for a ll bays. The web
stresses obtained for the rear spa r are 16 per cent
conservative compa red with the average of correct
web stresses in the rear spar for a ll the bays. The
Fokker method does not provide a ny way for determining
the stresses in the web members of the bulkheads.
An attempt was made to find correction factors and
special ways of distributing the loads along the spars
by which the Fokker method could be improved upon
and stresses closer to the correct values could be obtained.
In order to be of va lue, these factors and dist
ribu tions should have a direct relation to t he physical
proportions, such as number of bays, ratio of spar
depths, and other similar characteristics. There should
be just one set of constants for each general ty pe of
frame. Thus far no system of factors or load distributions
that is satisfactory has been determined.
38
From the foregoing it may be concluded that the
Fokker method in its present stage of development
does not have any practical value for this type of wing
frame. However, the method obtained from this
source for locating the elastic centrum is of quite great
importance. Even the method for computing stresses
may possibly be incorporated with other ideas for developing
a more accurate process.
COMPUTATIONS FOR SMALL GROUPS OF
MEMBERS
Four ways were investigated to see if a group of
members could be found whose stresses are independent
of the stresses in other members of the frame, so
that the Method of Least Work could be used for computing
the stresses in just a few members at a time.
If possible, this would give a starting point from which
the stresses in all other members could be obtained
by static equations alone. The four ways tried are as
follows:
1. A section was taken across the middle of the outer
bay and the stresses computed in the wires and web
members of the spars at that section by the Method
of Least Work on the assumption that the torsiona1
moment is resisted by the shear action of the stresses
in these four members. This group has only one
redundant member.
2. The Method of Least Work was applied to just
the members meeting at one joint. The upper joint
at the outer end of the front spar where there are two
redundant members was chosen.
3. The stresses in members of the end bulkhead (A)
and wires and members of the front and rear spars
touching at that bulkhead were computed by the
Method of Least Work on the assumption that the load
could be divided among them directly. They have
only one redundant member.
4. The stresses in all the members of the outer bay
were computed as if they were acting alone. This
group gave only one redundant member.
One or two wires were necessarily included in all
the four groups of members investigated. The computations
for the first three groups gave the stresses in
the wires as only 37 pounds, the correct stresses for the
upper wire being 2,539 pounds. either did the first
group give any close agreement for stresses in any
other members.
In group 2, for the members meeting at the upper
joint of the front spar at the outer end bulkhead,
stresses were obtained for the web members of the spar
and bulkhead which agreed closely with the correct
stresses, there being only 11 pounds error for both
members. However, the stresses in the upper chord
members of the spar and the bulkhead meeting at this
joint are directly related to the stress in the wire and,
consequently, were also small and of opposite sign to
the correct stresses.
For the members in group 3, the computed stresses
give no particular agreement with the correct stresses.
Those obtained for the spar web members and the
wires are the same as those determined by group 1.
The stresses in all the members of the bulkhead are
exceptionally small.
By group 4, which includes all the members of the
outer bay and the end bulkhead, the stress in the
upper wire was computed as 353 pounds, which is only
14 per cent of the correct value. Neither was any
close agreement obtained with the correct stresses
for any other member in this group.
It is to be concluded from the results of this set of
localized computations that the stresses in all the drag
members are directly dependent upon the rigidity of the
two spars. The function of the drag members when
the frame is loaded torsionally can be explained as
follows for the conditions of case A1• When loaded
with the drag members out, the front spar deflects
clown under its down load until a state of equilibrium
is reached, and the rear spar deflects up due to its up
load until it comes to equilibrium. The vertical
distance between the two spars increases from zero
at the supports to a maximum at the end. The
function of the drag members is to resist this tendency
of the spars to spread apart. The greater the distance
from the supports the larger are the stresses required in
the drag members to resist this spreading. Since the
drag members decrease the deflections of the spar
members, they also decrease the stresses in those
members by a proportional amount. When the whole
frame is acting the drag members deflect until their
stresses are large enough, so that when combined with
the stresses in the spars a state of equilibrium is
reached.
The fact that the stress in the outer wire, as computed
for group 4, which has more members than the
other groups, is much larger than that obtained for
any of the other three groups illustrat.es how the stress
in any one member is related to the stresses in all the
other members of the frame.
EFFECT OF ASSUMING CERTAIN MEMBERS
TO BE RIGID
Computations were made to determine the effect of
assmning certain members to be perfectly rigid. In
other words, it is assumed that certain members have
an infinite modulus of elasticity and do not deflect
under load. Those parts of the coefficients in the
Least Work eq nations which would be contributed by
these members in precise computations will be equal
to zero. The effect of this assumption on the redundant
stresses is a measure of the effect on all other
stresses, since those which were chosen as redundant
for the Least Work computations happen to be the
most critical members in the frame for a torsion
loading.
First, the redundant stresses were computed on the
assumption that the web members of both spars and all
the bulkheads are . rigid. Second, the redundant
stresses were again computed, assuming that just the
members in the bulkheads, including the chord members,
were rigid. In both cases the redundant stresses
were increased by not more than 3 per cent, which
would decrease the stresses in the spars by small
amounts that could be neglected.
By making use of this finding the amount of labor
involved in th. Least Work computations can be
lessened only slightly. However, it is possible that it
39
will be valuable in helping to develop some more
rapid method of computation which will be dependent
upon several other approximate assumptions.
THE TWOPIN SUPPORT METHOD
The most useful idea thus far discovered toward
meeting the needs of a more rapid way of computing
stresses for tor ional loadings is a method developed by
Mr. De Port, now of the Engineering Di\rision , for
figuring the stresses in the rear end of a fu selage that
is loaded in simple torsion. The method i based
mainly upon the assumption that the reaction perpendicular
to each bulkhead from one bay to its
adjacent bay are all equal to zero. The torsional
moment is t ransmitted from one bay to another only
by reactions that act in a plane perpendicular to the
axis of the fuselage. De Port derived formulas for
dividing the load among the four panels of the bay.
In these formulas the bulkheads a re assumed to be
rigid, and the constants depend upon the relative
proportions of the two adjacent bulkheads. The
stresses in the longeron members and wires of that
bay are computed for the individual loads acting in the
plane of each panel.
It was found that De Port's formulas could not be
used for dividing the load among t he four longitudinal
trusses of the wing frame under consideration . This
is due tot.he fact that the bulkheads are not rectangular.
However, a way was discovered by which the s tresses
can be computed for all the members, includ ing those of
the bulkheads, using only one of the assumptions made
by De Port, and no formulas are necessary. The same
assumption is made that the horizontal reactions
perpendicular to each bulkhead from one bay to another
are equal to zero. Just the outer bay is considered
first. It is separated from the other bays, and static
equat ions are written for all t he joints. When the
horizontal reactions are assumed equal to zero there
are enough equations that can be solved simultaneously
to obtain the stresses in all the members of the
bay, the vertical reactions, and the drag reactions.
These vertical and drag reactions are used for the loads
on the next bay, its stres es and reactions are computed
in the same manner, and so 011, for the remainder of the
frame.
The stresses in the twospar experimental wing frame
were computed by this method for the loading used in
cases A1 and B1. These were found to agree exactly
with those computed for case B1, by the Method of
Least Work, excepting for the slight inaccuracies due
to sliderule precision as previously explained. Since,
when the frame is supported by only the two upper
pins as in case B1, there are no horizontal reactions, the
above assumption is correct for this supporting condition,
and the stresses in the whole frame can be
computed by statics alone. The way in which the
loads are divided among the different parts of the frame
depends solely upon the dimensions of the members
and the shape of the bulkheads, and this division is
done automatically by the static equations. Furthermore,
the same torsional moment can be applied to the
end bulkhead either as one or any number of couples,
and when using the above given assumption t.he
stresses in the spar members and wires, the vertical
reaction s, and the drag reactions will be the ame
for all cases. Only the stresses in members of the
bulkhead are affected by the manner in which the loads
are distributed over the bulkhead as long as they are
applied in the form of couples acting in the plane of
that bulkhead.
Since with a twopin support the stresses can be
computed by static equations alone, the stresses for
this condition are independent of the size of wires.
Therefore, the stresses for ca e B2 with X'28 rods
are the same as those computed for case B1 with o.
1032 rods.
COMPARISON OF DIFFERENT TYPES OF DRAG
STRUCTURE
The method just de cribed can not be use
puting the stresses with a fourp in support. However ,
the stresse obtained this way are conservative for the
members of the drag structure. Therefore, it would be
practical to use this method of computing stresses for
more quickly comparing the relative merits of different
types of drag structure. From the stre ses thus computed,
t he deflections are read ily determined by use of
the work formula
s SL
d = A E                                 (3)
d= Downwa rd defl ection' at front spar plus
upward defl ection at rear spar, or the
algebraic d ifference of the two deflections.
S=Stress in member for equal loads on front and
rear.
s= Stress in member for unit loads on front and
rear.
L = Length of member .
A = Area of member.
E= Modulus of elasticity.
The same loads must be used for all d ifferent types
of construction compared, and the design of the spars
must be kept the same. Then, under these conditions,
the drag structure with which the frame has the smallest
torsional deflection, or smallest value of d, for the same
total weight, would be considered as the most efficient.
The abovegiven deflection formula is for general use
in obtaining t he deflections for any magnitude of equal
loads. However, for the purpose of comparing diffe rent
types of drag structure for equal concentrated loads at
the outer end, unit loads alone could be used, and
formula (3) would become
s2L
d= A E       (4)
I n order to avoid the inconvenience of working with
large decimals, a value of 10 or 100 pounds could be
used for the unit loads. For a frame like the one in
Figu re 1, which is of uniform construction along the
span, the stresses in the spars and wires are the same
for all bays and would only need to be computed for
one bay. The stresses in the three inner bulkheads are
also the same, but a re different from those in the end
bulkheads.
After the most efficient type of drag structure has
been determined, it will be desirable to find out what
40
would be the best relative proportions to use for t he
izes of homologous members of the drag structu re in
d ifferent bays for equal uniform loads along t he two
par . This can be done by distributing any convenient
amount of un iform load along the spars in the form of
concentrated loads at the bulkheads like the way in
which drag loads are distributed along a drag truss.
In other words, the uniform load over o ne bay would
be divided equally between the two bulkhead points.
Then, when computing the t resses, the reactions of one
bay would be added to the concentrated loads, applied
at the same points as the reactions, to obtain the loads
on the next bay. In order to determine t he best relat
ive sizes to use for the drag members in each bay, the
to rsional deflections at the outer end bulkhead should
be computed by formula (3) and compared with the
total weights of the frame for several different systems
of dividing the material among the different bays.
This procedure would determine, particularly, whether
the most efficient torsional stiffness could be obtained
by making all the homologous drag member the same
ize, by making the inner drag members the la rgest,
or by making the outer drag members the largest.
No special types of d rag structu re have as yet been
compared by this method, but that is to be one of the
things carried out next under th is p roject. However,
the effect of different izes of tierods is a good illustration
of the importance of designing t he drag structure
properly and of how effectively this method of p rocedure
might be used.
GENERAL DESIGN PROCEDURE
In designing a wing frame of the type shown in Figure
1, the front and rear spars would be designed for
high and low incidence loads, respectively . T he most
efficient type of drag structure to go with these spars
would next be determined by the method j ust desc ribed.
The type of structure selected could then be so designed
as to give the amount of torsional stiffness
required with a twopin support. This would be done
if it is desired to avoid the large amount of work involved
in computing the stresses for the whole frame
with a fourpin support by the Method of Least Work.
In the twospar framework studied, the computed
torsional deflections with a twopin support were only
one and twothirds times those for a fourp in support
with No. 1032 rods. With %'.28 rods the ratio was
only 1%'.. Therefore it is reasonable to conclude that,
until a more rapid method of computing the stresses
for a fourpin support has been discovered, a wing
tructure designed to give the required torsional stiffness
with a twopin support would not be too excessively
conservative for practical considerations.
After more experience with using the above method
and in comparing the computed deflections with deflections
obtained in tests on the completed wing has
been gained, reliable correction factors can be determined
for predicting the actual experimental deflections
for a fourpin upport from those computed for a
twopin support. From con ideration of the tests
made with different sized rods and of t he computed
deflections, a ratio of 0.90 would be a reasonable one
to use until more experimental data is obtained.
The degree of torsional stiffness required would have
to be determined by aerodynamic considerations.
RECOMMENDATIONS FOR FURTHER STUDY
1. More tests should be made for the purpo e of
fi nding a way to locate t he posit ion of the elastic cent
rum more directly by experi ments. Recommendat
ions to th is effect a re made in the last ar t icle of Part
I. In addit ion, the rear load might be applied more
accurately if a la rge ballbearing pulley could be
obtained, say about 12 inches in d iameter. A better
way yet would be to use a horizontal balance a rm upported
by a k ni feedge at the center. One end would
be attached" by a cable to the outer end of t he rear
spar, and the load platform would be suspended from
the other end of t he arm. Means should be provided
for varying t he vert ical posit ion of this a rm as the
frame deflects, so t hat it will be horizontal for every
load. Also, in order to keep the applied load exactly
ver t ical, it would be desirable that the balance as a
unit be capable of being moved horizontally.
2. F urther study would be practicable to see if a
way can not be found for taking into account the effect
of t he slope in the upper horizontal truss of the wing
frame upon the location of the elastic centrum, both
horizontally and vertically, in a method for determining
the location more q uickly analytically than by t he
Method of Least Work.
3. Several different types of drag structu re should
be de igned for the twospar wing frame and compared
by t he method worked out to get some idea as to which
type would be the most efficient for resisting simple
torsion loadings. It would not be of any account to
use net bending loads for this purpose, since t hey are
resisted almost entirely by the spars. Following a re
three variations which might be tried out :
(a) Use of steel tubes for drag bracing in place
of wires.
(b) Diagonal Warren trusses in place of wires.
When the e are used the bulk heads might
be lightened considerably.
(c) U e the original bulkheads and fasten sheets
of corrugated d ural covering to t he top
and bottom of the frame, leaving out t he
wires.
4. A study could be made to determine the advantages,
if any, of having three or more spar s. In th is
connection tests should be made on the t hreespar
framework a lready constructed. Only imple torsion
loadings would be necessary in these tests.
5. Further study would be practicable to see if ways
can be fo und whereby addi t ional assumptions can be
made, so that the stresses in a redundant frame can be
computed in much Jess t ime than by t he Method of
Least Work, and so that the approximations will be
much closer to the correct stresses than can be obtained
by the TwoP in Support Method.
A PP EN DI X
EXPERIMENT AL DAT A AND COMPUTATIONS
TABLE 1. Measured deflections of twospar framework. Fourpin support. Drag bracing No. 1032 swayed
tie rods
Reading No. Up load,
rear
DATE OF TEST, NOVEMBER 5, 1925
Down
load ,
front D,
Deflection, rear Deflection, front
A, Di
1        
Inch Inches Inches Inches Inch Inch Inches In ches
Q _   (*) 15 15 0. 00 0.00 0.00 0.00 0. 00 0. 00 0.00 0.00
L.            2 ___ ________ __ _____ ________ ___ __________ JOO 100 .08 . 22 .35 . 50 .03  . 11 . 18 .25
150 150 . 11 . 31 . 53 . 70 .07 .25 .32 .45
3_                    200 200 . 15 . 42 . 60 . 94 .11 .35 .48 .70
4_            300 300 . 27 . 67 l.10 1. 47 .20 .51 .80 1.10 [, __ __ ________________ ___________________ 350 350 . 32 . 78 ]. 32 1.80 .24 .61 1.00 1.36
6.            400 400 . 38 . 93 ]. 70 2. 14 .28 .74 l.20 1.64
g7 _______________________________________ 400 300 . 42 1. 05 ]. 78 2. 48 .19 .50 . 76 1.02 400 200 . 48 1. 20 2.08 2. 85 .04  12  18 .22
9 _                   400  100 . 54 l. 34 2.30 3. 22 +. lO .18 . 32 . 92
10 .                    300  100 .50 1. 20 .2.05 2. 89 .10 .18 .30 . 45
ll ___ _____________ ___ ___________ ____ __ ___ 200 100 .37 .88 1.48 2. 00 .05 . 08 .18 . 27
12     100 100 . 22 . 51 .80 1. 01 .02 . 10 . 12 . 16
13 ..      15 15 .09 .18 . 24 . 32 .02 I . 04 .06 .10
TABLE Z.11i[easured deflections of twospar framework. Fourpin support. Drag bracing, 7:1,28 tie rods
(unswaged)
Reading No. Up load,
rear
DATE OF TEST, NOVEMBER 7, 1925
Down
load,
front D,
Defl ection, rear Deflection, front
A, Di A i
!  
Inch Tnches Tnches
1                    (*) 17 17 0.00 0. 00
23 ._•__._ ___________________________________ 100 100 12 . 22 200 200 . 21 . 40
4_ __ _________________________________ _ __ 300 300 . 25 . 48 5 _______________________________________ 400  400 .32 .62
6       500 500 . 38 . 75
7g ______·_________________________________ 17  17 .00 .01 500 400 .42 . 89
9_   500 300 . 49 1. 06
10 .      400 300 . 45 1. 00
11 .                     400  200 . 44 . 98
12.             300 200 . 40 .85
13.                 300  100 .38 . 82 [4_ ___________________ ___________________ 200  100 .31 . 67
15.               200  17 I . 31 . 69
16   17  17 I . 01 . 03
(*) + Denotes an upward direction, and is to be understood when no sign is given.
 Denotes a downward direction.
(4 1)
0. 00
. 3.5
.64
. 78
1. 02
l.19
. 03
l. 47
1. 79
1.68
1.65
I. 43
]. 41
l. J4
1.18
.05
Inches Inch Inch Inch Inches
0.00 0. 00 0.00 0.00 0.00
. 45  . 02 .02 .02 .06
. 81  . 08  II .16 .25
.99 . 15  . 27  . 41 .62
l. 32  . 20 .40 .63 .90
I. 58 .28 .50  . 90 1.26
. 05 .00 .Ol  . 02 .03
1. 96 .19  .31  . 48 .66
2. 43 .03 .02 .06 . 09
2. 27 .03 .00 .00 . 03
2. 24 . 04 .18 . 32 .44
J. 92 . 04 .14 . 22 . 32
1.90 .11 . 30 .52 . 74
I. 50 . 09 . 25 . 42 . 58
1. 58 15 . 38 . 64 .91
.07 .OJ .03 . 03 .04
42
TABLE 3.Measured deflections of twospar framework. Fourpin support. Drag bracing, No. 1032 swaged
tie rods
DATE OF TEST, NOVE;MBER 14, 1925
Up Down
Deflection, rear Deflection, front Angle of twist
Reading No. load, load,
rear front D, c, B, A, Dr Cr Br Ar A,Ar Radians
 
Q_ _____ ______ ______ Inch Inches Inches Inches Inch Inch Inches I nches Inches l_ _ ________________ 15 J5 0.00 0.00 0. 00 0.00 0.00 0. 00 0.00 0.00 0.00 0. 00000 100 100 . 17 .30 .50 .68 .06 .13 .19 .26 . 94 .01296
32 ___ ________________ 150 J50 . 21 .44 . 73 .99 . JO .22  . 33 .46 1.45 .02000
150 100 . 23 .49 . 82 1.13 .05  .08  . 10 .13 I. 26 . 01738
45 _________________ 200 200 . 30 .63 1.04 1.42 .14 .30 .45  . 62 2.04 .02810  15 15 . 01 .00 .02 .02 . 02 .02 .00 .00 . 02 .00028
6?_ __________ ___ ____ 100 200 . 08 .12 . 21 . 27 .16 .40  .65 .93 1. 20 .01655 200 JOO . 30 . 75 1.12 I. 57 .00 .00 . 02 . 05 I. 52 . 02095
89 __________________ 250 250 . 37 . 79 1.30 I. 78 . 19  . 40 .62  .85 2. 63 . 03625 250 100 . 37 .83 1. 4.2 2. 00 . 04 . 08 . 17 . 23 I. 7i . 02440
10_    15 15 .00 .00 .00 .00 . 00 .oo .00 .oo .00 . 00000
1I2L _ ______ __________ __________ 300 300 . 37 . 76 1. 23 1.68  . 26 . 55 .86  1.18 2.86 . 03940 13 __________________ 100 300 . 02 . 03 .04 . 03  . 25  . 62  1.04 1.50 1.53 . 02110 300  JOO .42 . 97 1. 66 2. 35 . 05 . 13 . 25 . 36 1.99 . 02745
1154  _________________ 350 350 . 51 1.11 1.85 2. 52 .29 .62 .9 1.33 3. 85 . 05310 15  15 . 02 . 00 .04 .00 .00  . 02 .02 .06 . 06 . 00083
16_  350 300 .49 1.08 1. 81 2. 48 .22 .49 . 75 1.08 3. 56 . 04910
J 7     300 300 .42 .89 1.47 1. 99  . 24 .53 .82 1.18 3.17 . 04370
1 150 2li0 . 16 . 28 . 45 . 55 . 21 .50 .80 1.21 I. 76 . 02425
19 ~: :: : : :: : : :::: :: :: 250 200 . 34 . 70 1.16 I. 57 . 15 .31  .46 .69 2. 26 . 03120
220L_ ______________ ___ 300 20J . 40 . 87 1. 45 2. 00 . 12 .25 .36 .53 2. 53 . 03490 22 _______ ___________ 250 350 . 34 . 70 1.13 I. 48 .30 .68 1.08 1.57 3. 05 . 04210
23 __________________ 350 250 . 45 .98 I. 62 2. 25 . 19 .39 .58 .83 3. 08 . 04240
350 200 . 47 1.05 I. 77 2.46 .08  . J6 .22 .36 2. 82 . 03890
24.              15  15 . 02 . 03 .07 . 07 .00 .00 .00 .07 . 14 . 00193
25_       150 350 .00 .02 .00 .05  . 31 . 76 1.26 1.89 1.84 . 02540
2267__ ________________ _ 350 150 . 50 1.12 1.92 2. 68 . 02 .04 . 10 . JI 2. 57 . 03540 28 __________ ___ _____ 15  15 . 02 .00 .03 . 02  . OJ  . 03 .05 .14 . 16 . 00220
29 ___________ __ _____ 300 150 . 42 . 92 J.58 2. 19 . 02  . 04  . 03 .07 2. 26 . 03115
3Q ________ ____ ____ __ 300 300 . 41 . 82 1 I. 33 I. 79 .28  . 58  9 1.28 3. 07 . 04230 15  15 . 01 .01 .00 .00 .00  . 02  . 04 .06 .06 . 00083
TABLE 4.Measured deflections of twospar framework. Fourpin support. Drag bracing, y,{28 tie rods
(unswaged)
DATE OF TEST, NOVEMBER 23, 1925
Up Down
Deflection, rear Deflection, front Angle or twist
Reading No. load, load,
rear front D, C, B, A, Dr Cr Br At A,Ar Radians

Inch Inch Inches Inches Inch Inch Inches Inches Inches
1      17 17 0.00 o.oo 0.00 o.oo o.oo o.oo o.oo o.oo o.oo 0.0000
2         100 100 . 08 .15 .21 . 29 .04 .07 .09  . 11 .40 . 00552
3.        150 150 .11 .20 . 31 . 40 .07 .13 .16 . 21 . 61 . 00842
45 ____ ____ ____________ 200 200 .16 . 31 . 45 .59 .09 .16  . 21  .26 .85 . Oll72
150 200 . 05 .08 . ll .14 . 11 .25 .39 .52 . 66 . 00910
6   100 200  .06 .13  . 22 .32  . 15 .34 .55  . 78 . 46 . 00634
87 __________________ 200 150 . 21 . 42 . 65 .89 .01  . 00 .05 .10 . 79 .01090
9 ___ ___ __ __________ 200 100 . 25 . 54 .85 1.18 +.08 . 19 . 34 .50 . 68 . 00938
250 250 .21 . 41 .60 . 79 .11  . 19 .26  .32 1.11 . 01530
10_       200 250 . 11 . 20 .28 . 35 .12 .29 .43 .57 .92 . 01270
1112 ______ _______________ 100 250 .11 .22 .38 . 54  . 19  . 46  . 75 1.07 . 53 . 00731 250  200 . 25 .50 . 77 I. 04 .04  . 04 .00 .03 I. 01 . 01393
1134 ______ _______________ 250 100 . 31 . 68 1.10 I. 53 .10 . 28 . 50 . 73 . 80 . 01103 17 17 . 00 .oo .00 . 00 .00  . 02 .02 .03 . 03 .00041
1165 ______ _______________ 300 300 .24 .44 . 67 .89 .14  . 26 .36 . 46 1. 35 . 01860 200 300 .03 . 04 .04 . 04  . 19 .44 . 67  . 93 . 97 . 01338
17        150 300 .06  . 15 .25 .38  . 21  . 50  .81 1.15 . 77 . 01066
1189 _____ __ ____ _________ 100 300 .15 .34 .55 . 78  . 25  .59 .97 1. 39 . 6J .00842
20 __ __ __________ ___ _ 300 200 . 32 . 66 I. 07 1. 49 . 01 .09 .19 . 32 1.17 . 01614
2J ____ __ __ ____ _____ _ 300 150 . 33 . 70 I. 17 I. 62 . 07 . 20 . 38 . 59 I. 03 . 01420 22 __ __ ___ ___________ 300  100 . 36 . 78 1. 30 1. 83 . 12 .32 . 59 .88 . 95 . 01310
23 ___ _______________ 17 17 .00 .OJ .02 .05 .01 .02 .04 .05 . 00 . 00000 24 ___ _______________ 350 350 . 3J . 61 . 94 I. 22 . 16  .29 .38 .47 I. 69 . 02320 25 ____________ __ ____ 250 350 .14 . 24 . 35 .40 .21  . 44 .66  . 90 1. 30 . 01792 26 ___ ______ ___ ______ 200 350 . 04 .05 .05 .oo  . 23 . 51 .80 1.10 I. 10 . 01518 27 __________ __ ______ 350 250 . 35 . 71 1.14 1. 52 . 04  . 04 . 02 .10 1. 42 . 01960 28 __ _____ _________ __ 350 200 . 40 .86 1.40 I. 92 . 06 . 18 . 36 . 57 1. 35 . 01860 29 ____________ ____ __ 150 350 .06 . J 7 .27  . 44 .26  . 59 .96 1.36 .92 . 01270
30 __ ___________ ____ _ 350 150 . 40 .88 1. 46 2.00 .09 . 27 . 51 . 78 1. 22 . 01682 31 ___ ____ ____ _______ 17 J7 .00 .03 .03 .06  .03  . 06 .06  .08 . 02 .00028 32 ___ _______________ 400 400 . 35 . 69 1.07 1. 39 .18 .35  .48 .60 1. 99 . 02745 33 _____ ____ _______ ___ 200 400 .02  . 09 .16  . 29 .29  . 64 1.02 1.44 1.15 . 01585 34 ___ _______________ 400 200 . 45 . 98 I. 62 2. 22 : 01 . 22 . 45 . 71 I. 51 .02085 17 17 .oo .02  . 02 .06  . 03 .05 .06 .08 . 02 . 00028
43
TABLE 5.ilfeas ·ured deflections of twospar framework . Fou1·pin support. Drag bracing, 1.24 tierods
(unswaged)
Reading No.
l     
32 •__._. ______________ _
4.       
56 ______ ______________ _
8i _________________ _
g _____ ____________ _
IQ ______ ___________ _
IJ _____ ____________ _
1132 _.._ _____ ___________
14. • .       
15 ...     
16 ...      
17     
1189.. : _______________ _
2210 ._._.__ _______ ______ _
2"2        
23 ___      
24 .         
25 •. .        
2216 _________ ________ _
2298 ._ ________________ _
3Q _____ _______ _____ _
31.        
32.     
33 __ _ 
3354 _. _. ____________ ___ ·_
36         
3373 _________________ _
39 ___ ____ ________ __ 4Q _______ ______ ____ 41_ ___ ______________ _
42       
4443 ._ __ ______________ _
45.         
46 ..         
47         
48 •.       
5409 ._ ________________ _
5521 _. ._ __ _____________ _
5543 _______ __________ ____
5565 .. . . . _____________ 57 ___ __ ________ ____ _
58 _______ ____ ___ ___ _
59 __ ___ ___ ___ ______ _
6() __ ____ __ _____ ____ _
6612 ._ __ _________ _____ _
63 ______ ___________ _
Up load,
rear
18
JOO
150
JOO
150
200
150
100
200
2CO
18
250
200
150
100
250
250
250
18
300
250
200
150
100
JS
300
250
200
300
300
300
300
18
J8
350
250
200
350
350
J50
350
350
 18
400
300
250
200
18
400
400
4CO
18
450
3CO
250
18
450
450
18
3CO
250
2CO
18
Down
load,
front D,
DATE OF TEST, DECEMBER 2, 1925
Deflection, rear
A, Dr
Deflection, front Angle of twist
Cr Br Ar A,Ar Radians
20
JOO
J50
;:::;: ;:::;: Inches Inches ;:::;: ;:::;: Inches I Inches Inches · 
0. 00 o. 00 o. 00 0. 00 o. 00 o. co o. 00 o. 00 o. 00 o. 00000
. 11 . 23 . 36 . 49 . 00 . 05 . 09 . 13 . 36 . 00497
.13 .29 .42 .58 .OJ .co . 02 .05 .53 .00731
150
100
200
200
200
150
100
20
250
250
250
250
200
150
100
20
300
300
300
300
300
20
300
300
250
250
200
 150
100
20
20
350
350
350
250
200
350
150
 350 I  20
400
400
400
400
20
300
250
200
20
 450
450
450
20
.06 . 13 . 17 .23 .06 .10  . 15 .20 .43 .00593
. J 6 . 37 . 58 . 8J . 05 . 13 . 25 . 36 . 45 . 00620
.15 .32 .48 .65 . 04  . 05  . 05  . 05 . 70 .00965
. 09 .20 .28 .38 . 07 .13 .J9  . 25 .63 .00869
. 00 .Ol . 00 .04 .ll .22  .36 .51 .47 .00648
. 17 . 40 . 62 . 85 . 02 . 07 . 16 . 24 . 61 . 00841
. 22 .50 .81 1.14 . 09 . 23 .41 .59 .55 .00758
.00 .02 . 02 .03 .oo .00 .oo .oo . 03 .00041
. 17 . 38 .56 .75  . 06 .()9 .ll . 13 . 88 .OJ213
.12 . 26 . 37 . 48 .C9 . 17 .25 .33 . 81. .01117
.03 .08 .09 . 10 . 13 .27 .41 .57 .67 .00924
. 06  . 10  . 20 . 30  . 16  . 35  . 58  83 1 . 53 . 00731
.22 .52 . 82 1.13 . 03 .10 .20 .30 .83 .01144
.23 .55 . 87 1.22 . 07 . J9 . 34 .50 . 72 . 00993
. 27 . f,;J J. 04 ]. 47 .13 . 32 . 57 . 82 . 65 . 00896 .oo . 02 .02 .03 .oo .oo .oo .oo . 03 . 00041
. 19 .39 .58 . 78 .IO .18 .24 .30 1.08 . 01490
. 15 .32 . 46 .61 .12 .2"2 .32 .42 J.03 .01420
. 04 . 10 .11 .13 .17 . 33 . 52 . 73 . 85 . 01187
. 04 . 05 . 12 . 21 .18 . 41 . 65 . 93 . 72 . 00993
 .12 .22  . 40 .58 .22 . 50  .82  1. 16 .58 . 00800 .oo . 00 .01 .02 .02 .03 .04 .06 . 04 . 00055
. 18 . 38 . 56 .75  . 11 .19 .27 .34 1.09 . 01502
.14 .30 .43 . 56 .12 .24 .35 .46 1.02 . 01406
. 11 .23 .31 .41 .11 .22  . 32 .41 . 82 .Oll30
. 21 . 48 . 73 1.01 .04  . 05 .03 .oo 1.01 . 01392
. 25 . 59 . 93 ]. 31 . 04 . 13 . 25 . 38 . 93 . 01282
. 29 . 68 1. 11 1. 57 . l1 . 28 . 49 . 72 . 85 . Oll 72
: &~ I :b~ L~~ 1 : b~ : i~ : rig :gg 1:gg :66 : ~~
.00 .01 .OJ . 02 . 00 .01  . OJ  . 02 . 00 . 00000
. 21 . 45 . 67 . 9J . 13  . 23 . 31  . 39 !. 30 . 01792
.09 . 18 . 22 .28  . 18 .37 .56 .78 J. 06 .01461
.01 .02 .08  . 14  . 21  . 47  . 75 1.04 . 90 .01240
.29 .65 1.03 J.43 . 02 .08 .19 . 30 1.13 .01560
. 31 . 73 !. J7 1. 65 . 08 . 22 . 4J . 61 1. 04 . 01435
.09 . 16 .31  . 47  . 24 .55 .88 1.24 . 77 .01062
. 36 . 83 !. 35 ]. 92 . 14 . 36 . 65 . 94 . 98 . 01350
. 2J .44 .64 .87 .14 .25 .34  . 43 1.30 I .01790
.00 . 02 .02 .03 .02 .02  . 03 .04 . 07 .00097
. 24 . 51 . 74 1. 00  . 16  . 29 . 41 . 51 1. 51 . 02080
.14 .29 .6 .50 ~20  .w ~m ~a 1.n .01~
.05 .IO . 09 .09 .23  . 49 .77 1.06 1. 15 . 01585
.03 .07 . 18 .29 .26  . 58 .93 1.30 1.01 .OJ392
.oo .02 .02 .03 .03 .03  . 04 .05 .cs 
.29 .68 1.07 1.48  . 02 .00 +.07 . 15 J.33 .01832
. 34 . 78 ]. 28 ]. 79 . 05 . 17 . 24 . 54 l. 25 . 01723
.37 .85 1.41 1.98 . ll .30 . 54 .8J l. 17 . 01612
.00 . 03 .03 . 05 .02 .02 .03  . 04 . 09 .00124
.25 .52 . 79 1.06 :20  .36 .51 .66 ]. 72 .02370
.08 . 16 . J9 .22 .26 . 54 .84 l.16 1. 38 .01902
. 00 .02 .08  . 17  . 29 .62 J.00 1.39 1.22 . 01680
.00 . 02 .02 . C3 .03 .04 .06  . 08 . I I .OOJ52
300
 250
. 37 . 89 l. 35 1. 89 . 03 . J2 . 27 . 43 J. 40 . 020J 5
. 39 . 90 1. 48 2. 08 . 09 . 24 . 4 7 . 7J J. 37 . OJ 890
440~0 I
400
20
.OJ .03 . 03 .06  . 02 .03  . 03  . 04 . IO .OOJ38 .u .25 .n .o ~m ~n ~M ~~ 1.29 .OJ™
.04 . 08 . 07 .06  . 23 .50  . 80  1. JO 1.16 .01600
 .04 . 09 .20  .30  . 26 .59  . 95  1. 33 l.03 .01420
. 00 . OJ .00 . 02  . 04 1 .06 .07 .09 .11 .OOJ52
44
TABLE 6.Measured deflections of twospar framework. Fourpin support. Drag bracing, h24 tie rods
(unswaged)
DATE OF TEST, DECEMBER 4, 1925
Down
Deflection, rear Deflection, front Angle of twist
Reading No. Upload, load,  rear front D, c. B, A, Dr Cr Br Ar A,Ar Radians
     
Inch Inch Inch Inches Inch Inch Inch Inch Inches
64 .         18 20 65 __________________ 0.00 o. 02 o. 02 o. 04 0. 02 0.03 0.04 0. 05 0.09 o. 00124
JOO 100 .11 . 24 . 37 .51 .02 .01 .OJ . 04 . 47 . 00648
66.   150 150 . J3 . 28 . 43 .60 . 04 .05 .05 .03 . 63 . 00869
67. " 200 200 .16 . 35 . 53 . 72 .06 .08  . 09  . 09 .8J .011J8
5689 _____________ _____ 250 250 . 18 .38 . 58 . 78 .09  .1 4 .J7 .20 .98 . OJ350
70 ______________ ___ _ 300 300 . 19 . 42 . 64 . 87  .11 . J9  . 25  . 30 1.17 . OJ613
71 _________________ 350 350 . 22 . 46 . 71 . 95  . J3 .24 .32 . 40 1.35 .OJ860 72 __________________ 400 400 . 24 . 5J . 77 1.03 . 16 .29  . 41 .51 I. 54 .02125 450 450 . 27 . 56 . 84 I. 14 . J8 .35 . 49 .61 I. 75 . 02415
73 _____ ___    18 20 . 00 .03 .04 .05 .03  . 04 .05 .07 .12 . OOJ66
74 _   450 450 . 25 . 55 . 82 I. 11 .19 .36  . 5J . 64 I. 75 .02415
75.               400 400 . 25 . 54 . 82 1.10 . J6 .29 . 41 .50 1.60 .02205
76.   3.50 350 . 23 . 49 . 75 1. 00  .14  . 25 .33 .41 I. 41 . 01943
77 _  300  300 . 2J . 45 .68 .92 .11  . 20 .26 .31 I. 23 . OJ698
78.      250 250 . J9 . 41 . 64 .85 .09  . 14 . 18 .20 1. 05 .OJ450
79.  200 200
I
. 17 . 37 . 56 . 76  . 07  . JO
. II I .11 . 87 . 01200 SQ ___ ______ _______ __ 81 _ ___ _____________ 150  150 . 15 .32 . .5 1 . 68 . 04 . 0.5 05 .02 . 70 . 00965 2 _________________ _ 100 100 . 14 . 31 ..1 1 . 65  . 00 .02 .07 . J2 . 53 .00731  15  15 . 00 I . 03 . 04 .06 .02 .03 .04  . 05 , JI .00015
'
TABLE 7.Measured deflections of twospar framework. T wopin support (upper pins). Drag bracing, No. 1032
swaged tie rods
DATR OF TE T, NOVEMBER J6, 1925
Down
Deflection, rear Deflection, front Angle of twist
Reading No. Uprelaora d, load, front D, c, B, A, Dr Cr Br Ar ArAr Radians
  
o_ _____ ______ ______ Inch Inches Inches Inch<S I nch Inches Inches Inches Inches J _____________ _____ J5 J5 0.25 0.50  o. 71 0.95 0. 23  0.49 0. 74 1.06 0. 11 o. 00152
JOO JOO .06 .14  . J9  . 26  . 3J  . 65 .95 1.35 J.09 . OJ503
2.  J50 J50 . JO . 16 . 27 .3J .37 . 74 1.07 !. 49 I.SO . 02480
3_    200 200 .17 . 29 . 45 . 57 .42 .85 1.24 1.71 2. 28 .031'10 4_ _ ________________ 5 __ ________________ 250 250 . 37 .66 .99 I. 27  .49 .97 1.40 1.89 3.16 . 04360
300 300 • 41 . 75 J.11 1.41 .55 1.09  l.60 2.17 3. 58 . 04940
6.       350 350 . 66 1.22 1.80 2. 30  . 63 1.24 1. 79 2.40 4. 70 .06480
'     J5 15 .22 .45 .64 .88 .23 .50 . 75 1.08 .20 . 00276
 350 350 .58 I. 07 1.58 2.01 .62  1.24 1.80 2.43 4. 44 . 06120
JQg ___ ________________ 300 300 . 53 .98 I. 43 I. 82 .57 1.10 1.60 2.15 3. 97 . 05470 250 250 . 40 . 72 !. 05 !. 34 .50 .98 1.41 1.91 3. 25 .04480
II .    200 200 . 27 . 47 . 70 .88 .43 .86 1.24  1.68 2. 56 . 03530
12     J50  150 . 10 . 14 . 23
I
. 28 .37  . 76 1.10 1.53 1. 81 . 02495
1143 _. _________________ 100 100  . 01 .07 .09 . 12  . 32  . 66 .96 1.35 1. 23 . 01695  15 JOO .38 . 74  1.09  1.44  . 30  . 69 1. IO 1.61 . 17 .00234 15 • • __ ____.__ __ ___ ___  15 15 .24  .50 .72 .96  . 24 .51 .77 !. II .15 . 00207
TABLE 8.Measured deflections of twospar framework. T wopin support. Drag bracing, 'U,28 tie rods
(unswaged)
Reading No.
J ____ ________ ______ _
32 ._ ___________ 
4 .  
55 ___ ________________ _
7      
8.    
Upload,
rear
17
100
150
200
250
300
350
 17
Down
load,
front
 17
 100
 150
200
250
300
350
17
DATE OF TEST, NOVEMBER 20, 1925
Deflection, rear Deflection, front Angle of twist
__ D_, ___ c_, ___ n_, __ A_, ___ n_ r ___ c_, ___ n_, _l __ A_ r _ A,Ar Radians
Inch
0.00
. II
. 14
. 16
. 27
. 28
.37
.00
Inch
0.00
. 20
. 25
. 30
.48
.51
. 67
.00
Inches
0.00
.30
.37
. 43
. 71
. 75
I. 00
. 00
Inches
0.00
.40
.48
.56
.92
.96
I. 28
. 00
Inch
0.00
.01
.06
 . 08
.10
. 14
 . 17
. 00
Inch
0. 00
.02
 .09
. 15
.17
.25
.31
. 00
Inch
0.00
.03
. 12
.2'2
.22
.36
.43
.00
Inch
0. 00
.04
. 16
.30
. 28
. 48
.55
.00
Inches
0.00
.44
. 64
.86
I. 20
1. 44
!. 83
. 00
0. 00000
.00607
. 00883
. 01188
. 01655
.01985
.02520
.00000
•
45
TABLE 9.Measured deflections of twospar framework. Threepin support (two upper, front lower). Drag
bracing, '!428 tie rods (unswaged)
DATE OF TEST, NOVEMBER 23, 1925
Upload, Down
R eading_ N_o_. ·irear }~ggt __D_ , ___c_ , __B_ , __A_ , ___D_ ' ___c_ ,_ l_ _B _' _ __A _r_ A,Ar Radians
Deflection, rear Deflection, front Angle of twist
l __________________ 17 2 ______ ____________
3 __________________ 100
150 4_ __________ _______ 200
5 __   250
6     300
7      350
s ___      400
9 _____________ ___ __  17 IQ _____________ _____ 400 11 __________________ 400
12.       17
13 .     350
14 .    300
15 .  250
16 __________________ 200
17   150
1189 _. _________________ JOO  17
TABLE 10.M easured
Reading No. Upload,
rear
Inch Aches Inches Inches Inch Inch Inch Inch Inches
17 0. 00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0. 00000
100 . 18 . 37 .52 . 67 .01 .06 . 09 . 16 . 51 . 00703
 150 . 22 . 42 . 63 . 81 .01 .01 . 03 .08 . 73 . 01008
200 . 28 . 52 . 77 1.00 . 04 .03 .02  . 02 .98 . 01351
250 . 34 . 65 . 95 1. 23 .06 .07 .07  . 03 1. 26 . 01740
300 . 39 . 75 1.09 I. 41 .08 .11 . 14 .12 l. 53 . 02110
350 .4~ . 82 I. 20 ]. 54 .12  . 19 .25 .27 1.81 . 02495
400 . 51 . 98 1.43 l. 84 .14 .23 .31 .34 2.18 .03010
17 ' . 00 .02 .00 . 03 . 00 .01 .00 .00   400 . 49 . 95 1. 36 1. 70 .17 .28 .37 . 44 2. 14 .02950
400 . 55 1. 06 1. 52 1. 92 . 14 .22  .29 .30 2. 22 .03060
17 . 00 . 03 .05 . 01 .00 .02 . 01 .02    
350 .46 . 87 1. 28 I. 61 .J2 .19  . 25 .26 1. 87 .02580
300 . 42 .80 1.16 I. 48 .09 . 12 . 15 . 13 I. 59 .02190
250 . 35 . 65 . 96 I. 20 .08 . 10  .1 2  . 10 I. 30 .01792
200 . 30 .58 .85 1.06 .05 .05 . 04 . 00 1.06 . 01462
 150 . 26 .48 . 72 .90 .02 .00 . 03 .09 .81 . 01118
 100 . 22 . 42 . 61 . 76 . 02 . 07 . 11 . 20 .56 . 00773
 17 .00 . 02 . 04 . 00 . 00 . 01 I .00 . 00 . 00 .00000
deflections of twospar framework. Threepin support (two upper, front lower). Drag
bracing, &24 tie rods (unswaged)
DATE OF TEST, DECEMBER 8, 1925
Deflection, rear Deflection, front A ogle of twist
Down
load,
front
~~~l~~~1
Cr Br De Cc Br Ar A,Ac Radians
 1   
l ____ _____________ _
2 _________________ _
3 ____ _______ ___ ___ _
4.      
56 _.._._ ______________ _
7         ,
L::::::::::::::::I
9.      
10 ..   
18
100
J50
200
250
300
350
350
400
450
18
20
100
J50
200
250
300
350
350
400
450
20
Inch
0.00
. 14
.20
.21
. 25
.32
.29
.35
.36
.34
. 00
Inch o.co
. 27
.39
. 41
.47
. 62
. 56
.68
.69
. 71
.00
Inches
0. 00
. 40
.59
.62
. 71
.92
.82
1. 02
1.03
1. 04
.02
Inches
0.00
. 54
. 78
. 81
. 93
I. 22
1.08
1.34
1. 35
1. 35
. 02
Inch
0. 00
. 02
.03
. 00
.03
.03
.10
 . 05
.09
.J3
.00
Inch
0.00
. 04
. 07
.00
 . 05
.03
.17
 . 07
 . 16
 . 25
.00
Inch
0. 00
. JO
.13
.02
 .04
. 00
.22
.07
. 19
 . 34
. 00
Inch
0.00
. 15
. 21
.07
. 00
. 07
.24
.03
.20
.39
.00
Inches
0. 00
.39
. 57
. 74
. 93
1.15
1.32
1. 37
1. 55
1. 74
. 02
0. 00000
. 00538
. 00786
. 01021
. 01282
. 01587
. 01820
. 01890
. 02140
. 02400
. 00028
TABLE 11. Measured deflections of twospar framework. Twopin support (two upper pins). Drag bracing,
&24 tie rods (unswaged)
Reading o.
l_ ________________ _
2 __ _____________ __ _
3 _________________ _
45 ______ _____________ __
67 _____ _
g _________________ _
lQ9 ._ ________________ _
Upload,
rear
 18
100
150
200
250
300
350
400
450
18
Down
load,
front D,
DATE OF TEST, DECEMBER 9, 1925
Deflection, rear
A, De
Deflection, front Angle of twist
Cc Br At A,Ac Radians
;:::;: ;:::;: Inches Inches ;:::;: ;:::;:I;:::;: ;:::;: Inches 
20 o.oo 0.00 0.00 o.oo o.oo o.oo o.oo o.oo o.oo 0.00000
100 . 21 . 41 .61 .84 . 07 . 19 .31 .43 . 41 . 00565
150 .26 .50 .75 1.01 . 07 . 18 .30 .41 .60 .00828
200 .25 . 48 .73 .w . 02 . 06 .13 . 20 .79 . 01090
250 .28 .54 .80 1.08 .Ol . 00 . 05 .09 . 97 . 01338
300 . 33 . 63 . 94 I. 27 . 02 . 01 . 04 . 08 1. 19 . 01640
350 . 37 .70 1.04 1.40 .05 .05  .03 .01 1.41 . 01943
 400 .40 .78 J.16 1.55 . 07  .10 .09 .09 1.64 .02260
 450 .42 .83 1.22 1.64 . 12 .20  . 23