Extended tables of stiffness and carry-over factor for structural members under axial load

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Material Information

Title:
Extended tables of stiffness and carry-over factor for structural members under axial load
Series Title:
NACA WR
Alternate Title:
NACA wartime reports
Physical Description:
18 p. : ill. chiefly tables ; 28 cm.
Language:
English
Creator:
Lundquist, Eugene E
Kroll, W. D
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

Subjects / Keywords:
Airplanes -- Design and construction   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: The tables of stiffness and carry-over factor that were presented in NACA Technical Note No. 652 have been extended to include the stiffness of a member with the far end subjected to a moment equal and opposite to the moment applied at the near end. These tables are of use in solving problems in the stability of structural members under axial load as well as in the usual applications of the Cross method of moment distribution when the effects of axial load in the members are considered. The values presented are for structural members in which the cross section and the axial load do not vary along the length of the member. The interval between successive values of the argument is small enough to make interpolation unnecessary in engineering calculations.
Bibliography:
Includes bibliographic references (p. 7).
Statement of Responsibility:
by Eugene E. Lundquist and W.D. Kroll.
General Note:
"Originally issued February 1944 as Advance Restricted Report 4B24."
General Note:
"NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003806524
oclc - 124082647
System ID:
AA00009414:00001

Full Text


ARR No. 4B24


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


WARlTIME REPORT


ORIGINALLY ISSUED
February 1944 as
Advance Restricted Report 4B24

EXM'IDED TALES OF STIFWESS AND CARRY-OVER FACTOR

FOR STRUCTURAL MEMBERS INDER AXIAL LOAD


By Eugene E. Lundquist and V. D. Kroll

Langley Memorial Aeronautical Laboratory
Langley Field, Va.


UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE LIBRARY
P.O. BOX 117011
GAINESVILLE, FL 32611-7011 USA


WASHINGTON


CA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of
apne research result to an authorized group requiring them for the war effort. They were pre-
.jij held under a security status but are now unclassified. Some of these reports were not tech-
Ij edited. All have been reproduced without change in order to expedite general distribution.


L 255


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IIATIONAL ADVISORY COrIITTEE FOR tLr.ROUTUTICS

ADVANCE R TTTRICT'D TEPOFT

EXTELIMED TABLT C 3 IFF:!'t "T. /-. CAREY-OVER FACTOR

FOR STRUCTURAL ir.qsBES UIiDER AYXIr.L LOAD

By Eugene E. Lundquist and .1J. D. Kroll


S'.T i i'RY

The tables of stiffness and carry-over factor that
were presented in "-ACA Technr.ical 'cote 'O.. 052 have been
extended to include the stiffness of a member with tne
far end subjected to a mnament e:inal and opnosite to the
moment applied at the near end. These tables are :f uise
in solving problem's in the stability: of structural members
under axial load as well as in the usual applications of
the Cross method of mti-ent ,Jl: tribllt ion when the effects
of a::ial l)nd in the nenber. are c:nidered. The valuess
presented are for structural members in v.h-icd. The roS.s
section and the ayial load do., not vary along the lenr -.t
of the meinber. he intervr l betvwe ien succ s i v atis
of the arctuient i'r nr'all r.nour h Lo :ia:-e interpolation
unnecessary in engint-rrin- cal-,cl&tiolis.

I'TRL.VCTTOi'

The tables of sti.'fne and carr.-over factor ,i'ren
in reference 1 have bcen founr.d uisful in the api;.lications
of moment-distribution r.et:.:'s to a rnm.ber of structural
problems. Recent studies have Lho," the de.sir"bility of
including In theFe tables an ad,-iiti o-l s.tffne.sF that
was not forrnerl:,' presented. .[.. -rpzrrt pa; er has
therefore been writtenn to incl'.Ide in the tales value:.
for this additional stiffness as vell as th' material of
reference 1 and con-equently supersedes reference 1.

The method of moMEnt castril--tion devrlope- 1 y lardyy
Cross (reference 2) can frequently oe used to advantage
in the stresr analysis of continuous beams-, conl-iUOLLu
frames, and rigid joint trusses, -co. of -hich occur in
aeronautical structures. in r-e re:re rC james summarized
the principles of the Cro-s ;eth.:od ind rhn-"ed how the
effects of axial load in the ner-ber- mray be con 'ideredi in
a moment-dIs triblution e nal i.?*. I refer-r.ce 4, it was
shown ho':T the principles of no.ment distribution nm:,y br-










2


used to solve problem-n in the stability of structural
rreimbers under axi.l loed.

In the stability calculations as ,ell as in the more
usual type of m.orent-ristribution analysis in which the
effects of axial load are considered, it is desirable to
have tables giviing the stiffness and the carry-over factor
as a function of the arial load and properties of the
r.member. Such a set of tables was presented in refer-
ence 1.

TI the use cf the principles of moment distribution
as applied to the stability of Vates referencee 5) the
Ftiffress of' a rate with the far eir'-e subjected to a
moment equal and opposite to the moment applied at the
near edge _.as found useful in stability problems in which
the assembly o' plate was s,-tinmEtrical about a plate.
Because the equations that nust be satisfied for the sta-
bilityr of platen are identical in the:'r general form to
those for bars uniier ayial load, a correspj.onding stiffness
for bars is desirable. .'.s is the care for plates, this
stiffi'ness is designated 33., and values .re presented
he r ei'n.

The tales appt'7 to m. rters for which the cross
sec.C non uiid a, ir l ln 1. d. not ',ry longg t -e le.r.th of'
the uFfnbSer. '.f. in-t.erval et'.'een .uccssive values of
the argu'n-'rt 'r- r,'adc '- uff ci entlv tall thi.t Interrola-
tion v, would be uinrec7' r..ry r ot -ngineer'ng calcule-
tions.


"Sl.iP DL?


E modulus .f ela.ticity

E eff-ctive -odull's of elastic 1y for stresses beyond
the ela.-tic rnrce-

I rorn.o'er't of iner'-in of' crosz sfe-tion about centroidal
rlis perpend'.cr.lar to pl].ne of bending

I. length of rc-ember


F axial load in meter (absolute value)












LL
L est 1
a = 6 -- -


,' For c'i.rZz-s ion imbrs
1 T oot
~3 c5 3 3
(T L




csch --

.T\2


T L
1 coth I




:Tg
E T
p=5 -




-- = _f.f
/







Effecticje values of a and p arue obtined by substi-
tution of (L/'j) for L 'j.
e .V













The symbols used to designate the stiffness and
carry-over factor for the different types of restraint
at t.ie far und are given in the following table and
correspond tc those .ised in reference 5:


Carry-ovor I
StiffnrFss factor ,Conditions at far end

S i C Fr e.i fixed against rotation.

S C Fr and elastically restrained
against rotation.
iT Ti
S= 0 No res're&int e'ainst .otation
at f9r end.
.1 | rjr
3- I = -1 ar en. siajected to a r..omrnt
S equal '.:rnd opposite to that
S__ a.:plieL at inea&r nd


'.e quca.ntitt-s ., C, SIT and CII of tais paper
are tir q';aLit.Is S C 'r and C re:pe-tively,
of r3IrE'nct-s 1 crd 4.





T".e fr oi- .:'enir.ir-Lns of carry-over factor and
stiffr-'ess ie the. s-aie as ch-.c's Fiven in reference 4..

,rr'r-'-on; er '- .etor.- I" & ..' .ib.r i-. on uiyielding
supr'ort- *- enf &' er. ( :c:r-ent is applied at the near
erir, r')- .'i. o. ti. .*".net ..oped at the far end to
the 11'. 'i '-,U Gl 3 t the ;icar ~n. is c llOci tns carr'y-
ov." r cto r.

i ':- If a remo r .,1 .':' l'. l in. supports
at ear" .. tc re i-_ t at ni.Ce -: n-c .-Esa-'- to produce.
a rot- ti.-. o ofe-fo'urth radien ,. tat en.d is called
the st-r, !'fic r .7










The stiffness of a member commuted according to this
definition is one-fourth that computed according to the
definition used by Cross (reference 2). In the Cross
method, the relative stiffness of the members is of im-
nortance and not the absolute value. The foregoing
definition was selected so that the stiffness of a member
of constant cross section with no axial load and fixed at
the far end would be EI/L instead of lEI/L.

Sign convention.- The sign convention used in this
report is the same as that used in references L and 5.
A clockwise moment acting on the end of a member is
positive. An external moment applied at a joint is
considered to act on the jcint; a counterclcckwise
moment acting on a joint is positive.


TABLES


Tables I and II do not give directly the values of
stiffness but give, instead, values of stiffness divided
by a factor EI/L. The actual stiffnesses are therefore
obtained by multiplying the values in the tables by EI/L.
The value of C is read directly from the tables.
S
Formulas used in calculation.- The quantities
SIT IV
-I -L-, and C are tabulated for values of
EI/L MI/L
(L/j)ep, in tables I and II for compression and tension,
respec-tvely. The stiffness S' and carry-over
factor CI that apply 'vhen the far end of the member is
elastically restrained are dependent on the restraint at
the far end and can be computed by the formulas given in
reference 4 or 5.

In the elastic range of the material comprising the
members, E = E and = 1. Aove the elastic

range, however, it is nr.cessary to use a reduced modulus E
which is called the effective modulus. A method for ob-
taining the effective modulus from the accepted column
formula for the material is described in reference 4.









6


Thc forr.ulas used In calculatir- the second, third,
fourth, e-.1 s -venLL- columns In tables T .n-d II are:


c = + (1)


(2)




S- F 1



.1 I L-t3
LI/L EI/L [T C~



(4)
ET/L EI/L ,1 + C


Equ.iptions (1) and (2) .ie.r. first nrmcsented by James
in re rience, Eqi.ai-'on (3) fo'lovia ,'romi equation (15)
of r .. ,.-,nce .[.;, sr:.I e.iu-tion f(.), fr -n: ,.Equation (1,) of
rujfer ~enc 5.

Tve fifth snd six'h colui-ns ir. tables I ar. TI were
obtai:-Led fro:. t e -'d.-:n?. co] :.n c, -s i i t.cated by
their her. '. .n s- h -: cc n,,r, E ..,v- inr.lucel Ps a on-
v.nirn-,e be':.'.'i2 e t' ey are of u :-' 'n so.e stab- lity
calculation-: (re : fro-n.?e 1).

Accursc.y cf ttabtle-..- Th'; t abl,-s of ref.frence 6 were
used in t-..: ,r par' ti,: Lf Ln--es -I en II. All VE.1uos
pres.ent.dc eh 'e 2 C:uL'tate-' to t nu.:'..r of _1co C gciv.n.
E":uatLj*-. (1), (2 ), ia.i. (5) v.,r- u.-cd to calculate the
suc',n2, thiri' s*c1." fourth cOluj.sd r-- p'.z ,Livaly. The
fiftn coluiqn vc.s obtained by .'.n; c,'.-C second column.
The ci;.:Lh colur.n vwas c"t..inrcd i :, s.-1'rl.:.g th3 fourth
colui'n Lrd "-ul t i lyin t:, t')e fir.... co.l- n. _-The sixth
coil rn w, \ s tiv.ere fo:: .,'cade to .:iir .-. n altl pr'eceding&
cclu.'nsz. All v:-.1u. s wre t, L ].,ted to Es..x signif'i-ant
fir-ur*.s nd hlt. di'fe'-ence: s for t. last col'Uu:'1 wexe
taken; rn sor..e cas- dif'f ren'es as .i..-h as the fourth
order .";re used. 'The difi irrun:3s wvre studied and,
where trr Irulcr ties occurred, to.n'- tsoles were checked.












An independent check was also made of a series of values
.IV
throughout the tables. The values for --- were
E T /r,
computeJl from equation U!,) and were independently checked.


Langley I.-emorial Aeronautical Laboratory,
National Advisory Oomrmittee for Aeronautics,
Langley Field, Va.




1. Lundquist, Eugene S., and b},roll, W. D.: Tables of
Stiffness and Carry-Over Factor fr Structural
Members under Axial Load. T.Y. To. 652,
IIACA, 1938.

2. Cross, Hardy: Analysis of Continuous Frarics by
Distributing Fi;ed-Lnd Moments. Tr-ans. A. .C...,
Vol. 96, 1'92, pp. 1-10.

5. James, Benjamin Vd-lie, : Principal .-ffects of A::ial
Load on ?"omen-it-Distribution rnalysis of Rigid
Structures. T.il. Po. ., ACA, 19~3.

l. rundquist, Eur.'ne E.: S c:ilit;y of Structural
'mojibers ur.d--r Axial Ioead. T.' o. l17,
?,ACA, 1957.

5. Lundquist, Eugene ., Sto';ll, otf-id.e Z., and
Shue tte, Evan {. : Prr.incip, ]'s of crlomen.it Distri-
bution Applied to Stabili-t: of Structures Com.nosed
of T-ars or Plates. 1. .A ARF- No. 3T)06, Nov. 1943.

6. "-ayashi, r.iIchi- S3ic-n- und : ichr tc-llige Tafeln
der Freis- und Hyperlvelfiu.'.tionen unJrd deren
Produckte so'vie der Cam'rsfun.tion.
Julius Sprin-er (Berlin), 1'926.















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TABLE II.- TENSION

C S11 5 C 2 f202 SIV
V/eff EI/L ZI/L fI/L)2 EI/L
0 0. 00000 0.750000 1.ooooo 0.250000 3.;'o0oo 0.500000
.1 .2Q9757 .750512 1.00036 .249757 .2956 .500422
.2 .L99001 .751998 1.00133 .249002 .2LQ665 .501666
.5 .29776o .752488 1.00500 .247765 .29252 .5:o4
.496055 *75764 1 00552 246049 216675 .506
.5 .:95851 .7612 1.00651 :2458o9 .247957 .510374
.6 .41167 .767818 1.01194 .2L1245 .247012 .514911
.7 .807 .77+16ct 1.01623 .2!'200 .2L1?% .520252
S.8 .48519 .71 3 1 1.02116 .25 759 .2LL798 .;26586
.Q .h4057 .7829?5 1.02672 21055 .2l59 "' 5 .02
1.0 .476246 .79W82 1. 0291 .22:..I10 .21785 .540987
1.1 .471556 .808515 1.0,5971 .222%65 .240.76 .5.,428
1.2 .466550 .C19215 1.0 712 .217651 .25666 .55Mb08
1.2 .461194 .86070, 1 05513 .212700 .2'67%9 .568510
1. .455575 .299 1 .0672 .2075468 .5hib .579118
|1.5 .9b99 .055921 i 072d9 202229 2527" 5 .502141
1.6 .43554 .869586 1.082b2 .196775 .20653 .6023576
1.7 .85726 .b5915 1.09290 .191219 .226,.96 .614969
1.8 .430802 .89db67 1.10571 .1tc559 .226081 .628260
1.9 .124167 .14429 1.11 05 .1--91~ .223697 .642080
2.0 .417*08 .3 50555 1 12689 1 71229 21251 .c56516
2.1 .410548 .947214 1.1''92 .1Th550 .-16751 .671522
2.2 .40610i .9,643dO 1. 1520, .b12901 .21t205 .c7071
2.5 .356616 .962024 I.It53 .157505 .21b621 .705145
2.1 .'9588 1.00012 1.17906 .151779 .211006 .79723
2.- .524, 1.01865 1.19525 l. 6,40 .20ab6 3c7d2
2.6- .'.7 o02 1.0575 1.207..5 .1,i1002 .-20'57 .75430)
2.7 .56480 1.05853 1.22287 .15777 .0o0 .77226
2.8 .561092 1.07646 1.25827 .l'0677 .200;b9 .7.?0646
2.9 .5r,553 1.00642 1.25406 .12570D .197697 tbOq.20
3.0 .347676 1.11668 1.27022 .120678 .195052 .d2859L
5.1 .530871 1.12722 1.2 -67 .116195 I.2'78 .-4120
3.2 .534149 1.lSdOA 1. '050 .111656 .1- 7L .p1'79 1l
5.5 .327520 1.1'79 1.32076 .107269 .187122 .886158
3. .2090Q 1.20057 1 .6.26 .100534 .164C29 -06b9;
5.5 .51456&6 12-198 1.55606 .0C6hL19 .181965 .929491
3.6 .508215 1.24L558 1.37L16 .095-0214 .179.29 .9505,64
5.7 .302062 1.2bi7 1 C'25r .091241- .176 29 .97 16 8
5.b .295990 1.26754 1. 11i7 .0876098 17L466 .975477
9 .2900L2 1.50976 1.45007 .0,41242 .172042 1.01529
:0 .284221 1.5521 1.LU921 .0807L 15 it?658 1.037, 1
4.1 .276L529 1 .3566 1.46659 .0775765 .167518 1.05955
4.2 .272b67 1.97731 1.L4620 .074510U .165022 1.01?77
4. .2t75 i.ooo9 1.5002 .071579 .162771 1.10457
4.4 .2622.4 1.42297 1.2005 .O87666 .160566 1.127!5
4.5 .257063 1.-L597 1.9 828 .0660815 .158409 1.15026
4.6 .252022 13.4607 1.56870 .0635151 .156oo00 1.17555
4.7 .247110 1.L9225 1. 58950 .0610651 .15.236 1.19057
4.8 .242524 1.-, 1.610oo .0567210 .152225 1.21992
4.9 .23766 1. 5869 1.63101 .0564845 .1 0260 1.2-438
64.0 .2128 1.56252 1.5211 .0h456 .tlEt53 1.26696
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UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE UBRARY :,
P.O. BOX 117011 :
GAINESVILLE, FL 32611-7011 BUA



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