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1/19

Instructions for use

TitleFurther analysis of rectangular building fram es by them echanical tabulation m ethod

A uthor(s) Takabeya, Fukuhei

C itationM em oirs of the Faculty of Engineering, H okkaido Im perialU niversity = , 1: 219-236

Issue D ate 1928

D oc U R L http://hdl.handle.net/2115/37675

R ight

Type bulletin (article)

A dditionalInform ation

H okkaido U niversity C ollection of S cholarly and A cadem ic P apers : H U S C A P
http://eprints.lib.hokudai.ac.jp/dspace/about.en.jsphttp://eprints.lib.hokudai.ac.jp/dspace/about.en.jsp


2/19

Further Anaiysis of RectaRgular BuiEdingFrarnes by the M[echanicag

TabaxZation Method.

Prof.

ByWukuhei Takabeya,(Received February

7rqg'akztkakztshi.

25, 1928)

In a previous,paper the author described an exact solution of rect-angular building frames by the Meehanical Tabulation Method. Thismay be' very instructive to show how easily the siinultaneous eqttationscan be mechanically obtained, from which the redundaneies afe to becomputed.

The mefits of this rnethod consist in the large reduction in thenumber of redundancies and no mistakes can be inade in writi"g thefundamental table, on account of the systeinatic arrangement of thesttthx-number and the symmetrical property of the places where tliecoefllcients of unknown quantities are to be written.

With all its accuracy of the calculation, the method is very simpleand a knowledge of elementary algebra is sufliicient for tbe caleulationof the stresses of higher structures,

While the solution of these simultaneous equations is generallytoo long to be used in the actual design of a high building we 1iavedescribed in the sunimary and conelusion of the former paper how ithas great value as a standard calculation for checking the accuracyof another approximate rapid method.


3/19

220 Fukuhei Takabeya.'But we can now say, for this solution after repeated trials, that it

has indeed great value also as a practical calculation of the stresseso,f high buildings with speed and precision.

The solution of the set of 18 simultaneous eqqations for unknownslope op, solved in tlie previous paper, i.e. the problem of a depart-ment store building symmetrical abottt the vertical center line of tlieframe, six-stories high and six-spans long, carrying loads concentratedsymmetrieally and distributed uniformly, "'as effected by repeatedtrial and checked comparing witli the true value.

This repeated trial is accurate enough to obtaiii the approximatevalue of redundant stresses.

1. Recapitulation of the Result Obtained inthePreviousExampie. 'Fig. 1 represeiitsa six storied symmetrical bent with six equal

spans. Ail the systems of load are symmetrical and all legs of thebent are fixed at the bases, It is required to find the bending monientsat all of tlie joints.

Representing by #==l:l and by p, two tinies the sum of g of allthe members which intei'sect at tlie joint r, e.g. in Fig. 1

Pi == 2 (& + gi + g'i]

2 {98 + 55 + 88,5)

461S iiz3,P2==2(gii+gi+&+gr2)

'2 (157.8 +' 55 + 55 + 156)

847,6 in3etc.)


4/19

Further Analysis ofthe Rectangular Building Frames.

we obtain the following table of general equations to be usedminiiig the unknown slopes for the bent in question (Table

'ISABLE 1.

ill'

1).

221deter-

g-lP.i LeftFIandMemberofEquation 'oEa"-"lltlP.S[Aitu--pt /.aili

tA.t.m -Cocfficientsof(JnknownSlopesg)

=t--- -..T=OcrgndMn

dzl l(p) l;opt [{Pnlep4 i'-op,slepd]t-lltt"

rp7 riiI

1paHIopLilepio'epn .,,I rp11 1epSiirpM rpIG peIT [--"'{P]s=y..Y.koti.catso.-.ocaDV

i' F Ii11 Pi e, : il t L C122iillg, P2 8,

9i4,; tI]t 0'

' l g L3 g, P3 g.,l l ' E 0-; i41,i 8, P4 g, lgl L ' 0]I

'5al: g:a 16I P:] ij;, I 8:1i' ogJi poI g,, Tt

/C6s

7]Ir 1 g,i P7 4,1 6f E Cisi '8I 1 E g: 18, ,,I gl 6" : IO

' L /giI// g;I ctr;,g Po g,. .ig 0rl101 : i g, Pie 'E'fio Ig;, 011 E g,, P]iltit gl,! '0l2 ije lgi] Pl:] trILI2[ C]2il,

13 EliqiP]: g,: i8'i'r` C13,14

14 t g;,l1 Ltt.]: Pl" g,, ijf4 015 g;e gi4 Pl:, g,, 016 p t'ir} Fic, 6,fi 0lli7' l

'

'TH 1 gl, g,, P17 gi7 oi8i,l I

L gl, gi7 Pis Cls,lv

Substituting each value of p and g in Table 1 gives Table 2.The figures given in the right-hand column are coefllcients of

.00001X as indicated at the head of the column.These simultaneous eqttations have been solved by process of elimi-

'atioii and the fesults are as follows:"` Erratum in Memoirs Vol. I. No, 3, Appendix, for 0.0001 read ,00001.

di -


5/19

222 Fukuhei Takabeya.

'Il)ABLE 2.Left-HandMemberofEquation 1,:eo".pu'-Uarl -o

'CoefieientsofUnknownSlopesg vs:6,UouoLLaij'?elJD

'rpiop2ep:l rpi oic, ep7I elrpo ep]o ep11 epI'a ep13 ep14 rplr] g)]{i 1epiTi9is

"vlm'

1 li4e3 SJ; .t8,5 lor,s2 l:s.si M7.6i55; 1 156 l

j 03 ssIB47,6i l t o4 i56.736 55 102 o5 illj'61,ii,55!736f' 10-i.. 06 ssslii 551 432 72.5 106.g7 i/l 7.).,5r 55 72.5 : 10658 1102 55 530 uC5 5.l 09 IioeI , 55 530 5J'1 i'Fx/ 010 I' .Ir.] i402i5j L 011 53 55 co2.5)- ss

'i[]l 0

12I

4T55400i11 72.5 liii llt76513 L 362 ,sslF 'iIs3.s

14 ' t I ss .-sl366lissIlss1 015 l l'jj8I liiii55l3661-")'

1 I0

i6j F iilJ51/ '38ein,s o:11 l

/

'I.s1,lll7i-,sl'saoy7.s

18 ]lll e )ii t's3sl/1l,l/7x:)'l2621allgJ-

opi=-1.902 23,=-L-,110 09,= .006 6ifop,= ,002 72qs=-,081 71q6==1.720 ffq7==2.014 slors= -.171 21ep,== ,014 24

opio= 029 95qii==-.es6 24i2 =1,982 7sopi3--1,914 58rpi4=-.175 12op,,::= .005 13epi6= .188 98epi7==-.928 95is=4,444 89


6/19

Further Analysis of Rectangular Building Frames. 223

4

2

P, AA 1)L P,P, A Pt P,Roof e,,=7;.5 grc=77,S E=!77.5 t erat ]s 17 l6Coti :: '1 nrl 1,, IJ, I), 1., 1), P, i)z 1, I)t

tlL .cbyt-4.=J5 e=ss lk,:

13 14 15Ct" te coo A] rt n" st ll Pl 1,t'1}, 1)t 1)!P: P, P, P,th. .l.

nt.S"`-SS

s-Eie=S5 t=5S 1 o"t!/l 112 11 10Cc e ct etN "" /ftLr, dl ll Pi I)t ilt Irt pt 1) P, Pt p,pt ttt L.L'A t' 1 I elt}t' t' e,r-55 g--ssL. e,--.iJ} b ' '"': ,c. t'1 s"1e et' ret' - Tl 1)1 P,Pl /)t 1)iII P, P,1].gt- t -- s- alde lp toTtl. F.L. t,=S5 g,=55 e=ss '

t n6cl s.t"tut tt InTr-[/ lr rl[i P: l), i'l I]: I): t)T ' f); AP, op- u - zatlld I:L, e:r5,i g,=55 g=ss t/,l - '"t't- +;'-e's -S.-e't .:'-o'' -;tNift- -:'-o'" t;'-e'F. :'-e" ;,'.e'- -s'-t' .tt:tfi-,t.+ -s'-t--; x. eqct'e tt'.es

11

)Jet th.-[" ig 'F II'1'--- 'u. va

1 ll III'Z de nvaUppeT,si:Tfllcc 20'-.O'' 20'Nov 2ot.. J 2o'NO'' OO'.liOFtoffoeti[lg

!20'Nott-- ';iELEVATION OF FRAME.

P,==8200#, P,=7450#,tu,== 255#/l.fr, tu,==200g/lft.

Fig. 1.

theThe nioments at the ends offniidainental equations

the columns

Mrs=2Et"o(2op,-+ops-3pt)-C,sn4,- =- .;:)Eg,{2op, + ep, -5pa] + C,,

and girders given by

))... ....... (1)j

are indicated in.Memoirs Vol, I. No. 3 p. 187, Table 3, xvliere the" 7}7tal Ado7nen"' nieans the total sum of the illoments due to con-centrated loads and uniformly distributed loads.

The accuracy of the computation has been checked by the staticalcondition of equilibrium of the joint-moments and the percentage oferror for the least moment is giveii in the rigl}t-hand column of thesame table.


7/19

224 Fulcuhei, Takabeya.2. Computation Procedure by Repeated Trial.

The soltttion of the set of 18 simultaneous equations is too lon.crto be done, by process of elimination, in the actual design.

For the process of eliminaVion we are forced to ,use a calculatingrnachine of high capacity and the neglect of a small decimal part incalculation effects the res"lt with a considerable percentage of error.

The successive approx{mations by rePeated tfial give good resttltsin the case of ottr problem. Each equation lias generally one termwhose coefficient is relatively large compared with the others,i.e,thevalue of p is in general much larger than that of g.

If we therefore obtain an approximate value of ep for each joint,the substitution o these values in each equation in tlie terrris withg, and the solution of the equation for the terrri with p, consideringthe op with p to be unkllown, leads to a much closer approximation.

Thus for joint 18 tlie last equation of Table 2 gives:53,5(]ri3+77.5epi7+262epis=1195,

ifi which we have roughly approximate values, opi3, opi7 and epis withcoeflicient ,00001 as

opi3= 2,266ep,,=-0.970.opis=: 4561es

Substituting rpi3 and gi7 in the last equation and soiving for episwe get 4,385, a value mttch nearer the true value 4.444(SP with coeffi-.ient ,00001.This same procedure is carried out for each joint and thus a fuIIset o new valttes fQr {p is obtained and the process repeated untilthere is no practieal percentage of error when tlie accuracy of thecompiitation is checked by the statical condition ofequilibrium of thejoint-moment, i.e.:

:M;0.* The computation of these approxirnate values is treated in section 3.


8/19

Further Analysis of Rectangular Building Frames.

Table 3 shows the values of op obtained by the repeatedin Table 4 we tabulate the joint-moment compttted by


9/19

226

'

Fukuhei

TABLE '4

Takabeya.

(Continued)Total moment Percentage.theleastof errorfor1110111ent

M17-14 - 3556.1'

M16--17 tttt.tt57615.2' rm.rr .t.t.t. -

M16-c -58286.0'

M16-15 670.2n M15-14 -tt.tt.ttL- 52777.3 "tt. tttrv ttttt.M15-c - 53200.6

M15-16ttttt

348,7'

A,I15-10rmtt.

.t

76.5

..tt2.4

i"" ..t%t-tt-tt-

..- "I14-13 tur 57532.4 t.-M14-15 rm

-- 54175.7M14-17 L - 2236.6

t-

M14--11' ny - 1112.6 tL O,6 o/70

M13-14 - at--'

".t.t.

43181.5ttt .".nttt .tttt tTtLL

-M13-lg veL -t 22130,8M13-I2 21068,5

t-

M12-11 L t.t.. - t..ttt.42968.9 tttt .tttttt ..t.-M12-13

wu

21320.0M12-7 mt21665.5


10/19

Further Analysis of

'II]'ABLE

Rectangular Building Frames.

'4 (Contintted)

Tota l.,/t,,liionient PercentageoferrorfortheleastmomentM11-12 57385.0"rl11-10 -54446.0

unM11-14 -1229.4 O.5%orvI11-8

nv Ttuntwu10--11 NM10-c -53064.7M10-15 123.5 O.6%M10-9 196.3

-uuzttM9-8 52839.2ttt -m

M9-c -53150,5M9-10 155.0 2.7%'M9--4 160.6

-tttttttwwme M89 -54127.057818,1

M8-11 -1529.0M8-5 -2168.5

-"di7-8 -42645,5M7-12 21759,4

227


11/19

228 Fukuhei

'I[SABLE 4

Takabeva.v

(Contitttted)

Totalnionient PercentageoferrorfortheleastmoihentM7-6 20913.3

-ttttt..-M6-5t.tt. ua-43835.2

"ff6-7 19973.3M6-1 23g,81,2

t.ttttttt.-.u-tumttrf.-.TTttttt"M5-6 57578.0'

M5-4 -53685.4M5-8 -1729,9M5-=2 -2158.2

.r7...rrtt M4-5 t-t.t .tttt53013.0nvL-. rrttwwtu

M4-c -53213,2M4-9 102.5

'

M4-3

tttt.

95,1 2.7%t.tttttrmt.- -. tttM3-2 52967.6

ttt- .7tuatttt-tma-1vr3-cIE -53194.0M3-4 122.4 '

M3-IIIttt

100.9 3.0%ove-.rr.tt..--

rvI2-37tttt

-5ss05.0M2-1: 57851.1

t


12/19

Further Allalysis of Reetangular Building Frames,

TABm 4 (Continued)

229

Total' moment PercentageoferrorfortheleastmomentM2-Il -1702.6M2-5- -2341,5M1-I r18585.7 umM1-6 24528.6M1-2---=-

MI-1

-43095.1ttfi.tt'9292.8 -t

II-2 -851.3MIII-3 r50.4

53246.8 '

Mc-4 53237.1Mc-9 53268.5Mc-10 '53311.4Mc-15

-...--53243.5

Mc-16 60481,9

firsttliey

3. Method of Finding the ApproximateValue of op.

For the approximate valcies of g), which are to be used in theapproximation, a rough computation is sucacient oT in other wordsniay be assuined.


13/19

230 Fnkuhei Takabeya.As it is convenient to get the value as nearly as possible to thetrue value we have calculated the valties of op as in the following.

Since this procedure is not absolute, other proper methods of findingthe approximate value of ep can be applicable here, as the object isnothing btit to find the approximate value of ep. When the rnoreapproximate values are applied at first, the more aeenrate results areobtained with less effort of repeated trial.

i) A,mproximale Ultlzte ofopis.To obtain the approximate value of opis, we assume the bent as

shown in Fig. 2 and tlie equation to be used in determining the un-known slope opis is:

Pt P' P',,, g?is=C=-Z518T

-tO"

?-st-

#=77.5g =53.5.

z13

From

with coefficient'

To obtainbent as shownunknown slopes

2o'tvb" -

Fig. 2.(2) we

2Epis PisN 17 wliereI P== 2CE=01195 in3 .....,,.(2)I Pis=2(gi7+g'i3)

=::262 in3get therefore:

epis := 4,561Of .00001.-t

'ii) APProximale Vdlues of epi7 and epi6the approximate value of opi7 and rpi6, we assumein Fig. 3 and eguations to be used in detei'miningepi7 and epi6 are obtained from Table 5.

thethe

* We oniit the coeflicient in the caleulation note below.


14/19

c

Furtfher Analysis of Rectangular Building Frames.

TABLE 5.Igeft-HafidMemberofEquqtion

Cl)16 opl7

Right-HandMemberofEquation(KnownTerm)

P16 g o

E R]7 -E'op.B

231

18R

vts= 4.561 e=77.5

otl

mL7i

A17

(g)=35

Fige

6=77.5

, 3.We get the following two quations

epi6Pi6+epi7gla'0,opi6g+ep]7Pi7=-iSg,whereP16=p17=380 i7ig=77,s 7n3,

{Pis = 4.5bl.From these tvgro equations we get:

op,,=:: 0.197,op,,=-0.970.

iii) Amproximale Vkelztes of opi3,To obtain the approximate values of

assume the bent as shown in lig, 4.determining pe is:

op--i=: f.,(2)-g)ctg').

z

fl'OM

P,

16 ' t4=77.5(g)=:'35

TS

Table 5 :

'Cre

--..-I-.----1----

op12,9)13)The

op7, q6 and pai.epi2, ep7, 6 andequation ,to be

(3)

ep1,used

,(4)

welrl


15/19

232 Fukuhei Takabeya.where opet is the kllown value computed directly previously.For epi3 we have: 'c

(J)i3= ilig(7)-(J)isgti3] ' g,P., P. P,,--"==2.266 '' cili

epi2::=2250 (g)and for op7, q6, opi we assumed the same value

ie rp7--op6:==rpi=2,250 Fig.4,iv). APProximale Vtzlues of epis, opi4, rpii, opio, opg,

s, ops, op4, op3 and op2,To obtain tlie approximate values of remaiiiing op, Nsre assuniebent as shown in Fig. 5. '

EF

The

A

P----ewth--..P:,PzP!' 4' P2I).aPt

gt PtPtPxg:tl

r(g)'

et

lntt=---- l l---------eefty va

equatlons to be{J)B= -

Fig, 5.used in determining epgpAgpa+{pE6rpa-ep,,Tgire:

)

are :

.....,.....,(5)

N B

the

epa=pc2hg2

ep.ig2+(pE,C'g-(pig}'pep,2-g2

e


16/19

FurtherAnalysisofRec[angularBuildingFrames. 233where g)A, opE, opE are the known values computed directly previously,' Applying (5) gives:

qis=0.018{p,,= -0,250

With these values of epis, epi4 and also epn previously coii]puted,we get:

ep,,=: - 0,289epn'L:0os7

For ops, ops, op2 vgJe assumed.the same value as opii and for opg, ep4, op3the same valiie as wio, as the former find theniselves in almost thesame boundary condition as the latter.

1e eps=eps=-ep2=-0.289ep,:=: op4== op, =0.os7.

' Sz"nmai).,andClonclztsioiis.

The pfincipal conclusions to be Grawn from the investigation ofthe rectangular building frames by the Mechanical Tabulation Methodhave already been given (Page 183).

The solution of simultaneous equations tabulated mechanically isvery much simplified by the method of repeated trial. We have ex-plained the solution with respect to the numerical example treated inthese Memoirs Vol. I. No. 3.

Together with some additional facts, the method of finding thefirst qssumed values of rp will be hefe restated briefiy but more general-ly than described in the previous section.

For frames iii symmetrical condition 'vs,ith inultiple span of evennumber, the slopes at the joints in the center line of tlie buildingare zero, thus in Fig. 6 opD== opa==ope:=epr==O; wliile in frames of thesame kind with multiple span of odd number, the slopes of the jointsin one vertical plane nearest left to the eenter line of the building,must be eqttal and opposite to those of the joints in another verticalplane nearest right to the center line of the building. Thus in Fig. 7


17/19

234 Fukuhei Takabeya.

n Ba

Cb

t

'

z

n d

A

e

f

'

rDgg

7 n

rp.=e

rpdt=O.

Aa bge

N

e

Bc

c g c,topc=-epc1cl1 ,rpc = - rpc

t

Number oE Spans=2nFig. 6.

x{I Bx,Brp (given)

(i)

op (given)

a(ii)

N

va

Ba

9 (given)

1R'1

w 1t

Nurnber of Spans=2n+1Fig. 7.

C D fl B C'Repp=O op(g{ven)(iii)va va va(given)rp(giVeri) rp(given)Ctu

P1,:=crva nv(iv)

Fig. 8.rr, etc.

-v

c'

1 CPc = - rpc'

(iii)az'

pCgiven)R CLa b c cge(g .IVell ) 9c-- -

(iv )/Lx 7

gec'

(Pe= -ep ct1epc= - ep

and in generalop = - peI.

For frames in symmetrical condition with multiple span ofor odd number, we calculate for both, firstly the value of epArespect to the frame in Fig. 8 (i Secondly we calculate, withvalue of opA, the valtte of op. with respect to the frame in Fig. 8

e

evenwiththis(ii)"


18/19

FurtherAnalysisofRectangularBuildingFrames. 235

We find likewise with the above computed value of epA the values ofopB and opo with r'espet to the frames in Fig. 8 (iii) and (7rii)., forframes with span of even and odd nuniber respectively,

In a similar way, with the known vaiues of opa, opB, opa computedpreviously, we proceed to earry out the calculation for the frames inFig. 8 (iev) and (iev),, respectively corresponding to the given frames,with span of even or odd number.

The computation in a similar way is to be continued until oneobtains the approximate values of all op's.

In Table 3 for the values of op wliich are firstly computed as ex-plained above, we ttse the term assztmed evalztes of. ep and fof the valuesof op coTiiputed by the method of approximation of i}epeated trial weuse the term aPzSroximate valztes of op.

The principal object is to obtain, even roughly but easily, theassumed values of ep, which can be hereafter gradually corrected, eras it were, refined by the test of repeated trial which lea'ds to a mttcheloser approximation. This process must be repeated ufitil there ispractical agreement of the condition of equllibrittm or until thedifference is small enough to satisfy the desired accuracy of the given

''In the pfoblem treated above, the percentage of error for tlie leastmoment has been computed and is shown in '])able 4, where we seeno large pereentage of error greater tlian 3.0%. This error is notsconsiderable to the purposes of praetical design and if we distributethe error of discrepancy over all the joint-moments, which niust satisfythe condition of equilibrium, we can obtain thereby an even less per-centage of error.

It is notieeable that the proeess of elimination for the solutionof simultaneous eguations may perhaps be considered as a good methodof finding ttnknowns, notwithstanding that we often find there someinevitable error of accumulation caused by the neglect of some snialldecimal part in ealculation. The process of repeated trial is generallyin the strict sense the method of correctiflg the computed results


19/19

236 Fukuhei Takabeya.The percentage of error for the least moinent due to tliejoint-momentsshown in Me7noirs, Vol. I. No. 3, Table 3 may become zero by theprocess of repeated trial.

The method here proposed however, starting upon the approximaterepeated trial, lends itself w611 to an exact solution. A completesolution of the 18 simultaneous equations by process of eliminationrequires, in our experiments, at least one month-involved labour offive continuous hours a day in virtue of the necessity of using a cal-culating machine of high capaeity; the inethod of repeated triallioweveT solved the same problem with only two day's work with amachine of low capacity; the rapid computation by sliderules mayalsogivegoodr'esultsonthesameproblem. -

It is worthy of note that for all the calculation of secondarystresses the Mechanical Tabulation Method is the most exact andsimple method to get the simultaneous equatipns to be used in deter-mining the redundancies, the saving in time and labour being veryconsiderable, and it gives results, by the process of repeated trialwith speed and precision, for all values substantially identical withthose of the exact inethod of elimination.' This procedure may be well applieable to the investigation ofbuilding frames in asymmetrical condition with rnore or less consid-

fieration to the different boundary eonditions., Sapporo, January, 1928.g

f takabeya.1

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