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Title Analysis of rectangular building frames by the mechanical tabulation method

Author(s) Takabeya, Fukuhei

Citation Memoirs of the Faculty of Engineering, Hokkaido Imperial University = 北海道帝國大學工學部紀要, 1: 155-191

Issue Date 1928

Doc URL http://hdl.handle.net/2115/37667

Type bulletin (article)

File Information 1_155-191.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

Analysis of RectanguZave Building twames

the MeclaanicaZ Wabulation Method /

' By

Prefl Wukuhei Wakabeya, Kbgwkz`kak2tsrki. ' '

(Received Ma,rch iS, i927)

by

'

' The proper treatment of frames with stiff connections, composed of

rectangylar elements, is of constantly increasing importance in structural

design. When loads are applied to fi'ames of stiff connections at thejoints,

the members of the structure are subjected to deformations accompanied

by seconclary stresses. For the analysis of these stresses there are several

methods which are however so involved and so tedious that most building ,ordinances are accustomed to mal<e certain assurnptions for the determina--

tion of the redundant stresses. ' For such structures it is highly desirable that the most convenlent

methods of analysing the stresses and rigorous calculation should be

developed.

We are hopeful that an easy and accurate method for obtaining joint

moments of framecl structures in stiff connection will eliminate current

assumptions and iead to a rigorous design with speed and precision, con-

tradicting the prevailing impression that the analysis is very tedious.

In some problems of eiastic beams the method of area moment may

often facilitate the caiculation of the statically indeterminate stresses, As

its application to the investigations of the stresses in rectangular building

IS6 Fukuhei Takabeya. '

frames and continuous beams the slope.defiection methodi) proposed by

Prof. WiLsoN may be recommended.

The merit ofthis method consists ln the large reduction in the number of

unknowns. The solution of the equations is very much simplified therefore.

This paper offers fundamental tables, derived fi'orn the slope deflection

methocl, tabulated mechanlcally by the rules upon which our mechanical solu-

tion is based. These standarcl tables Iead us to the solution of statically

indeterminate stresses with strict accuracy. . The advantages of the new method are simplicity and easy understand-

ing of the principle as well as superiority in speed.

Assumptions upofl which the Analysis is based, and

the Cenventionai Notation aptd Signs utsed.

The analysis in this bulletin is based upon the fbllowing assumptions:

I). The connections"between the columns ancl girders are perfectly

rigid. . 2). The length of a member is not changed by direct stress, and the

deformation of a member due to the internal shearing stress is

･ zer'o.

3), The settlement of the fbundations and temperature change are not

considered. ' tt/ 4). The vertical deflections of the ends of all girders are relatively

equal to zero and the horizontal deflections of the tops of all columns

of a story are equal.2)

' '

I) The prlnciples of the slope-deflection inethod were gkren firstly by O. MoHi{ and

the equations of the slope-deflection for a member acted upon bY'forces ancl couples at

the ends were deduced by Manderla in I878,

The application of these equations to high buiidings has been made by sever,al writers:

W. M. WILsoN and G. A. MANEy, "Wind Stresses in the Steel Frames of OficeBuildings'', University of Illinois Bulletin, No, 8o, Igl5.

W. A･I. WiLsoN, F. E, RrcHART and CAMIuo WErss, " Analysis of Statically Indeter-

minate Structures by the SIope Defiection Method '', University of IIIinois Bulletin, No.

Io8, Igl8,

2) Provided the change of span length caused by bencling is not considered, the

proposed method may be'applicable without assurnption (4). Some special problems, in

Analysis of Rectangular Building Frames by the Mechanical Tabulation Method. Is7

According to these assumptidns' all the columns and girders, in strained

state, vvhich intersect at one point are gubjected to an equal change in slope

and the moment at' aii end of a column is expressed as a,function of the

changes in the slopes' and of the deflection of one end of the column

relative to the other end. Lil<ewise the moment at an end of a girder is

expressed as a function of'the changes in the slopes of the ends of the

girder. Generally if a member AB restrained at the ends in flexure is

subjected to any system of intermediate loads, the moments at the ends of

thememberisexpressbdbythewellknownequations,vis., '

' llllb`:2atab(2{C'a+SC'b-3iCtabj-(J;tb,

(I), AtiZa =2E8ab(29b + 9a-3vCt(tb] + Cla,

wherewedenoteby . fi4b, JC. the resisting mornents at A and B,

./ . E themodulusofelasticityofthematerial, g., gb the changes in the slope of the tangent to the elastic curve

at A and B, ' / ghb==7, in which f==moment of inertia of tlie section of the member

AB;l=Iength of the member AB,

pt.b=::-, in whlch o"----deflection of the end A from its initial position,

' C;,b = 21A.. (32-l]; CL.= 21A, (21-.3R], in which A=area of the m6-

ment diagi'am of a simple beam AB due to tlie intermediate

loads; R=:distance of the centroid of the area A from the

end B.

The conventional signs of the quantities used in the equations in this

bulletin are as fo11ows:

which thetreated by

this (4)-

besonderer

change of spanthe same i･yriter.

This paper is:

Ber[icksichtigung

length forms an essential part in worldng stress, have been

To make this point of view n]ore clear we have clared to write

Zur Berechiiung des beiderseits eingemauerten Trttgers unter der Lttngslcraft, Berlin, ig24.

IS8 Fakuhei Takabeya.". 'The 'resisting moment on a section is considered positive when the

resisting couple acts in a cJoc]<wise direction upon the portion of the member

consiclered and by this conventional rule each resisting moment of,the

portion of the member considered must always be indicated witl} the sub-

scriptse.g.1di[,b,nl.i . ･ The moment of an external force is positive when it tends to cause a

clockwise rotation.

The change in slope is considered positive when the tangent to the

elastic line of a mernber･ has been turned clocl<wise, measured from its

initial position. The deflection is considered always positive to the rotation

in the clocl<wise direction from the initial position of member. Here we

give the definition of " defiection " for the movement of one end ofa member

relative to the other, measured perpendicularly to the initial position of

We denote the intersections of the neutral axes of the columns with

the surface of foundation on the same level by I, II, III, IV, etc., and the

yalues of e of the columns of the first story by 8b 4th 6m} Cv, etc.,

beginning at the left and reading toward the right.

The intersections of the neutral axes of the girders with the neutral

axes of the columns are clenoted by i, 2, 3, 4, etc., beginning at the left

and reading towarcl the right and upwarcl from the last intersection at the

top of the first story, then along the girders of the top of the seconcl story

toward the left. From the last intersection at the top of the second story

read upward again and toward the right and so on (Fig, i).

The values of e of girders and some special columns in the way of

reading are denoted by subscript of the letter oftheintersection from which

the member considered begins in the direction as mentioned above (Fig. i).

The values of 8 of columns are denoted by ? with subscript of the letter

of the intersection upon which tlie neutral axis of the column considered

stands. e.g,, 6',, 6'2, 4',, etc. (Fig. 2).

gk 'k gk k

n"t d ga `

mgi agh-

1 2 3 f

'agmmec

!¥'

3'"

-I

ne"/

m-g, E. E, Jk.M-1

Analysis of Rectangular Building Frames by the Mechanical Tabulation Method･ Isg

' 'ptgv%-,.gs S,ll [ii.lt,-;'I SiE-i--T: /i

l'ltll.ISimi･-1-li/ gi"il"igi,,',:- l/

Ldii-ISdyi .t-l

Fig.L' Fig.2. Fig.3. 'intersWtcet adtei}?tg,gb.¥ P(FlgW.03)ti,MeS the sum of g of all the members which

Po･=2(er--i+&t+E"r+#'?n],

(2) p, ==2k,.-,+e,+e,,],

p,==2(g,-1+6,].

For all columns in a story we denote two times the sum ofe by

' ' , IFoi-IIIrlie,=fi.i`s(2S+tOl`Y.iF6m.+......+6,.]

For the second story:

(3)i.., li,il.2'1,i,I.i'`,',.i[III"g'3"･""'+6'--i+g-)

1 X,=2(6t.,.,+6'..,+.･････g',,.-i+g,,)

Itii)e dpe.10.i,e..thg...l9.itd; te'm C divided by 2E by p ,g .

'

'

I6o Fukuhel Takabeya. ' 'the total horizontal shear due to wind loads in a bent at the first, second,

third and n`h stories by PT2;, ng, PV'k, and PZ,,;and we put: ･ '

tl pmlk, i ･ -ngth2, ･' , (s) pl75rk3 e3.= 6E' : PZekn 9n== 6E '

where rki, k2, k3, and h, denote lengths of columns, in the first, second, third

and n`h stories, measured from neutral axis to neutral axis of the girders.

' ' ' . ConditioptofEq"iNibr,iumafld'E)ergvationof

GeneraE Equatioms.

' We treat in this bulletin a multip16 storied bent with any number of

spans, The stresses by wincl loads and any system of vertical loads, in all

of the members, are required to be calculated.

Since the moments at joints are expressed as a function of the slopes

and the deflections of one end of the member relative to the ether end, we

firstly worl< out the slopes and the deflections by solving the equations

obtained by such conditions ofequilibriuni as are explained in the foilowing.

With those values of the slopes and the deflections the moments can be

then computed.

To obtain the equations fi:om which the unl<nown slopes and deflections

can be determined, we have two 1<inds of staticai conclition ofequilibrium:

one of which is of the equilibrium under the action of the moments at

every point of intersection of the neutra! axes of girders and columns,

considering the point of intersection as a free body; the other 1<ind of

condition is of the equilibrium undgr the action of the moments at the two

extremities of all the eoiumns in a story, considering together all of the

columnsinastoryasafreebody. ' From the former condition ofequilibrlum, each joint gives one equation;

.s

Analysis of Rectangular Building Frames by the Mechanical Tabulation Method. I6I

from the latter each story gives one equation,i,e. for an n-storied bent with

m-points of intersection of the neutral axes of girders ancl columns, the

number of equations obtained, from which the slopes and the deflections

can be determined, is fn+ii. The unknown quantities in the obtained

equations are the changes in the slopes and the horizontal deflections of

the columns. The changes in the slope at a joint, by the first assumption,

are equal for all of the members which meet at thejoint; and the horizon-

tal defiections of the columns in a story are egual by the fourth assumption.

We obtain therefore m+n equations in which fiz+n unl<nowns are contained.

Solving these equations we can determine the slopes and the deflections,

and then the moments requirecl.

Sibzglla SZoried Bent zevicb eviay Mtinber of' .Sipa7zs. Aiz.7 &!stein of l72?rtical

Loaals. Ltzg:s l71ixeal at cbe Bases.

Fig. 4 represents a single storied bent with any number of spans.All legs of the bent are fixed at I, II, III,......ALf, and Afr. This bent

carries any system of vertical loads. It is required to find the stresses in

all of the members.

Representing the horizontal deflections of i, 2, 3,...... n-i and n by

a and yi= 2 , we obtain fi'om equation (i):

(6) -2I4il.R=2alt(2so.-3t2i],

' ' (7).2TCI..-1=:2ff,-1(290ot+SOr-ll+C;ur-1, , '.

(8)nC.+i=2EEI(297+9r+i]-Cw'+b ' '

(9) ntL,,,.=2nt.(gO.-3itt,]･

123 r-1rr+ln-2n-17tJs

l'

h,

e,=it'

.Itt

･?1

ent

.v,.

en=h,

k- ii -"- le -teis J

Er't fi:i'i eei:Ilti]'

i I .111 ･t/--

3-L,,i-'

i R

t-1Fig･ 4"

`

R+I-Q ,J-

- c#-i

eNJt

,Ni-ili'li!,Ykiiklg,M

e.

162 Fukuhei Talcabeya.

For

(IO)

(II)

(I2)

For

nel,

the condition of equilibrium at joints:

1(I+21a12==o,

%, + ,evE.+ .nig = o,

l

. .ntC., ,.-i + -t7t(., R+ jlC･, .+i= os

:

. jlfC,-1,,,m2+.il(1-1,A,-J+.ilC,-1,,,==o)

Atfll, oi-1 + ?ICt, N== 0'

the condition of equilibrium for the bent as a whole:

,+Jq,+14,.+14},,+...IC,.+U.,.+･･･+n41,.+ua.T,?L =0 '

or

(A==OX)

(i3) Z (jldi;,R+n4}e,,]=o,

(k-.D

The expressions of the moments n(2, n43, n%4,-..･..Vra;,H2, ,,"i, and JC,-i,n

are obtained from equation (8); ?I45i, -0852, J?lt43,･･････-n(f;,"],o,-2, and -nV;,,."i fl'om

equation (7); .Zlali, nAib -nd5llb･･････ICIHi, rv-i and 1(1,, N from equation (6) and

uaA, 11}i2, ?Z4}n3,･･..･.n(iv-4,,.-i, and filv,. from equation (g).

Substituting the expressions of these moments in equations (Io), (ii),

(i2), and (i3) gives the equations of Table I. Since we obtain n+i

equations from the Table we can determine therefi'om n+f unknowns,

9i, 92, 93,･.･..t9. and pti.

SVnglle SZon'ed Bent

Hordebntal

All legs of the

III,...･･･N-f, ahd AE

at the top i.e. at i.

zevicb afay Nuneber of" .Sipans, derrp,iiq{r a Cbncenlralea

Load at the 7bp. Lags Eixed at the Bases.

same bent as shown in Fig. 4 are fixed at I, II,

The bent carries a concentrated horizontal load tui

e

Analysis of Rectangular Building Frames by the Mechanical Tabulation Method. I63

Since horizontal deflections of all the topjoints are equal, we represent

D,them by e, and Yi= k,･

For the expressions of ld;,R and uaR.. equations (6) and (g) are

applicable. For nC..,.-i and ?ldL...i, putting C=o into equations (7) and

(8) gives:

nC. ,,-,== 2El3, -,(29, + 9i-i),

(i4) ua. ,..i =2Lg (29 + 9i+i]･

For equilibrium at i, 2, 3, 4,......7z-x, and n, equations (io), (ii),

and (i2) are applicable. ' ' For the bent as a whole to be in equilibriutn

(}-..,X)

(Is) : (llll..+1diIkh･)+(z,irk, =o･

(x=.b

The expressions of moments ndil2, Aalii, Ii2il3, n(l2,･...･.Al,"i, ,v and A(t,,t-i

are obtained from equation (i4); and nVlb ld}i, 214im 14h2,･･････ItCl,, N, ancl fllivT, n

from equations (6) and (g).

Substitutlng the expressions of these moments in equations (io), (Ii),

(i2), and (Is) gives the equations ofTable II, from which we can determine

9 and pt.

Si)iglir SZon'ed Bent zevitth cxiay Mtmber of .SPafzs, thi77ing a74,

.Si)yslefnofvarde'calLoevttsanaevClpncentvateal.Elbrip"onafal "'

Lond at the 7bP. L<gtg Fitxed al the Bases,

' Under the iaw of superposition of stress and cleformation, the previous

two cases are applicable here. We represent the horizontal deflection of

6, 'i, 2, 3, 4,......n-T,and n by 6i and tti== rki･ .

For the expressions of nl, R, ntZk.. ACL..-i, and nl...+i, equations (6),

(g), (7), and (8) are applicabie.

a

't64 Ful<uhel Takabeya.

For eguiiibrium at I, 2, 3, 4,......7i-i, and n, equations (Io), (II),

ancl (i2) are applicable.

For the bent as a whole to be in equilibrium, equation (IS) is

'

Table III gives 7z+f equations from which n+f unl<nowns 9i, 92,

93,.･･.･･9n and pti can be determlned.

boztbte S)fore'ed Reclaii{g7dor BuiZ2211iag' nufne of aay .IV2dmber of kS£Pans.

.Lezl:s iFllxecl aaf the Bases.

Table IV gives the genera! equations to be used in determining the

slopes and the deflections in a double storied rectangular building frame,

whose legs are fixed at the bases, with any number of spans, carrying any

system of vertical loads.

In Writing equation (i) of Table lV, wh{ch is ,

iO,SO,+g,SO,+6',g,,-.3e.,,Lt,-..76t,Y,=P,,,

'pi, the coeflicient of sp,, is placed in the column uncler sp,; 8,, the coeflficienE

of g,, is placed under g,; 6i,, the coefHcient of g,,, is placed under g,,;

-38h the coefficient of#,,is placed under #,;-.gg"',, the coeflficient of y,, is

:le9gebderUonfdeerquYaii'onall9 Pi2 iS piaced in the coiumn heaclecl " Right-hand

' In writing equation (2) of Tabie IV, which is

8i9i+P292+E'29C'3+Elik9?i-i-.)'g"intZim.)'8tL,St2==.Z)g3-P2i,

c, the c6efiFl61:'ent of g,, is' piaced in the c6iunili under gi; p2, the coefificient

of g,, is placed under g,; 6,, the coeflllcient of g,, is placed under g3; g-',,

the coefflcient of g,,,rmi, is placed under g.-t,-38fl, the coeflficient of pti, is

p'iaced under pt,;-36t,, the coefficient ofpt2, is placed under pt2; ancl lea3-P2i

is p'lacedi in the column headed " Right-hand member of equation".

In a similar way we can write the equations firom (3) to (7i+2) of

Table IV, from which we can determine n+2 unl<nowns i･e･ 9i, 92, 93･･･.･･

g,."landy2.'･ . Table V gives the general eguations to be used in determinln.cr the

Analysis o,f Rectangular Building Frames by the Mechanical Tabulation Metho[l, I6s

'slopes and the deflections in a double storied rectangular building frame

fixecl at the bases with any number of spans, carrying wind loads.i)

In wrlting equatlon (i) of Table V, which is

iOi9i+E'iS02+6'iSOer36ijCti-36'iXt2=O,

'p,, the coeflficient of gi, is placed in the column uncler g,; q, the coeflflcient

of g2, is placed under g,; 8',, the coefficient of g,. is placed under g,,;

-n"gb the coeflflcient of fl,, is placed under y,;-36t,, the coeflicient of y2,

is placed under y2; and the rlght-hand member of this equation is zero,

because the value in the column headed " Right-hancl mernber of equation"

tt In writing equation (2) of Table V, which is

g"i9i+P292+62g3+6'2g,,-i-3C",-36t,y, =o, ･

6i, the coefficient of gi, is placed in the column under g,; p,, the coeflicient

of g2, is placed under g2; 6,, the coeflicient of g,, is placed under g,; 6t,,

the coefficient of sct,,,-i, is placed under g,,-i;-36th the coefficient of pt,, is

placed under pt,;-36t,, the coeMcient of g,, is placed under pt2; and the

i,i.g.hdth,hda?,dRll.Ilgebill..?,e.r.dOf.ge.l'gbe,q,".aftiO,"q.i.S,iZ.e.r?,) l,ec.aip"hse,,ihe value in the column

In a similar way we can write the equations from (3) to (n+2) of

Table V, from which 7z+2 unknowns can be determined.

Table IV gives the general equations to be used in determining the

slopes and deflections in a double storied rectangular building frame fixed

at the bases with any number of spans, carrying any system of vertical

ioads and wind loads. In a similar way as explained above, we obtain,

from Table VI, as many equations as there are unl<nowns.

' ' Idetzaiple SZoiied Reclaagtzlar Bztzua7<g' iFlaanee of a7u, IV2t7feber

of .SPans, Lags filixed at the Bases. ' Table VII gives the general equations to be used in determining the

slopes and deflections in a triple storied rectangular building frame fixed at

i) In this bulletin wlnd loads are assumed to be horizontal and

side of the building.on one vertical

concentrated at ]olnts

,

I66 Fulcuhei Tal<abeya.

ghned bwaiSnedS roiatdhs.anY nUMber of spans, carrying any systgm of verticai ioads

For a rectangular building frame, any number of stories high and any

number of spans long, we continue further to tabuiate, in Table VII, the

coeflficients of unl<nown slope,s and ratios of the deflection to story height,

considering carefu11y the rules that exist, in the Table, on the symmetry

of the places of coefllcient of g and y, and the systematic arrangenient of

the suffix-number.

' ' Benl with Le:g:s' .Etifageal at the Bases.

If the,legs of the bent are hinged at I, II, III, IV,......1.4Ll and ?lf]

the equations of moments ua.R and vaR.. are expressed as fbllows:

'

Fig･ S･

(I6) iZlt(.,R== 2atR(2SO.+SOR-3,tti),

(I7) ZLtZ}e.."=2rkR(2SOre+gqi-3iCti),

and the leg is hinged at the base; therefbre; vaa.=o. From this condition

weget: . (I8) 9R ==in5vUi-O,5g･ Substituting this expression of soR in equation (i6), we obtain:

(I9) ..' IL(･.R==2EgR(T,59r-li5iLti]･ . '

Fcr expressions of moments at the ends of all the girders and the

columns, except the columns in the first story, the expressions, obtained

gAk

g;,

g,2 -/ r r+1 m-!

neflgn,mSM

gr-I gnIregKMsm-/

Analysis of Rectangular Builcling Fiames by the Mechanical Tabulat!on Method, 167

3"retha.;p.iSclae.bie,1.,ha,g:b.?.",t ,W,ilt,h,,,?,X:.d lgi',due to ve'ticai ioads or wind i.,d,,

nfl,+?lfI,+llV;.h=:o 'glves :

9,P',+9S,+9,,6',-If",6i-3YS',=:P]2,

where (20)

Pt,=2(6,+6',)+l,5<.. '

, For condition of equilibriu!n at r:

AIII. ,..-i+ ?itf;1. R+ -Zla;1. k+ fl41. r+i==O '

giVeS: g.m,e-,+g,p,.+g..,e.+g,6'.-l,sTtiGi-3ptS'r=Pr･r+i"-Pr･r-i'

+

where ･ (2i)

P'r=2(6r-i+&-+g',]+ln5gR･ .

For condition of equilibriurn at fn:

uae. m-i + ZIt4i, nt + nt4b. ptb+t = 0

' ttglves :

lil)"ll',iii.M-'i+SP"'tO'"'+90"L"iCl"ZMih51ti6M'3tt26M=-"iOM'"i-i'

(22)

P'm=2(g,･-i+4int)+I,56nf･

For the sake of convenience we sum up the above obtained expressions

for p' as fo11ows: ttt/t

P'i ==2(6i+8'i]+I,56n .

(23) P',-==2(8,-i+&+6',]+in55R,

p'., :2(6.-i+g.)+tE6fif･

'

I68 Fulcuhei Takabeya.

SV7agTle SZon'ea Be7zt zorith a7ay Msmber of' .SiPans.

' Legs l7ii(gea al cbe Bases.

A single storied bent with any number of spans, hinged at the bases,

carries any system of vertical loads.

Representing the horizontal deflections of the joints at the top by

" OlDi and pti=: rk, , we obtain from equations (ig), (7), -and (8):

fi(n r-1== 2ffr-1(?9?･+ 9o'-i] + a'. r-1,

{ fiie;1. . =2E6. I,59,.-i,5'Y,),

nC. ,..,=2Eg, (29. + 9, .i] - q, .+i･

For condition of equilibrium at joints, equations (io), (Ii), and (i2)

e

:For condition of equilibrium for the benE as a whole:

(iiE.lfo')

(24) :(?rtC.. R) -- 0. (k-.D

These conditions of equilibrium give the equations from whlch we can

determine unknown slopes and deflections.' These equations are tabulated

in Table VIII. ' ' Table IX gives the general equations to be used in determining the

slopes and the defiections in a single storiecl bent, whose legs are hinged

at the bases, with any number of spans, carrying a wind load at the top.

In writing equation (i) of Table IX, which is

g/ Pi'g, + 6,g2 -f,s8iet, = o,

pit, the coef)ficient of g,, is placed in the column under g,; 8,, the coeflicient

of g2, is placed under g,;-f,s8b the coefficient of pt,, is placed under g,;

and the right-h4nd member of this equation is zero, because the value in the

column headed " Right-hand member of equation" is cipher.

as--

'

Analysis of Rectangular Building Frames by the Mechanicar Tabulation Method. I6g

'

In writing eg.uation (2) of Table IX, which is

#,9,+PSt9,+e,g,-l,s6nv,==o,

#i, the coeMcient of g,, is placed in the coiumn under gi; p2t, the coeMcient

of g2, is placed under g,; 6,, the coeMcient of g,, is placed under g,;

-i,58ih the coefficient ofgi,is placed under pt,; and the right-hand member

of thls equation is zero, because the value in the column headed " Right-

handmemberofequation"iscipher. ' ･･ In a similar way we can write the equation from (3) to (7z+f> of

Table IX, from which n+f unl<nowns can be determined.

Table X gives the general equations to be used in determining the

slopes and deflections in a single storied bent hinged at the bases with

any number of spans, carrying any system ofvertical loads and a wind load.

In a similar way as explained above, we obtain, from Thble X, as many

equations as there are'unknowns.

' ' uaulmple SZoriea Rectwagzdor Bumei7rg thaime of aay IVTuf7eber tzlC .SIPans.

Ltgr .U7iggeal at the Bases.

Table XI and Table XII give respectiveJy the general equations to be

used in determining the slopes and deflections in a double storied and in

a triple storied rectangular buiiding frames hinged at the bases with any

number of spans. , For a rectangular building frame, any number of stories high and any

number of spans long, we further continue to tabulate, in Table XII, the

coeMclents of unl<nown slopes and ratios of the defiection to story height,

considering carefully the rules that exist, in the Table, on the symmetry

of the places of coefficient of g and A, and the systematical arrangement

of the suMx-number.

' Speciai Cases.

･ As a special cases of the above treated structure there are rectangular

building frames with an axis of symmetry of the building, carrying vertical

loads symmetricai about the vertical center line of the building. To this

I70 Fukuhei Takabeya.

case no horizontal clefiectio' ns eopae in calculation and therefore the equations

areverymuchsimplified.. , The general equations to be used in determining the unknown slopes in

triple storied bents, whose legs are fixecl at the bases and the number ofspans

being 2m and 2m-･i are given from Table XIII and Table XIV respectively;

to the hinged condition Table XV and Table XVI are applicable.

. Further, to the symmetrical loads about the center of each span,

Tables XVII, XVIII, XIX and XX are applicable.

For fixed condition and span of even number, the writer has already

treated in the Memoir, Vol. i, No. 2, ig26.

Continuous beams of any number of spans, carrying any system of

vertical loads, may be considered as a special case of the rectangular

building frame. There may be three types of continuous beam; the first

of which is defined as a continuous beam .fixed at both ends and supported

simply at the intermediate points; the second of which is defined as a

continuous beam hinged or supported simply at both ends and supported

at the intermediate points; and the third is the combined type ofthe above

two; i.e. at one end of the beam it is fixed and at the other end it is

supported or hinged.

These three types of continuous beam are, in this article, treated

firstly assuming that all of the supports 1<eep, even after ioading, their

initial position on the same level. Fig. 6 represents the above mentioned

types of continuous beam and the condition of equilibrium at supports gives

'the equations to be used in determining unl<nown slQpes at the supports.

'

c e,･ e. e.. (a)

St l2 r' lt o il+1

tho i 2 nin n+Sb)

ct "".ogb2"]2tir nn+(tC)

Fig. 6.

t

Analysis of Rectangnlar Building Frames by the Mechanical Tabulation Method, Izr

' Forconditionofequilibriuniat,interniediatesupportr, . ･

-nC. .-1+ nC. ,･+1=o gives: ,' tt , l9'-i6r-!+gP･tOr'i'S9r,rtSi''.?'Or-r+irmPr･o-i, '

' ttt tttt (,s) .Iblr.-b{e･-i+e'i, ''

1'' for.o"+i=t"i'+i' ' ,

'.'. IAv-i=t"i-i '' ''1

,... 1... t. At support next to the fixed end, II o+?ra;2=o gives:

(26) so,P,+SP2G,--LPi2-1>io･

At support next to the hinged end, 1(o+n<2==o gives:

soipit+{e2E'i=.ibi2-.pio-oxspoi,

' (27)' where. P,' =f,s6,+2C.-'

Atsupportbeforethefixedend,?Ztllt,,n-i+Aili.et÷i==ogives: i

(28) 9oi-Sv-1+9nPn=A,･oi･+1-2t･7t-i' ' ' ' At support before the hinged end, A4I,.n-i+nC,.,,+i==o gives:

tt// 9oz-16oi-1+'9?,P,"=O,Mt+1.,b+Pohet+lm""Porm-1,

' (2g) where P?t'"=26?i-i+･ix5'87}･

By proper combination of these equations from (2S) to (2g), correspond-

ing to the type ofcontinuous beam as shown in Fig. 6 (a), (b), and (c), we

obtain Tables XXI, XXII and XXIII; or vice versa, from these Tabies we

get general equations to be used in determining the slopes at supports and

with the values of slopes obtained we can determine moments at supports.

To speak more generally the moments,in a continuous beam n]ay be

I72 Fukuhei Takabeya.

adpepVfi.IOdP.ed bY the settlement of the supporting parts as well as the loads.

Assuming that the beam remains in contact with all supports even

afrer the loading and settlement of the supports, we obtain, almost in the

same way as explained previously, the general equations, which determine

unknown slopes, for a continuous beam fixed at beth ends and supported

at the intermediate points on difl?)rent leveis, carrying any system ofvertical

loads, These equations are given in Table XXIV. Here it is noticed that

the settlement of supports must be glven or estimated for materials of the

supports or properties of foundation used, and therefore the ratios of the

vertical deflection to span length must be, in this case, 1<nown quantities fbr

the calculation of redundant stress. .

' '

Some Examples on Application ofthe

Proposed Method.

The fbllowing examples may be convenient so as to make it possible

to apply the proposed method easily.

' ExAMpLE i. (Table XXVIII).

Triple Storied Bent of Six Spans. Legs Fixed at the Bases.

Cross-Sectiens of all Members Diffl]rent

/･/ Each Story Height Diffl]rent

Any System of Vertical Loads on Girders

Wind Loads Assumed at Joints on one Vertical Side

' ExAMpLE 2. (Table XXIX).

Five Storied Bent of Triple Span. Legs Fixed at the Bases,

Cross-Sections of all Members Different

Each Story Height Diffbrent

Any System of Vertical Loacls on Girders

Wind Loads Assumed at Joints on Ope Vertical Side

' ･-Ana}ysis of Rectangular Building Fraines by the Mechanical Tabulation Method. I73

ExAMpLE 3. (Table XXX).

Symmetrical Eight-Span Bent Four Stories High.

Legs Fixed at the Bases

Any Systetn of Symmetrical Vertical Loads on Girders

ExAM?LE 4. <Table ×XXI).

Symmetrical Seven-Span Bent Five Stories High.

Legs Fixed at the Bases

Any System of Sym[netrical Vertical Loads on Girders

ExAMpLE S. (Table XXXII),

Triple Storied Bent of Six Spans.

Legs Hinged at the Bases .Cross-Sections of all Members Dif¥l:rent

Each Story Height Different

Any System of Vertical Loads on Girders

Wind Loads Assumed at Joints on One Vertical Side

ExAMpLE 6. (Table XXXIII).

Five Storied Bent of Triple Span, Legs Hinged at the Bases.

Cross-Sections of ali Members Different

Each Story Height Diflerent

AnySystemofVerticalLoadsonGirders ,.Wind Loads Assumed at Joints on one Vertical Side

ew ExAMpLE 7, (Table XXXIV)

Symmetrical Eight-Span Bent Four Stories Heigh. Legs Hinged at

the Bases.

Any System of Symmetrical Vertical Loads on Girders

ExAMpLE 8. (Table XXXV).

Symmetrical Seven-Span Bent Five Stories High.

Legs Hinged at the Bases

Any System of Symmetricql Vertical Loads on Girders

di

'･I74 ''' FukuheiTalcabeya.

Solution of Equations and General Formuia,s for

'･ SpeciaXCases. ' The general equations obtained from the Tables of this paper are

solved by determinants or by process of elimination. To the calculation

of a large number of unknowns, the solution by determinants is very long

and the process of elimination is preferable, however it may'be sometimes

painstaking work. ･' As an exampie.of the SOIUtiOn Of BENT wlTH Axls oF SyMMETRy oF THE

a bent in symmetrical condition Mr. Y. STRUCTURE AND ANy SysTEM o)' ' SYMMETRICAL VERTICAL LOADS,YANo, assistant of the author's Institute, ' I I･has designecl and computed the stresses of

a symmetrical six-span bent six stories

high, carryingasystem of symmetrical "load.

This numerical example is added at

the end of the note, the writer gratefu11y

acknowledgingindebtednesstohim･ ,i Loads To the special 6ases as shown in ' i l

Fig. 7, the general formulae fbr unl<novvn ! nv tquantitieshavebeenintroducedasfo11ows:i. ipmi i Dividingeach,equationbytheco-l pmemcient of the fir'st unl<nown of tlle equa- 'l ･ , 'tion,weobtain equ.gtions whose firstterms 1

in the left-hand member are plus unity, Fig･ 7･These equations are written in tabular fbrm

as shown in Table i, in which a, b, ancl c are given in actual case by

numerical values.

E :

r

'

,

1 .Z 7, x. vt F2z Z Zn Z1

1

:i! 1

!

ICQntinuous Bearn with

' ' r.

1

' any System of Vertical

Analysis of Rectangular Building Frames by the Mechanical Tabulation Methocl･ I7S

TABm I.

ii'

Left-HandMemberofEquation(CoeflicientsofUnknowns) v==.p-e-

rpBgxgs

watnEU

xo .xrl

.Y2 X3 I4 Xs -Xn-4 Xn-3 In-2 Xn-1 .X

n

1' 1 altt

C,

2 b, a2 C.

3 1 'b. a3 C,

4 1 ･b, a4 C,

5 1 b, as C.

t ×4 ×' i

×4

n-2 1bn-3 .a

n-2 n-2

n-1 1bn-2 an-1 Cn-i

n ' 1 ,btt-1,att ,C,,

n-1 1 bn Cn+i

For

fo11owing

(3o)

convenience of calculation we

:

CAo==J,

alzlxi7ai-bi== A, wwbp

A.,.- a2 -b.,.- a2 - " airbi " A,

A3. a3 b3-

use the notations as

b 2!

a3 - b3,A,

denoted in the

A,,:

:

;

:

:

:

:

"

a.- ai -" bi -b2

a4a3

-b,-b4== P

a2ai - bi

: : :

ma b,

q - 4,A,il

:s:

:

,

t

I76

Then

A7v :

Fuicuhei

a Ol,

Takabeya.

･ an-2 - -b,,-2 ' - ' - ' ' a3 'r nv b3 a. -' ai-Zi -ba

general expressions fbr unknowns of

q -qA2k, -G

A, -q -q A,

- -Chml Aoi.-3 - a, An--2 - C;t+1

an-1--

4, tU,i

An-i-4.

L

we obtain

c

Table i:

i

(3i)K ;kfib =

K

M,-2 : : xh'

Iiil:

For '.In moments

given by

formulae

Tab!e

the moments at

number of

To the '

number

puter who can

A,,-1 ' Aot

x-i== C;t+i--- bei#;kr;},

= C;t- cVnalY;Lh boe-lj Ylt-b

q-a,2¥4-b,X5,

q-a,xfyi-b2]tG,

4-a2u)kG-biXl･

continUous beams it may be more convenient to choose unl<nowns

than those chosen in slopes, i.e. the equation of three moments

CIapeyron determines unl<nown moments directly. And the

introducedabove'areapplicabietothat, '

XXV gives the general equations to be used in determining

supports on the same level for a continuous beam of any

spans, carfying any system of vertical loads. (Fig. 6b).

mvestigation of stresses of a continuous beam with a great

of spans, Table XXV is of the utmost importance for the com-

i therefrom easily worl< out the moments by the process of

'

When we further snbstitute 4,=i=o in Table XXVI i,e.

dividing the last equation by m.Hi and then substituting

'in it m,,-i=co we obtain Table XXVII which gives the

general equations for the redundant moments of a conti-

nuousbeamfixeclatbothends. . ' ' Properties of Proposed Tables and

Rules in the Tabulation.

Petermination of CoeMcients.

For coethcients of unknown slopes there are two

kinds, one of which is expressed by 6 and can be deter-

mined by f:l for girders and l:k for columns; the other

1<ind is denoted by p and can be determined by equations

(2), (23), and the equations given in Tables.

More detailed instruction about g has been given

tional Notations and Signs Used".

Next, coefllcients of the unknown ratios of the

story height, i.e, ict aiways takes minus sign, and one l<ind

pt is expressecl by 6 with multiplier -3 fbr a bent with

multiplier ,being-fff for pt, (at the top of the first story)

Khinged legs; the other 1<ind is expressed by-X except- 2

of p, fbr the equation obtained by condition of equilib'

story of a bent with hinged legs. (Tables VIII, IX, X,

and XXXIII). ''

Analysis of Rectangular Building Frames by the Mechanical Tabulation Method･ I77

' "elimination for the numerical values of the coeMcients'of the unknown

From Table XXV we can derive･Tabie XXVI and Table XXVII;

e.g. on substituting ffe,--o in Table XXV we obtain Tabie XXVI which

gives the general equations to be used in determining the moments at

supports on the same levei for a continuous beam fixed at the left end

and carrying any system of vertical loads. ･ ' --

Fig. 8.

' above in " Conven-

horizontal' deflection to

of coeflicient of

fixed legs, this

of a bent with

, the coeflicient

rium for the first

XI, XII, XXXII,

I78 ･ Falcuhel Takabeya, ' The right-hand member of equation, denoted by P and 4, is given by

equations (4) and (s). This is' the variable term due to the condition of

applied loads: P due to any system of vertical loads; and e due to wind

loads assumed horizontaily at joints on one vertical side. g takes always

ininus sign and fbr a bent with hinged legs a, takes multiplier-2.

' Diagonal Line of p and C Other Properties of Table.

The coefficient p finds itself in each Table on a continuous line which

may be expressed as a dlagonal line of a square forrned by columns of

'

p

e

e･

Fig･ 9･

' t. t tt t.unknown slopes g and rows of general eguations, g being tal<en horizontally

and number of equation vertically; we name this sguare " Great Square ".

As shown e.g. in Table VII, p arrange themselves, along the diagonal line

'of the Great Square, in order of the suMx-number of p, fi`om the top Ieft

corner of the Great Square to the bottom right corner of the same; in

other words, p arranges itselL fi'om pi to p., reading downward to the right

along the diagonal line of the Great Square. In the same Table we find

,

Left-HandMemberofEquation ight-HanaMember

CoeMcientsofUnknownSlopesg CoeMclerrtsofpa VerttcalLoad x}risld

Load2 g-'

fa e e 8' '

2 iC"

'6.

t 6･ g'

e'

't' f' 2 e･

'

cz･･ 6･

8, 'Pe

6'

'4- ?6･ 6･

6' 6･ 9

Analysis of Rectangular Bui]dlng Frames by the Mechanlcal Tabulation Method. I7g

lines of e on both sides of the p-line, and the arrangement'of the suffix-

number is the same as in the case of p; while the line of e' with the

mark (') is piaced intersecting at right angles with the above mentioned

three lines of p and e, arranging themselves, in this case, towarcls the

diagonal of Great Square,･ one part descending leftward and the other part

ascendi.ng rightward; these two parts meet together in the cliagonal of the

Great Square and the line of 6' with the mark (') may be also considered

as a diagonal Iine of a small square which is one part of Great Square.

We call this the "Small Square ".

On considering the diagonal ofthe Great Square as an axls ofsymmetry

of the Table, we find the arrangements of 6 in stepped form in the position

of symmetry; and on the prolonged line of the diagonal of Great Square

X is piaced in order of the suffix-number. For the arrangement 6 in

stepped form, 6 with sufllix of large letter firstly comes and next 6' with

the marl< (t) comes in the same relation as shown' in the diagonal of Small

Square, in other words, as the horizontal projection of the 6i-line ofsmall

Square 6i finds itself in the colurnns under g and as the vertical projection

of the same e' fincls itself in the column under pt (See e,g, Tabie VII),

' ' For rectangular building

Fig. Io,

order, the right-hand tcrm ofP taldng minus

For fi'arnes with wind loads, each equation

eguilibrium for a story as a whole has q in

equation, e taking minus sign, regularly

'downwards. ',

frames with any system ofvertical l6ads, each equ,ation,

given by conclition of equili--

brium at joint, has P in right-

hand member; the orcler of the

suffix-number ofP is indicated

in Fig. 8 in the form ofa zigzag

Jine, of which there exist also

systematic relations in strict

sign,

' , given by condition of

the right-hand member of the

ordered and reading vertically

I80 Fukuhei Talcabeya.

For special cases of rectangular building fi'ames, the properties of the

Table are almost the same as explained above, and wili not be repeated

tt

chanical solution of such a

problem may be applicable

to the investigation of

trusses with stiff connec-

tion and the secondary

stresses may be mechani-

cally determined in a simi-

lar manner to that explain-

ed above. Fig･ iit To facilitate the under-

standing of the existing properties in the proposed Table, Fig. g is of

importance, where the regularly ordered lines ofeach quantity mentioned above

are shown graphically. From this figure we can deduce a relation between

the number of stories and the number of diagonals of Smali Squares.

The general type of figure of Tables due to loads and number of

stories is given in Figs. io, u, I2, I3 and I4.

Fig. Io shows the general type of Table for a single storied bent urith

anysystemofverticalloadsandwindloads. ' Fig. ii shows the gene-

ral type of Table fbr a double

storied bent with any system

of vertical loads and wind

loads.

Figs. I2, I3 and I4

show respectively the gene-

ral type of Table for triple,

quadruple, and five storied

bents with any system'of

vertical loads and wind loads, Fig. I2.

e

Analysis　of　Rectallgu互ar　Building　Frames　by　the　MechaniGal　Tabulation　Metllod。工81

Fig．13．

Left・Hand　Member　of　Equation

CQef五cients．of　Unknown　Slopes幹

騒騒

四獣灘灘総

Fig・14・

I82 Fukuhei Talcabeya.

' ' AnalyticaKndexofTabies, .

The Foilowing index may be convenient so as to mal<e it possible to

locate the desired table in this bulletin quickly.

FuRdarnental Equations. - ' General Equations for the Moments at the Ends of any Member

Values of Constants C and H to be used in the Equations of Table A.

ForanySystemofVerticalLoads(TableB) i For Loads Symmetrical about Center of Member (Table C)

Tables of General Equatiens.

.Bent of Single Story with any Number of Spans.

Legs Fixed at the Bases (Table I, II, ancl III)

Legs Hinged at the Bases (Table VIII, IX, and X)

Bent of Double Story with any Number of Spans.

Legs Fixed at the Bases (Table IV, V, and VI)

Legs Hinged at the Bases (Table XI)

Bent of Triple Story with any Number of Spans

Legs Fixed at the Bases (Table VII)

LegsHingedattheBases(TableXII) . Bent of Triple Story with an Axis of Symrnetry of the Building

Legs Fixed at the Bases

Number of Spans 2m (Table XIII)

Number of Spans 2m-i (Table XIV)

Legs Hinged at the Bases

Number of Spans 2m (Table' XV)

Number of Spans 2m--i (Table XVI)

' Bent of Triple Story with an Axis of Symmetry of the Building,

Carrying any System of Symmetrical Vertical Loads about the

Center of each Span,

Legs Fixed at the Bases

Number of Spans 2m (Table XVII)

Analysis of Rectangular Building Fr'ames by the Mechanical Tabulation Method- I83'

Number o'f Spans 2m-i (Table XVIII)

t t.t Legs Hinged at the Bases

Number of Spans 2m (Table XIX)

Number of Spans 2m--i (Table XX) '

Continuous Beam with any Number of Spans, Supports all on Same

Level.' ,' ' Beam Fixed at both Ends (Table XXI and XXVII)

Beam Hinged at both Ends (Table XXII and XXV)

n BeamFixedatoneEndandHingedattheOther(TableXXIII

and XXVI) -･ Continuous Beam with any Number bf Spans, Supports on Different

Lev'els. ' '･ '･ .BeamFixedatbothEnds(TableXXIV) ' ･ ' ttTables of Examples. tt Triple Storied Bgnt of Six Spans in Asymmetrical Conditlon (Table

.XXVIIIandXXXII) . .. Four Storied Bent of Eight Spans in Asymmetrical Condition (Tabie

XXXandXXXIV) , FiVxexSitll?riaenddBll2111txOifii)Three SPans in Asymmetricai condition,(Tabie

Five Storted Bent of Seven Spans in Symmetrical Conclition (Table

XXXI and XXXV) ' ' Summary and Conclusions.

The General conclusions to be drawn fi-om the i`esults ofthe investiga-

tion described in this bulletin may be summarized as foliows:

i). The general form of the fundamental Table is memorized with

little effbrt and can be easily tabulated by arranging 6, p, P, a,

and X' with suffix, in order of the suffix-number, in the colurnns

2), No mistakes can be performed in writing the fuiidamental Table,

on account of the systematic arrangement ofthe suMx-number and

I84

3)･

4)･

s)･

6).

Fukuhei Takabeya.

the symmetrical property of the places where the coeMcients of

unl<nown quantities are to be written,

For complicated calculations lt is sometimes very dicacult to

avoid some mistakes when thgre is no way to checl< the obtained

results; our systematic arrangement of the suMx-nurnber and the

symmetrical property of the Tabie are of great help in this

diraculty.

The accuracy of the calculation is the same as that of the slope

defiection method, because our solution has been derived from'it,

tainodnstheexrceefoprtetfihoesenl.fit?hOed iSssxkr;;tia.ocncsu.rate, it having no approxima-

The 1<nowledge of elementary algebra ls suflicient for the calcula-

tion of the stresses of higher structures except in finding values

of load terms P; for the load condition usually appliecl, there are

Tables in current use, which give the values of P.

As in the computation by the slope deflection method, the calcula-

tion is long to ' be used in the actual design of a high building;

it has however its greatest value as a standard calculation for

checking the accuracy of another approximate rapid method.

The properties of our Tables are of great help as a process to

check general equations of equilibrium obtained by slope-defiection

method.

Analysis of Rectangular Building Frafues by the Mechanical Tabulation Method. I8s

APPENDiXe

' ' Msmaerical dethztlambn of SZambady inclktereninate SZresses.

As the numerical example of proposed method, Mr. Y, YANo, assistant

of my institute, has designed, after the regulations of the Japanese Govern-

ment, a department store building of reinforced cbncrete. This skeleton

building is symmetrical about the vertical center iine ofthe frame, as shown

in Fig. is, six-stories high and six-spans long, carrying Ioads concentrated

symmetrically and distributed uniformly. In calculation the average value

of the moment of inertia of the central section and the end-sections has been

tal<en as that of each girder-section. With such values of moment of inertia

there are indicated in the figure the values of 6 for each member of the

'

Table 2 gives the general eguations to be used in determining the

unl<nown slopes' due to the concentrated loads and unifbrmly distributed

loads; and the numerical values ofthe constants in these equations are given

there. The figures given in the right-hand column are coeMcients bf

.oooi as indicated at the head ofthe column. The' Se simultaneous equations

have been so!ved bY the process of elimination and the results are as

' 9i=I･90223S3200 9,,=:･029'9589920 'SC),==-･IIO09Il744 9,,:-･236248470S 9,=･O0663450I9 9,,==I.98278I0470 9,==･O0272S6734 9,,=L9I45803200 9,==-･0817I37493 9,,=:-･I7SI202IIO 9,==L7205323630 9,,==･OOS13337S5 g,==2.ol48I6I2oo so,,:.I88g8ISS63 9s=-･I7I2I22482 9i7=:-･9289375437 9,=.OI42468460 9,,=4･44489SSIOO

The moments at the ends of the columns and girders given by the

fundamental equation (i) are indicated in Table 3, where the "Total

Moment" rneans the total sum of the moments due to concentrated loads

and uniformly distributed loads.

I86･ ' FukuheiTakabeya, , '

The accuracy of the computations has been checl<ed by the statical

condition of equilibrium of the joint-moment and its percentage of error for

the least moment is given in the right-hand column of the same table.

. TABLE 2. TABLE OF TI{E E9UATION USED TO DETERMINE [I]HE UNKNOWN SLOpES OF THE SYMMETRICAL ?2illl-IEu?AlaNte[IIXB-ySTeBYyBAENgY. SHOWN IN Fig･ IS

geLeft-Hand

meono.-

CoefficientsofUnknownSlopesg

ga..9

.:'"..g.-

fi.mV'."tt--Stno,ee.

l9, 92 93 9, 9, ,9s 97 ･sos 9, 9io SPII ･91t 913 'SPIg 9is spifi 9,i 9}a

1 -8 55 88. 06

,o. 55 847e655' 156 o

3 55 847e6 156 o

4 156 736 55 102 o

,5 Z56 55 736 55 '102 o

6 P8,5 55 432 2.5'

06

7 72,5 400 55 2. V06

8 Z02 55 530 Jr5 53 o

9 102 55' 530 53 o

10 53 402 5Jr 38 0

11 53 55 402 55 38 o

12 2,5 55 400 za5 06

13 z2. 362 55 53,5 106

14 38 55 366 55 35 o

15 38 Jr5 366 35 o

16 35 380 7Z5 o

17f

35 7Z5 380 77, o

18 3.5 7.5 62 19--mm-t--ut mu

Analysis of Rectangular Building Fremes by thel Mechanical Tabulation

'

TABLE 3･

TABLE oF JolNT-MoMENT

(Moments are Expressed in Ft.-lbs.)

Method,

MI8-I7

MI8-I3

MI7-I8

MI7-I6

MI7-I4

IM

I6-I7

MI6-c

MI6-I5

'MI5-I4MI5-c

MI5-I6

MI5-io

M,I4-I3

'MI4-I5MI4-I7

MI4-lr

MI3-I4

Total moment

- 28goo

28goo

6g78o

-6622o '

-356o

5762o

'

-582go

67o

S28oo

-5322o

35o

7o

57559

-542oo

-2240

-IIIO

-4320o

Mc due to con-centrated load

-24790

24790

5g85o

-568oo

-30So

4942o

-Soooo

S8o

46igo

-46sso

3oo

6o

50340

-474io

- ig6o

- 97o

-377go

Mudue to uni-form ]oad

-4IIo

4IIo

9930

-942o

-5Io

82oo

m82go

9o

66io

-667o

5o

IO

72Io

-67go

- 28o

-I40

-54io

?ercentage of errorfor the least moment

I87

'

I88 Fulcuhei

TABLE 3

Takabeya,

(Continued)

TotalmomentMc

duetocon-centratedload

Muduetouni-formload

PercentageoferrorfortheIeastnioinent

Mr3-I822i30 ig36o 2770

MI3--I2

21o70 !843o 264o

Mr2-II

-43ooo -376io -53go'

MI2-I3

21320 i865o 267o

MI2-72i68o i8g6o 272o

MII-I2

5o2ro 7I9o

MI!-IO

-S4470 -4765o'-682o

MII-I4

-1230 mIo70 -i6o

MI!-8-I70o -I490 -2ro

MIO-IIS277o 46i6o 66io

MIO-C

-53o9o -4644o -665o

MIo-I5

I20 I!O IO

MIO-9200 170 3o

M9-85286o 4624o 662o

M9-c-53z7o -46Sio '- 666o

M9-IoI50 I30 20

M,9-4

i6o I40 20

M8-9-54i5o -4737o -678o

Analysis of Rectangular Building Frames by the Mechanical

TABLE 3 (Continued)

Tabulation Method. I8g

TotalmomentMc

duetocon-centratedload

Muduetouni-formload

Percentageoferrorfortheleastmoment

M8-7578So 5o6io 7240

M.

8-u -IS40 -I35o -I90

M8-5-2I6o -I8go -270

M7-8-4264o -3730Q -S340

M7-I22x8oo I9070 2730

M7-62o84o I823o 26Io

M6-S

-440Io -385oo -55io

M6-7ig78o I7300 248o

M6-!2365o 2o6go 2g6o

MS-65753o 50320 72Io

-t-------rmmrrmmmm

M5-4-536go -46g7o -672o

M5-8-17!O -I490 -220

MS-2

-2I30 -i86o -270

M4-SS3040 463go 66so

M4-c-S3230 -4656o -667o

M4-9

IOO 9o IO

M4-39o 8o 10

M3-252g8o. 4634o 664o

-tttttt-rm

I90 Fukuhei

TABLE 3

Takabeya.

(Continued)

TotalmomentMc

duetocon-centratedIQad

Muduetouni-formload

Percentageoferrorfortheleastmoment

)il

3-c-532ie -46sso -666o

rvI

3-4I30 I20 IO

M3-6IOO 9o IO

M2-3-5384o -4709o

'

-675o

rvI

2-I5788o 5o63o 725o

M(5o)2.87%TotalM(4o)2,63%cM(io)4,64%u'

M2-b-I74o -I52o -220

M2-5

-23SO -2o6o -290

"･I

I-bI864o i63oo ,2340

MI-624450 2I390 3o6o

rv!

I-2 -43090 -376go -S40o

Mb-I tmnv.rmtt--..

932o 8i5o

Ttpt.ttttttt..vt-ut

!r7o

undn

runu -di"-87o -76o

tt"

-!IOTnvm-I71

b-3

=I==-

5o 45 5

rv

Mc-3..ttt--L- imtmN-ut-5327o 466oo

tun667o

Mc-4t.t-

5326o.rmuntttt.-uaum

465go 667o

Mc-9nvMmtt-tt

5329o 4662o 667o

tt

5335o 4667o 668o

LT.rmt--7-tt5328o 466io 667o

Mc-i66o48o 5i88o

tt86oo

-$. :.ttttlYk's.

,//･･･

t /tt///tttt/ttttttttt/t/tt '''/"//"1'1/1"'1111111//'11/11111'11it'1'

" t't ' ''/t tt''''' 'i"

tttttttttt ''''''''1111111t'11""' ,,,1."te.i'/et"

.1.-

E

Analysis of Rectangular Building Frames by the Mechanical Tabulation Method. Igl

'Roof(n='77.5Eiti==77.5E=775

PlPbP:P]PlPlPiPiPE,,,Sil.

18no l7' 16

E=55.

PoP･,Po--- Pn-P;P21

P,,P,P..---

rJthF.L,

sLoiswri

pge[1LvNt 13gg l4coon 15･

C=t55

P2P2P2 P!PtP2I

P2P2P!'

c)th

4thF.L,

J

"otswHRgell.)t7 12eeTl.Kdi(T--55.ll, ll 10

qtt55

P2Po.P! PL,P..P! P.P.P.'-

l-

3rdF.L,

"olNWri 7ecOr,' 8, 9'E=55

P2P!P,

1

Pg.P?P!

'PoP"P･,-r+

2ndEL,

"7:.5.' 4E=55

P2P!P2 P.P,,P.-;t P4PoPo--"

3 fi.gr r,e-"S- t)bee6plj(rseg) fit--t,5Nf5.Hr fiey(r

gtNON

eelltv 1eq8diTl"b

2--co

Vppersurfaceoffooting

20tNot, 201-v,eot, 20t-,,ett 20tN,,ot, 201A.,,ett 20t-ott

1201A-or,

EZ2?･vaXion (zle .Plaame,

Fig. I5･

k

,.,e/].ft

-,oti].ft

tet#ll.ft

toa#tl.ft

e=:s2ooe･, 4 =7aso",(Oi==2s:5ftll,fr,

(o2 =2oo"llLr{Z.

l8. ･17 16C16 17 18･

l3'

12'

15C'

･15 14i 1314

11

8ro

10,C 10 11 12

7 8' 7943

5 66'-

1br 2b

b

1'

b b

,

Bentth7(g' ua. Diagzranz

cthee to the Cbncenlrated LoaalL

sbateofnaz. ,o,oooofr`S

u

'

k

Fig, i6.

Bencin4{r 1de, Dz'agram

cthte lo the U)zz]i(2prnz Loaal

S2rateofsw. iopoofl･"

Fptq tw

tif.t.

`

General

FUNPAMENTAL EQEJATXeNS

TABLE A

Equat{ons for the Moments at the Ends of any Member

NVhere

If end s

If end r

RelatioA

x

Aiay xSl)1islefn of l7}irlical Lo`i(ts

is hinged,

is hinged,

betweefi

kl. q"+ 2Values of Cand ff for Various Loads are given in Tables B and C:

N.llTfrS S'h... 9r 7. 9e

l

I.-.-- lr " 11･,=2nt,･(29r+9s-3ptJ

fidll" = 2ntr(2sos IF g･v m3ti ,f

c- 6r=T,, A==

nl･, ==3ntr(9,･- pt)

eq,=3Egv(9smpt)

C and ffL

a,.=q･,+---q-,

2

c

i-)l.-.-li.IS"-1]o"

l 0tsr :･ /t )- C･,

}+ q,

B

4.

" lil's'

+ 4,.

.jt

ttn.

.

m

-:'i'i-1.,i' ii

'/ i'I""'i'nv'1

hJ.,1

Values of Constants C and

Aay

TABLE B

H to be Used in the Equations

･NS),ste7･n of l7?2'tical Loads

of Table A.

ConditionofLoading

CjiS Csr U,-s ffs)'

R'tdi･abstItlrs

Le.--l-"i

l)ab2

l2

Pev2b

l2

i:e(a+,b] Pab-']Jll(2a+b)

pc pcHiP2Cl2(J2ab(b+c)+6b2c+4c2(a+b)+c3) fP2Cl,.(f2ab(a+c)+6a2c+4c2(a+b)+c3)

sl2(4ab(a+2b+p'c)+2c(a2+2b2)+4c?(a+b)+c31 sl2(f2abC+8a21?+4ab2+4aSc+2b2c+4c?(a+b)+c3)

P-i,:.C-tb

rLWsllilLe-l-di

$pecialCases

a=0･fP2Ci2(6b2+4bc+cgl JP2Ci,(4b+cl PsCl2,(4b2+4b.+,21' PsCl2.-(2b2+4bc+c21

b=:o.fP2C;2(4a+c] PC2f212(6a2+4ac+c2) sl2(2a2+4ac+c-l, sl2(4a2+4ac+c-l

ex=b==o,21! Pl2 Pl2 Pl2

c:=l.f2 J2 8 8

zelc 7erC

evc zerc I2ol2 f2ol2,(2obc(a+b)+sc2(a+2b)+3oab2+2c3]6ol

6ol2(fOaC(a+C)+i5b(2a2+c2)+4oabc+3c31[soabc+4ob2c+2o:Ci3.g7ei:.ebOa+bO],,} (fooabc+2oa2c+2ob2c+2sac2+3oab2

+6oa?b+4obc2+8c3)SpecialCasesoa.,.o.7VC"(iob2+sbc+c2]

zerc3 "zevc'e Zevco. f."tW'i`/bto

,,

'aj"-lpmo-pt

3ol22ol2(5b+c) f2ol2(40b2+.)'sbc+7c-] 3ol2t5b2+Jobc+2cl

b..o,70C3 (sa+2c]6ol2

6ol2(fOa(a+c)+3c-) i2ol2(iOa2+2oac+7c-] J2ol2(2Oa2+2sac+sc2]

a=b==o,wl2 zevl2 7

c=:l.3o 20zol2

I20/

f7evl2

l5

iS:yx(l-x)2de f fbl2

SpecialCase

l2Sl',)ix2(l-x)de

212 212-S.)ix(l2-rf-)dx.

'l.･b.La-v'i,iisur't-e-X"'":,sttva-l--S

ci==o,b=:l.

fl Ii..S:yxg(l-x)tlx I Jwom-SiJUA(l---v)(21-x),lx Si.lx(l2-.t'2)dxl2j.JiX(l-x)2dx212 21u?

(iiesIs" gedegiyi lx,

"t,,.,. s,.!t./.o.s'

t

es,,.igg i'

Values of Constants

Loacls

TABLE C

C and U to be Used in the Equations

LSIymmetf!ical aboztt Clgnler of .ll4lamber

of Tabie A.

ConditionLoading

of C'8 ==Csr Elrs=ffs)'

t 1)l-Plr-T, 2-twfi 3 Pl

r'l

i6,Lt,----l---twi

s 8

Pa(l-a) 3

Feva P 7'.apt

l 21Pa(l-a)

'

l t

r

-l----"Is Special

l

Case

21 2!?l

a= P93

(a +b)f 3(a +b)f

l

l2t21'ia(a+2b)

+P..(a+b)2

212(2P,a(a+2b)

+Pa(a+b)2

f'il

vas

lelelSpecial Case

i5Pl=4

P,==R.

p7.i[LIJ'f6

32

f 3l

2[]l,a(l-a)+ll,b(l-l?))J

21

(P,a(l-a)+R.,b(l-b)

idaj-b･

la1)1

P2P2･b-.L

Special Cases

pat. 31'af"

.f2･a==bI)(p,--P,

l--sa Jl-sar{"l-"-M.-,J

s IP 211

ba=-=-l21P 3.Pl1..tza.ttl..na

l:......e."

2) 2Pi=l)o-'

5

:. y ･y"

TABLE C (Continued)

ConditionofLoading

Crs=Cs,' Hrs==Hsr

P6Cl(6a(l-a)+.3bc+4c2] ff-(6a(l-a5+3bc+4c21 '

,atq,bLctat SpecialCa,ses

f)a==o･Pil2(3b+4c) Pil"'3b+4c)

lllp:1I

pl 2c 3a(l-a)2)b=o･31+2c2

2crl

1"l--v-iglpV

3)a=b=-c=:ll-.Pii.i2 31 Pl25oo

Pl24)a==IJ=o,2c==l･l2

Pl2

8

iW2Cl(6ab+4bc+2ac+c2) im7srrcr'l-C6ab+4bc+2ac+c2 )

s

SpeciaiCases

wc2 7gf2 (,b+c)"r,alc:ICIa'twLdileleMvtb-m-t,leb-:-ajSlJL.-.--･lny

zorc6az2)a+c==b.I21+f2ac+sc2

u3sV-IC-6a2+i2ac+sc2

3)a=o,b==c.----S.w5gl62 5L"zevl2

64//i' iii/il/liiii'l/11illlii/il'/ll//.///･i･//i'illl,,/･,g･

,iiil'111L.,,il,/,fii,li,i-,.i･tttff'

TABLE C (Continued)

Condition of Loading

,pt

o) tto

,it.stl,E.ji･ci ,i2jl.eq)1

f-"d" 1' Lpt---b--..,l

t..--.I-'

s

rgr,, ,IS vaX.Il lH;V- -1- ltlptal+l .t-ala

Illl !.tsb-pt'tm,-b--'i

t"l-

a.,== q,,

zevc

l21

(6ab + 4ac + 2bc + c2 l

I)

2)

3)

Special Cases

zevc2 (2b + c)a =o. f2l

a+c=b. :IC(2a2'

2} + 4ac +C J

l2 la==o) b==c=== 7ev 2 32

Ll

SZYX(i- x),ihr

aho,

Special Cases

ib-

f, fS]/X(i

-Dde

.H;ts==4i'

wc81

(6ab÷4ac+2bc+c2l

zvc2XZ- (2b + ,j

hl?lilevl-C (2a2 + 4ac +c

3 7ul2 64

2)

321

-s b

.yA<l-:v)de'

"

321

' e!:/x(l-x),ix

',/fig'////･'illlillll･i/1iiiii･･i,･t･･,.

.....i･･,,,..x･pt""tsY

TABm I

General Equations for a Single Storied Bent with any Number of Spans.Legs Fixed at the Bases.Cross-Sections of all Members Different.Any System of Vertical Loads on Girders.

12 3 di-1 r r+1 n-2

h,

-,t-xe

..tlgl=

mt l, pt le .Jgll, J

t'fttr'.'-l `="rft'

e.-=-1'L

.xv XR-I R R+I Lrft-l de lr J

r-1

n-1 n.Cn-s

eff-,

1,7t

v

N -LigII l Nk' Il rllS

tl-2 n-.1

en.,

Left-HandMemberofEquation Right-HandMemberofEquatlon

CQefficientsofUnknownSiopesg "

9, 9s 9 P. 9s 9` -be 9,1..4 9p-3 :4-!9faLi 9,n A,

VerticatLoad

1 Ps e, -3.:, Pi2

2 e, pt e, "3e,, P23-P2,13 e, Ps e, m.IEnt PS.4-P3.2

4. g, Pg e. -3e,. P4.s-P4e5 e.,

P5 e, -3g, Pa'6-Ps,4

ll:{ e,,-" Ppt-s e,,"4 -3eA･-IIt Pn-3,n-e-Pn-3,jl-4

st-e e!i-3a,-f e,,-y -3en-n Pit-e,n-!mPst-2,n-3

u-l eL-2 p#-l elt-1 -3eA=i Pn-in,mPn-1,n-2

tee,,-1 A, -.Ie.･ -Pn,,d-1

tt+I e, etl enl elt. er ef, -)b-6",-ir e.I'-II ervLli eN-l e" -X, o

e/"g･iii(i/l'i;/'ili'illll'liEilii/･i,;///;

'{S!;//1-ti･k･･.S}),rv"

TABm II (Figure of Table I)

General Equations for a Single Storied Bent with of Spans.Legs Fixed at the Bases.Cross-Sections of all Members Different.Wind .Loads Assumed at the Top of one Vertical

any Number

Side.

Left-HandMemberofEquationRighVHandMember

ofEquationg'stugra

CeefieientsofUnknownSlopesg pWindLoad

9iPe Y"1 P, 9, 9s -+, 9.-4 9n-s 9H-s .lt-i 9,, Bl

1 P, e, -3.e, o

2 e,.Pe cfi -3eii o

3 e, Rs e, -.yent o

4 e, p, e, -3eJL, o

5 e, P, e, -3e,o

n--3en-4 En-s e.-, -36N-ut o

lt-.een-s Prt.E e.., -3eA,-lt o

n-i 6tlLl O"-t e.", -3eN-i o

n e.-r Pfl -.3eN o

ft+l

'

e', elt eul en, e, ert 'eN-Il' eA,-lt eS-II eh'-t eiv -xt -e,

.

e

'I'ABLE IIg (Figure of [E'abJe X)

General Equations for a Single Storied Bent with anyLegs Fixed at the Bases.Wlnd Loads Assumed at the Top of one Vertical SideAny System of Vertical Loads on Girders

Number of Spans.

Left-scandMemberofEquation Right-HandMegnber'efEquatioit

=.9-tg5vm

CoeMcientsofUnknownSlopeSg p

{)nl Y2 9, 94 9s 9, --j" SC?fz-4, 9n-3 SPvi-2 {e,,-l 9n #IVerticalLead WgndL3ad

1 P, 6, -3b:, Pi2 o

2 e, P2 e-

,,

'

--3b'li P2.3-P2.i o

3 8e P3 #3 -.g6tll P3.4-P32 o

4 e, S)g g.,

t

-L;gfpr l)4.s-2I),B o

5 ea P5 6nF1 --3gr P5.6 P5.4 o

･[,IIll

Il!

x-

71-.9t

8n--4 PnH.,t -Cli-3 -3grv-IJi l)n-3,n-2-Pn-3.n-4 o

ll-2 8n-3 Pn-2 e,,-2 -3t-N-tl Pn-2,n-i---Pn-2,n-3 o

71-l 11･.i i en-e Pn-! e.H-1 -36Nml Pn-1.n.-.Pn-a,n-2' o

ll

'

1g.-1 Pn -3e.

.------Pn.n-i o

ll+rlltg'tlblf

iilligtr]ll,I

1e,-S;r,

fpl

iffSv.IA.-ett

6N-J, 6.v -fX"1 o -- ei

･･s

ll,iiiii.Iiii･･11ili,i,IIillillilliiiiliil,i.i,,11;･

.1･/,11,･/t/111･･lt/. ･t,･/t･//･/･/･.

TABLE IV

General Equations for a Double Storied BentIJegs Fixed at the Bases.Cross-Sections of all Members ancl each StoryAny System of Vertical Loads on Girders

svith any Number of Spans,

Height Different

lt #-r nLe

' b' 3 IJ

ft va .7."1va va. veva Tl

.l ll tll R

,tls+3 tn-F2 anU

Oi"-･2 tit-J ?tt

nte9ti jZ,!}i-i .",f'

Left-IlandMemberofEquation Right-HandMemberofEquation

CoefftcientsofV'nknownesiopestp Coeff.ofIt

SOI 9e ifs So, 9i - ta-3 m-" m-1 '9ntvam+ m+t m+: 9m+i ri 9:IJi 9n-3 P".! P:i-l yn,i tlt tl.nVerticalLoad

,1 Pi e, e', -.;e, -3c", Pi2

2 e, LOO. ee e', -.{E,, -3es, P"o,3-P2.t

3 ti: p: e, e', -.;e,. 'Jg".3 P3,4-P324 e', e- j#:4 I e.,

ma3F;iv -3g,, P4,s-･P4s

I

x q " y v

JtLu?1

em.s Pca-s Fsm'g 6.-s p-3iH-!t -3.e'.-e Pm-2.t,t-l.Ptt,-2,m-3

1/s-i l e'rn-! Pm-t e.-1 e,:", .-Sk..t. -Ji6'.-t Pm-ltttt-4Pnt-l,::-.p

･in k e..1 Pm e., -34. -3g-.. -PVL},i-t

tsH-t .(`m' Pm- Emit -S.t. -Pm+i,nt+2

fn)ct g;-1tu+

PAfV g.... -3e+.-1 Pm+2,.m+1-.IbJ:t+2,ttt+3

ilt','J' -Itl-e

filt

m+! m+cP

."-3e!n,.t Pnt+S,tnt2-Pm+S,nl+4

L s

n-3 et, g,,-4 Pt,-3E,,-3 -3e'4 Pnt-3,n-4-lbn-3,""a

tt.e e', gn-s P,,-: e.-E -3e,, P"-2,"-3-Pn-e,n-1

itpl e',e,,-s Al.1 ep-t -3"'t P,i-i,,J-amubts-i,n,

is e',e..t P,, -3.='i P",ti-.i

u+t e, 6tt eui e.. er -."b e-ll F.-tt e"t-t eN d"'l o

ll+2 F.X e', ET, ?`l

els nb A-s eA., e:-, e. .e. .jpt-L

a., ,i-

".s pa e's g'. ets e, e', mlt o

r'

tg

.F

..,,.li'･ .･,-,//"l"'illilll･

t,1"1,.;s,･../.1.,ill/"

,st.'"'/" , ･t;･ ･., ････

//'f'/jili., le

TABLE V (Figure

Generai Equations for a Double StoriedLegs Fixed at the Bases.Cross-Sections of all Members and eachWind Loads Assumed at Joints on one

of Table IV)

Bent with any Number of

Story Helght DifferentVertical Side

Spans.

Left-HandMembero£Equation

Right-HandMember'

ofEquation'

Coethc{entsofUnknownSlopesg Coeff,of/2

q, trn? 9s .9e Ys -÷-

SOv--3 spm-r- SPn,.-t 9m 9,,,+ 9,nt2 ns+3 9",+4 -- so."4 9n-a sp,i-s 9"rri 9}- IJI i2i,

WindLoad

1 Pi 6, e', .F'

'.)hlp-'

-3g, o

'2 6, P, e, g""'

Li-.ie,,

=.-.tM.)SL}

o

3 e, .Ps e, . 6',, -.,,6,,, -3g-3 o

4 e, P. 4, e', -.36tlt '-36',o

/

v

1]t--2 e.{-, P,n-2e,.-, eS.-2 o"--JbM-Il

Orft･)r)t:2-Jb' o

M-I e-.-.2 'Pn･-1 ent-t g.',Hi -.,.6M-t -3g'.,-1o

fSl 6ni-t Pm e., -36,, -3g-}e o

M+l e., P]n+ g'Fi+t -3e. o

th+ eA-, e.+ Pmt2 e,.+2 .3e'.,-1o

m+3 e.',-2 6m+2 P;ri+3 Emt3 -.76!m-! .ox

lt

1

li-.)- e'<. l g,i-4R,,..3

e,,-3 -36,., o

n-2 g".a

6"-s Pn-! 8,s-a -3e',o

n-I e', e.-2'P,i-1' 6"-s -3g"'2 o

iz g', li g.-1 Pd,, -36,, o

7t+t e, 6iz .elll en. e.- eleD e'

V-11

g]t-Il eAt-t .eJTt 1 -x,'-g,

il:t-2 6', g"2

Gi, 6'g 4', e;,-, e;.-a gA.-, g. FJns eJ.',-t :n-!'g;,.,

"-- g,or g', g}., e", -Xt -e2

TABLE VI (Figure of Table EV)

General Equations for a Double Storied Bent with any Number of Spans.Legs Fixed at the Bases.Cross-Sections of all Members and eaeh Story Height DifferentW･ind Loads Assumed at Joints on one Vertical SideAny System of Vertical Loads on Girders

Left-HandMemberofEqua.tion･"'' rdgh"HandMember･-''･OfEquation"'

CeetheientsofUnkrtownSIopesg Coeff.ot#

et 92 9s 9, % T-",9m-3 9n`-s 9m-1 9m 9nt+i m+2 m+3 9nb+g ･- 9,t.4 9n-u3 '9tz-s 9,,.t P,, t21 Ft

VerticalLoad WindLoad

a P, e, ?1 -36, ---3S, Pt2 o

2 e, ,e.:

ele'e -3e.r -3g',' P2S-P2.i o

3 g'.. P, e, e', -.SiU

.-3E'. Pa4-P3e o

4 e, p" 6, 'E',

-3e. -3e', P4.5-P43 o

'

'

n-i e.,-.3 PTn-2.tr

m-!'g:J-.

-3eM-Jt -3g'.-, Pm-2.m-2..P"t-2.m-3

-t ,Em-･2 P,."1 6.-, eL-1 -3eM-i 7lt)-)hm-1 Pnt-l.m--.-Pm-1,m-2 ottt ltttl/t

fn/tt

'tnt-.1 'Pm ･,'e... /-.... tttttt/t/tt/tttt/ttt/tttt

-3e.'' -3g'n: -Pin.nt-t o

111+t e. Pm"' g.,+i '3gA.'---･Pm+1.m+-P

o

)11+2 tteA-, g..1 Pnt+e g..!Pnt÷2.m÷1-Pm+2.m+3 o

fll+3 e:-em÷S

P:n+36-+E' -3e.., Pm+3.m+2'ny'bm+3.m+4

e

' tt/

t.tt

'

'

"

n-3 ,e',ttft-4

e,L-3 -3g,, Pn-3.n-4-Pn--3.n-2 o

Pl-n2 e], e,,-,P,,-s g..t -3e,, Pn-2.n-3-Pn-2.n-1 9

le"T2'e'l e.-E 'P7Lel

e,,-1 -3e', Pn--1,n-･2-P#-1.n･ o

,ll e', elt-1 P,, MJeT, Pn.n-2 o

ll+t e, E.t ent 'enr er --ebE"tl ept-II, 4.nt-1 egt -.lrr, e -9,

n+2 6Jt e･, gts e, e, -'1m-.S sc-2 6A..i g. e. :--1 ;l

n-Sle:-,

."--.- e, e, e,, ?' g', -Xe o

.ttt

t t ttttt /ttttttttttttt ttttttt

ss}si,,tt,.,･,･,,,･

TABLE VIg

General Equations for a Triple Storied Bent with any Number of Spans.Legs Fixed at the Bases.Cross-Sections of all Members DifferentAny System of Vectlcal Loads on Girders ･Wind Leads Asssumed at Joints on one Vertical Side

n+teL.tn+!et,-en+:ert-.).-4 t.2erJtr-lh-'r-1

e:.: e;,, EL',., c.+";.+s e.+1 .,.f;"'`.--et,-se.-1Cm

l i) i" si --ewpi-Jl )S-t ,s.r

de

Left-HandMemberofEquation vaght-HandMemberofEquation

tt

CoeMcientsofUnknowhSlopesg Coeff.oftt

v"t 92 '9, - Vta-2 '9m-1 'p. gem+: :n-･2 m{･: > 9n-: 9."1 9v 9n+1 .9s,-2 9#+- - Vt-2 9r-1 Frlt[ rc' tt:'

VerticalLoad WindLoad

1 iel E,'ei,

"-.lhl -se, P12' o

2 e-1P2 e.- -li

g,,' -")51t -d-.1' CL' Pes---'P:･J o

3 e.-P. e, 4', -.rg'

ttt -.lt g"r, P34P3-. o

yl x x ' / " ･ "Xt>･L

Vi-Je-tn-! Pni-1 e,.-1 e;L-1 ' -5;'

"-l-"---JSm": Pm-1.mPmHJ,tn-2 o

f･nnhM.: n. hgm "ffJ." -.--

JVN'

r-Pm,m-1o

Vll÷Ie., P..t em+1 e., -3g:,, h-

-"-.)sm+1 -Pm-1,;n.2 o

JnH-2 :)hnl-i

5." P..2 'gs,+e -Jc.+!'t'-J.m-] ":T-J-thiC Pn:-2,#1+t-Pnt+2.m+3 o

rjl+S -"gm-e Pvt+: Em+1 -Ig.-","Jbin-e -.;g-"

ta+: Pnt+3.,n+2"Pm+3,nt4 o

' .7V" x x>a. / ･ "

×" 5

te-l 4'.. e,,-t iOn-1 t)nml e;-, -se, -.l;-..1

Pn+i,ttww2Pn-.1,n o

lle.-1 Pn En -l-3;, Mptg-

ptPn.n-1 o

n+2e. th+ e,,+1 -3e. Pn+J.nt2 o

lt+2･ 4-, etL-s ･P#+: E.,, -4-3;t4.1 Pn+2,n+3-Pn+2.,Ml o

n+3-

e,:., :-Tt+:Pr.+3 -C,l+3

' -.le'..: Pn+3.n-4'-Pn+3.,:+e o

" / ×" x>{. "

x B

)'-xeJ"nF--.

e.-: Pr-i er-1 -se'..e Pr-･l,r-PF-i..-･e o

rtr:.:+i ttt -Cr.1

'P. 't'-.)Nttttl Pr,r-1 o

r+t e, en enr->

EM`I; e.1;-t e,u -N, o -e,

1'+2 ･e', e.- er, -ib･t'5m-Lb

6,'.-, -･c,.

e,. eJ-i""1

elhui-e .e-- E,, -+

g-e e, -.lr.. o.t -- et

r+.;

'

E:ig -1Cjn+r

trt

StJi+: 'Bb.'E,f., eii'-i E., e. --!

b--1e,',.,

-e- e:., ;Jm'bt

e:･Li -yX"3o -9;

h

.h.. 't'/t'/.'t., ,

,･･l･sg-･ "1; '

v'

TABLE VIII

General Equations for a Single Storied Number of Spans.Legs Hinged at the Bases.Cross-Sections of all Members Dlfferent.Any System of Vertical Loads on Girders

Bent with any

t23

t tt ttl

tltrtz t"-t' n

rntVLII .iV:･t N

Left-HandMemberofEquation Right-Hand'MemberofEquation

doeMcieptsof'U'nknownSlopesgti

9, 9e 9,] Pd 9, 96 di 9n-i '9"-s ti"b P:i-1 '9- tll

Vertical'!,oad

1 pi e,. -1ifeJ P12

2 e, pt' e, -lfen P23-P2,i3 e, P3' -1ifeln P3.4-PS24 e, ps e, -l,5en, P4.5-P435 e, ps e, -i.fEy PS,S-P5.4

4 xft'-l E.Fi RA., e,,-3 -i,5e,v-ni "Pn-e,ti-2mPn-3,nH4

il-2 e.-, p#-! e,,-" -ltie.vrg Pn-2.n.1-PnL2,n-3

n-l gl:m! ptJI- e,,-1 -1.se.yJt Pn-t.n,rmPn-l,n-g

it e.-1 'F; .ife. -Pn,n-1

nt.r e, 6,. Ent e. e, 6s･,-lbe

e'N-IV

es'-ui eA'-I,t e'

.l'-I

･eis, ..Y,

7 o

tll/r' ''i

tL''"i't'･k･E.1,

9KX,1'

TABLE IX (F{gure of Table

General Equations for a Single Storied Bent withLegs Hinged at the BasesCross-Sections of all Members DifferentWind Loads Assumed at the Top of one Vertical

VIII)

any Number

Side

of Span.

Left-HandMemberofEquationRight-HandMember

ofEquationec

.9tsscrra

CeethcientsofUnknownSIOpesg pWindLoad

9, tP,. 9, 9d 9s qe e 9n-4 tPn'ts 9,t-e SP,L-t 9,i Pi

1 P; 6, -fsE, o

2 e, PS 6a -li5-ltil o

3 e, P3' e,---

iffeIJI o

4 e, pG Eg -l,selr o

5 e, pg e, -i,sGvo

x

"

jl-.I e.-a P,:-3

i

e.`3 -I,5eN-uf o

n-2e.-3' 'Pn-2 En-2 lrfe.v.II o

tt-l e..2 'Pn-i g.-1 -I,5e.vHl o

n e..1 ･p.-ltig.y

o

il+I e, ibU ent ef,･ e, e., O.EALtt' e,Hfll 4A'-II eh'H1 eJ-" -Y;t

7 .2et

TABLE X (Figure of Table VIII)

General Equations for a Single Storied 'Bent with any Number of Spans.I.egs }{[ingecl at the I3ases

Cross-Seetions of all Members DifferentWind Loads Assumed at the Top of one Vertical SideAny System of Vertical Loads on Girders

./

aL･9ts5crtu

.t.

Left-ElandMemberofEq"ation

'CeeMcientsofVitkmowitSiepesg ?iz .VerticalLoad

ca1'9,,･ 9n3 9n4 yt,, 9(s ' SOel--S epn-3 Sl?,,-2 qe,--1 9n gei

1.fi

gi i'

-I,5gi Pi2 o

2 e, ps 4T,

4

-x,s.e,, P23 P2.i o

3 < P3' 6.x -t,S8m P3.4 P32 o

4 g, Pf, g., p-is5gm P4.5 P43 o

5 -6, f)g- g'

,stttttttttt ･-- g,sgy ttt ltt

P5.4 .o

.

/te

x v

n-3 6n-4 PA-3 eelm3 1,5e.v-llI IPn-t3,n"2-.IPn-3.nm4 o

7Z--2l

e.-3 PA-2 g,,-2 ---IJgN-uo

7Z-l'

8n"fi Ph-i 6n-i -l,5Cv-f Pn-1.n,--"-'".Zf?n-1,n-2 o

71'gn,ml ptv

--i,5gN

'----Pn.n-Z o

7Z+f 6,･ g,, glll 6.,. 6v 6p'i

b6N-tr e,-IJI gN-ft 6N-I eN -X,

2-･29i.･'

si･.

:"/giil,'//h, ,.･ ,,

･･,i,i,,i,,,k,l,,i,i,,..,.. ,,ii, ii･i･i･･････

e

TABm Xg

General Equations for a Double Storied Bent with any Number of Spans,I.egs E[inged at the Bases.Cross-Sections of all Members and each Story Height DifferentWind Loads Assumed at Joints on one Vertical SideAny Sy$tem of Vertical Loads on Girders

el'lt=l･st-E3 , V'i+3,v?-2tn-ltr

1 le 3 711-2 7tt-t 7

tUlfl ,lf-ugt-lna

"e

'Lef"HandMemberofEquation Left-HandMemberofEquation'

Right-Hand

MemberofEqqation

Coethcientse£UnknownSIopesge Coeff.of#

P, 9e 93 ge4 P, . 9m-: m": 9n-1 9m 9m-･ 'm+!4m+3 Yv,+4 - .9,,-4 9n-: Y--2 ,9,,-1 ･F:c pt: "L'

Windi･Load

l pii' e, 'i',

plif4. -3g", P12 o

2 e, P'e e, es.- -tutg. P23-P2.i o

3 e.. e, g'3 '-liem. -3g"s' PS.4-P32 o

4 e, p)" e, e', "I,se. -3e', P4.5'P4S o

ve-2 g,n-a p#.-t e...t eJktts-t -3e.u.II -3gT.-, 'Pm--2.m-1.-.Pm-2.m-3 o

)t--'l Ie.-2 P#'ri e.-1 e,-1 -3E.-, ep-J-""-1 Pm-1,m-Ptn-l.n:-2 o

e..1 p- e,. -3g..

-3g'.. -Pm.n:-1 o

m+t e. 10m･e g.:+1 .eptJ)m -Pm+t,n:+2 o

M+2 --F..-1

"m+S IF)#s+h- e..s '3E'm-i Pm+2,m+1MPrn+e.nt+3 o

fJl-T3nt+r･ 9m+s e.t3 -3g"'..2' Pm+e.,n+2-.D"i+3,nt- o

iz-3 eu 'eit-4 'Pi-: 6,,-: -3g-. Pn--3.n-4-Pn-3,n-2 o

't-2 e, en-s' -34', Pn-2,n-3-Pn-2,n-i o'

1-l' e'e ,g,,-2 ,o,t-; e)lrl m3g"e Parl-,n-2-Pn-1.v:. o

ll e, ea-l Pa; -3C"1 2>n.n-1 o

n+1 e, -gtl ellt e.' g. rib ak-ni ei.LII eM-l ent-X,

2o

'-eei

ll+2 e', g', e', ?. . g,:.-3 e,:,-, e;Lel e,,, e,,. eA-, 8,:x-s g:,-: e e, gk '41, -Tg.- g', -Xe o -e,,

g.,' ,lil'l .h", s

'"'"･････,//,.,,{,i,.,il･,r/-lll.ll/i{,l,//.,:,'.l･;･ig

rs

s

a"retlllTVt""-, '+S q.s

-" r--

TABLE XEIGeneral Eqttations for a Triple Storied Bent with anyLegs Hinged at the BasesCross-Sections of all Members DifferentAny System of Vertical Loads on GirdersWind Loads Assumed at Joints on one Vertical Side

Number of Spans.

eq ecLl ec-Sn q-: n"lq.tn:i - -st+a

.e', ET. e',

r C2 C3 q

-e.,.-1･q-t

m

->el e. ernm

l il fll

e:.,

.s em

st+:

+:

r-t"m+i

e..,

ek.,

:e.-:,

Eu-ll

M-lr

'

Mi

eA., 'e.

'-ag. m

g' .eU-t JV e M

Left-HandMernberoiEquatien Right-}tandMemberofEquation･

CoeMcientsofUnknownSlopesg Coeff.oip

yn, 9t 9s - 9nt-: 9m-i 9m 9m+1 9m+t 9nt+s " 9..t-9.-i

9# 9n+ ･9n+s Pn+3 ) 9r-! 9r-t 9r. A, g, P3VertiealLoad WindLoad

1 Pt e, e, -tse, -se', pre o

2 e, ps 6, e.- -i,5gtJ -3"": P2S.P2t o

3 e, pl e, e', -tifelil -.le, P3`P32 o

" x x y " " "

x '

tn-x e.-, g."1 e.- E-tTi -tff".l -Je'.-, Pm-Ln:Pm-1,m-2 o

ftt e.., PA e., -tifE.u -35. Pm,m-1 o

,M+I e..'Pnt-1

e..., e., -3e. -t'C- Pnt+2.m+2 o

lll+2 e:., e..1 Pth+2 Em+: -t..-2

-se'.:, -.lc-...: Pm+2,m+t-Pm+2.m+3 o

nt-la3 e-e e:.eP-+s Evt+3 E:"s -..e.-, -le'..,' Pm+3.m+2Pm+3.pt-4 o

/ x x / " "

' x -

lt-'t e'.-'

e.., Pn-1 e.-l E;-, ",.-7ta: m),c7'.-i' Pn-j.nk2-Pn-1,n o

ttet, 8.-, P: e. -3e, -.1.c'

tPn.n-i o

Jt÷tEp P-a+i e:i+1 rm3C". Pn+1.n+e o

lt+2 e.:, e.tl Pn+: e,,+e -3el,J, Pn+2,n+3-Pn+2,n+t o

n+3 E,:"t e.+! fJn+s glL+:-.rE'.r: Pn+3.n"Pn+3,n+e o

" ' / x x ･

x '

r-x'

E:.e e..t Pr-i E.-1 -3e'.., Pr-i,Pr-i,,-2

r E,:.t 'e.-t R. -.IE'.,., -Pr,r-i o

r+t E, e,, elJt

>eN-tt ett-l elv =.Y,

2o .-2ed

r+2 e, e, E', " E:-, e-, g. e. eJ-nt-1

:-t -4F- e', e:' e', '- x.' o -ar

･t-+-{

'

'

e:.1 e:., e:･..

"e.'-t e,t-, e. e. e,t-, g;,-,

-s.- e., 'mtt e;,, -x, o -tls,s'

-.

e

/ix'tag'..-sS..., l,

TABLE XIIIGeiieral Equations for a Triple Storied Bent

Symmetry of the Building･.Legs Fixed at the Bases.Any System of Symmetrical Vertical Loads of SymmetryNumber of Spans 2m

with

about

an Axis of

the Axis

lt-l-2 }t-t-2 tt+.9 .r-2 r-t rfr'tl it-l ll-2 lt+3 JII+2 m+ 7Xil+I'

l 2 3 flt-2 fll- flt fit

zl llz

,'.ii

?;Lw,'z

M-lz

,"hS`!i'L L1

Left-HandMemberofEquation.s- Right-HandMemberofEquation"'wo-s.r

CoeMcientsofUnknown'Slopesg

9i get 9ts - V'",-e 9ni-1 9et vi"+ 9m+a 9",t3 - 9nee 9n-1 sc>,, V}nt 9.+2 Pit+3 -."bijb, 9r-! Pr-1 9r

VerticalLbad

1 pi' e, ?t'

Pi2

2 6, p! e, 6'2 P23-P213 6.,

P3 e, e6 ,P34-P32* x S)"ajL ,i$t"

N>,$, .t

Vib-t Gn,-e P,,.-I :7't-: 47in-i

,

P,}i-i,,nlhP"i-i,"t-2

OJt g.,-i Pn,eT}L PiJi."i'-PM,",-i

Jlt-l-Ig"

,,,P"L+t e.., A. P,n+i,nt'+i'nyPm+i,m+2

7n+.) 6:.-, E,nH Pfit+g 6-+! gr..,Pni÷2,}n-1tuP-"k2,,ut3

lti-.{ 6f.., g.,.2 P,ftt3 e,Jtf3 ,e:,,.3 Pni+3,m+2-P,n=F3,m-4

,et .z7vt x x /x

il' (g""

,,

Gn-2 Pn-:6si-i C-:-i Pn-i,n-2-P,t-L'n

11 e'l.

e.-1 p,,e,, Pn.n--1,

ll+r'

et, R･,,+1 e.+) Pn--1,n+2

lt-? e;", 6,,,, Pet+2 et,e Pit+2,,i3-Pnt-2.,i+i

tt'}"" el., e.+2 Pn+3 e.+3 Pit-3.,t-.l-Pn÷3,n+2

" ,,7f' S>hal･ x×

r-lg:,,, e..2 Pv-t

ftbt-Lt Pr-t,,-Pr-t,.-"P

r 6,'.,,, er-1 Pr Pr,r'-Pr,r-i////////////////ll//1111//if"tl/I/g'/,;ti//?,

"'za-.,Q' /t'.;..nt.tF-

t

'gt/tli,tS,-.ili,..xi.///,,s

TABLE

General Equations for a Triple Storied BentLegs Fixed at the BasesAny System of Syrnmetrical Vertical LoadsNumber of Spans 2m-i

XIV with an Axis

about the Axis

of

of

Symmetry of

Symlnetry

the Bullding.

n+1ft+21t+3r-2r-l .r

¢rl

is tt-l lt-2 nt+.ittH2 m+ ni,l+-I '

I 2 3 fl-2M-l fu

l ui't

Ml--"-

"Right-HandMemberofEquation

CoeMcientsofUnknownSlopesg

9, 9, 93 --9pt 9,a-2 9m-t 9]n 9n,+i 9mt2 SO,ttl3 - Y7ttrs ,9,,-t 9,, 9,ltt 9fita 9,,..: -ilp` 9-2 9r-1 9r

1VerticalLbad1 pi e, ?1 Pi2

2 6, PL' e.. 6'･t : P23PM3 e, P3 6, eg P34'-P32W x ">tsSL /

Ni>hts,

Vl-l e,.-! Pm-1･ttlUt

e;,,-i Pm-1,m-Pm--1,m-2

ftl 8m-i to;tt e.1 Pni,m'-P,,iimN

11t+I 6nt P:l+S e..1 :ttl P"i+i,,n'+i'-Pm+i,m+2

fli+2 el}nm1 e.,flP,tt÷2

ein+e 4fn-2 Pnt+2.nt+1-Pni+2,nt+3

)ll+5- e;l-2 g"

,n+! Pnt+3 6,,,+3 e,in+sPtn+3,in--2-Pnt÷3,in+4

,9, "'}' x x ,eii

' xn-i

6', 6n-? Pti-1e-,,-l gA.r,-i Pn-1,n-2Pn-I,n

ll 6,l' 8n-i Pti

e',, P,t,n-1

ll+l e, P,l+1 6."1 Pn+1,n+2

it+,2 E:,-, eiFl1 P,,+2 6,t+r Pn+2,n÷3ptPn+2,n"i

n+,-.-

8,t.! g.+s Pn+3 e.+, Pn+3,tt+4Pn+3.,t+2

" / x N>,,,

x1'fl.

g":,,2'

8r-E ･Pr-1ch

T-'i Pr-i.rPr-i.,-2r

tt

g;.+1 ,e.-t P; Pr,r'Pr,r-i

P"', :2(8.",+e.,+6,,,)-l-gof7n-f7t',

tO;-+i = 2(6m + #nt+i + 6'm+t) +6 of 7n + f,

p:･ =:2(6'.+i+6.-i)+6ofr-r'.

71Zt+.l't,

TABLE XV

Ge"eral Equations fQr a Triple Storled Bent with anLegs Hinged at the BasesAny System of Symmetrical Vertical Loads about theNumber of Spans 2m

Axis of

Axis of

Symmetry

Symmetry

of the Building.

n+lnS:2tl+.3r-2.;t--tr

't

.rt

n lt-I JILf.; iiHe mt M-l '

l 2 3 ltt-2 lttsl ill

t 11 IU -f nl-1 M Mt

Left-HandMemberofEquation Right-}IandMemberofEquatien"..OJHstv",

CoeMcientsof'UnknownSlopesg

spt 9t 9,{ " 9m'-e 9nt- 9m 9"fl 9m+o 9v,ts - Y7n-! 9,l-1 9n 9n+i 91,+: 9,,+a " 9.-g r-1 9rVerticalLbad

i e',

?1 Pi2

2 e, p,I e, e'2 P2,3hiP2i3 e,, e, 6g P34-P3ew x>esy x /

N>",s,

-11b-I 6m-s Pnf-iut-1

6ttJi Pn:-1.,n-P,n-1."t-2

vt e."1 P,I, 4,,, Pnt,"t'-Pm,"t-1

Nn+t e. p,,,+t e.., tfi1

,nl+lPm+1,tn'+1'-Ptn+1,nt+2

"t-2 e:-, e"l+1 Prn+set;t+s e:,,e Pni+2,,n+1-P,n+2.,it+3

Iii-ltt; e;.-a g.÷! Pn,+3 e.f3 ler.., Ptn+3,nt+2pPm+3,tii+4

w / 'x NNtsaj, /x

n-te', 4n-2 PflH;

6f--1 -1gai-1 Ppt-1.n-2Pn-1,,t

lte'L 6n-i P,, e, Pn,n-1

tt+ie,, P"+1

estt.1 Pn+1,ti+2

lt+2 e., 6,,-t Pn+2 e.+2 Ptt+2.n+3Pn+2,n+l

Jt+J'6S,., eN+2 Pnf3 e.+, Pn+S,n÷4P,t÷3,tt+2

" / x ×

xr-t e.-E A-rl er- Pr-1.tPr-1,r-2r 6:,,, le.-1

Pr Pr,il'-"-"Pr,r-i

.i,orl.igg,g,,, .),i･''･･･..

(" tl, .. "W)'

TABLE XVIGeneral Equations for a Triple Storied Bent with anLegs Hinged at the Bases,Any System of Symmetrical Vertical Loads abouttl2eNumber of Spans 2m-I.

Axis

Axis

of

of

Symmetry of the

Symtnetry.

Buildillbcr･

tl-l=l ll-F2 tl-f-.S ･r",2 r-l9rrl

il-t n+3 M+2hti+F

litttTJ'

' '2 3 M-2 M-l fd

't I) lll' -I M-t Mt

i

Left-HandMemberofEquation Right-HandMemberofEquationg.esgcrpt

CoeMcientsofUnknownS!opesg

vni 9",. 93 b 9m-R 9m-! 9m 9m+ qn,+2 9ntt3 piip' 9ttm2 tPu-1 so,, 9."1 9P,,+2 ,9u+s "igt' 9r-2 9r-i 9rVerticalLoad

1 p; e, g, PJ2

2 e, p6 6r e',,, P23-P2i3 F.,

pi e, es P34-P32v Xkeq N>htk y

X>tSL

m-i e,,,-2 p,r,-ti"t-1

e,'it-1 Pnt-i,m-Pm-i,in-2

71! eTtb-1 R',, e., Pfwn'-Ptn,tn-2

11t+l 6nt Pfn+i 6m" ";l+1 Pm.+1,vl'+i'-Pm-Ff,mt2

Vl+2-(,,

eltnH1 E,n+i Pnt+2 4m+2 6:,.,Pm+2,nt+1-Pm÷2,,}t+3

in+3'6f,,-!

g-

vt+2 Pm+3 e,.+3 8,'a+3 Pt"+3,in+2-Ppi+3,nt+4

W .,,7ft X>>k, x /x t

tl-f6,, En-2 Pn-1

e.,-1 g",:-l Pn-l.ii-2-Pt,--1,,:

it6,, e,,-,

PTtFJn PII.11Jl

il+Je,, R"+1 e.+1 PiiH-l.n+2

li+2 g., e.H P,,+2 E"+`v' Pnt2.n+3-Pn+2,n+1

lt+38,l-a e.-2 Pit+3 e,,,, Pn+3.n+4rmPn--3,nt-2

" / x x･

xr-t

e,'.+2 6,,-, Pr-L :sl.-1 'Pr-2.rPr-1.r-2

g,t.+1 6,,", P;' PW'-Pr,r-i...,,

ttttL,t/1'1

/{(il,'s ll!l' IF " 'S r. i..s

'"'ii"/ttt,..g'i'i,l'ktSs"ew

PIL ==2($.,,-i+g.)+i.5 6,u+6 of 77･z-7iz',

iOllb+i:==2(6.,+ 6.,+i - 6t.+!) +6 of 7n ･+ f, 7iz' + i', `

io:･ =:2(gt,.,÷i+g.Hi)+6 of r-rt.

k･･i･IXIIt･･･//t･･,/z:/',,i･,its2}3s

TABLE XVII (Figure of

General Equations for a Triple Storied Bent withLegs Fixed at the BasesAny System of Symmetrical Vertical Loads aboutNumber of Spans 2m

Table XIII)

ati Axis of Symmetry of the

the Center of each Span

Builditig.

Left-HandMemberofEquation ,Right-HandMemberofEquationg'kgtu

'CoeMcientsofUnknownSlopesg

y, F2 9s - 9nt-s9m-1 q. 9ntfl 9n;+ 9vttu - 9n-! 9n-i 9n ll+i 9,z-r 9n+3 " ger-2 9r-1 .9'r

VerticalLbad

1 Pi 6, g, Pi

2 e,PLi e. 6'e P2-Pi

3 ""

"" Ps 6, 6S P3-P2" x X>,SL "`" v

xil"-rJ' lemna PTn-t

tt;-1

g;N-i Ptn-i"Ptn--2

Vil e.-1 PnL e., Pnt-P,n-･t"t+t e,. Pm+t f)m+l 1:+1 P.,.,-p,..211tte 61z-i 8nt+i PTn+2 6m+! g:.". Pm+2-Pnt+3itt+3 .e:.-2 e,.+2 Pm+3 emfs et'n+s P"s+3--P,n-:4s5' / k x / g

!S>ajL

lllmI6'z ,enH2 P"-1

.e.-1 e,CHt Pts-1-Pnlt 6'1 6n-i ppt

6,lPn

li+l6,: p.+t

.e,il･t P}i+1

'n+e'

gTi,-1 e,,+1 PTi+2 gn-e Ptt+2'1)n+i'

tt-t-3 e,-a'

,gnn ttn+S .e.+3 Pn+3'Pn+2" 7of' '

'x'tsx.

' ･v'x

r-tE4,.,. Ept-･e' P--1 r-i xPr-1Pr-2

r'

' tttt

g:+, 6r'-i･Pr' PrPr-･i,,

(The similar problern has been tried by the author. This memoir, Vol. I, No. 2, Ig26).

TABLE XVIII (Figure of Table XIV)General Equatlons for a Triple Storied Bent with an Axis of Symmetry of the Building.Legs Fixecl at the BasesAny System of Symmetrical Vertical Loads about the Center of each SpanNumber of Spans 2m-!

Left-HandMemberofEquation Right-HandMemberofEquationao-esNgra

CoeMcientsofUnknownSlopes･g

se, 92 9.a --:)- 9m-t 9"tLI 9m 9tn+1 9mi2 9ntt3 - sO,,-e 9,,-t 9,, tl+1 9t-2 9ni3 - 9r-! 9r-i 9r

･,VerticatLoad1 Ps g, crt Pi

2 6, P2 g, e',J

P2Pi3 6,

P3 6, ei P3P2w X>･K, X)tsesL / '

x t

7i"-I 6nt-! Pm-1oets..1

6,'n-i P,n-i-Pm-･2fll gmul 'Plli e.,, PM-PM-i

IXt+l 6vt P7n÷1 6m+i :t PrntlPtn+2"1+2 6,1..1 ,g.+1 Pn,+2 6m+2 g;..2 Ptn+2Pm+3m{.3 e:,-, e.+"- Pm+3 e.,+3 eS.+3 ,Pni+3pPtn+4

" / x x /XN

>hes,

IV-lg,, .e.-2 Ptt-1

efi-1 -1gn-1 Pn-lPn

ll6,, e.-i p"

e,,

Pt+1'6,, Pn+1 6,,+1

'

Pn+l

lt+2 6e',-i e.+t Pn+2 e.+, Pn+2Pn+12t+3

6,t-s 6n-2'Pn-3 e.+, Pn+3Pn+2

" / x s)>S-

'>>'N

r-t '

6;n+2 6..,. Pr-1 6rrl '' P.-,PtL.r 6;,+1 6r-{

p･"' ''Pr' Pr-i

K- Ckpm-....IS"gk p:n =2(6,n-i+S.+gif)+6 ofm-Lrz',

P:t+ix:2(#.+cr.+i+6r.+i)+6 of m+i, mt+7,

R. :2(6'..i+e.-.i)+6ofr-r'.

TABLE XIX (Figure of Table XV)

General Equations for a Trip]e Storied Bent with an Axis of Symmetry of the Building.Legs Hinged at the' BasesAny System of Symmetrical Vertical Loads about the Center of each SpanNumber of Spans 2m

Right-HandMemberofEquation.stsg･ra

CoethcientsofUnl<nownSlopesg

Se, 92 93 " 9nt-2 SPm-1 9m 9Tntl 9m+2 sP,.+3 " 9,,-2 9th.i 9,･ 9..1 9ii.e･ 9:,-e -jtv- 9r-2 9r-i 9r

VerticalLoad

1 P; E,?1

2 e,pE e, 6',, AbPt

3 G,}P3' e, eg P3P2w x Sr>ih, ,a,it

N)>K'

'

Vj-l 6m-g Pn't-ttw-1

g";,-, Pn:-lmPnt-2.1,i 8m'-1 Pn't E., p,.-P,.-inNl

6,,,P,n+1 e..1

'

P:e+ iPm.1Pm+2M+2 6k-, 8m+i efi4+2 gn,+e 6r..! Pm+2P,n+3'tl+3 et,-, g.+2 Pm+3 6:n'+3

'

6//., Pm+3-P,n+'4sigt / XtststsL x /

X)>esL

t--re", en-2 p.,Li 6..1

Jefi-1 Pn-iPntt

ty1 6n-i PneTi p.

IZ+i･ e, P,i+1 e.+1 Pn+,

ll+2 e;,-, 6,,+i Pn+2 e.+2 P.+2p,,+ln+3 6,l", e,,-2

Pn÷ae.+3 Pn-Y3Pn+2

e / '>tses4, ht)S,

-1 S>)SL

r--II･ 6:n+2 e.-2 Pr-1 e.-, Pr-1Pr-2'r ,e:,, e.--i

.Pr PrPr-i

..y

TABLE XX (Figure of Table XVI)General Equations for a Tripl' e Storied Bent with an Axis of Symmetry of the Building,Legs Hinged at the BasesAny System of Symmetrical Vertical Loads about the Center of each SpanNumber of Spans 2m-i

J

Right-HandM.emberofEquationg.ssg'crta

9, 9t 93 ---)pt 9m-2 9.,-1 9m 9mti so.,+s 9,,,.f.3 - 9.,,-2 9e,-i 9,,･ 9,,-1 Pti+2 9nta " 9re 9r-t. Pr

VerticalLoad

l p: e,

eg,' Pi

2 e, P.'= e, '?2 P2-Pi3 C'

2P3' e, 6g P3-P2

* x S>teiL, ,se`i

N>haj,

lib-,Jr e.-a p,r,-tlnH1

'e;.-i Pm-i-Pnt-2fit e.-.t p#, ,e. P.-P.-i

-IItLt-t

e,. e?n+tm+1.

pt il

/}.+sP"i-S-.Pili"2

Vl-2 ,t,:,-i gF"

ut.+1 e"t+fF.rn+! gf.+!

m+3/6e,,",

g-,ttt:, Ptn+3 e:Tt"3 e,',,.: Pnt+3'P,nt4

" .!7vt x>,a, x A/

X>tajL

tt-iE,,

'

e."2 Pn-i 6.-, gzC-, Pn-i .p,,,lt ?1 6n-i Pn

8,, P)i

lt+I-di

'e,,P,v+!

6fz"i Pn+1

lt+2 6I,.", g,,"1 P#+2 #M+! P,t+2-Ptt+til+.,i e,t-! g.+2 Pn+3 6.,, Pn+3Pn+2w ,,,,f'

Nts,e,, Nr>ts,

xr-r

'

C-ttttt2 Er-e Pr-L

e.u.1 P,--iP.-2"r '6:,.,

, 6r-1 pvr PrPr-1

tt･s･･y

io;,, --2(6,.mi+e.)+i,sgu+8 of 7n-m',

ioj'n+ii 2(8.+4.+i+8t.+i)+g of 7n+i, 7nt+it,

p;, =2(g'.,.i+e.-i)+gofr-r'.

.

TABLE XXI (Fig. 6a)

General Equatlons for a Continuous Bearn Fixed at both Ends.Supports all on same LeveiAny Number of SpansCross-Sections and Span Lengths all DifferentAny System of Vertical Loads on Beams

Left-HandMemberofEquation･Right-HandMember

ofEquation

CoeMcientsofUnl<nownSiopesp

9t 9t F3 V4 9, 9, . 9fl-` 9n-3 9,,--. 9.-L 9nVerticalLoad

1 Ps e, Pi2-PiP2 6, Pe e, P2,e-P2,1

3 6, P3 e, P3,4-P32･4 e, Pd 6, P4.5-P4,3

5 e, Ps e,. p.1,-p,,,

lt-.]t e.-, Pn-s 6,,-3 Pn-3,n'e-Pn-S,n-4

ft-2 6n-3 Ppt-, 6s-i Pn-2,n-1-Pn-2,n--3

'll-t e..2 Pn-1 6nHt Pn-1,lj,-Pn-1,,i-2

n e.-, PT; Pn,n--XmPn,n-1

TABLE XXII (Fig. 6b)

General Equations for a Continuous Beam Hinged at both Ends,Supports all on same LevelAny Number of SpansCross-Sections and Span Lengths all DifferentAny System of Vertical Loads on Beams

.･･../1/t-/,-

' ･･1･, 11ili}$i'1'll

...,..il-1･S.,･･ tttt･ittt'

'Left-HandMemberofEquationRight-HanqMemb.e'r･iofEquation'

CoeMcientsofUnknownSlopesg

9, 9e Y3 94 9, 9,. - 9n-4 9n-3 9,,-2 9v-t 9,:

1 ･p; e, Pl,2-PIP-O,5･PO,1

2 e, 'Pt e, P2.3-P2.1

3 ee P3 e, P3.4rriP3,2'

e, P"g,' P4,5-P4.3

5 e, Rse,, P5.6-P5,4

ft-3 e.ne4 Pn-3 gttH3 lb,t-3,n-2-Pn-s,ttL4

lt-2 e."3 Rn-2 g.-2 Pn-2,n-i-PnL2,n-3

n-i e,,-, Pn-t 6n-t Pn-i,n,-P,t-i:i:-･2

ttetllli PTt O,5P"+1,n+Pn,n+t-Pn,n-i

TABLE XXIII (Fig.

General Equations for a Continuous Beam Fixed atSupports all on same Level.Any Number of Spans.Cross-Sections and Span Lengths all Different,Any Systsem of Vertical Loads on Beams.

6c)

one End and Hinged at the other.

Left-HandMembepofEquation Right-HandMemberofEquation

gusscrpa

CoeMcientsofUnknownSlopesg

9, V2 9, 9" Ps 9s . Pfi.4 9n-s 9n-2 9,,-L 9nVerticalLoad

1 P, e, Pi2-PiP2 e,

Pe e, P2.3-P2,i3 e, Ps e, P3.4-P324 e, P, e,- P4.fi-P4S

5 e, Ps e, PS,6-P5,4

"

lt-3 eptL4 Pn-3 e,,-s Pn-3,n-'2-Pn--e,n-4

n-2 ' e.-3 P,l-S 6n-2 Pn-2,n-1-Pn--2,n-3

IZ.l 6.-2 P"- eit-1 'P,i"1,,;,-Pn-1.n-2ve e.. P,., e,5Pn+1,llt'Pn.n+1-Pn,nLl

nt o

TABLE XXIV

General Equations for a Continuous Beam FixedSupports on Different Levels. Any Number ofCross-Sections and Span Lengths all Different.Any System of Vertical Loads on Beams.

at bothSpans.

Ends,

L.Sil t,

et 2ti

e

tt ',. ,.,';'.'tiG

de .... I.･ e..i"" S"'

.34

D,

L

,ll--Y#vi

n

3,1

e,..t

'41-1- e.

4

fiXTt

n+t ee,

Left-Hand.MernberofEquation Righ.t-HandMemberpfEguation

'CoeMcien･tsofUnknownSlopespVerticatLoad SettlementofFoundtition

9, 9, r3 9` Vs ieG --bb. Pn-4 P."3 9,.rr! 9,,..1 9"

J Pi E, Pi2-Pre 3e,(a,-e,):'t,+3,e,(a,-ao:l,

2 e, Rt 8, P2,3-P2.i 3g,(b,-o"o:1,+se,(a,-a,):l,

･3 ･e, Ps･ e, P3,4-P32 .Ig,<e,HD,):l.-+3e,(6,-o".,):l,

1.4. e, ,P, e, P4,5-P4S 3e,<e,-a,):4,+.le,(e",-o'.):t.

.5 {{ 1pi e, P5.6-P5.4 3eKa,-6,):l,+:3e,(S,-eL):l,

,lt:{ ke

41-d p,,-: e.-, Pn-3,,s-2-Pnr3.n-4 3en.-(SN-s-O".-):t.g+.i'C-"T:(Si,-2-Ba-3):t#.-a

･n-a g,,Js Pst-s e,,-, Pti-2,n-J-Ptj-2.#-3 3e,ini(e"tt-e-6.-s):1,t-s+3eit--p(fiti-tHOar2):larre

,fe-t c",,-s Pn-t ertul 'Pn-1,n,-P"-1,n-e. 3e,,Ls(6.-!-b,,.t).:t,,ue+3e,,-a<o".-Si,JD:t,,.i

n lF)pt .7e,,-i(B.-e",,nt):l"-T+3e.(Oi+i--O.):t"･

TABLE XXVGeneral Equations for a Continuous BearnSupports all on same Level,Any Number of SpansAny System of Vertical Loads

Equation of Three Moments es E, e, -- --Sza e.

OI ?3 "r II Ji+t

Lef-t-HandMemberof' Equation(CoeMcientsefUnknownMements)

.'SMt! sf. MM n4, M. jfcr lt`s . lt

P-ln-;

jfrt-qtJ-t M$}.:n-1 M"-:,Tl nf'

4111:+1

Right-HandMefnberof'Equatioil

12(11to+1

1 -2ntoHio--2Hi2

2 tlts

2nttts1 -2mlH21-2H23

3 fltt til!+I1 -2m2H32-2"s4,

4 flts

211ts+I

1 -2msH43-2"4Jr

5 fitA

2fit"+l

1 -2m4Hs4-2lls6

6 flis

2tlt,.+,I

1 -2msH6s-2H67

･ × × ×

n-3 tn"-l tn+I)"-l .1 -2mn-4Hrt-3,"-a-2Hn-3,n-2

n-2 tlttL-: M+Ill-1 1･ -2mn-3Hn-an;3-2H,,-2,n-1

tt-r tttH-: 'tn+?'e-S

1 -2Mn-2Hn-1,n-2-2Hrt-1,n

n ttts-1 fn+.t)--t

-2mmlHn,tia-2pt,,,n+I

['ABLE XXVI

General Equations for a Continuous Beam Fixed atSupports all on same Level,Any Number of Spans.Any System of Vertical Loads.

Equation of Three MomeiitS. "athzei..,r.. L...re

l2 3

one End.

e. e.

nr jt ti+tLeft-Hand.M.ember'ofEquation(CoerncientsdfUnknownMoments)

retsi-

Afn nf. AfY jf" nf. 1lfer M:e .jf"lpt" jf'ny-I,tl-t nfd-Iri`S 'jt

r:-t"

jftt,n+t

`Right-HandMember

ofEquation

1 2 1 -2H12

2 :ltl

?"lltftf

1 -2ntiH2i-2H23

3 gltt

2ilttt't

1 -2tn2ll32-･2H34

4 sit:

2t,ttsPJ

1 -2tn3H43-2H4s

,5 ttltl2vtifl'

1 -2nt4Hs4-2lls6

6 tns

2t)tfi+i.1 -2tnsH6s-2lls7

× XK ×4

it-3 vltti"- tlt+ln-- 1 -2Mn-4Hih3,tt-4-2Hti-s,ti-2

ll-2 flt"-4

2,11t"ji1:tL"

1 -2inn-3"n-2,n-3r2Hn-2,,t-1

tl-t 11t"-: ttt+rn-e l -2tnpt-2Un-i,f,-2-2H,i-i,,:

ItVt#-1 tlt÷1).n-S -2Mn-IHn,n-1-2ffn.rt+1

"-va

ks;.gL$)ii

In Tables XXV, XXVI, and XXVII

2 For ` (f7z+f)' read `2(rz+i) '.

and q, ,. rz',s==q-,s+ 1 2

6i 6o . e, 7no==:g, f7Zl==61, M2=:X"'

e.+17nr-- 6. '

TABLE XXVIIGeneral Equations for a Continuous Beam Fixed at botli Ends.

SupportsallonsameLevel - .Any Number of SpansAny System of Vertical Loads' Equatlon of TIiree Moments

-- Z-t:SLz-ny-£2di=r;2 3･ -ws･ rx6 E r it-t tt

'Left-HandMemberofEquation(CoeficientsofUnknownMoments)

i/,

nelr xTif,, fif3a IJf-s M. nfor 11di:hs

---ept nftl-","-S

vaVt-S,fl-S n4II-e,II"t fiK/n el`te+ict

Right-HandMemberofEquation

1 2 1' -･2H12

2 ?itl

2vnl+r)1 -2mlH21-2"23

3 11eg

2M2rlTt)

1

4 fles

2lil3+l)

1 -2m3H43-2H4s

5 ill"2li14+t)

'1. --- 2m4ffs4-.2Hs6

6 Vlt5

2Vltsrt'l

1' -2msH6s-2H67

t "XS)>,L x ×

'rt-LSfltn-4

2･(fn+r)n-4 I pt

2"zn-4Hn-3,n-4-2Htt-3,n-2

ft-2 fVttl-:

2fn-l-l)tl-s

1 '2Mn-3Ht;-2,n-3-2Hi:-2,n-i

lt-l

.

7nn-!

2M+ln-:

1 -2Mn-2Hn-i,n-2-2lln-･i,n

71

.

1 2 -2Ht,,n-1

.ttt tt ttt t ttt tu tt tt''

/ t't t"f'' '"1/tillltr'.t,t.,t//tt

/ ok.' ll ,, ･t ,,tt/t･

"., ･x'/･Y'

TABLE XXVIIIGeneral Equations for a Triple Storied Bent of Six Spans.Cross-Sections of all Members Different.Each Story Height Different.Wind Loads Assumed at Joints on one Vertical Side,Any System of Vertical Loads on Girders,

Legs Fixed at the Bases.

IS

l4

t6' t7 t8 Ip 2e 21

4l3I2Ilte' P 8

reJ4S6 7

rva/Tfla?zm7/!Ziurl

1 il Ul iV V V? Vtt

o.･es

'-eri.)

c",=crra Coeff.oi/2

9i 92 93 gp, 9s 96 97 9s 9g- 9io SOIt 9m 913 9N qth 9ie 91? 9ig 91g 9pa spe Pi pt2 tl3 VerticalLoad WindLoad1 Pi q ?i

/

=38, -36', Pi2

2 g, P2 g, 6'i l-36,, -3g'2 P23-P21

3 e, 'P3 6, 6', -.;gtlt -3.e,,, P34"H-P32

.4 g, P4 F., 6', -3en･ -3g', P45ndP43

5 ea P5 6, e', i d36y -36',,

P56-P54

6 6, PG g"

66', -.3e,, ---

36', P67MP657 6, p7 e,' -36rn r3< -3 76

･8' e, Ps 6s #}s -3e, -3e's -P89 t.t

9 6', e, Pg e 6', -3e'6 -36', P98-P9,iO,

10 S's, 6,. Pio 6io 61o -36', -3e',, PIO,9dPIO,1!

11 6', 6,, Ptl 6,, 61, -.gg,, -.?.6'tl Pll,IO-PII,l2

2 et, 6n P12 61,., 6:z -3e'r, -.l.ft'i?

P12,U'dPl2,l3 -

13 6', 8,, P13 6,3 6;,/

-36', -3e',r, P13.I2-dP13,l4

14 6', e13 P14 e, -36',pt.

.)6,, ･f)14,i3

;5 ei4 Pi ei5 -36,, P15.16

16 gg;, ei5 Plfi gta -.)'g-'

i3P16,i7"it}16,15

17 e{.. .Ce} P17 6,, -3E',, P17,18ndPl7,16

18 6i, .gi7 PIS 6is -.s6'lt P18,19-P18,17

19 gl, 8is Plg gi9 -3e',, Pl9,20-Plg2s

2e e', 61, P2e gbo -36', P20.21-P40,19

21 6ts G. R21 -"P21.20

2 gT gn eIII 6IF 6v 6Vl eVII

-.Yi -ei

23 6', e', g,i g', 6's g,, g, g, 6", 6'a 6', 6', 6'h e',pt

Xot- -ai

4 6's g,,, 6'to e, .e'i2 el,, 6,4 6,, 6I, e,l, 61, q, e', 6's.-- X3

.,...1,t,tt.... ,,..,.//t-

'a"1'/'t't'1"'･j･si/1....;itt.t//

"ts･･,･

,leQttt..pa･,tr･-･di-"

'ErABm XXIXGeneral Equations for a gei･ve Storied Bent of TripleCross-Sections of all Members DifferentEach Story Height DifferentlilYS'y'ds}1,Ota,d.S .AfSSvU,M,teiE.?tLJ.O,id",tS.O." GOi",de,>i,ertiCal Side

Span. Leg,s Fixed at the Bases.

f7 l8 X9 2oRight-HandMember

ofEquationCoeficientsofVitknownSiopesg Coeff.of/t

9i 92 93 SCIg.9s 96 9r 9s 9g qlo 9n 912 9i3e914 91s 9ie if17 9is 91g 92e Ft, y, pt3 gLtla StjnVerticalLoad' WindLoa

g f)t g, e', "-3gf -3e', Pi2

2 8, Pe 6o. .g',. -3gJi -3?"/ P23-P21

3 6o- P3 e, g',, -.;g rn-j'g'3 P34-P32,Il, .6, P4 g, -36 ---.9-

--P43

5. g, P5 S5 e',, -.gg, --p'g.-･

"'-P56

6 8}, 6, P6 #6 9, -36', -36'e,' P65-P67

'

7 -g'.- '.e, Pr 8, ?7 36'g, =;?T P76'P78

8 #!1 e,･ Ps' 8s -36'g --3ga P87.t

9 ,e, Pg g, e'. -38's

pt-.?gglP9,IO''

le c,bT'

gb Pio g,o e', -.;?r --;6'io' PIO.lidPIOP

11 e',1 g"

roplL411i 71 -.;e', ofit-.)b11 .Pll,12-Pll.10

12 ?r. 6uPig .E.,, -34',1 --3gie 'P12,11

i3 S12 R13 el3 8',3 3g'm -j'

t' lr, "P13,14

X4 'l. 4is ie14 gi4 gl, "el.)S11

38',,' P143'P14,15

15 11 g14 Pli, els g:., Ay3glo

.)"g'i5 P15.14--P15,I6

16 6･b 6iri Plfi ei6 --I?Q "-oi-P.16 P16,25

k7 gt, P17 gi7"

Oeb

-P)16Pi7"8

k8･ 61, eiT S!s e,s 3g-'ltt f)l8J9"P18.17

19 el, gt,- P19 e,,, 'h..;gt P1920-P19J8.

2g ?!a gi9 ,p,,tr -J-6',, dP20,19

2i6I 8tt exlr E"t -X, -ei

22 g, e', e',･ e, .6,- 6', #ea''- e',t. -M. -92

23 8', 6', ･e', e, 6s fi)

b78', 6', -X, -e3

24 e, g,, II 6,i 6m el, 61,, 6', -X, -ege

25.8,', g,G･ 6,',･ ei6 e,s e,g gi'G g,1

･- X, "- g.r

li}

l5 t4 f3x6･

9 fo If X2

7 6 5i2 3 4

itM IC. ny7.zZ)z7E

:.-tt.. /t.t,

.tpttttt.

,-' "' /iiii,

･,1･' k

l ll .lll lpt

ttt"

TABLEGerieral Equations' for a Symmetrical Legs Fixed at the Bases.Any System of Symmet]'ical Vertical

XXXEight Span Bent

Loads on Girders,

Four Stories High,

LefVHandMemberofEquation ,l6l5'i4tlIIn"'X'..Rlght-HandMember

ofEquationCoeMcientsofUnknownSlopesg10 II te 121

9t 9! P3 P, 9s 9, 9T Ps 9p 9m 9" 91t Pis yn" P]s /V[o VerticalLoad98l

7 6 5 5'

1 Pt e, E･, Pi22 n' 4 4'

2 e, Pt e, e', P23-P2!

3 6, Pa e, et., P34--P32 7t. xn vl x 7. x .J

4 e, P, e, P44'PP43

5 e, Ps E, e'., P55`-P56

6 e+, e, P6 e, e', P65-P67

7 e,, e, P, e, ,E', P76-P7S

8 g't e, P6 e,' Pel

9 e, P, e, e', PgJo

10 E', e, Pto e,, 6'[o PIO,ll-PIOS

11 ei. e±n p:] .e. ebll Pll,te-Pll,IOttt

12 e', eLl P,t e,t Pl2Je'-P12Jt

13 6,! Pn'e13 P13J3,'P13J4

14 e･,, e13 l)t- e,, lbu,13-P14,ts

15 e'tti ell Pts ete

16t et, e,s Plg P16,15

E

,///////1illlllll'11il,i.iiili.iiiiii)i,,

Eit...,,.,/... ..lpa.i.-"

TABLE ><IXXI

Genera] Equations for a Syminetrical Seven-Span Bent IJegs Fixed at the Bases.Any System of Symmetrical Vertical Lbads on Girders,

Five Stori'es High.

Left-HandMemberofEquationknt7i8iPeei-eot

Right-HandMemberofEquation

g';Nlgtu

CoeMcientsofUnl<nownSlopesgt514 i3 rj}

et 9t 9, 9" Ps 96 ･97 9s 9e 9io 9,, 9ie ,91: 9Al 9ts 9,fi 9n 9,- P:g 9ts VerticatLoadiou I2

iret1 ?l e, . e', P12

l6

9'8t

76 s 5j2 e, P, ee e', P23"P2t

41?3 43 6, Ps e, e', Pe4-P3e

4 e, FS e, P4'-Pts1Iiw

5 e, Ps' e, .e', P55'-P56 va va ' xcatr

6 Er3 e, p, ee crG P6S-P67

7 e', ea P: e, ei, Pf6-P7S

8 'e,, " Pa e, P87

9 e, Pe e, e, P9JO

10 ,t'1 ot Pse 'el, e'1, lbao,i-i-Pio,o

11 e', e,, Pll e,, e',, Pll,lg-PllJa

12 e･, e,, 'Pfio. ell P12,12t-Ple,tl

13 en pfi en e'13 P13,ie'-P13,14

14 e+ls e13 P,, el, S'l- Pl"3-P14,15

15 e',S e,, As C"

i5e'L5 'P15,l4-P15,t6

J6 e], e15 Pts "a P16,l6

17 esG P17 eiT Pl7,le

18 eils'ei71 PiS fiIS. P18,19rPle,17

9 e',t[ gt, Pte- e,, P!g,eo-Plg,te

20 e'13 ,e,? pfe P2ogu-P2o,lg

toa =2(83+ 64 -t- S[r) +8 of 4-4'･

pl'2--2(8's+6,,+8,2)+goff2-f2t,

tObe' =:2(?,, + 8,,) -l- 6 of 2o-2ot.

Pg -- 2(6, -lr 6, + g',) +6 of s--,sJt,

Pf3=:2((9,2+41',,+5'i,)+(lr of J3---.rLilt,

t

TABLE XXXIiGeneral Equations fol a Triple Storied Bent of Six Spans. Legs Hinged at the Bases.Cross-Sections of all Members Different.Each Story Height DifferentWind Loads Assumed at joints on one Vertical SideAny System ef Vertica! Loacls on Girders

tSt6t7t8I92o2tLeft-eeandMegnberofEquationt3 I2 ll to p 8

o'sesptcrpt

CoeficientsofVitknownSlopesg Coeffof/iRight-HandMegnber

ofEquationr4

t e 3 4 5 6 7

91 92 g)3, 9n4 9s 96 97 gns 9g 9Plo 911 9i･..iSOi, 91., 9,,, 91G 9,7 ifiS gplg 92o 921 Yi StLl pt'3 VerticalLoad WindLoad1 Pi #1 g', -I,5#l -3?i Pi2

2 e, P2 g, )2 l,sg"rr -36･.-, P23.MP2i

3 6, P3 e, 6', -4sgm -36', P34'Pee.4 6, P4 8, 8'a- 'x,s6fp -.g6', P45dPus'5 e, P5 e, ?,5 1,5g"pr -36',

'

P56-P546 6, P6 g, #t6 "g., -36', P67-P6Jr

7 #6 I07 e, l,s8va --78, '768 6, Ps #s 6's

--.9#7 -38's -P89

9 #16 .e, }09 e, e', -:-3g'6 -36', P98-P9,iO

la 8'.s #9 Pio e,, 8;o -36', -3e',, Pio,g-lploJi

fi･x e', 6ie Pll gil 61', -3#'a -3e,'ii IPu,io---iPll,12

2 g,, eii P12'6ir, 1{,. -.g6'3 -3#',2 lp12,u---･lp12.13

a #Jo ,ei2 P13 #13 ei3 -3e'2 -3g',,3 IPi3.12-'P13,z4

14 6', 6,, 10M e,g v6', --3ei4 P14J3

15 gi4 Pi5 6,, ' -3gi4 P15,16

16 ei3 'ei5 P16 ei6 -3?m P16.17--P16,15

'17 6i.n. 6,, P17 8,7 -.9e',2 fa17.18-IZZ6

i8 iglrii ei7 PiS g,, -3e'il Pis,tg----2>ls,17

19 6;o g,s P19 gi9 -3g,, P1920-.P19.i8

2g ?9 6,, P20 So -3?g P2022-fa20.19

2K ?s 8. P2i. '-P212e'

22 4l 6XT gur 4rv, gv 6Vl eVJI

/

mX,-

2-2ei

'23･ ?i e'e ?3 g', #5 8', g, 6, 6', ?,5 g', 6'r, ,8･.. e', -X, --- 92

/

't･3,',l.'lli2i:･･,./i""i'i',//':2':-･g//,//,,l/i,･;'h･-･ljt.I',"ytt

2･4. e', le', 6Uo q." e;, 8,, eq4 el, 6:, 61, 6;o 6', 6's -X, "93

.s･･'""'ny'" )i.t/im･ ･i::1'//l/S/･･･

t//g"

11 11/.

s"

TABLE XXXIffIcG,e.",e,Iasi,,Etiq.U.a,ti.Of"S.i{orMa,.Fbig,e, SDt?.fti,e,d,.?ent of Tripie

Each Story Kelght DfferentWind Loads Assumed at Joints on one Vertical SideAny System of Vertical Loads on Girders

Span. Legs ffinged at the Bases

Left-ffandMernberofEquation Rigkt-ffandMesuxber

efEquatienCoeMcientsefYnknownSlopesg- Ceeffof#9i 9293 9, 9s 96 Y7 9s 9g t?to 911 9tLi 913

14g)ts {Pl6 9n17 9ts 9ig9. pt]Y2 g[E3 Y4 S2t) Vert･icalLoad WindLoa

1 Pf 8, e', I,sg"i -3g""'t Pi2

2 g, gi 8, e', -gs.e,-36', Pee-P2i3. S･.- Ps' e,, g, '

ptse,.,-3e', P34-P324 g, pa 6, -gsen -.g6, -P435 6, Ps 8s g', -36, -3g"'s -p6 6', g, peie, ?e. -3g'3 -3e', P65--P67

7 e', C-

6P7 e, e', 's 3?2 -.38S7 P76-P78

8 epl 6, Ps e, -3?i --7e, l P87

9 e, P9 6, e', -.gg, -3e', P9JO

10 ?7 e, Pto 6io }1r-

.ye'･7 -L?e)lo IIPio.u-Plog .

11 6', 6io sOll 6,, Sl' -3e', "ft7-pblt 1)II,i2-IPIuo

X2 4r, eii P12 gale -3e's -:-.lbtL) -#l2,ll

13 l 612 P13 g13 6'is -.gg"1,,, 3gi,,, -flpi3.l4

14 g', ei3 P1414

F,4 -.,lg'. 36',, 2)14.i3---2IPI"s

15 ` .)1

#14 P15 e15 gl,s, 36'ie .,.6',, Pl5J4-P15J6

i6 g, gi5 P!6 6,6 =9?g -3g"i6

P16J5

X7 e, P17 4,7 -.9g"i6 P!7J8

gs q, 6n Pis e,, -3g"ils P18.!9'P18,17

X9 ek e,, tOie gi9 3?i P19,20MP19.18

2e ela 6,, p,ro -3#'13 -'-P20,19

2gel. #fl ethr 6A' pX,

'

-2 -2el

22 6', #l2 g,, g"

,e41 6,,, es2 g',

-Xn. -92

23 8', e', 6', 6s 6s S'7 e', 6',.-X,

t.

-qs

24 6', #'10 711 e,, #12 8:.･ gk e', -Y,1 -- 9e

25 6fa#,G ei'5 G,6 giG 6,E g"il g,1 -Xs -es

l7

z6

9

8

i

i8.

ij

JO

7

2

x9

x4'

fx

6

.?

2o

BI2'

5

4

f fi MIY

ttt.//.tt."/1/'//1111/''t/

s""' g･ge,111･

s'･ ' //.lil/iil'l

/ttttt t ttt tt../ptlt/ .//.ts.....tt/tt../tt../..t//t/./

11 .,

///

l#s

TABm XXIXIV

General Equations for a Symmetrical Four Stories High. Legs HingedAny System of Symmetrical Vertical

Eight-Span Bent at the BasesLoads on Girders

Left-HandMemberofEquationll6Isr4I3lti

g･"g Right-HandMember

ofEquationCoeMcientsofUnknownSlopesg IO 11 l2 t'tevpt 9i 9s 93 '9" 9r. 9a 9; 9s 9, 9:g 911 91: Pm ,9u {Plfi 9ie VerticalLoad

i9,

8l 7 6 s 5'..s P'l e, et, Pi2

'2 e, P'e 6, e", IP23-P2J2 4 ,4'7

3 g P', e, 6's .P34ndPS2

･-4 6, R'4 '6, P44'-Pas

･5 -6, nfi 6, e', Pss'-Pse

6 e', e, Ps ea, '?G IPfi5-P61

.7 e'e e, Pi gr e},. P76umP7S

.8 e', e, Pe e, 1P87'

9 e, Pg eg e]p9,10

10 e', 6, Pie eso fil

b!e:bio,n-Piop

11 e', 6io PEI eii 6'11 PJI.12-Pll,10

12 e', 6,, Pn 6,, P12,12'-Ple,21

13 eie Pi3 ei3 P13,IS'-P13,i4

14 eJ,, e13 P,A e,4 .P14,13rP14,IS

15 e'to ei4 P15 ei5 ･P15J4=P15,16

16 6', 6,fi PLe .Pl6J5

'

(iilllilli'ieigiees'y"i

TABLE XXXV

General Equations for a Symmetrical Seven-Span Bent Hinged at the BasesAny System of Symmetrical Vertical Loads on Girders

Five Stories High. Legs

Left-HandMemberofEquationtsi:t718l92ol2ot

Right-Hand'Member

ofEquationCoeMcientsofUnknowhSlopesgt6rs t4 r3t13

91 9. 93 P, ･9s 9, 9, 9s 9g ･9i'o YOIL gpn 9ia 914 9m '･9ie P:t opIS 9tggese VerticalLoad9.to It l2

l

t2t

1 pi .e, e', P12

87 6 5 5'2 e, pi e, e', P23-P2i

l2 413 43 e, ps e, et3 Pca-P32

4 e, RS e, P44'-P49

5 e, P,' e, 6･, Pss.-Pse

6 e', e, Rfi ee e', P65-P67

7 e･, e, P; er e, Pt6-P78

8 e', er Ps e, Pe7

9 e, P9 e, ?9 P9"O

10 e', eg PID e,, 6:io PiO,11'PIO,9

11 e', e,o Pll eii e'iL Pll,l2-Pll,10

12 e', e. Pi".. eir P12,l2'-P12,11

13 em Pi'i ei3 '63s Pl3.13"P13,14

14 e',, e13 Pi4 el- .6',, P14.IS-P14,15

15 e'le e14 Pls ei, ?i5 lbls,i4-Pls,t6

16 g,, ei5 Pao E,G PIS,15

17 e,a P17 6tT P17,18

18 e',s el; Ris ets Ple,19-P18,17･

19 e,- e,s Pig eig lbig2e-Pig,is

20 6',, el, Pfo PleO'21'-'P2e,19

P3 =2(e,+4},) -Pi,56.,.+g of 4-4t,

iO;2=:2(g's+ 6n + 8,,) +8 of .r2-i2',

PEo--2(e',,+g,,) +6 of 2o-2o'.

ge'i

tlP5 =

eP13 ==

2(& + 6, + 6',) +g of 5-5',

2(C2+8ir,+e'i3)+6 of f3-i3',