0perational method for clapeyron's theorem · 2018. 2. 11. · no.18 operational methocl for...
TRANSCRIPT
0perational Method for Clapeyron's Theorem
Norikazu YOSHIZAWA* and Bennosuke TANIMOTO**
(Received December 22, 1964)
SYNOPSIS
Various metkods for the analysis of coneinuous beams have been proposed
and investigated. They are compelled to treating simultaneous equations
involving many unkRowns, which have been recognized to be uRavoidably
necessary in the analysis of solid mechanics.
This paper takes a set of bending moments at any two consecutive
supports of a continuous beam as the matrix consisting 2-by-1 elements,
whick is referred to as the "eigenmatrix."i>
Then, one of these eigenmatrices can be shifted from one span to its
adjacent one by a certain shift operator matrix which is defiRed by the span
length and the rnoment of inertia of cross-section. By such a shift operation,
the bending moment at any support of the continuous beam can be obtained
readily and systematically, dispensing with simultaneous equations.2)
The infiueRce from external loads, settlement of suppots, and temperature
change is also expressible by a 2-by-1 shift operator matrix. For these
factors, it will be necessary to make a simple modification to the fuRdamental
operators.
This method will be of great advaRtage in the analysis of complicated
continuous beams.
INTRODUCTION
The generalized Clapeyron's theorem is written in the following form
for any consecutive two spans of the continuous beam illustrated in Fig.Ia.
M}-iCr,ii, + 2M}'(f'E', + ft1) + Mr+it?1
:k
**
Assistant of Civil
Nagano, Japan.Professor of Civil
Nagano, Japan.
Engineering, FacttIty of
Engineering, Faculty of
Engineering,
Engineering,
Shinshu
Shinshu
University,
University,
2 N. YosmzAwA and B. TANiMoTo No. 18
..
. 6(%r-i + l,-r) + 6E(.eq.zZtW,-r=i - Wr+1 -lr
Wr)
nv
F 3Ee(£r,-iATr-i + k'l.ATr)・<1)4), 7)
r-2. r-1
1l
r ・r-Al r+2・tt).L2
w?r-1
TWr
1Wr-i-1
wTr+2
l
'
l'r-2 lr-'J --I lr
l,r','f
(a)
Mr--1
r--1 lE
E
iVfr
r
-hr-1dTr--1
-ir-Jl
Afr-f-f
<b)
t
i
mArLS
lli1
i
tE
Qtr-1 Br-1
IiHA・ rlli
il・ ll ll li・
li il m
Qtr .8 ,・
(c)
Fig. 1.
In the above equation, the right side represents the infiuence by the external
load, the settlemeRt of support, and the difference in temperaturerespectively. ")
IntroduciRg the ratios
,ll, lr l'r l'r ==: I7 J.'lr' crr == ;',q' ler :lt,-,' (2)
Eq.1 yields
Mr wwi + 2Mi・ (1 + kr) + M} +ikr =r - 6 (%r mm ili, + Ur'/-. kr) ' ' '
- 6 ,.Lk, (a, wr-i - ao tv, ' wr {- wr÷i) + 3Es(Ihr.--',ATr-i + tll}AZk,)' (3)
No.18 Operational Method for Clapeyron's Theorem 3
and, for simplicity, using the symbols
6%r-1 6Ur lr"i' K}'= lr' l Ktr-1 ==
1,. (4) t s,=6iE.i.,e'fe,, z}. .. llL;, '") p= 32Es, )
the above equation becomes
Mr-i + 2M)' (1 + ler) ÷ Mr-t-ifer
= - (K'r-1 + K}-fer) - 6r(arWr-1 rm arWr -tVr -l- Wr+1)
+ B(ZV-iti Tr-i + Z3'ti T,'k・r)・ (5)
This is the fundamental equation adopted for use in the present paper.
CeNNECTION CONBITION
In order to derive the basic relation between bending moments at any
consecutive supports of a continuous beam, we shall consider the case in
which all theinfluences represented in the right side of Eq.5 equal to zero.
Taking out any set of three consecutive spans as shown in Fig.2, the
Clapeyron's theorem yields two equations
Mr-i + 2Ml' <1 + hr) + Mr+iler = O, l r (6) M・ + 2Mr+1(1 + kr") + Mr+2fer+1 = O・ 1
Mr-1 Mr Mr';-l Mr・+er-1 r r-f-1 r-2
X--ir-1rma'Y' ir 'mmmmS-l'i lr+1 -X'E
Lrs Nr・Fi Nr -l -l:'r
Fig. 2.
The above equations can be written in the matrix form
-1, 2(1+fer)nt -Mr-i- - ler, Oin -Mr+i + "O, 1 rm .M)・ " -2(1+ler+1), ler+1ww -Mr+2ww
which may also be written in the compact form
Cr{Nr-i, N?'+i}m-O,
= o, (7)
:v)
**)
The notations in this paper
It is assumed here that the
its netttral axis.
are indicated in Appendix III.
beam has a symmetrical cross-section with respect
<8)
to
4 N. YosHizAwA and B. TANiMoTo No.18in which
cr ==i Ioil 2(i t fe")I' I2 (i +k'" 'le,.i), fe ,O+ il l' (9)
and
-M)'-1- -Mr+i Nr-i == , Nr+i= ・ (10) ta -M.2mEq.7 or 8 is the connection equatioR for the two physical quantities Nr-i
and Nr+i. The physical matrix of the form of Eqs.10 will be referredto as
the "eigenmatrix" of the span considered. Contrarily, Eq.9 is one of the
operational matrices, with which definite operations are to be performed on
elgenmatrlces.
SMIFT OPERATOR
Eq.8 will yield the shift formulas
Nr+i == LrNr-i, Nr-i=L'rNr+i, (11)
in which the shift operators itqr and L'r represent
1 rm fer+i, O- -1, 2(1+kr)" Lr == - krkr +wwi mny 2(1+ kr+i), ler- -O, 1 -
1 n -kr+1, -2(1+kr)fer+1 - -rm k"ler"i .2 (1 + k?' +i), - kr +4(1 + le.)(1 + k..i) - ' (1 2)
and
Ml, '2(1+kr)- nv kr, Oh Ltr = m -O, 1 m -2(1+ kr+1>, fer+1nd
rmm fer + 4(1 + kr)(1 + kr+i), 2(1 + kr) fer+ie
-r , (13> - -2(1+fer+i), -kr+1 -respectively. It can be verified that
LrL'r =E <ee being the 2-by-2 unit matrix), (14)
which proves, from the practical standpoint, the correctness of the compu-
tatlons.
Lr is the rightward shift operator, or briefiy the right shiftor, since
Eq.11a suggests that the physical quantity Nr-i can at once be shifted
rightwards by premultiplication by the shiftor Lr. By similar reasoning,
No.18 Operational Methocl for Clapeyron's Theorem 5
L'r is the left shiftor.
Lr and L'r are, in the broad sense of words, the "ratios" between the
two physical quantities Nr-i and Nr+i・
VARIeUS FORMS OF FVNDAMENTAL SOLUTION
As a simple application of the preceeding reduction, a continuous beam
subjected to the external edge moments E!Jt and EI]t', which are given quanti-
ties, is taken as shown in Fig.3.') It will be found necessary to classify the
analysis of the continuous beam into several groups by the number of spans,
i.e, by the odd and even numbers of the constituent spans.
wtl' Wl
{/1 2 3 r-lr r-l-1 n-1 n n+I N) N i2 -r-dir-ilirU inrm'il irt /7 ' K ll
1. Continuous
When afour forms of
elgenmatnx
Solutien 1
In this
Ni ==[
in which DZ
Taking
N3 == L2Ni,
' -oddmnumber spans-Ilrmllllll..l
even-number spans-
Fig. 3.
Beam with Odd-Number Spans.
continuous beam consists of odd-number spans, there will be
solution, corresponding to the way of selection of the standard
Nr・
(Fig. 4).
case, the eigenmatrices are taken to be
tLII' N3"IIuaM,I・ Ns=:[MM:6,]・ ・Nn-i==[Mx'IJI (is)
and Ml' are the given external edge moments.
Ni as standard, Eq.11a yields the following shift formulas:
' Ns=L4N3 == L4L2Ni, '''''', Nn-ixLn-2Ln-4'''L4L2Ni・ (16>
*) To denote
used. The the letter n
support and
the support nttmber of a continuous beam, two letters r and n areletter r represents any support number and can be any integer, while
represents the support number at the extreme right or the next must be an even number: n = 2i (i = 1, 2, 3, ・・・).
6
I
Nl r2 N3 r, nt ny- tt:fTL Nn-1
t. Fig. 4.
The shiftors L.2, L4,・・・, g.n-2 are giveR by Eq.12. Then substituting
Eqs. 15 into Eq. 16c,
-anl-1rm -EMM = E..n-2 ecn-4-'' L4L2 ,
E)]t, -Mb-which can be transformed to
[Sz,I -t- I(lI M}i-i == Ln-2'''L2 egoZI -i- Ln-2'''L2 iOiI MIi,
from which it follows tkat
--1 IM#-i[ :== ILn-2inn-4 k2 I?], I-oil-
.M -DV 'O- × -Ln-2Ln-¢''L2 + ・ -O- -E}IIZt-
This is the desiyed final equation, and the present problem (Fig.4)
been solved dispensing with simultaneous equations.
Selutiosu 2 (Fig.5).
The eigenmatrices are taken to be
N2 == -ca- , tw4= -M"- , ..., N. ww2 .., -M]'-2- ,
.cam mMsm mMllve1rmand then the rightward shift operation yields
Nn-2 "" Emn-3 Ln-s'''Ls L3 N2-
E{.Ttt swt lv' ILd': ftflv 2Yflf lt(s ILIn-.2 2Lfn-1 -'qk
N2 r3 N4 r., --M[:n;I?II..3 Nn-2
Fig. 5.
N. YosmzAwA and B. TANiMoTo
r M-p M:, IL(; M,; IVfh-・2 IVf,t..i x.
No. 18
from
(17)
<18)
(!9)
has
(20)
(21)
No.18 Operational Method for Clapeyron's Theorem 7
To take boundary conditioRs into accoRRt, the Clapeyron's theorem for
the extreme right and the extreme Ieft censecutive two spans is writteR as
.,,-,M. ",),l]Il)[.(ii,?,:k,i]-:, il,kSi,'h,el .,, ,,l (22)
which can be written in the matrix forms
Li, 2P;Tl kL.3i(l]'Nfe;'l,k#J Ytilel-?'.. o. I (23)
' Eqs.23 are the necessary boundary equations which are expressed in
terms of the given external moments MZ and Mi'.
Substituting Eq.21 into Eq.23b, and referringto Eq.23a, the final solu-
tion is obtained as follows:
1vaMhl = - 1 Li, 22((1 -+i" ill'ii)Li)JE.nww3Lki-s'''L3I -ibtlllSn.i]' <24)
Selution 3 (Fig.6).
Using the foliowing method, the eigenmatrix at any odd numbered span
is determined directly. For example, taking Ns as standard, the right and
leftward shift operations from Ns yield
-mz r Ni= == Lr2Lt4Ns, -as- (25) :s) -ant-1" Nn-i= = Ln-2Ln-4'''LsL6Ns・ EEnt
w}' sz]}. f/"I'lr----ecl)-ll?-"-------III>---<ll!l---{IIII------------- N')
.- -- x
Writing
x- Ni -e-;T, N3 - Ns LL, L[`
Fig.
the above equations together,
[ee'i,,I M,t- -gl 2M-
・== +-Mn-1nv O, 1 - 0
-EMt .- wwM?Ii- -O, O-
L6
6.
ua・
M}l-1
- ----e- Nn..1 Ln--2
Lt2Lt4
Ln-2Ln-4'''L6
Ns, (26)
8 N. YosmzAwA and B. TANiMoTo
from which
ff M6- - o, o--i--wrZ- - Lt2L'4,
Mh 1,O O -- ca O,1 O -ecn-2Ln-4-''L6, HM.-i- - o, o- --mzt- This is the desired final solution. Eq.27 requires to compute
inverse, although this is of a simple form.
So}utieR 4 (Fig.7).
Taking N6 as standard, the right and leftward shift operations
Nn-2 = kn-3Ln-s'''ingL7N6, pa2 == L'3L'sN6・
tTLt"t2 M' itt' 7Mi M"Li2 .-bNI"-'i .'"XS(l t
No. 18
(27>
a 4-by-4
yield
(28>
N"- NL) - N4 'ntE-= N6 -m---N,n-2 s-- l-g L・r) L7・ Ln---3
Fig. 7.
Substituting Eqs.28 into Eqs.23 yields
Li, 2(! t2£,-ll) lr2il.-fe,2il.e,1(lel.'Ig5..ewSsl'1.,,-, -Ei'lii・fe.-,. 1 (2g)
Writing these equations together, the following final equation is obtained:
-M6- - L2 (1 + fe,. >, k,j ttmt3k., rs --iN EII)t -
=- ' (30) -M7m -Ll, 2(1 + kn-1>] gmn-3Ln-s'''L7 - rmEI])l'kn-1-
2. Continuous Beam with EveR-Number SpaRs
As described in the preceding article, there are also the following cases
of solution regarding the selection of standard eigenmatrix:
(a) Ni for the first span,
(b) N2 for the second span,
'Y') There are the following relations between the right
Ln-2Ln-4'''L4L2 = [Lt2Lt4''`Lln-4Lin-2]-1,
[Ln-2Lh-4'''L4L2]-i = Ll2Lt4'''Lrn-4Ltn-2.
Combining these relations with Eqs.25, we can get the
and left shiftors :
same result as soltttion 1.
No. 18
Solution 5
ES}--
ll 1 S2,.ll x N
X Ni r.2 N:i ri ---Lr.2 Nn-i --
Fig. 8.
Taking Ni as standayd, the rightward shift operatioR yielcls
Nn-i " Ln-2Ln-4'''L4L2Ni・
The right boundary equation is written to be
Ll, 2(1 ii- len>] Noe-1 '= - Tl'kn.
Substituting Eq.31 into the above equation,
mz- nto- Ll,2(1+kn)jLn-2Ln-4'''L2 -l- Mli =-℃}'len, Ol
from which the followiRg final equatioB is obtained:
rm --1 -orm ML) nd- - Ll, 2<1 + fen>JLn-2Ln-4'"L2 -1-
-mt- × Ll, 2(1 -Y len>J Ln-2Ln-4'''L2 +wr'kn ' uao-
Solntion 6 (Fig.9).
'
yr]lt m Mn--1 Mn. 's l.b M2
Operational Method for
(c) N2r-i for an arbitrary
(d) N2r for an arbitrary
<Fig. 8).
M2 M,3
23
Clapeyron's Theorem
odd numbered spaR,
even numbered span.
Mn-2 Mn"1 Mn n-2 n-1 n
and
' s n-l-1
K. N2 - L3
N.1・ - Ls
Fig.
" l;nl:?7-i Nn
9.
-1
9
(31)
(32)
(33)
(34>
10 N. YosmzAwA and B. TANiMoTo In this case, the standard eigenmatrix is taken as N2,
shift operation yields
Nn "= kn-ikn-3-''gsL3N2,
which is transformed to
Ol + M}i = Ln-iLn-3'''ksL3N2・ E)]'l' O
The left boundary equatien is written to be
sw't + L2(1 + k,), fe2] N2 = O・
Writing Eqs.36 and 37 together,
-sw'e- -O' -2(1+k,), fe, -
o+I A(L,+ N2=O, -Ln-iLn-3'''L3 EI]tt O
from which the final equation becomes
HN,un -Mhum -2(i+k2), k2, orm-i+Dt
-Mh ==- 1O - Ln-iLn-3'''g-3 it, IVC, om rms}ntSolutien 7 (Fig.10>.
wrt /--b' M2 M3 M・t Ms Mb' Mn..J (2----41)--<3 4 s 6' n,rmi
and the
.
fidrn
n
No. 18
rightward
・ffnptK xn+1 )
1
'
x S' NI T.2 N3 XT4 Ns r6 -n'rt,.2 Nn-1
Fig. 10.
For example, taking Ns as standard, the right and
operations from Ns yield
Nn-1 =Ln-2Ln-4'''LsL6Ns, Nl = L'2L'4N5' ,
The right boundary equation is 1 ' Ll, 2(1+kn)] Nn-i+EIiVt'kn=O. "
Substituting Eq.40a into the above equation,
-
leftward
(35)
(36)
(37)
(38)
(39)
shift
(40)
(41)
No.18 Operational Method for Clapeyron's Theorem
Ll, 2(1 + len>] Ln-2Ln-4'''a.sL6Ns + wrl'kn = e・
Eq. 40b is transformed to
-o lh -v Ml, - Lt2Lt4Ns -l- =: O.
1OWriting Eqs.42 and 43 together,
- L'2Lt,i
1M5.+ Ns+O -O- -Ll, 2(1 + fen>j Ermn-2Ln-4'''L.6- -MZ'kn-
from which the final equation becomes
-M6.- rmo, '-irm Wl- - Lt2Lt4
Mk ==- 1, O mM6rm -O, Ll, 2(1+fen)]Ln-2S.n-¢''L6- -MZ'knrm
SglutioR 8 (Fig. 11).
E)Jt -F-' M: ltf;s ?Vl.: M,-, Mn /1 "--.-llll>" t5 n
= o,
4 71+1 1rc -A K X"' NL' '"(Z,e Ni -cr,s -rmM Lnhi- Nn
Fig. 11.
Taking N4 as staRdard, the right and leftward shift
' NnmLn-ILn-3'''Lstw4, N2=L'3N4・The left bottndary eqttation is written to be
E!)Z + L2(1 -i- fe,), le,J N2 == O・
Using the same procedure as described in the preceding soltttion
and 47 may be written in the form
-L2(I+k,>, k,jLt3- MO- rmMZ-
A(C,-g- o =o, N,i + l - Ln-iLn-3'''g..s se}t o
g-}x,
x x 1 '/
11
(42>
(43)
(44)
<45>
N.u
operations yield
(46)
(47)
3, Eqs.46a
(48>
12 N. YosHizAwA and B. TANiMoTo No.18from which, the final equation becomes
-M,- -L2 (1 + k,>, k,j Lt3, o- -i -swl -
M, =- 1 O. (49) - L"-iLn-3'''Ls,
Mi, o EMt3. Continueus Beam with Clamped End.
The clamped end of a continuous beam (Fig.l2a,and 12b) may be
considered to be equivalent to the adjacent imaginary span which has an
infinitely great amount of momeRt of inertia as shown in Fig. 12c, and 12d. 5)
kt 23 :n--1nntJl' 't (a) I i (b) 1 Jo =- ,x) . In +i =-,oo , e I Ni 2 .3,i bt-1 nN" n'f'1 ' n.t.2 'i'L'"' lo r'L'rm' ti 'z,Ll"rm t2 r"31)L'j' l'`"Cilltn ic'Si''' zn "' llL lng'j
Fig. 12.
To derive the basic equation of the clamped end, assume that there are no
external Ioad, no settlement of support, and no difference in temperature in
this system.
C!amped at the Left End.
By Eq.1, the Clapeyron's theorem for the span li and the imaginary
span lo in Fig.12c yields
M6 tP- + 2M, (k' + -fl!,)+cai'- - o, (so)
from which it at once follows that
ca -=-2Mi, (51)and the eigenmatrix for the span li becomes
N,= llMthlll=I-II M,. (s2)
Two unknown elements are reduced to one, and the problem can besolved. ')
ttt t t tttt ttt
'it) Writing down the three moment equation for the spans li and l2, the support moment
Mk can be represented by the clamped end moment M!. Thus, in the case ofacon-
tinuous beam with clamped end, we can express the entire support moment as a function of the clamped end moment.
No.l8 Operational Method for Clapeyron's Theorem 13
Clamped at the Right ERd.
By the same procedure as in the preceding derivation, the eigenmatrix
Nn in Fig.12d is represented by
H-2- Nn= M}i+i・ (53) - 1-
EXTERNAL LATERAL LOAD
When a continuous beam subjected to external lateral loads, theClapeyron's theorern (Eq.5) takes the following form for any two consecutive
spans in a contiRuous beam:
M}e412rXitiiM+'(1,#1''M"r'+IXjk+1-:-1`5K']IJ+''KKr+1feEr'Li),l `5`'
in which
6 6X K'r-i =: l,rm, %r-1, K} =: 7.- Ur,
(55) *'*)
66 K'r= 7;J' %r, Kr-= l,+i Ur+i'
Eqs. 54 can be rearranged to the matrix form
1,i; 2`i ', k"'][t'M'I " I, ,, //'k,.,), kP.J[MMI::
-'K' r-1 + K}kr h x- , <56) -K'r -t- K}+lkr-ww
which can also be written in the compact form
Cr {Nr-i, Nr+i} =-Kr, (57)in which, Cr, Nr-i, and Nr+i have been defined by Eqs.9 and 10.
Kr is a 2-by-1 matrix which is determined by Ioading conditions as follows:
- K'r-i + Krkr rm
Kr= ・ (58) -K'r + K)'+lkr+1-
Eq.57 yields the shift formttlas
*,k) The valuescollected in
in Eqs.55
Appendix Iare for
defined
typicai
as the "Load Term'' inloading conditions.
this paper, and are
14 N. YosmzAwA and B. TANiMoTo No.18 gl:.Ikilew.i:ligei/tl,,l (sg)
in which the shiftors Lr and g'r have been given by Eqs.12 and 13.
Pr and P'r are the shiftors wkich represent the influence of the external
lateral load, and are defined by
1 - '(K'r-i+Kl'kr)kr+i rm P" == ferkr+i -2(K'.-i + Kl・fe,)(1 + kr+i) - (K'r + K}'+ikr+i) kr- '
(60>
-2(K'r + IL'+ikr+i) <1 + fer) - (K'r-i ÷ K}'fer)-
- -(K'r -l- Ki'-t-lkr+l) -
1. Contin"ous Beam with Oee-Number Spans Subjected tD Lateral Load.
When a continuous beam consists of odd-number spans, there are also
four cases of solution as described in the previous article.
Solution l <Fig.13).
Evzt x N '
Nl rr.,p2 N3 -E;/;ig7p,t --pm r[;,:IT-X;.,.pa.2. Nfi--1
Fig. 13.
Taking Ni as standard, the crightward shift operation yieids
Nn-i = Ln-2gdin-4Ln-6'''insL6g..4L2Ni + itoi-2Ln-4itn-6・''LsL6gmu4ge2
+ ±n-2g..n-4Em)t-6'''gansX6P4 +'''+ #ptn-2gen-4 + Pn-2
== kn-2#-n-4'''L2Ni + tcn-2 Ln-4{'''gig6(LP2 + P4)
+ P6'''1} ÷ Pn-4 + gen-2・ (61)
The above equation can be transformed to
bOt,I + Ici, l M}i-i = Ln-2Ln-4'''Lil-Moil + IOiIM{・i]
+Ln-2 Ln-4('''ge2'-')+gen-4 +Pn-2, (62)
from which it folSows that
!ve
"""ll 2 3 4
:[tt.
.I i tl-2n-1]
][II,iR" ..ss
nl
N
No.18 Operational Method for Clapeyron's Theorem 15
IAMa,Z[ -ww ILn-2Ln-4'''L2 IO, -- ・ I'gII.-ij-re Ln-2Ln-4・・-L2 IEi,iiZI
-}- SOz,1 rm ILn-2 ILn-4 ('''pl'') + pn-4I + pn-2I] (63)
This is the final equation for the present case. Comparing this with
Eq.19, it may be seen that these equations are quite similar, the only
difference being in the additional term due to loading condition postmultiplied
by Eq.63.
Selution 2 (Fig. 14).
t/ ll"" 2 3 t Yi "l n-,?S nnv2 nww1 l ; XN
K.: k N2 -m-'" --in - Nn-・) L3, P3 Ln tl, Pn-3 - Fig. 14.
The standard eigenmatrix is in `Lhis case taken to be N2, and
rightward shift operation yields
Nn-2 == Ln-3Ln-s'`'LsL3N2
-i- Ln-3Ln-s'''LsP3 +'''-i- Ln-3Pn-s + P"-3
"= Ln-3Ln-s'''LsL3N2
+ Ln-3 [Ln-s('''P3''') + Pn-s] + Pn-3・
The right and left boundary equatioRs are written to be
Li・ 2(i de-ew,,,nz2,i.; .E.MIk)tsi ne- :."i;i:2 k:i,l,iJifenrei)・ ]
Referring to Eq.64, and using the same procedure as described iR
vation of Eq.24, the final equation iR this case is obtained in the
-M,- nv 2(1+k,), k, --i N2 "" .m- , Mti - Ll, 2(1 + kn -1)] Ln-3Ln-s'''L3 m ww ev. × [?"i'lelli + (I'if52' tK",it2k'.-i)
then
the
form
the
(64)
(65)
deri-
16 N. YosmzAwA and B. TANIMOTO
Ln-3 [Ln-s('''P3'''> + ge n-s] + Pn-3 + Ll, 2(1 + kn-i)j
Comparing this result with Eq.24, the term due to loading
added in the postmultiplied matrix in the above equation.
Solution 3 (Fig. I5).
9.T} /-'1 2 iTg;IIR/'//,; ,-, 6 "-2 7i-f nvi (
No. 18
(66>
condition is
DLIti..ss
xnS
x
v Nl - L' L,, P,£,
L
N3 - LZ P,f
Taking any odd numbered
the final equation becomes
-M6- va
ca Mlz-1
Solution 4
S.}] L'A
! !1N
×
- #.p'2L'4,
- Ln-2'''L6,
-- Ml '
o + o
-EInt
(Fig. 16).
2 3 ,v 'IT
N- .- o Lci, Pa
Fig. 15.
elgenmatrlx,
o,
1,
o,
o,
Ln-2
6
o
o
1
o
-1
- -... Nh-1 Ln--L), Pn-2
for iRstance
L'2Pt4 + P'2
Ln-4('''P6''')+gen-4
7
- /1
, Ns as staRdard,
+ Pn-2
n-2 n-A tr Fll"i
, gAIJI
-- xn N
1
(67)
×
s
Taking
N6=
N2 t)/sp;'sNiitPtps-N6L;pi-i.M.,JtiMl;,..:,N'i'-fL'
J Fig. 16.
N6 as standard, the final equation becomes
-M,M - L2(1+k,), k,JLt3Lts --i-AL- -Ll, 2(1 + kn-i)I Ln-3Ln-s'''g-7 ww
/
No.18 Operational Method for Clapeyron's Theorem 17
× ISztfe.-S",Z" (k"'ll-; 522?'ikn")
+L2 (1+fe,), le,.1 -Lt3P's+Pt3M ww
rm ' - -・ (68) + Ll, 2<1 + hn-i)] Ln-3[k.n-s<'''P7'''> -l- Pn-s] wwi- Fn-3
2. CentiRueus Beam witk Even-Number Spans Subjected to Lateral Lead.
As in the preceding solutions, the final equation for this case will also
take the form postmultiplied by a certain matrix due to external load.
Assuming that, there acts an arbitrary series of external loads on tke
continuous beam as shown in Figs.8, 9, 10, and 11, solutions for each case
can be derived using the same procedures as described previously.
SolutioR 5.
TakingNi as standard, and shifting to the rightward direction as shown
in Fig.8, the final equation becomes
Mli m ut li, 2e+ kn)JLn-2Ln-4'''L2I?lrmlrm'
× ILI, 2(1 + len)jLn-2Ln-4'''L2 1swoZ" -+ EM'len + <K',t-i + Kltfen)
+ Li, 2(i + fen)j ILn-2[Ln-4 ('''p2'l) + pn-4] -t- pn-2I] (6g>
'Solutie" 6.
Taking N2 as standard in Fig.9, the final equation becomes
-M3.M M2 (1 + fe2), fe2, O- -1
M, =- 1 - Ln-iLn-3'''L3,
-WZ÷ (K,,+ Khk,) -
×-OM me -. (70> - Ln-1 [Ln-3 ('''P3''') + Pn-3] -i-Pn-1 EIIyzt
18 N. YosmzAwA and B. TANiMoTo No.18SolutioR 7.
Any odd numbered eigenmatrix is taken as standard in this case. Taking
Ns as stan(lard, the final equatioR is written to be
-ma"rm MO, --1 -g.'2g,.t4 ua == - 1,
th.ua- tO, Ll,2(1 + fen)] g-n-2・・・L6"
rm rmEIIrl- - (Lr2F'4 + Pt2) xo
-SM'kn + (K'n-1 + Knkn)
. <71) + Ll, 2(1 ml+ kn)J g.,n-2[E-n-4('''P6''') + Pn-4] + Pn-2
SDIution 8.
Any even numbered eigenmatrix is taken as standard in this case. Taking
N4 as standard, the final equation is written as follows, and may be com-
pared with Eq.49 which was derived for the unloaded case.
-M,M rfL2 <1 + k,), fe,]Lt 3, o- -1
- Ln-iE-n-3'''Ls,
- E"l +(K',+ Kl,k,)+ L2(1 + le,), fe,] P'3 -
xO, - -. (72) - Ln-i[Ln-3(''-Ps-'')+Pn-3]+ Pn-i
-E"Zll ' -M3. Clamped End of a Continugus Beam Subjected to Lateral Load.
Using the same procedure as described in the previous fundamental
solutioR, f.he clamped end of a continuous bearn can be analyzed for given
loading conditions.
Clamped at the Left End. tt In this case, the additional term due to loading condition
-6(S..''E2) (73)
No.18 Operational Method for Clapeyron's Theorem 19
is-to be added in the right-hand side of Eq.50, and the following relation
is readily obtained:
ca =-2M,-K,. (74)
In addition, writing down the three moment equation for the spans li
and l2,
Mi - (4Mi + 2Ki) (1 + k2) ÷ Mhk2 ww- ma (K'i+ Kafe2), (75>
the third support moment is obtained to be
i
1- -' Mh := k-. (3 + 4k,) M, + 2(1 + fe,) -K, - (Kt,+ K, le,)m . (76)
de -
In a similar manner, all support moments of a continuous beam caR be
represented by the clamped end morneRt Mi, and therefore the problem can
be solved if the right boundary condition is given.
Clamped at tke Rigkt End.
By the same procedure as descrlbed above, tke following relations are
obtained:
M}t =-2MLt+1-K'n (77>
L M}t-1 == (4 + 3len) Mi+1 +2(1 + fen) K'n- (K'n-1 + Kikn). (78>
In these equaeions, M)t+i represents the bending moment at the rigkt clamped
end.
SETTLEMENT OF SUPPORT AND DIFFERENCE IN TEMPERATVRE
Taking the infiuence due to settlement of support and difference in
temperature into consideration, the shift formulas can be represented by
N,'+i=Lr Nr-i+ ger+$r+Tr, (79)and
Nr-i == L'r Nr+i+ ge'r+$'r+ 'E"'r, (80)
tny WEhqilYh6oL;'esapnedct!.:I&yh.aVe been defined by Eqs・i2 and i3, and p, and pr.
20 N. YosmzAwA and B. TANiMoTo No.18 Sr, S'r, Tr, and T'r are 2-by-1 matrices which are defined by the
following formulas:
Shiftors for Settlement ef Support.
un - o"r (arWr-1-ctrWr - Wr -- Wr+1) kr+1 - lSr := ferkr+i ' o-r+i (evr+iwr - crr+iwr+i - wr+i + w7 +2) kr ' (81)
- -im 26r (arWr-1- arWr- Wr+ Wr+1) (1+ fer+1)-
-' O"r (arWr-1 - arWr - Wr + ZVr+1) + 20r+1 (ar+IWr ' ar+IWr+1 M
S,, .. -Wr+i+ZUr+2) (1+fer) . (s2) - -tir--1(crr+IWr-crr+IWr+1-Wr+1+Wr÷2) ,-
Shifters for Temperature Chaitge.
- (a-idT}-i+Z)'dZkr)kr-Fi - T" i= ferfePr+i m2(Z}-iATr-i mF Z)'ACZZxfer(1+kr+i) ' (83)
rm +(Z}ATr + Zi・+iA Traler+D ferm
-(ZL'-iATr-i + Z}ATrfer) ve 2<ZL・ATr + Z)'+idT}+ikr+i) (1 + kr)-
(Z5・ATr -- g・"riTr+ikr+i)
Using the same procedure as described in the case of external lateral
load, the infiuence due to settlement of support and difference in temperature
can also be taken into the analysis of coRtiRuous beams.
PARTIC(JLAR CASE
In the case of a continuous beam with equal span and equal cross-section,
the shiftors take the following simple form:
- 1, -4 Lr == , (85) - 4, 15-
15, 4 ec'r =: , (86) ww- 4, - 1-
"r == [4(K・;-IK"ki.)'-liillllt,+Ki.,)I' (87'
No.18 Operational Method for Clapeyron's Theorem 21
-4 (K'r + KU'+i) - (K'r-i + K}')-
- -<K'r+K}J+i) -
- Wr-1 + 2ZVr ' Wr+1
-4Wr-1 - 9Wr + 6Wr+1 um Wr+2-
- Wr-1 'i- 6W}' um 9Wr+1 + 4Wr+2
S'. =b , <90) - -Wr+2ZVr+1'Wr+2 -
1 Tr -- 2PZtiT , (91) -3
-3 T'r=: 2PZdT , (92) 1
GENERALIZED SOLUTION
As in the previous investigation of a continuous beam with simply
supported ends, the common form of final equation consists of
(1) n-by-n inverse to be determined by the beam construction,
(2) n-by-1 column matrix representing the edge moment, and
(3) n-by-1 column matrix representing the loading condition
(the number n representing 1, 2, 3, or 4).
Then the generalized solution can be written in the form
N=:= R[M+Q]. (93)
Here R represents an n-by-n inverse matrix and is designated as the
"Premultiplier." In the same manner, M is the "Edge Moment Matrix,"
and Q is the "Load Matrix."
When the both ends of a continuous beam are clamped, the final equation
takes the form
N== RQ. (94) In the case of a continuous beam with overhanging end (Fig.17a), the
load on the overhanging part can be reduced to the edge moment effect as
shown in Fig.17b. Therefore, the final equation is the same form as Eq.93.
22
li n-1
IIiix
lIi;
N. YosllIzAwA
P
tn
and B. TANIMOTO
n+1
17.
kin
Q
I1 n-1
No. 18
.Ev}i -- -Pl 7t
-,e・N
nN [ [-=; ln-i;1 lft t
(a) Fig.
The generalized formulas for all
Table I. The values of ee, gkafag, and
Ta'ole l. GeReralized Foxmulas
../ i lr,l -1 --1 (b)
ds of continuous beam are shown
are summarized ilt Appendix II.
for Continueus Beam
in
Type of Beam
Simplets-Simple
Free .vFree
SimpleNFree
(? e. '-
L " tt(ll "i
-) 1:/"',
tp=tt7 :/t it//t
Formulq
・1
N= R[M + Q]
Clamp -Simple
Clamp tvFree .-.4
Clamp -wClamp
ii]i N=R[{Nfi + Q]
N=RQ
Corresponding Table
Table III, IV
Table V, VI
'
Table VII, VIII
EXAMPLES
Practical applicatioRs of the present method will
ing examples.
Example 1.
DetermiBe the bending rnornents at the supports
with seven equal spans when the middle span 6)uniformly distributed load q.
1 2 3 . 4: "・ ll Mi ii ": ql 6
be given in thefollow-
of a continuous
alone is loaded
7' 8'
beam
by a
e・
ll・
a
b--
From Table II,
a
7@l
Fig.
is
m.-7l
18.
th x th
the load term obtained to be
No.18 Operational Method fer Clapeyron's Theorem ` 23
1 K4 =K'` == -2i-ql2. (95>
Substituting the above value into Eqs. 87 and 88,
p3 = Ir Z,1 - -l}qi2 I- ?I, pt,- Iww4il:1- {}qi2 1-il,
--K, - 1 --1- -4Kt,-K,- 1 - 3- P` == -4K, - K,,ww = mu4nqi2 - 3-, P'4 = - - K,, - =T z-gi2 -mi-, (96)
ps = I-4KKIil -- -l}qi2I-X, pts =: ImmoK'[ = -{}qi2Iegl,
and
P2 == P'2 == P6mP'6xO. (97)Then the shiftors Lr and L'r are given by Eqs. 85 and 86, and reduce to
--1, -4 ww -15, 4- .4, 15- --4, -1wwUsing the Solution 2 in Table III, the eigenmatrix is obtained as follows :
ww2(1+fe2), k2 --i -4 1 --i )R==- =- -- 1, -4 -2 Ll, 4] -Ll,2(1+k,)JL,L,- . - 4, 15-me
-- 780, 1- l 2911 ' me 209, -4-
(99)M -= o,
- (Kr,+ K,le,) -Q- (K', + K,fe,) + Ll, 2(1 + le,) ][LP3 + Ps]
oo = -1-qi2wwLi, 4Jll[-1:. 14s 1 lm Oil+I-ill- -- nv1-qi2 --4i-'
24 N. YosffizAwA and B. TANiMoTo No.18from which
N2 = IMAzl -- ew[ffwa +Q] = t/ilt:4 i'II・ (loo)
By the right shift operation, the unknown support moments are de-
termined as
-M4- ql21ma--1, -4"--1" - Oe- ql2--15rmN4 == mMtsm = LN2 + ge3 := 2'g'4" - 4, ls - - 4- + -m71 rm ": 2gtll4 rm- ls- '
rmMbrm ql2- 4'N6= -M÷-=LN4 -t- Ps= 2{tgZ4 nd -1rm' (lol)
The check calculation for the above obtained values may be carried out
by the left shift operation as follows:
rw7=nvillZ,- == gsl2, rm -g-, .... ・i
ql2 -15 Ns == E-' rw7 =: Egtli4 4, O・ K・
(102)
ql2 4 N3=: g-' ews+P'4 =: Egtz4 mls , O・K・
ql2 O Ni= E"' ee3 == twt l4 ml . O・ K・
Example 2.
The maiR pile of a cofferdam is strengthened by wales and struts as
shown in Fig. 19a. Calculate the bending moments at the strengthened points
of the rnaiR pile. The wales are arranged with ten equal spaces on the main
pile.
No.18 Operational Method for Clapeyron's Theorem 25 Support Moment (xql2or xe.Olwbff3)
11O IO -- e.O16 666 67
9 -e.O1443371
-8 -O.025 598 59
7 -O.e3317231
6- -O.04171228
O.6q-5 -O.04997857
O.7g-4 q -O.058 373 44
e.8g-3 .i -O.e6652766
e.9q-2 -O.07551591
1, .- O.081 408 71 --- g = wbff---.i main pile
(a) (b) Fig. 19.
The main pile is subjected to the hydrostatic pressure and can be con-
sidered as a continuous beam with the lower end clamped and the upper end
free as shown in Fig. 19b.
Using Table II, the load terrn is written down easily as follows :
l2 qla ql2 ' Ks= 6-d (8qa + 7qb) == '6d (8 + 6・ 3) = 'grt × 14.3 = 14. 3R,
K'i -= g-?o-- (7q. + sqb) =g6/-o-2 <7 + 7. 2> -- -/-i8- × i4.2 -- i4. 22, I
KIi == 12. 81, K'2= l2. n, Kl, =R.32, Kr, ,11. 22, i (103) "C)
K, = 9. 82, Kt, =: 9. 72, Ks == 8. 32, Kt, =:: 8. 22,
k:. :Igil :16,:- 21Iil kr. ZI]il ".17,i. :'.i'1.1 )
Substituting the above values into Eq.87, it follows that
- -(Kt,+K2,) - -27 -4 (Kt,+ Kl,) - (Kr, + Kb) rm m 84 - (104)
--21- --15 rm -9 ps, =2 , P, =2 , P, =A . -66 rm -48- L 30-
11wale
-"J
/ strut
10---'
9
N.,
8
@"
76
byn
i
54
ny"Y
I "a
N 32
.--"N l
.t 7' xX' st' t ・' ・・Z tt,' ./t
'
;') For simplicity, the notation 2= ql2/60 is adopted for use in this example.
N. YosmzAwA and B. TANiMoTo
Here tke shiftors Lr and L'・r are given by Eqs.85 and 86 and
stant value for the entire span.
The load on the overhanging part in Fig.19b may be reduced
the edge moment effect on the support 10, which is given by
1 2}]rl' = - 6-t ql2 = - 2.
Then, using the Solution 9 iR Table V, it follows that (n ;= 10 in
R =: I L,L,kL, I-i[ , 1-glli= [I-;: -l 4s 14 I-;I , I-g l[ 'i
O, -1 1 nv 70 226 , - 70 226, - 18 817m
ooEvg=: ==R , EDtt -1 -o -Q=: L4 - L3Pr L2P, - k..P6 - Ps rmK,-
M-2911, -10864' nt O lt -- 209, -780- --27-
- 10864, 40545m ]4.3- " 780, 2911-m 84rm
- I' lgl -,ggl I- i,il - I-il -,4,I I-i5,I - Im,,9I]
rm- 91 911. 2- =1 ' 343 021. 5
from which '
MMiM 2 -- 343 020. 5M N= -Mb- == R[M+Q] = 7o 2'i6 " 6os17.3 '
Using Eq. 74,
' -M,- - 1- -O- 2 -- 343020.5- Nl・== = Ml- = . m-2- -Kl- 70 226 rm-3!sigo.snd -Mlem
No. 18
have con-
to one of
(105)
this case)
(106)
(107)
(108)
No.18 Operational Method for Clapeyron's Theorem 27
Shifting to the upward direction, each support moment is determined sue-
cessively as follows :
-M3ww 2 -- 280 318. 3n N3-- -M4- ww- LNi + P2 = 7o 226 nt- 24s g6o.o- '
wwMsrm 2 -- 210 587. 7- N, -- M6 = LN3 + P4 == 70 226 - 17s 7s7. 2-'
un (109) N7 = Iilk]= LNs+ p6 = 'Jii7o&2'ww2'6"[: lg97 s767o31 s5",
Ng=` IMMb,,I = LN7+ ps = r7/ot'i'26 [I 6,O, ij],7I gl・ ?
Comparing Eq.109d with Eq.105 or 107, we can see the correctness of
computations. Thus the computation of tkis method can be carried out with
an automatic procedure. The values ef support moments are given in Fig.
19b.
Example 3.
' Find "he bending moments at supports of the continuous beam wlth
variable cross-section showR in Fig. 20.
l/1
21
q.
;st
/I
!f'y
f
Lttf
II,
ilii
tk
E l'
!・lf
z ''r
f.;l
2l-A-
3-- A4
.¥)..mutmugt-l-,7
.:/tt
l 21 3.l--
and
Frorn the given
therefore
From Table II, the
Fig.
configuration of
fer =:1 (r ==
load
20.
the beam,
2, 3, 4, 5,
- 1, -4
Lr- . - 4, 15-
term is obtained thus
it
6)
IS
'
evident that
(110>
(lll>
28 N. YosmzAwA and B. TANiMoTo
Ki :K',= {i- ql2 =ft,
Kh = Ki2 = Ks =: Kts=: K6 == Kt6 = pt,
K3 :Kt3 = 4pt,
K, == Kt4 ::= 9pt. i
Substituting the above values into Eq.87, it follows that
-2 ge2-nv pt, 3
-- 13rm I ge4 == Jt, 42
- 10 ge,=- pt. 7 38
UsiRg the Solution 21 in Table VIII,
R = [L,L,I-!I, I- 4i 3fe6I rm -i
..el-11 -,g--2llI, IvegI--i= -,3iol,,gl ,I
-O M -4Kt, - (Kt, -F K6)-
Q==L4k + -L4P2-Pa -Kl- - -Kt6
-- 1, -4rm2 rm o- - 2M M- 1, - 4- -- 2- 13-
== St -l- ,Lt- LS- H. 4, 15- rm1th -- 1- - 4, 15rm- 3m 42nv
- 31rr Ft, 129m tttttt tt t t t tt ttt tt ttttt ttt tttttt tt tt tttt tt t t ttttttttttttttttttttttttttttttttttt t tt tttt ttttt tt t t tt t
'i`) For simplicity, the notation pt = tql2 is adopted fgr use.
,
t'e
No. 18
(l12) ")
<113)
(114)
No.18 Operational Method for Clapeyron's Theorem 29
from which
N - [MM,i- =RQ =2・3a4o II1284g[. (11s)
The eigenmatrix for the first span becomes by Eq. 74 as follows :
Ni - ItX -IwwIIMi - },1 - ,st,, I-ve gg,il・ (ii6)
The unknown support moments are determined by the right shift operation
as follows:
-1207 No =m LN,+P2 == pt , . 2340 -6214 ' -"L (117>
. -4357 N, =LN3 + P4 -- rmi3'4o 242 '
The values above obtained may be checked by the following shift operation
N6 =- IillSI -- LN4+ps- ,g,, l' ll lgl IZ2ilil + 1- :gl pt
242 pt ' =Tl}'g'iifO -12gl' O・K・ (!ls)
CONCL{JSEONS
In conclusion, the follow}ng notes are g}ven :
1. The exact solution for all kinds of continuous beam can be obtained
by simple matrix algebra.
2. The bending moment at any support of a continuous beam can be
determined directly.
3. Compared with the moment distribution method, this method has a
kigher efflciency and perfect exactness. The eficiency will grow greater and
greater as the system become complicated.
4. The computatioR can be designed with the simultaneous checking
methed.
5. The calculated result is checked arbitrarily and effectively by the
right or left shiftors.
30 N. YosHizAwA and B. TANiMoTo No,18
ACKNOWLEDGEMENTS
The authors express their sincere thanks to Prof. K. SATo, Faculty of
Engineering, Shinshu University, for his lnspection of the manuscript from
wider point of view and for various useful suggestions. Their thanks are also
due to Prof. A. HAKATA, Faculty of Engineering, ShiRshu University.
REFERENCES
1) B.TANiMoTo, "Eigen-Matrix Method for Beams and Plates," Proceedings
of the A.S.C.E., Structural Division, Oct., 1963, pp. 173-215.
2) B.TANIMoTo, "Operational Method for Continuous Beams," Proceedings of
the A.S.C.E., Strttctural Division, Dec., 1964, pp. 213-242.
3) J.S.C.E., "Civil Engineering Hand Book" (in Japanese), GmouDoH Co.,
Ltd., 1964, pp. 65-69.
4) N.YAMAGucHi, "Applied Mechanics'' (in Japanese), ARusu Co., Ltd., 1952,
p. IIO.
5) I.KoNism, Y. YoKoo, and M.NARuoKA, "Theory of Structures" (in Japanese),
MARuzEN Co., Ltd., 1963, pp. 95-105.
6) S. TIMosHENKo, "Strength of Materials," Part I, 3rd] Edition, D.VAN
NosTRAND Co., Inc., 1955, pp. 204-209.
7) RoBERT B.B. MooRMAN, "Analysis of Statically Indeterminate Structures,"
JoHN WiLLEy & SoNs, Inc., l955, p. 153.
8) A.HiRAI, "Steel Bridge, III" (ln Japanese), GiHouDoH Co., Ltd., 1956,pp.
38-51.
No. 18
Table Il.
Operational Method for Clapeyron's Theorem
APPENDIXI. LOADTERM
LOAD TERM FOR OPERATIONAL METHOD
31
Loading Condition KE{iiill,:-za
i
Fl/l'.,'j'i,, ,l 1,' 1 ¢:e=tr"r=:i
'
4 .ww' t ""'
rr a --'b'1
-c-- t"
di. 1ntptllg
L--- t --'"
/il I・・ ,,
tg4rr --
ql,',.
ant
q--tl--"" 7t -ma'm
gg[III'i'l:,d,'/・i
as pi. ,irill' 1[ :l/,,
"U. e"L---- t -'
f,b (i+b)P
-II- gl2
oq4-tl--o."
(21-a)2
q (d2 - b2)(212 - b2 - d2)412
7
um ql2
60
g/" (i +3)(7 nv 3?,)
o6qnvoai, (3a2 - 15al + 2ol2)
qa26r6i2 (12a2 m 45al + 4el2)
l2
65(84a + 7qb)
l£t,(l3 - 2a21 + a3)
Kl
`flb, (i+a)p
-li- qi2
oq,-X (2i2 - .2)
1£/i2 (a2 - c2 )(212 - a[ - c2)
8- ql260
fli/Ilo (
a1+- l×,-3tA)
qa2
6el2(1012 - 3a2)
4qa2-6Cti}l2
(512 - 3a2)
l2
6-t(7qa + 8qb)
f/ (i3 - 2a2t + a3)
Note: The following relations are seen between the load
Method" and the "Slope Deflection Method":3)
K == - 2IL,B, K, = 2RZBA.
terms in the "Operational
APPENDIX II. VARIOUS FORMS OF SOLUTION FOR ALL KINDS OF
CONTINUOUS BEAM
The solution's' for all possible cases of continuous beams are collected and
classified ln the following Tables III, through VIII.
Using together with Table I, the analysis of all kinds of continuous bearns
can be carried readily and systematically.
32 N. YosmzAwA and B. 'IIANiMoTo
Table M SIISffPLE・-SIMPLE
No. 18
Shifting ProcedureIIlt
N R
ywt Abe
(-f2 Nl .....ip
Solution 1.
ii l'ig i?l, ll
-x・vl / ii -!
rmM,- mu
mlVCt-1 -
I:lliI
i[
LnT2Lnrm4'''L2
[ Ol --
'
-- 1-
- o-
-1
pe- t )tab-
("L2, N2・ ..- Solution 2.
Si,,li
-" x l' li/ liNY L i L
-ML-
-Mh-
'-Mk rm
ua
M2
-utt-1 -
111Ii"
1-Ii
I'
- 2(1+ks), fes --Ll, 2(1 + knrmi)jLn-3Ln-s'''L3nt
-1
(.Lr----<1 6 ) s" N- ig Ns- Solution 3.
t l
i
ll:
i
sR -qs-- N6 S Solutjon 4. s
t
-Mh-
-Mirm
- o, 6''- LtELt4,
1, O
O, 1- Ln-2'''L6,
- o, oT
-1
- t2(1 ÷ k,,), ksi Lt3Lr,., -
rmLl, 2(1 + fen・-i) Il-n-r)Ln-: '''L7-
-1
Tab}e IV SIMPLE--SIMPLEshifting pr5cedure t
it
i NI
wr
Nl ---v. - Solution 5.
yw・
i I[ua]
I
IEE)tm. ' ".c ' SJ}t
'N.- N2-- Solution 6:
i
A41,
ua
Tfuf}trm
R
Ll, 2(1+ fen)Jtn-2'''LE
nt o-
-1-
-1
L
IIl
Illl
(-ptim-- .t2> 4e-Ns -pt V Solution 7.
'
$.}t! 's)xt' Ap.- ---K(.il---GiL--g>-- .".m-L>
'iv ee4- Solution 8.
il
- 2(1 + fe,,), k2, O-
1 ww LnrmILn-3''"L3,
o
-1
-Mle-
ca
-M5-
-ua-
ca
mM)1rm
o, . LtsLk 1,
- O, Ll, 2(1 " fen)jLn-2'''L6-
-1
-L2(1 + le,・), k,jL'3,
- Ln-1'''Li,
o'1
o
-1
No. 18 Operational Method for Clapeyron's Theorem
(fer Odd-Number Spans)
33
M
- Ln-2Ln-4'''L2-mt-
"e-
ww o
+ EDtt
' EM-nvswttkn-i -
-- swt
o o- S!nt
' E[{} nv
-EI}ltfen-1 ww
Q
-I Ln-2[Ln-`("'P2"') + Pn-4]+ Pn-2 l
'(K,, + K,le,)
-(Ktn-2 ÷Kn-ilen-1)
+ Ll, 2(1 + kn-i)]IIILn-3[Ln-s("'P3''') +Pn-s] + Pnth3]-
Ln-21
Lt2Pt4 + Pt2
Ln-4 ("'P6'-') + Pn--4I + pn-2
-(Kt, + K,le2)
-(Ktn-2 + Kn-ilen-i)
"+- Ll,2(1 + fen-i)J ill
+ 2L(1 -f- k,), fe,][Lf,Pts -t- P'3]
Ln-3[Ln-s("'P7"') + Pn-s] -t- Pn-3
(fer Even-Nllmber Spans)
M
Ll,2(1 + fen)llLn-2"'L2
'EV}- × -l- EPIIlen pmo-
wwEVt -
o
s]zt
m owrt' kn
-EM -
o
mt
Q
(Kt.-1 + Knkn)
+ Ll, 2(1 + fen)1 l Ln-2[Ln-4 ('''P2'") + Pn-4] -l' Pn-2 l
-1
(Kt, + Khle2)
Ln-1[Ln-3("'P3''') + Pn-3] + Pn-t l
' -(Lt2Pt4+Pt2)
-(K'nmi -1- Knkn) ÷ Ll,2(1 + fen)A Ln-E("'P6''') -t- Pn-2 L
--1
(Ki, + K,le,) + L2(1 + le,), le,j Pi,
Lnml [Ln-3 ('''Ps'--) -t- Pn-3] 'tnv Pn-i 1
34 N. YosmzAwA and B. TANiMoTo
Table V. CLAMP-vSIMPLE
No. 18
Shiftin.cr Procedurei
11
N 1 R
(-
1 2
Ni -----i,.
Solution 9.
.eq ,,,I
n>ls. 7
rmM, -
-Mn-1-
E
:i
Ln-2Ln-4'''L2 IM
-2'
-1
o
I } lMi I i I
i
Solution
123 "n N2 '
Solution
10a.
10b.
i [ sJe' I
-dis- /nX l・il---/ I'
I
l・
[M,]
"M,rm
ca atgb-l
iil
'I
-2Ll, 2(1't-knnti)JLn-3'`'L3
3 +4k, - feo -
2, 1, O rm (3 + 4fe2)
-rm-rs. ' O, 1O, Ll, 2(1 -1- kn-i)jLn-3'''L3 -
-]
-1
sJeiWtg-'-e. s--"")
So}.ll. IioR. 11.
,l i
al 89' -.s
n
・-,-- Ns---e.-
Solution 12.
EUzt
x}
/
-M,
M!o
M,-M},rm1 rm
・ '-M, de
ca
mMb rm
l
t
=' 1
Lt2LkV6Lts,
Ln-2Ln--4'"Lie,
-1,
2,
o,
o,
o o-1 o
M =2,
3 -Y 4k,
le2
- O, Ll,
- Lt3LtsLt7
'
2(1 + fen.-i)ILn-3'''Lg
-1
Table VY. CLAISfP-vSIMPLE
Shiiting Procedure
Solution 13a.
-'"K Mt'
Nne7 Ni op
Solution 13b.
i
l N
If1:
1:I
[M,]
-M) un
Mhne!
rmM;1 m
R
i
H
X;lt
SV--ag:i> N2 ,ab s- solution 14' - m,
S,L-Gve-=ctPi) - Ng ---b.r s"- Solution 15.
i'
Ii
ii1
l11
-M2-
M,
Mi
-Mh.
-?va -
ca
.-Mieth
iii
lil,
Ll, 2(1 + kn)jLnm
rm 1' Ln-2'''L2
--2
O1,
--1 N IM2・-・L2
-2
M 1, O -1 'T O, 1 2(1 + k.n) .
1, O, 2, (3 + 4fe2)o, 1, o-, le2 .'- Ln-1"'L3 o,-l, Lt2L14Lt6Lts 2, O, Ll,2(1+,len)jLn-2'''Lio
o
o
1
e!
S"lti
-Nsi. -- I soiution i6. 1 11
uaMl
ma
ua
-Mb-1E
:1
:':' 2,'
3 + 4fe,
fe2
o,
o,
o,
o,
1,
o,
rm -1
-1
-1 -Lt3E.lsL17
-Ln-iLn-3'''Lg
No.18 Operational Method for Clapeyron’s Theorem
(for Odd-Number Spal裏s)
35
㎜0㎝
@一號
@「
[顕~’々η_1]
0 -
0_購~~んη司_
ン
000
一
一 〇 一
9_田~’々アz..一1_
Q
Lπ_2L,~_盗…L2
…〇一
一1 一
一・司1・一(…・・…)+P司・・。一・
L…(・+・・一1)・ト・・一・L・・一・(…P3…)÷P司・ら一・
一κ1 } + 秘一3Lπ一5■■.L3 _叢(1十々2)κ1 一 ゑ (κ’1 + κ2々2)_ 十κ’π_2十・K麗_1んη_1
瓦 ゑ(κ’・+嗣一十橘)κ・
(K1アz_2十瓦~一.1んη_1)+L1・2(1+婦1)」「しか・[L樋(…P・・一)+P ・葺P囲L
一 工OKOO
一 }
十
L’2[L’4(L’6P18÷P’6)+P’4]+P’2
Lπ_2[Lπ_4(…Plo…)十Pπ_4]十Pπ.一2
一 一ノ(1 -2 1 -L’・(L’・P’・+P’・)一Pノ・病(1十々2)κ・「聾(κ11+κ2々2)
(K1・一・+K・・一・㈲・L…(・+・・一1)ll…↓・一・(……)・町訊一・= [.
(for Eve鷺・Nu紬er Spa駐s)
i釦~1々・]
0
0
戴}~’んπ
ン
000勢
一
}0
0
分〕~ノ々π
㎜σ 一..
0
0
1璽’_
Q
0κ’π_ユ十Kπ々π÷L1,2(1十んη)」 L,置_2L7z_墨…L2
_ _ _一κ1_
+L,、一2[L。一曇(…P。…)+P,掃]+P,囲
一L2z._含Lη_4…L2
一 1..6漏.
一L7z_2[L,z_4(…P2…)十P7ト4]一Pπ_2
_一 P(L
(K’π_1土《πん2ゆ
-」…
κ11(’1十」κ2々2 2(1十んの」K1
}
弓2 π
んP
十
う
ら P
(
ゆの
ヨ
々 ㎜
L} }
《
L
「
一
1一・…㌧
}0- L’2[L’墨(L’6P’8+P’6)+P’4]+P’2+
_κL・κ勉一・・魚璽デL…(・殉・}}{一・(…ヒIG…・性・1豊・
徽 一十 2(1十秘)κ1 K/1一κ2島
病一L’3(L’5P’7 十 P’5)一P13
ん3 _
一Lπ_1[L7~_3(…Pg…)十Pπ_3]一Pπ_1
一
36 N. YosmzAwA and B. TANiMoTo
Table VXI. CLAMP-CLAMP
No. 18
Shifting Procedure IN Il R
N1.
.Solution 17.
ll
'i I,
i i
I,
-M, rm
rmMh-
j
Iil1
!t
Lnrr2Ln-4'''L2 '1 -2
'
- 2-
-1
-1
N2. Solution 18.
I'
l・
I!
-Ml-i
hth-
1l
i
Ii
Ln-3Ln-s'''L3
-23+ 4k, le2
'
-' 4-3kn-i
2
-1
.- N7-Solution 19.
ilt
II・l・
ll
-Solution
Ns-20.
1Il
1
j
-Ml rm
ua
ta
rmMg-
-M,M
th
ca
.-
. .Mb.=.:,
l
iI
1, O,-2, O, e, - 2,
e, 1,
- Ll2Lt4Lte
-LnrELn-4'''L6
-1
- 2,
3+ 4k2im-・- t k2 o,
rmo,
e, - Lt3LtsLt7 o,
4 + 3len-i,
- Ln--3'''Lg- 2・
-1
Table Vgll. CLAMP--CLAMP
Shifting Procedure INl.
l1
R
Nl . Solution 21.
, mM,i
l rmMh.,
Np -ip- -i
Solution
l
ii
l・
i
ml
IiI
mI,
Lnm2Ln-4'''L2-1 -2
'
-- 4 - 3kn-
2
-!
22.
NM,
-Mh+1
"1 78 fl+1 ?-
--s- N7 --ep-
Solution 23.
MM,
Mn+i
ta
..Mh -l
M,
Mh+1
va
-Mb -
1
1
:!/
1!
i
Ln-itn-3"'L3
-23 + 4le,
fe2'
- 2M
-1
-1
.Hs-xg)-cuEFI -Ns-Solution 24.
! 1, 2, -2, O, e, (4 + 3kn),
- o, -21
-Lr2V4Lt6
- Lnm2Ln-4'''Ls
-1
I!
[III
Ii
-2, O, 3 + 4fe2 hn k2 , O,
e, - 2,
- e, 1,
-U3LtsW7
-tn-iLn-3'''Lg
-1
No. 18 Operational Method fer Clapeyren's Theorem
(for Odd-Number Spans)
37
Q
Ln-2Ln-4'''L2I2,]- -Ktn-1 nv
e- Ln-2IIILn-4("'P2''') + P"m"II m Pn-2
2(1 + kn-i)Ktnmt - (Kln-2 + Kn-!len-i)-
rm Kln-1 ' 2(1 + fe2) m Ln-3"'L3 le2
o-K,
Kfn-1
o
mK,
K, -Kt, + K2fe2
fe2- [Ln-3('''P3''') + P,t-3]
+
Lt2(Lt4Pt6 + Pi4) + Pi2
Ln-2[Ln-4('''Ps''') + Pn-4] + Pn-2
K, ww Kt, f.K2k2 - 2m(1fe+. k2) K,
Ktnrm2 + Kn-lkn-1 m 2(1 + lenml)Ktn-1
Ktn-1 ww
+
Lt3(LtsPt, -f- PIs) + Pt3
Ln-3[Ln-s('''Pg-'') +Pn-s] ÷ Pn-3
( for Even-Number Spans)
Q
Ln-2'''L2-o'
-Kl-+
-2(1 + fen)Kln - (Ktn-i + Knkn) -
- Ln-2[Ln-4('''P2"') + Pn-4] - Pn-2
Ln-iLn-3'''L3 Kl, + K2k2
le2
xi,
2(1 + fe2)
k2
-' -rm Kln-
+K,
o" Ln-i [Ln-3("'Ps'") + Pn-3 l - Pn-i
o KlKln-i ÷ Knlen ff 2(1 + fen)Ktn
+Ln-2l
Lt2(W,Pt6 + Pt4) -i- Pt2
IILn-4('''Ps''') + Pn-4] + Pn-2
- K,Kt, + K2le2
k2 Kln
o
2(1 + k,) Kl ke +
Ln-i [
L!3(LtsPt7 -I- Pls) + Pt3
Ln-3('''Pg"') + Pn-3] + Pn-i
38
r
n
lrM,・
L・
na・
EAr
Ur,
Wr
E
hr
A7';・
l'r
evr
fe7.
K・,
6r
p
NrGr
Lr,
Ex)i,
N. YosmzAwA and B. TANiMoTo No.18
A]PPENI])IXIII. NOTATION
The following symbols kave been adopted for use in this paper:
== support number countered from the extreme left support;
==r any even number representing the number of constituent span of
continuous beam ' ,== span length of r-th span countered from the extreme left span;
== bending momeRt at r-th support of continuous beam ;
= moment of inertia of cross-section of beam in r-th span ;
-- standard moment of inertia ' ' = section modulus of cross-section of beam in r-th span;
== modulus of elasticity ;
= area of bending moment diagram of r-th span calculated asa simple
beam for its loading;
E8r = reactioB at the left or right support of r-th span calculated as a
simple beam subjected to the moment diagram load A. ;
=: settlement of r-th support;
= coeMcient of thermal expansion;
= depth of cross-section of beam in r-th span ;
== difference iR temperature between the upper and lower sides of beam
in r-th span;
" ii・i. Ir,see Eq. 2a;
==
7l:L"---l,see Eq. 2b;
= 'l'/-,'/t',,See Eq. 2c;
K'r = load teym, see Eqs. 4a, 4b, and Table II;
6EL== '7r. kr, see Eq. 4c;
=: -23-Es, see Eq. 4d;
= "eigenrnatrix," see Eq. 10;
= connectioR matrix represeRting the connection conditions between
two eigenmatrices Nrffi and Nr+i, see Eq. 9; .ec'r = 2-by-2 matrices called the "shift operator" betweeR two physical
quantities Nr.i and nvr+i, see Eqs. 12, and 13;
= 2-by-2 unit matrix;
ept' == given external edge moments at the left and right extreme
No.18 Operational Method for Clapeyron's Theorem 39
supports of a continuous beam ;
Kr = 2-by-1 matrix representing external loading conditions, see Eq. 58;
Pr, P'r = shift operators consisting of 2-by-1 matrix, representing external
loading condition, see Eqs. 60 ;
Sr, S'pt == shift operators consisting ef 2-by-1 matrix, yepresentikg influence
of settlement of support, see Eqs. 8!, and 82;
Tr, T'r == shift operators consistin.ff of 2-by-1 matrix, representing influence
of temperature change, see Eqs. 83, and 84;
R = "premultiplier," see Eq. 93, Table I, and Appendix II;
M -- "edge moment matrix," see Eq. 93, Table I, and Appendix II;
Q = "ioad matrix," see Eq. 93, Table I, and Appendix I;
2 = tedl, see Example2;
" ,.. Elll;", see Example3;
L J = row vector; and
{ }= column vector.
PAGE
8
13
13
13
19
20
21
29
LINE
8
5
7
Eq. 54a
Eq. 77
Eq. 83
Eq. 92
5
ERRATA
FOR
Spans
beam subjected
a M>+1 + ・・・
K'n
4ATrkr(1 -F fer+i)
'
operatlon
READ
Spans.
beam is subjected
the
A4ml + ."
Ktm
ZrATrkr)(1 + fer+i)
operatlon: