Embed Size (px)
Citation preview
Comp uters & Structures VoI. 21, No.3, pp. 563-580. 1985Printed in Great Britain
0045-7949/85 $3.00 + .00Pergamon Press Ltd.
LATERAL LOADING DISTRIBUTION BETWEEN TREELEMENTS OF A TRREE-DIMENSIONAL CIVIL
STRUCTUREALBERTOCARPINTERIand ANDREACARPINTERI
University of Bologna, Istituto di Scienza delle Costruzioni, Bologna, Italy
(Received 19 Aprii 1984)
Abstract-The problem of the lateralloading distribution between the resistant elements of athree-dimensional civil structure is analyzed. The bracings (shear-walls and frames) are sup-posed to be axially undeformable and interconnected through floors undeformable in theirplanes, so that only three degrees-of-freedom per floor are considered.
A generai formulation is presented and the internalloading transmitted to the single bracingis expressed through pre-multiplication of the externalloading vector by a distribution matrix.
A numerical example is eventually shown, in order to illustrate the use of a FORTRANcomputer program, which translates the algebraical formulation into a sequence of elementaryoperations.
1. INTRODUCTION
The probIem of the Iaterai Ioading distribution be-tween the resistant elements of a three-dimensionaicivii structure has been anaIyzed under its variousaspects in recent years. Wynhoven and Adams [1]studied the reduction of the ultimate Ioad carryingcapacity of a structure when it is subjected to Ioadscausing torsion. CouI! and Irwin [2] presented asimpie coefficient method for assessing the load dis-tribution in three-dimensionai multistory shear-waI!structures. Heidebrecht and Stafford Smith pro-posed a simpie approximate method of anaIysis forthe behavior of open thin-waI!ed shear-walls [3] andof talI wal1-frame structures [4], subjected to tor-sionai moments. Stamato and Mancini [5] andGluck and Krauss [6] used a "continuum mediumtechnique." to analyze the three-dimensionai inter-action of walls and frames, whereas MacLeod andHosny [7] proposed the use of a discrete matrixmethod and a frame computer program. More re-cently, Haris [8] presented an approximate matrixmethod to determine the Iaterai Ioad distributionand the deformations in high-rise buiIdings due towind Ioads. Haris considered only one degree-of-freedom per floor corresponding to Iaterai defor-mation.
The object of the present paper is the extensionof the anaIysis by Haris [8] to the most generaI case,when three degrees-of-freedom per floor are con-sidered. A generaI formulation of the probIem ispresented and the internaI Ioading transmitted tothe singie bracing is expressed through pre-multi-plication of the externai Ioading vector by a distri-bution matrix. This was defined by Capurso for awind-bracing system constituted by cantilevers ofthin-walled open section in [9]. A numericai ex-ample is eventually shown, in order to illustrate theuse of a FORTRAN computer program, whichtransiates the aIgebraicai formulation into a se-quence of elementary operations.
2. GENERAL FORMULATION OF TUE PROBLEM
A generaI formulation of the probIem of the ex-ternai IateraIIoading distribution between the brac-ings of a three-dimensionai civii structure will beshown. The structure is idealized as consisting ofNTOT bracings interconnected through floors un-deformabie in their pianes and the axiai deforma-tions of bracings are not considered. With these hy-potheses, the floor movement can be expressed bythree dispiacements: transiations in X- and Y-di-rection of the gIobai coordinate system origin (Fig.1) and floor rotation.
If N is the number of stories, the externai loadwill be represented by a 3N-vector F, whose eIe-ments are three elementary Ioads for each floor and,more exactly, two shears and the torsion moment.In the same way, the internaI Ioading transmittedto the ith element will be represented by a 3N-vec-tor Si and obtained from the preceding F through apre-multiplication by a distribution matrix.
Let Pi be the 2N-vector representing the shear-Ioadings, on the ith element in the gIobai coordinatesystem XY (Figs. 1 and 2), and mi the N-vector rep-resenting the torsion moments, so that
Si = {~J. (1)
The internaI Ioadings Si transmitted to the ith brac-ing and reiated to the globai coordinate system XYare connected with the same loadings S; reiated tothe Iocai coordinate system Vi Vi (Fig. 1):
*Pi NiPi,
*mi (2)
where the superscript * is used to indicate the Ioad-ings in the Iocai coordinate system Vi Vi, N, is theorthogonai matrix of transformation from the sys-
563
564
x
A. CARPINTERI and A. CARPINTERI
Uj
Pj
p*I
m,~., V'v'
Fig. I. Global and local coordinate systems. The Z-axiscompletes the righthanded global system X YZ and W; com-
pletes the right-handed local system U; V; W;.
tem XY to the system VYi, tjJi is the coordinate-vector of the origin of the Iocal system Vi V; in theglobal one XY, UZ is the unit vector in the Z-direc-tion (note that tjJi 1\p; . u, is a scalar triple product).The orthogonal matrix Ni is
[
cos w sin w]N· =
, - sin w cos w '
where each element represents a diagonal N x N-matrix and w is the angle between the X-axis andthe V,-axis (Fig. 1).
Equation (2) may be represented in the matrixform
s7" = A;Si,
where
Ai [N-UZ~ tjJi
~J.I is the identity matrix and O the null matrix.
mi,l
Pio N+ l
x
y
~Pi,N
Pi,2N
z
Fig. 2. Internalloadings Si (transmitted to the ith bracing)in the global coordinate system. Note that the highest floor
is indicated with I and the lowest with N.',//
The displacements Ci; in the global coordinate sys-tem XY are then connected with the dispiacementsCii in the Iocai system Vi Vi:
B7" = N;Bi,
*'P; = 'P;, (6)
where B;represents the translations and 'P; the ro-tations. Equation (6) may be represented in the ma-trix form
*Ci; B;Ci;, (7)
where
_ [N; O]Bi - .
O 1(8)
The internaI Ioadings si are connected with thedispiacements Ci7"through the relation
s; = K7"Ci7", (9)
(3)where Ki is the stiffness matrix in the Iocal system.Recalling eqns (4) and (7), we get
A;S; = K7"B;Ci;. (IO)
Pre-multiplying both the members by the inverseA:-1•
l •
S; = (Ai-I K7"B;)Cii' (11)
(4) It follows that the stiffness matrix in the global sys-tem for the ith bracing is
K; = A;-IKtB;, (12)
(5)where
K1 ~ [~' ;J (13)
The displacement 3N-vector Ci; ofthe ith elernentis connected with the displacement 3N-vector Ci ofthe rigid floors by the relation
Ci; = T;Ci, (14)
where the transformation 3N x 3N-matrix T; is:
_ [I C;]T; -O 1
(15)
The 2N x N-matrix C; is:
C; [. (16)
Lateral loading distribution between elements of a three-dimensional civil structure 565
Fig. 3. Degrees-of-freedom of the ith bracing in the local coordinate system.
where each elernent is a diagonal N x N-rnatrix and thenand (Xi, yJ are the cornponents of vector Ij!i. Equa-tion (11) can be re-written
or
where K, KiTi is the stiffness of the ith elernentwith respect to the floor displacernents. For theglobal equilibriurn we have
or
N N
2: Si = F = 2: Ka,i=l i=l
where K 2:~ I R is the global stiffness rnatrixof the rigid floors. Recalling eqns (18) and (20), weget
F = Ko.,
H
l
I
CD 0 CD @
~b; '%11' -:::%' /! ~ -:::!z
H
H
H
H
2
''l''
(22)
(17) Equation (22) solves the problern of the externalloading distribution between the resistant elernentsof a building. It is forrnally analogous to the equa-tion for the distribution of a force between differentresistant elernents in a piane problern. In fact, thedistribution rnatrix RK- I is the product of the par-tial stiffness rnatrix by the inverse of the total stiff-ness rnatrix, as well as in the piane problern thedistribution factor is the product of the partial stiff-ness by the inverse of the total stiffness. The surnof the distribution rnatrices is equal to the unit rna-trix.
(18)
(19)
(20)3. COMPUTER PROGRAM "DISTRIB"
(21)
A FORTRAN prograrn for the lateralloading dis-tribution between the NTOT bracings of a three-dirnensional N-stories building is presented, whereNTOT ~ 20 and N ~ 30. The bracings are shear-walls (open or closed thin-walled cantilevers) and
58(1) V
5C(1)
2
E * GEI8.12 * I
L;M E * GEIC
5C(1)= L H ;i=l
Q
58(1)= Li=l
N
M
w
Fig. 4. Equivalent shear-wall simulating a frame [11]: E = elastic modulus (or Young modulus) of thematerial; GEIBi = flexural inertia of the ith beam; GEIC; = flexural inertia of the ith column.
566 A. CARPINTERIand A. CARPINTERI
frames. The open thin-walled cantilevers must be 'consisting of converging pIane walls, so that thewarping function vanishes. For example, L, T orcruciform cross-sections are contemplated by theprogram.
The local coordinate system of the ith bracing isassumed with the origin in the center oftwist Ci andthe Vi V; axes are parallel to the centraI ones (Fig.3). Three degrees-of-freedorn per floor are consid-ered: translations in V,- and V,-direction ofthe localsystern origin and bracing rotation {}about the Wi-
axis.Since the external loadings on the building are
supposed to be horizontal and acting at the floorlevels, a condensation can be carri ed out [lO] andeqn (13) can be transformed as follows:
[K' O 0]U.~CON
K~CON = K* O . (23)V.I.CON
O O K~.l
Each frame is simulated by an equivalent shear-wall (Fig. 4) [11], which only presents fiexural andshear stiffness in the frame pIane V-W, i.e.Kt,i.CON and K~,i are null matrices. Therefore, oneshear-wall at least must be present in the consideredbuilding,
Then, the program transforms the condensedstiffness matrices from the local coordinate systemto the global one and the obtained matrices aresumrned. In this way, the condensed stiffness ma-trix K for the NTOT bracings in the global systemis obtained (this matrix is called MATK in the pro-gram) and system (20) can be solved.
Recalling eqns (7) and (14), it is possible to obtain
0.; = BiTio.,
that is the displacements of the ith bracing. Theproduct BiTi is called MATROT, while o.and 0.; arerespectively called ALFA and ALFAAST in theprogram. Finally, the internalloadings S; are com-puted for each bracing:
S; and K~CON are respectively called SAST andKASTCON in the program.
The Flow Chart of the program "DISTRIB" isshown in Fig. 5, while the "INPUT Data Organi-zation" and the "FORTRAN List" are reported inAppendices I and 2.
4. NUMERICAL EXAMPLE
A three-story building is analyzed; its piane isshown in Fig. 6. The building consists of four rein-forced concrete bracings and the fioor height is 300cm.
Bracing l is a closed thin-walled cantilever witha 15 cm constant thickness, Bracing 2 is an openthin-walled cantilever with a 20 cm constant thick-nesso Bracing 3 is a two-bay frame: the columncross-section is 40 x 40 cm for the upper story and50 x 50 cm for the two lower stories; the beamcross-section is 60 x 24 cm. Bracing 4 is differentfrom bracing 3 only because the beam span is 520cm instead of 450 cm. Ali the sizes of the buildingand the global and local coordinate systems areshown in Fig. 6.
The externalloading consists of four forces F =15,000 kg, one for each floor (see Fig. 6).
The "INPUT Data Format" is shown in the tableof Fig. 7 and the "RESULT PRINTOUT" is re-ported in Appendix 3. The displacements are mea-sured in cm, the rotations in radians, the forces inkg, the moments in kg cm.
(24)
REFERENCES
l. J. H. Wynhoven and P. F. Adams, Behavior of struc-tures under loads causing torsion. J. Struct . Div.(A.S.C.E.) 98, 1361-1376 (1972).
2. A. Coull and A. W. Irwin, Model investigations ofshear-wall structures. 1. Struct. Div. (A.S.C.E.) 98,1223-1237 (1972).
3. A. C. Heidebrecht and B. Stafford Smith, Approxi-mate analysis of open-section shear-walls subject totorsionalloading. J. Struct. Div. (A.S.C.E.) 99, 2355-2373 (1973).
4. A. C. Heidebrecht and B. Stafford Smith, Approxi-mate analysis of tali wall-frame structures. J. Struct.Div. (A.S.C.E.) 99, 199-221 (1973).
5. M. C. Stamato and E. Mancini, Three-dirnensionalinteraction of walls and frames. J. Struct . Div.(A.S.C.E.) 99,2375-2390 (1973).
6. J. Gluck and M. Krauss, Stress analysis of group ofinterconnected thin-walled cantilevers. J. Struct. Div.(A.S.C.E.) 99, 2143-2165 (1973).
7. I. A. MacLeod and H. M. Hosny, Frame analysis ofshear-wall cores. J. Struct. Div. (A.S.C.E.) 103, 2037-2047 (1977).
8. A. K. Haris, Approximate stiffness analysis of high-rise buildings. J. Struct. Div, (A.S.C.E.) 104, 681-696(1978).
9. M. Capurso, Sul calcolo dei sistemi spaziali di con-troventamento. Giornale del Genio Civile. Fase. 1-2-3,27-42 (1981).
lO. J. L. Humar and J. U. Khandoker, A computer pro-gram for three-dimensional analysis of buildings.Computo Structures 11, 369-387 (1980).
Il. P. Pozzati, Teoria e Tecnica delle Strutture. Val. 2,parte II, UTET, 361-377 (1977).
(25)
Lateral loading distribution between elemenls of a three-dimensional civil structure
READ: Master control card
(Appendi x 1- Section I)
N, NUMS, NUMF, H
READ: Shear-walls
(Appendix 1- Section Il)
l) E, PNI, GEIU, GEIV, GEIT2) X, Y, OMEGA
READ: Frames
(Appendix l - 5ection 111)l) 5C(I), I = l, N2) 5B(I). I = l, N3) X, Y, OMEGA
- Automatic element generation of KUCON, KVCON, KTETA(they are the submatrices of the condensed stiffness matrixKA5TCON in the local system) for the generic bracing
- Recording of KUCON, KVCON, KTETA on KTAPE
- Transformation of the condensed stiffness matrices KASTCONfrom the local coordinate system to the global one
- Sum of these matrices (MATK is obtained)
READ: F (external forces and moments)
(Appendix l -5ection IV)
FXI,···, FXN; FYI,···, FYN; M,' ... ' MN
ALFA (MATK)-l * F
FOR EACH BRACING:
ALFAAST = MATROT * ALFA
5A5T = KASTCON * ALFAA5T
Fig. 5. Flowchart of the computer program DISTRIB.
450 450
X
ooo '" Y<n...
o ® uo'"
o V<n o... o'"
I-750 300 750
'14 • I~ 'I
I- 900 900-111
Fig. 6. PIane of an example building.
567
1213
1415
1~:17!~
~C
1l-
---
----
-H
.15
'6:
11:I
.;U
le
3O
Oii
:'-'
-:"-
:---
"';-i-
'+-
t---;.
.-+-
Fig.
7.IN
PUT
Dat
aFo
rmat
for
the
cons
ider
edex
arnp
le(s
eeFi
g.6
and
App
endi
xI)
.
V\ o- 00 )- o l> ~ z "Ì tn ~ Il'
Q Q.
)- (j ~ z "Ì tl1 ~
APPENDIX I
Lateral loading distribution between elements of a three-dimensional civil structure
21-30 OMEGA Angle w in degrees (w isthe angle by which the X-axis has to rotate toachieve the U-axis)
INPUT Data OrganizationI. Master control card [312, F14.4].
Columns1-23-4
VariableN
NUMS
EntryNumber of storeysNumber of shear-walls(open or closed thin-walledcantilevers)Number of framesFloor height
5-67-20
NUMFH
II. Shear-walL input data. Two cards must besupplied for each shear-wall.(1) Shear-wall properties [FI5.4, F5.3, 3 FI5.4]
Columns Variable Entry1-15 E Elastic modulus of the
material16-20 PNI Poisson ratio of the
material21-35 GEIU Flexural inertia of the
cross-section about the U-axis
36-50 GEIV Flexural inertia of thecross-section about the V-axis
51-65 GEIT Torsional inertia of thecross-section (SaintVenant's Theory)
(2) Local coordinate system for the shear-wall [3FIO.5]
Columnsl-IO
VariableX
EntryX-coordinate (in the globalsystem) of the local systemoriginY-coordinate (in the globalsystem) of the local systemorigin
11-20 Y
Fortran Lisi
569
111. Frame input data. Three card sets must besupplied for each frame.(I) SC(I), where I = I, N[8 FIO.3]
SC(I) ~ E*GEICi ,
i~1 H
where E, GElC; and H are related to the M col-umns under the Ith floor (see Fig. 4).
(2) SB(I), where I = l, N [8FIO.3]
SB(I) f 12* E*GEIBi
i~1 Li'
where E, GEIBi and Li are related to the QM - l beams of the Ith floor (see Fig. 4).
(3) Local coordinate system for the frame [3 FIO.5]
Columnsl-IO
VariableX
EntryX-coordinate (in the globalsystem) of the local systemoriginY-coordinate (in the globalsystem) of the local systemoriginAngle w in degrees
11-20 Y
21-30 OMEGA
IV. Vector F of the externalforces and moments[8 FIO.3]. The loads are applied to the global systemorigin; they are two shears (X-direction and Y-di-rection forces) and one torsion moment (Z-axis mo-ment) for each fioor. The input consists of 3N ele-ments ordered as follows: N forces in X-direction,N forces in Y-direction, N Z-axis torsion moments.
APPENDIX 2
c 000**0*0*0.* ••••• *.*******.******* ••• 0**0**** ••0**'* •••• *•••••••••c ******************************************** ••••*.* ••••••••••••• *.c •• **c ** HORIZONTAL LOAD DISTRIBUTI1N BETWEEN S~EAR~ALL~ I~O F~A~ES **C ** IN A N-STOREYS THREE-DI~E~SIONAL RUILOING **C ** NU~S • NUMBfR OF S~EARWALLS (T4EY ARE ~PEN ~Q CLOSEO *.C ** THIN-WALLED CANTILEVERS. THE OPFN T4t~-WALLEO **C ** CANTILEVE~S ~U~T BE CONSISTING nE C1~VEQGING *.C.* PLANE WALLS, SO THAT T4E \lARPI''';oIJ'ICTI~'" *.C ** VANIS4ES) **C.* NUHF • NUM8ER OF FRA~ES .*C ** NTOT • NUMS + NUMF • TOTAL NUMBER OE ~~AéT~r,~ *.C.. N • NUHRER OF STOREYS *.C *. **c •• *••• ***.**** ••• *_ ••• *._*.00 •• *'-._-***_.****.*.**0*** •••••••••••C ****0 ••***._***.*****************.****************.**********.****CCCC
p ROGP AM D IS TR IB (IN P UT, o UTP UT, T AP E 5 = IN PUT ,T AP E 6 =fJtlT D 1fT, T ~D o l, T Ap E ~,STAPE3,TAPE4,TAPE7,TAPE8,TAPE9,TAPEIO,TAPEll,TAPé12,TAP~1~,TAPE14,TSAPE15,TAPE16,TAPE17,TAPE18,TAPE19,TAPE20,TAPE21,TA'E2~)
C NOTA BENE THf EOLLOWING DIMENSIONS APE VALT~ ~~~C NTOT LESS OR EQUAl 20C N LESS OR EQUAL 30
570 A. CARPINTERI and A. CARPINTERI
REAL A(9C,91),KAB(30,30),KAA(30,30),V(30),KUCONI10,3~),KVC~~(30,301),KTETA(30,30),MATK(9C,91),KASTCON(90,90),MATROT(Q"QOl,ALOAAST(902),SAST(90),SC(30),SBI30),VX(20),VY(ZO),OMEGA(ZOl
DIMENSION ITAPE(ZO)ITAPE Il) =1ITAPE(Z)-ZITAPE (3) =3ITAPE (4) =4!TAH(5)·7ITAPE (6)-8ITAPE (7l =9ITAPE(S)=10!TAPE(9)-llITAPE:(lO)-lZITAPE (11) =13ITAPE<1Z)=14ITAPE (13) =15IHPI:<l4)-16ITAPI:(15)=17ITAPEIl6)=18ITAP[(17l-19ITAPE (18) -ZOlTAPE (19)-21ITAPE(ZO)·2Z
INPUT DATA PRINTOUTWRITE(6,8)
B FORMAT(lHl,II,45X,17(lH*),1,45X,lH*,15X,1H*,I,4~X,174* TN'UT DATSA *,1,451,lH*,15X,lH*,1,45X,17(lH*))
READ(5,lD) N,NUMS,NUMF,4lO FORMAT(3IZ,F14.4)
INPUT DATA PRINTOUT~RITE(6,lZ) N,NUMS,NUMF,H
lZ FORMAT(7(/),10X,*N = NUMBFR OF STOREYS = *,IZ,I,lnx,*~U~~ - ~UMAER$ OF SHEARWALLS - *,I2,I,lOX,*~UMF • NUMBER OF FRAME" - *,I~,I,10X,$*H - FLOOR HEIGHT - *,EI0.4,111111)
IF(N.GT.30) GD TO 4460IF(NUMS.EQ.O) GO TO 4465NTOT-NUMS+NUMFIF(NTDT.GT.ZO) GD TO 4480NN'N*2NNN=N*3NL=N-l~l·N+l
THE ELEMENTS OF T4E MATRII MATK ARE INITIALIZEnMATK WILL RE UTILIZED AFTE~WARDS (SEE SUBROUTINE ~OT'T)
00 13 I=l,NNNDO 13 J-1,NNN
13 ~ATK(l,J)·O.C-------------------------------------------------------------------------------C-------------------------------------------------------------------------------CCCCCCC
C
CC
***************** ** SHEARWALLS ** *****************
DO i e s K=l,NUMSKTAPE=ITAPE (K)
READ SOME OUANTITIES REL.TED TO THE GENERIC S~E_RW'LLE - ELASTIC MOnULUS (OR YOUNG MODULUS) OF TYC .AT~RI'LPNI • POISSON RATIO OF THE MATERIALGEIU • FLEXURAL INERTIA ABOUT THE U-AXIS FOR T4E CR05S-SECTIONGE IV - FLEXlJRAL I~ERTIA ABOIIT T~E V-AXIS e'lR THE CR'lS,-SECTIONGEJT • TOPSIONAL INERTIA FOR THE CROSS-SECTI1N (SAINT VENANT THEDRY)
READ(5,14) E,PNI,GEIU,GEIV,GEIT14 FORMAT(F15.4,F5.3,3F15.4)
INPUT DATA PRINTOUT~RITE(6,17) K,E,PNI,GEIU,GEIV,GEIT
17 FORMAT(III,2X,3HK -,12,* (S4EA~WALL)*,1,2X,17(14-).4X,*E = FLASTIC1 M1DULUS (OR YOUNG MODULUS) OF THE MATERIAL*,9X,* •• ,F13.5,f,Z3X,*?2NI = POISSON PATIO OF THE MATFRIAL*,Z8X,*- *,Fl~.4,1,~3X,*GEIU _ F3LEXURAL INERTIA ABOUT TYE U-AXIS FDR THE CROSS-~ECTInN .*,E13.5,1,423X,*GEIV - FLEXlJRAL INERTIA AROUT THE V-AXIS F'lP THF CROS5-SECTtO5N =*,E13.5,1,23X,*GEIT = TORSIONAL INERTIA FOR THO CROSS-,ECTION*.617X,*-*,E13.5)
GU-E*GEIU/(H**3)GV=E*GEIVI (H**3)GT=E*GEIT IHGPNI-(l.) I( (2. )+(Z.)*PNI)C-------------------------------------------------------------------------------
CCCCCCCCCCCCC
CCCCCC
C
AIJTOMATIC ELEMENT GENFRATION OF KIJCON FOR HH, GENFRIC SHEARWALLKUCON - CONDENSED STIFFNESS MATRIX IN THE LOCAL conRDINATE SYSTEM
FOR THE GENERIC $4EARWALL WHEN THIS IS LOADE1 BY HORIZON-TAL FORCES IN U-DIRECTrON AT THE FLO~R LFV~L~
IIIIII
KUI KAA I KA8 I
([*GEIV/H)*I-----I-----I • STIFFNESS MATRIX I~ THE L1CAL COORDINATEI K8A I KBB I SYSTEM FOR THE ",ENEPTC ~4EARWALL WHE~ THIS
Lateralloading distribution between elements of a three-dimensional civil structure
cc
IS L~AOEO BY HO~tl,NT4L F~~CES IN U-DIREC-TTON AT THE FLO~R LFVF.L~
DO 20 I-l,NDO 20 J-l,NN
20 ACI,J)-O.cCC
SURMATRIX A IS OBTAINEO 8Y JOINING KBB ANO Kq4 4 - K B~
DD 30 l-l,N00 30 J-l,N'0 KABC1,J)-0.DO 35 I-l,NDO 35 J-l,N
35 KAA(J,J)-O.C SU8MATRI' A
AO,U-4.IFCN.EO.l) Go To 5bNL-N-lDD 4[' I-l,NLJ - Itl4CI,J)-2.
40 ACJ,Il-2.DD 50 1-2,N
50 ACI, n-e.5b ACl,Nll--6.
IFCN.EO.l) GD TO 75DD 60 I-l,NLJ=Nl+I
oD ACI,J)-6.00 70 1-2,N11-1-1J -N+ l l
70 AI I, J) --".C SU~MATRIX KAB75 KABO.l) --6.
IFCN.EO.l) GO To eeOD eo I-l,NLJ-I+lKABC I,J) o-b.
eu KABIJ,n-o.SUBMATRIX KhA
B8 KhACl,1l-12.IFCN.EQ.l) GD To 9900 90 1-2,N
90 KAAC l, Il • Z4.DO ge I-l,NLJ - 1+1KAACI,J)=-12.
9B KAACJ,Il--IZ.cC KUCON - CE*GEIVfCH**3)) • CKAA-KAB.«IKBB)"C-ll).K~AllCC ----------C l STE? l I C CKBB)**C-l) )*KBAC ---------- GAUSSIAN REOUCTION IS A?PLIEOCALL TRIANGIA,N,NN)
99 CONTINUECALL SOSTCA,N,NN)
C ----------C l STE? 2 I KAB* « (KBB )"1-1) )*KBA)C ---------- THE RESULT 15 ~EMnRIZEO IN THE FIRST N C"'Lr'M~~ nr KAB
CALL MOLTIPCA,KA~,V,N)C ----------CISTE? 3 I CKAA-KAB*« CKB81**I-lll*KBAllC ---------- THE RESULT IS MEMoRIZF.O IN KAA
CALL SOTTRCKAA,KAB,N)OD 10t; J-l,NDO 100 l-l,N
100 KUCoNCI,J)-GV*KAACI,J)CCCCCCCCCCCCCCC
1AUTOMATIC ELEMENT GENEPATION OF KVCON FoR THé GENFRIr. S4EARWALL IKVCON • CoNDENSEO STIFFNESS MATRII IN THE LOC\l C~ORqTN'TE SYSTEM I
FOR THE GENERIC S~EARWALL WHEN TH1S T, LO~nEq BY HO~IZO~- rTAL FORCES IN V-OIRECTION AT THE FLO~R LEVEL, 1
I
I KAA I KAB IKV - IE*GEIUfH)*1-----I-----1
I KBA I KBB 1- ST1FFNESS MATR1X I~ T4é LOCAL COORDINATE
SYSTEM FOR THE GE~ERIC SHEARWALL WHEN THISIS LOAOEO ~y Ho~T7.1NTAl .O~CES IN V-~IREC-TION AT THE FLOnR LEVELS
00 105 I-l,NOD 105 J-l,NN
105 ACI,J)-O.CCC
SUBMATR1X A 1S DeTlINED BY JoINING KBB ANO KqA 4 - KBB ~BA00 110 I-l,NDO 110 J-l,N
110 KABCI,J) -o.DO 115 I-l,N
GAS 21:3-M
571
572 A. CARPINTERI and A. CARPINTERI
DO 115 J-1,N115 KAA( I,J)-O.
c SUBMATRIX AA(1,1I-4.IF(N.fQ.1) Go To 128DO 120 1-1,NLJ -I +lAII,J)=2.
120 A(J,Il-Z.DO 125 1-2,N
125 AII,Il-a.12a AO,Nll--6.
IF(N.EQ.1) Go To 138DO 130 1-1,NLJ="I1+1
13U A(I,J)-6.DO 135 1-2,NII-I-1J -N + II
U5 A(I,J)--~.SUB~ATRIX KAB
138 KABIl,1I--6.IF(N.EQ.1) Go To 147DO 140 1-1,NLJ -1+1KAB( I,J)--6.
140 KAB(J,I)·6.SUBMATRIX KAA
147 KAAIl,1l=12.IF(N.EO.1) GO To 157DO 150 1-2,N
15C KAA(I,I)=24.DO 155 I=l,NLJ - 1+1KAA( I,J)--12.
155 KAA(J,I)--12.cC KVCON - (E*GEIU/(H**3)) * (KAA-KAB*(((KBB)**(-l»*KqA»)C
CALL TRIANG(A,N,NN)157 CONTINUE
CALL SOST(A,N,NN)CALL MQLTIP(A,KAB,V,N)CALL SoTTR(KAA,KA8,N)DO 158 J-l,NDO 158 I-l,N
158 KVCoN(I,J)-GU*KAA(I,J)CC ---------------------------------------------------------------------CC AUTDMATIC ELEMENT GENEPATI~N OF KTETA FJR TY~ GE~cQ,C ~~EAR~ALLC KTETA - TORSIoNAL STIFFNESS MATRIX FOR THE GENERJC ~~EA~WALLCC ---------------------------------------------------------------------DO 160 J=l,N
DO 160 I-l,N160 KTETA( I,J )-0.
KTETA(1,1)=GT*GPNIIF(N.EQ.1) GQ TO 166DO 162 1-2,N
162 KTETA(I,I)-GT*( (2. )*GPNIl00 164 I-l,NLJ=I+lKTETA(I,J)-GT*(-GPNI)
164 KTETA(J,I)-KTETA(I,J)1M CONTINUEC-------------------------------------------------------------------------------
CCCCCCCCCC
KUCoN,KVCON ANO KTETA OF THE GENERIC SHEARWALL IRF ~E~ORIZEO DI'lKTAPE. THESE ~ATRICES WILL BE UTILIZEO WHEN ~~TA (VECTOR GFDISPLACEMENTS ANO RoTATIoNS IN THE LOCAL coORnINATE 5VSTEM) ANOS (VECToP OF FORCES ANO MOMENTS FOR THE GENE~TC 8~A:.rNG) ARECOMPUTEO FCR E~CH BRACING
WRITE (KTAPE) «KUCoN(L,M),L-1,N),M-l,N),«KVCoN(L,~1 ,L-l,N),M-l,N$), (KTETA(L,M), L-l,N),M-l,N)
C-------------------------------------------------------------------------------CCCCCCCCCCCCCC
T~f PROGRAM TRANSFoRMS THE CONOENSED STIFFNES, MATRTC~S FROM THELoCAL COORDINATE SYSTEM To THE GLOBAL ONE. THE~ T4F O~TAI~FOMATRICES ARE SUMMED AND ~ATK IS OBTAINEO AT T4E ~~~ QF DD 185(MATK - CONOENSED STIFFNESS MATRIX IN THE GL1~AL COQR~rNATE SYS-TEH FoR T4F NUMS SHEARWALLS)
IIIIIIr
PEAD SOME QUANTITIES LoCATING T~E LoCAL cooR~rMAT~ çV~TEM OF THEGENERIC SHEARWALL
X,Y - COORDINATES (IN THE GL09AL SYSTEM) ~~ THF LOCAL SYSTEM ORIGIN
Lateralloading distribution between elements of a three-dimensional civil structure
OMEGA • ANGLE THAT THE X-AXIS OF THE GLOB~L 5v~TEM H4S TO ~OTATEIN ORD~R TO ACHIf.VE T~E U-AxIS OF THE LOCAL ~YSTEM
REAOI5,180) VXIK),VYIK),~MEGAIK)180 FQR~ATI3FI0.5)
x=vxlnY=VY(K)SINOMEG'SINO(OMEGA(K) )COSOMEG'COSOIQMEGAIK)1
lNPUT DATA PPINTOUTWPITE16,1821 ~,Y,OMEGAIK)
182 FORMATII,23X,*COORDINATES OF THE LOCAL SrSTEM O~Ir,IN.,lOx,tx -t,Fl10.4,lOX,*Y .*,FIO.4,1,23X,*0~EGA (ANGLE T~AT X-~YI~ ~A~ T~ ~OTATE2CLOCKWISE IN QRDER TO ACHIEVE U AXISI • *,F10.~)
CALL ROTATISINOMEG,COSOMEG,X,y,N,NN,MATK,KUCON,~V~ON,~TfT'1185 CONTINUE
IFINUMF.EO.OI GO TO 910C-------------------------------------------------------------------------------C-------------------------------------------------------------------------------CCCCCCC
cc
*************** ** FPAMES ** *•.•••••• * ••••• *
NUMSl·NUMS+l00 900 ~=NUMSl,NTOTKTAPE'ITAPE IKI
READ THE SCIII ANO ~B(II FOR THE GENERIC FRA~E, WYE~E I-l,NSCIII • SUMMATION OF E*GEIC/H
IE,GEIC,H ARE RElATED TO THE M COLIJ"I'lSIJ~OE~ T~E I-TH FLOORISBII) • SUMMATION OF 12*E*GEI~/L
IE,GEIB,L ARE ~ELATED TO THE o BEA"I~ nF T~E I-TH FLOORIREAOI',19D) ISCII),I-l,NI
lQO FORMATI8F10.3)REA015,1901 ISBIII,I-l,NI
IHPUT DATA PRINTOUTWRIHI6Jl'l7) K
lQ7 FORMATI6(/),2X,3HK -,12,* IFRAMEJ*,1,2X,1311H-J,8X,*~CIII, WHERE I1-1,N*,8X,*SCIII - SUMMATION OF E*,1H*,*GEIC/H*,I,57X,*(E,GEIC,H AR2E AELATEO TO THE M COLUMNS UNOER THE I-TH FLOOR1*.I,'3 •• *§~III, WH3ERE I-1,N*,8X,*S8III • SUMM~TION OF lZ*,3H*E*,*GEIQ/L*,1,57X,*IE,G4EIB,L ARE RfLATEO TO THE o gEAMS OF THE I-TH FL1QRI*1
DO 200 I-l,N.RITEI6,l'1cn I,SCIIl,r,S~(I1
1'19 FORMATII,35X,3HSCI,I2,3HI -,E12.5,20X,3HSBI,I2,1YI -,El?5J200 C ONTl NUEC-------------------------------------------------------------------------------
CCCCCCCCCCCCCCCC
cecCc
e
III1III
AUTOMATIC ELEMENT GENEPATION OF KVCON FOR THE GE~EAIC FAA~EKveON - CONOENSEO STIFFNESS MATAIX IN THE lO~4l C11~~I~~T~ SYSTEM
FOR THE GENERIC F!>AME. StJCH FRAME 1<; qMIJLHEO BY AN E-QUIVALENT SHEARWALL LOAOEO BY HORIZONTAL ~nAC~S I~ V-OI-RECTIDN AT THE FLOOR LEVELS
IIIIIJl
K V -1 KAA I KAB I1-----1-----1I KBA I KBB I
- STIFFNESS MATRIX IN THE LOc~L cn~AOI~~TE SVSTEM ~ORTHE GENEAle FRAME (SIMULAHn H AN EO'JIVAlF~T SHOR-WALL lOADED BY HORIZONTAL FOP~F.~ IN V-OIRECTION ATTHE FLOOR LEVELS)
00 204 I-l,N00 2C4 J'l,NN
204 AII,JJ·O.ccc
IA - r I(B~
I
r~8A I
TSUBMATRIX A IS OBTAINEO BY JOINING KBA AND KqA
DO 207 I-l,N00 207 J-l,NKABII,J)-O.
2(,7 KAAII,JI·O.SUBMATRIX A
AI1,11=4*SCI11+SBIIJIFIN.EO.l) GO TO 22500 210 l'l,NLJ-I+l
210 AIJ,JI-4*SCIIJ+4*SCIJI+SBIJ)DO 220 I-1,NLJ'I+1AfI,JI-2*SCII)
220 AIJ,I)-AII,J)225 AI1,Nll--6*SClll/H
IFIN.EO.l) GO TO 255DO 230 I-lJNL11-1+1J':H+I
230 A III, J 1'1 se II 1- SC III l1*61 H00 240 I-l, >.ILJ -Nl + I
240 AII,J)-6*SCIIJ/H00 250 1-2,N
573
574
250C
255
21>0
270C
275
280
290CCCCC
298
A. CARPINTERI and A. CARPINTERI
II-I-1J -N+ I!A(I,JI--I>*SC (Il l/H
SU8MATRIX KABKA~Cl,1l--6*SC(11 /HIF(N.EQ.ll GO TO 215DO 260 I-l,NL11-1+1KAB(II, II )=(SC (Il-Se (I! ll*b/HDO 210 I-l,NLJ·I+lKA8(I,JI--b*SC(II/HKAB(J,Il'-KAB(I,J)
SUBMATRIX KAAKAA(1,1)-12*SC(11/(H**2)IF(N.EQ.l) GO TO 298DO 280 I-l,IJLII-I+lKAACII,II)=CSCCI)+SCC1I»*12/CH**Z)DO 290 I-l,NLJ-I+lKAACI,J)'-12*SCCI)/CH**Z)KAACJ,I)=KAACI,J)
THE PROGRAM CALCULATES THE CONOENSED STIFFNE~~ ~ATRIX KVCQN CIN THELOCAL COORDINATE SYSTEM) FOR THE GENERIC FRA~E WHE~ THI~ 15 LOADED BYHOR1ZONTAL FORCES IN V-01RECTION AT THE FLOO~ L'VFL'. THE FOLLQWINGOPERATIONS ARE ANALOGOUS Tl THOSE PREV10USLY ~ONF F1Q TYE SHFARWALLS.KVCClN IS MEMORIlED IN THE 5'IRMATRIX KH
CALL TRIANGCA,N,NN)CONTINUECALL SOST(A,N,NN)CALL MOLTIPCA,KAB,V,N)CALl SOTTRCKAA,KAB,N)DO 300 J=l, NDO 300 I-l,N
300 KVCONCI,JI-KAACI,J)CC --------------------------------------------------------- _CC AUTOMAT1C ElEMENT GENERATION OF KUCON AND KTETA F1R THE r,ENERICC FRAMEC KUCON - C8NDENSED STIFFNESS MATRIX IN THE LO~AL ~~~RDTNATE SYSTE~C FOR THE GENERIC FRAME WHEN THIS IS L1AnF~ ~y HQ~IZO~TAlC FORCES IN U-OIRECTION AT THE FLOOR L~VEL5C KTETA TOPSIONAL STIFFNESS MATRII FOR THE GENF'T: FRA~EcC ------------------------------------------------------ _
DO bOO J-l,NDO 500 I=l,NKUCONCI,J)·O.KTETA( I,J )-0.
500 CONTINUEbOO CONTINUEC--------------------------------------------------------------------------- _
CCCCCCCCCC
KUCON,KVCON ANO KTETA nF TME GENERIC FRAME ARE ~F~OQIlEO OH KTAPE.THESE MATRICES WILl BE UTILIZED WHEN BETA CVECT1R OF OI5PlACEMENTSANO ROTATIONS IN THE lOCAL COORDINATE SYSTEHl A~~ S CVFCTOR nFFORCES ANO MOMENT! FOR THE GENERIC BRACING) AQE C1MoUTEO FOR EACHBRACING
IIIIIII
\oIRIT E CK TA PE) C CK VC ON CL, M ), L -l, N ),"I-l,N ), c CK VCON Cl, M) ,L -1, "l ) , M -1, N$),(CKTETACL,M),L-l,N),M-l,N)C-------------------------------------------------------------------------------
CCCCCCCCCCCCCCCC
THE PROGRAM TRANSFoRM5 THE CONDENSED ST1FFNESS ~ATQrCES FRoM THElOCAL COORDINATE SYSTEM TO THE GLoBAl OHE. THEN T~E Q~TAINEDMATRICES APE 5UMMED ANO MATK 15 OBTAINED AT THF E~~ OF no 900(MATK - CONOENSED STIFFNESS MATRIX IN THE GLORAL :n1ROI~ATE Sys-TEM FOR THE NTOT BRACINGS)
IItIIII
READ SOME QUANTITIES LOCATING THE LOCAL COOR~INATE 5YSTEM oF THEGENERIC FRAME
X,Y • COORDINATES CIN THE GLOBAl SYSTEM) QF THE L~CAL SYSTEM ORIGINOMEGA - ANGLE THAT TME X-AXIS OF THE GLOBAl ,Y~TE~ HAS TO ROTATE
IN ORDER TO ACHIEVE THE U-AXIS OF THE LOCAl SYSTEHVXCK),VYCK),OMEGACK)READC5,180)
X-VXCK)Y'VYCK)SINOMEG-SIND(OMEGACK) )COSOMfr,-COSDCOMEGACKI)
INPUT DATA PRINTOUTWRITEC6,182) X,Y,OMEGACK)CALl ROTATCSINOMEG,COSOMEG,X,y,N,NN,MATK,KUCON,KVCQN,KTETA)
900 CONTINUE910 CONTINUEC-------------------------------------------------------------------------------
c
Lateralloading distribution between elements of a three-dimensional civii structure
(-------------------------------------------------------------------------------( TH[ SYSTEM MATK * ALFA = F IS SOLVEOC MATK • CONOENSEO STIFFNESS MATRIX IN THE GLOq~l C10~OlN~TE ~YSTEMC FOR THE NTOT BRACINGSC ALFA - VECTQR OF THf FlOJR OISPLACEMENTS IN T~E GLOQAL COORDI~ATEC SYSTEM CTRANSLATIONS OF THE GLOBAL COOROINATE SY~TEM ORIGI~C ANO FlOOR ROTATIONS)CF- VECTOR OF THf KNOWN TERMS. IT IS MEMORIlE~ IN TYE C3N+l)-T~C COLUMN OF THE MATRIX MATK
NNNN-3*N+lREADC5,1111) CMATKCI,NNNN),I-l,NNN)
1111 FORMATC8FI0.3)C INPUT DATA PRINTOUT
~RITEC6,1112) NTOT1112 FORMATC9C/),BX,*VECTOP F OF THE KNOWN TERMS, I.~. EXTERNAL FORCES
$AND M~MENTS TO DISTRIRUTE BETWEEN THE *,12,* BR'CING~*,/)DO ll20 I-l,NWRITEC6,1118) I,MATKCI,NNNN)
1118 FORMATCI,15X,3HFXC,I2,3H) -,E12.5)ll20 CONTINUE
DO 1130 I-l,NJ-I+NWRITEC6,1125) I,MATKCJ,NNNN)
1125 FORMATCI,15X,3HFYC,I2,3H) -,E12.5)1130 CONTINUE
DO 1140 I-l,NJ-I+NNWRITEC6,1135) I,MATKCJ,NNNN)
1135 FOPMATC/,15X,3H MC,I2,3H) -,E12.5)1140 CONTINUE
"RITE Cb, 1145)1145 FORMATC/I,lX,130CIH$))
C RESULT PRINTOUTWR ITE C6.1150)
1150 FORMATCIH1,/1,45X,15CIH·),/,45X,lY·,13x,lH*,/,4~X,15Y*$ *,I,45X,lH*,13X,IH*,/,45X,15CIH*))
WRITEC6,1155) NTQT,CCCI,J,MATKCI,J)),J-I,NNNI,I-I.~NN)1155 FORMATC//1,10X,*CONDENSED STIFFNESS MATRIX CIN THE GLOBAL COORDINA
ITE SYSTEM) FO~ THE NTOT BRACINGS C NTOT •• ,12,+ ).,/I.l~20C/,lX,52C5HMATKC,I2,lH"I2,2H)-,E12.5,2X)))
CALL TRIANGCMATK,NNN,NNNN)CALL SOSTITCM~TK,NNN,NNNN)C-------------------------------------------------------------------------------C ALFAAST IS COMPUTéO FOR EACH BR~CING ALFAA~T • ~ATROT * ALFA
C ALFAAST - AL FA ASTERISK • VECTOR OF THE Dl~PLA~EME~T~ IN THE LOCALC COORDINATE SYSTEM CTRASLATIONS ~F THF L~C~L COO~DINATEC SYSTEM ORIGIN ANO ROTATIONS)C MATROT • ROTATION MATRIXCCCCCC
RESULT~
SAST IS COMPUTED FOR EACH eRACING SAST - ~~ST~~~ + ALFA~STSA$T - VECTOR OF T~E FORCES ANO MOMENTS I~ THE L~CAL COORnINATE
SYSTEf'IKASTCON • CONoENSED STIFFNESS MATRIX IN THE L~:AL ~O'ROI~~TE
SYSTEM FOR THE GENERIC BRACINGDO 4444 K-l,NTOTWRlTUb,2000) K
2000 FORMATCIHI,IC/),2X,*BRACING NUMRER *,I2,1,2X,17C1H-)1C THE ELEMfNTS OF THé MATRIX KASTCON ARE INITIALltED
DO 2215 J-l,NNNDO 2215 I-I,NNN
2215 KASTCONC I,J )-0.X-VXCK)Y-VYCK)SINOMEG-SINOCOMEGACK))COSOMEG-COSOCO~EGACK))WRITECb,2222) X,Y,OMEGACK)
2222 FORMATC2C/),5X,*X -*,FI2.5,lOX,+Y -+,FI2.5,20X,*OMEGA -*,F12.5)CALL MATROCN,SINOMEG,COSOMEG,X,Y,MATROT)CALl MOLTCMATROT,MATK,ALFA~ST,NNN)KTAPl- Il APE CK)REWINO KTAPEREAO CKTAPE) CCKUCONCI,J),I-l,N),J-l,N),CCKVCONCI.J),I-l,NI,J-l,NI
s , ((KTETAC I,J), I-I,N),J'I,NIDO 2224 J-l,NDO 2223 I-l,NKASTCONCI,J)-KUCONCI.J)11'1-1+1'1J N -J +NKASTCONCIN,JN)-KVCONCI,J)INN- I+NNJNN-J+NN
2223 KASTCONCINN,JNN)-KTETACI,J)2224 CONTINUE
CALl MOLTIPLCKASTCON,ALFAAST,S~ST,NNN)Io/R!TEC613330) K
3330 FORMATC3C/),12X,*VfCTOR OF THE FORCES ANO MOMENTS I~ T~E L'ClL ~OO$ROINATE SYSTEM FO~ THE BRACING NUM8ER *,12,/)
DO 3335 I-l,NIo/RITEC6,3333) I,SASTCI)
3333 FORMATCI,20x,3HSUC,I?3H) -,E12.5)3335 CONTINUE
DO 334(; I-l,NJ -1+NIo/RITECo,333B) I,SASTCJ)
575
576 A. CARPINTERI and A. CARPINTERI
3338 FORMAT(/,20X,3HSV(,12,3H) =,E12.5)3340 CONTINUE
DO 3345 l-l,NJ-l+NNWRITE(b,3343) I,SAST(J)
3343 FORMAT(/,20X,3H MI,12,3H) ·,EI7..5)3345 CONTINUE4444 CONTINIIE
GO TO 5000C
44bO446Z
ERRDR MESSAGESWRITE (b, 446Z)FORMAT(IHl,III,9X,*INPUT CATA ERROR+,9X,+N IS G~F4TFQ T44N 30+)GO TO 5000
4465 ~RlTf(o,4467)4467 FORMAT(IHI,III,9X,+INPUT DATA ERROR+,9X,*NUMS - 0+,I.~4.,.T~E FRA~
SES ARE SIMUlATED BY BRACINGS WITHOUT TORSIONAl STlcF~ESS*.1,34X,*A$NO THEREFORE NUMS HAS TO RE DIFFERENT FROM O.)
GO TO 500044RO WRITE(o,448Z)4482 F[1RMATllHl,III,9X,*INPtJT DATA ERROR*,9X,*NTOT IT'lHL N'I~~E~ OF 8~A
SCINGS) IS GREATEP THAN ZO.)5000 CONTINUE
STOPENO
SlIBROUTINE TRIANG(A,N,NM)REAL A(90,91)NZ-N-lDO l K-I,N2KI-K+IDD 2 I-Kl,NC-A(I,K)/AIK,K)DO 3 J-I(I,NM
3 AII,J)'A(I,J)-A(K,J)+C2 CONTINUEl CONTINUE
RETURNENO
SU~ROlITINE SOST(A,"bNM)REAl AI9(·,911NI=N+IN2·N-Iou 10(\ J-Nl,NM
100 A(N,J)-A(N,J)/A(N,NIIF(N.EQ.ll GO TO 600DO 200 K-l,NZI-N-KIl-I+lDO 30C J-Il,NOf] 400 l-Nl,Nr
400 A(I,L1-AII,ll-A(l,J)+A(J,Ll300 CIJ'HINUE
DO SUC l-Nl,NM500 A(I,L1-AII,L1/AII.!)ZOO C ONTINlIE6<00 CONTINUE
RETUQNE'IOSUBRDUTINE SOTTR(KAA,KA9,N)REAl KAAI30,30),KABI30,30)DO l l-l,NDO l J-l,NKAAIl,J)-KAAII,JI-KAB(I,J)~ETURNE'lO
SUBROUTIN E RolA T( SI NOMEG, COSOM EG, X, Y, N, NN, MA TK, K'JC'lN.KV<:'1N.K TETA)REAL MATK(90,Ql),KUCONI30,30),KVCONI30,30),KTETA(3~.~O)SINSIN-SINOMEG+*2COSCOS-COSOMEG·*ZSINCOS-SINOMEG+COSOMEG00 lO J-l,N00 i o I-l,N
lO ~ATKII,J)-MATKII,J)+CDSCOS*KlICONII,J)+SINSIN*KV~~NII,Jl00 30 J-l,NM=J+N00 20 I-l,NMATKII,M)-MATK(I,M)+SINCOS+IKUCON(I,J)+I(VCON(I,J)
ZO MATKIM,I)-MATKII,MI30 CONTINUE
00 110 J-1,NM·J+NN00 100 l-l,NMATK(I,M)=MATKII,M)+ISINCOS*X-COSCOS+V)*KUCONII.J)-IST~cn5+x+SlNSI
1N+V)*KVCoNI I,J)100 MATK(M,I)-MATKII,M)110 CONTINUE
DO 130 J-l,NI1-J +NDO i ac I-l,Nl'I +N
Lateral loading distribution between elements of a three-dimensional civil structure
1Z0 ~ATK(l,MI-MAT~(l,MI+5INSIN*KUC~N(I,JI+COSCOS*KVC"N(I.J)130 CONTINUE
DO 160 J'l,NM'J+NNDO 150 I-l.NL'I+~MATK(l,MI-MAT~(l,M)+(SINSIN*X-SINCoS*Y)*KUCoN(I.J)+(~n,c~,*X+SINCO
lS*Yl*KVCoN( I,J)150 MATK(M.l)-MATK(l,M)1bC CONTINUE
DO 180 J-1,N11-J+NNDO 170 I-l,Nl-I+NNMAH (l,M I-I1ATK(l, M) + (X-s INOM EG-Y*COSOMEG I**Z*KUC<JN (I. J) + Ix*CO S]I1EG
l+Y*SINOMEGI**Z*KVCON(I,J)+KTETAII,J)170 CONTINUE180 CONTI NIJE
~E TURt.·ENO
SUBROIITINE SOSTITlA,N,NMIREAL AIQO,nlNl-N+1NZ-N-1DO 100 J-Nl,NM
100 AIN,JI-AIN,J)/AI~,N)DO ZOO K-1,NZI-N-K11=1+1DO 300 J-Il,NDO 40l! l-N1,NM
400 A(I,ll-A(I,L)-A(I,JI*A(J,LI300 CONTINUE
DO 5CO l-N1,NM500 AIl, LI - A(l, III AIl .r Izoo CONTINUE
WRITE (6,7001700 FORMAT(61/),10X,*VECToR ~F T~E FlDOR OISPLACEME~TS I~ T~E GLOBAl C
SoOROINATE SYSTEM*,I,lOX,*ITRA~SlATIONS OF T~E Gl"BAl COO~OINATE SYSSTEM ORIGIN ANO FlOoR RoTATIONS)*,/)
N3-N/3N3Z-N3*2DO 720 l-l,N3WRITE(b,7l01 I,AII,NM)
710 FoRMAT(/,15X,*TRANSlATION IN X-OIRECTION FOR THF Fl"1~ NU~~E~ *,12$,2~ -,E12.51
7Z0 CONTINUE00 740 l-l,N3J =1 +N3WRITEI6,730) I.A(J,NMl
730 FORMATlld5X,*UANSlATlON IN Y-OI~ECTION FOR THO FlI'l'1R'1""~ER *,12S,ZH -,E12.51
HO CONTINUEDO 760 I-l,N3J-I+N32WRITE16,7501 I,AIJ,NMl
750 FORMAT(/,15X,*ROTATION FOP THE FlOOR NUMgER *,I2,?Y -,Fl!.~l71>0 CONTINUE
RETURNENO
SUBROUTINE MOlTIP(A,KAB,V,NlREAL A(90,Ql"KAB(30,30',VI30,00 3 l-l,NDO l J-l,NJN-J+N
AUXIlIARV VECTORV(J)-O.00 l K-l,NVIJl-VIJI+KAB(I,KI*AIK,JNIDO Z IP.-1,N
2 KAR(I,IR)-VIIP)3 CONTINUE
RETURNENO
SUBROUTINE MATROIN,SIN8MEG,COSOMEG,X,Y,MATROTIREAL MATROT(90,QO)NN~-N'3DO i o J-1,N"'~1DO i o l-l,NNN
10 MATROT(I,Jl-O.NN=N*2DO 20 I-l,NN
20 MATROT(I,Il'COSOMFGNNI-NN+lDD ac l'NN1,NNN
30 MATROTlI,!I-l.DO 4v I-l,NJ =l+NMATROTII,JI-SINOMEG
577
578 A. CARPINTERI and A. CARPINTERI
40 MATROT(J,I)'-SINOMEGP'X*SINOMEG-Y*COSO~EGDO 5(' I:l,NJ.I+N*2
50 :1ATROT(I,J)-Pp:x*CnSO~fG+Y*SINOMEGNl=N+lDD 60 I-N l,NNJ-I+N
60 MATROT(I,J)-PR fTlIP~1END
SUBROUTINE MOL T(MATROT,MATK,ALFAAST,NNN)REAL MATROT(90,90),MATK(90,Ql),ALFAAST(QO)NNNN-NNN+lDO 10 I-1,NNN
lO ALFAAST(I)·O.DO 20 Ial,NNNDO 20 J=l,NNN
20 ALFAASTII)-ALFAA5TII1+MATROTII,J)*MATKIJ,NNNN)\/RlTE(6dO)
30 FORMAT(3(fl,12X,*VECTOR OF THE DISPLACEMENTS IN THE l~CAL COORDINA$TE SYSTEM*,f,12X,*ITRANSLATIONS OF THE LOCAL COORDINATE SYSTEH ~RI$GIN ANO ROTATIONS)*,f)
NA-NNN/3NU·NA*2DO 50 I=l,NAWRITE(6,40) I,ALFAAST(I)
40 FORHAT(/,20X,*TRANSlATION IN U-OIRECTION FOR THE FL~nR ~U~~ER *,12$,2H -,E12.5)
50 CONTINUEDO 10 1-1,NAJ-I+NAWRITE(6,60) I,ALFAAST(J)
60 FORMAT(/,ZOX,*TRANSLATION IN V-OIRECTION FOR THE FLDJR N~~BER *,12S,2H -,E12.5)
10 CONTINUEDO 90 1-1,NAJ-I+NAAWRITE(6,80) r,ALFAAST(J)
80 FORMATI/,20X,*ROTATION FOR THE FLOOR NUMBER .,I~,?~ -,El?~)90 CONTINUE
RETURNENOSUBROUTINE MOLTIPL(KASTCON,ALFAAST,SAST,NNN)REAL KASTCON(90,90),ALFAA~T(90),SAST(90)DO 1(; r-1,NNN
10 SAST(I)-O.DO 20 ral,NNNDO 20 Ja1,NNN
20 SAST(I)-SAST(I)+KASTCON(I,J)*ALFAAST(J)RETURNEND
APPENDIX 3
Result printout
81UCING NUMBER
x. 600.00000 Y. 150.00000 r]/'tEGA • o. 00000
VECTaR aF THE DISPLHEf'1ENTS IN THE lOCAl COORDINATE SY$TEI1(TlUNSlATIONS OF THE lOCAL COORDINATE SYSTEM ORIGIN AND ROTATIONS)
TRANSLATlON IN U-DIPECTION FOR THE FlOOR NU"tBER l' -.798b6E-1')2
TRANSlATION IN U-DlPECT ION FaR THE FLOOR NUI18ER 2 • -.141of.,5E-02:
TRANSLATIDN IN U-DIRECTlON FaR THE FlODR NUHBER 3· -. "394 '5E-03
TRANSlA nON IN V-OIRECTION FaR THE FlOOR NUHBER l • -.4"t44QE-Ol
TRANSlATION IN V-OIPECTION FaR THE FLOQR NUI1BER 2 • -.26Q?BF-Ol
TRAN5l A TI ON IN V-OIRECTION FaR THE FLOOR NUMBER 3 . -.84~'3ZF.-OZ
ROTATlDN FaR T4E FLOOR HU/'IBER l' -.5151~e-04
RQTATION FaR THE FLaop NUI1SER 2 • -.Zbb9aE-04
ROT ATION FaR THE FLOOR NUHBER 3 • -.756Q6E-05
VECTOR QF THE FORCES ANO MOMENT$ IN THE tOCH COORDINATE SYSTE'1 FQQ r•..•E "RACING NUMER
SUI 11 -.48001E+04
SUI 2) .3103lF.+04
sue 3l .54b8CE+04
---------~._--_.
Lateralloading distribution between elements of a three-dimensional civil structure 579
SVI 11 • -.54343E+04
SVI II -.10449E+05
SV I 31 -.1l50lE+05
MI 11 -.38164E+07
MI l l .881Q3E+06
MI 31 .17732E+07
aRAC ING NUHRER 2
x. 750.00000 y. 900.00000 OMEGA. -11'.OOOO/')
'ItCTaR OF THE DISPlACEf'1ENTS IN THE LOCAl COORDINATE SYSTEM(TRANSlATIONS QF THE LOC4.1 COORDINATE S'fSTEI1 ORIGIN ANO ROTATtI)"IS)
TRANSlATION l' U-OHeCTION FOR THE FlOOR NUMBER l • .1sn'~t:;-Ol
TRANSLATION IN U-CIRECTION FOR TME FlOOR NUP1BER 2 • .10130E-Ol
TPANSLA nON IN U-OIRECTION FaR TME FlOOR NUH8ER 3 · .3~'3Q!E-n2
TRAN$lATION IN V-DIPECTlJJN FaR THE FlODR NU"'BER l · .614,~3E-Ol
TRANSlATION IN v-o lRF Cl I OH FaR TME FlOOR NUMBER l • • '33~16E-Ol
TRANSlATION IN V-DIRECTloN FaR THE FLOOR NUMBER 3 · .1,3&4E-01
ROTATlJJN FaR THE FlOOR NUI'tBER l • -.51575E-0,,"
ROTAT!!)N FaR THE FLOOR NUK8ER -.266Q8E-04
ROTATIQN FaR THE FLOOf! NUt'lflER: 3 • -.75bQ6E-oS
veCTOR OF THE FORCES ANO HOI1ENTS [N THE tOCAl COORDINATE SYSH" FI)Q T"4E 81UCING HU"8fR 2
SUI Il · .10325E+04
SUI 21 .28896E+04
SUI 31 .32942e .•.04
SVI Il .575bbE+OIt
SVC 21 .67565E+OIt
SV I 31 · .6QItQOE+04
MI Il -.15077E+05
MI 21 · .348~2E"'O"l
MI 31 · • 70054E"'04
RPAC!NG NUMaER
x • ovocooo Y· 450.000liO OMEGA • 0.00000
VECTOP OF TH' OISPlACEMENTS IN THE lOCAl COOQDINATE SYSTEMCTRANSlATIONS OF THf LOCAL COOPDIN4.TE SYSTP1 ORIGIN •• O A:OTA.TlI1N~ l
TPAHSlATION l' tJ-D!PECTION FaR THE FlnOR NUMEA: l · •7t,"~"':-02
TiaNSL4TtON l' U-OIPECTION FaR THE FlOOR NUl'19E R 2 · • 4o;~30F.-Ol
TRANSlATION IN u-OrHCTlON FaR T~E FLQOR NuHBER 3 · .16'HU-D?:
TRANSlATIOp.; IN V-OIPECTlml FaR THE nQDR NUHBER l ·-.11'503E-nl
TIUNSLo\TIQN IN V-~IPECTrON FO~ T"E FlOOR NUMBER 2 -.10QOQF.-Ol
HlANSl •• nON IN V-OIQHTJON FaR T"E FlODR NWIBER 3 -.1~41 ~E-O?:
ROTATION FO' T"E FLOOR NIIHBE'R l ·-.5151~H-04
ROTATJ,1N ~OP THE nOOR NUMBER • -.26698E-04
ROTHII')N FaR T"f F LOOA NItMRER 3 ·-.156Q6E-05
YECT8R OF THE FORCES ANO /'1QI"'EtHS IN THf l"lCAL COORDINATE SYSlF.'1 ':00 T4J: I\RA.CING ~U/l\"ER 3
SUI Il O.
SUI II •• C.
S'JI 31 O.
SV I 11 -.6'H46E+QZ
SVI 21 -.42433E+02
SV I 31 -.5306°1:+02
~( Il r.MI Il t.
'I 31 O.
580 A. CARPINTERI and A. CARPINTERI
BRACING NUME' ~
x. 15~O. 00000 y. 450.00000 DI1EGA. -10.00000
VECTOR: OF TH' OISPlACE/1ENTS IN THE lOCAl COORDINATE SYSTEMtTRAN$lATlONS OF THE lOCAl COOROINATE SV'SrEM ORIGIN ANO ROT.ATt"~S}
T1UN5lA TION IN U-OIRECT 10N Fa' TH' FlOOR NUI1BER l - ."i494BF-tn
TRANSl&TION IN U-DIRECTION Fa' TH' FlOOR NUMeER 2 - • ?<1Q<H)E-ot
TRANSlATION IN u-n I R EC TI ON Fa' TH, FlOOR "IUMBEA. 3 - .Q~,"'E-I'\?
TRANSU.TION IN V-DIRECTION FO' THE FlnOR NUI1BER l - -.qO'l'llE-Ol
TRANSlUION IN V-OlpECTlON Fa, TH' FlnOR NUHAER 2 -."',!7~eF-Ol
TRANSLATION IN V-OIPECTIOtJ Fa, T"' FlOOR NUMBER 3 - -.P..,Q'3E-Ol
aOTATION FO' THE F LOOR NUf'1BER l . -.51575E-04
RQTATION FOR THE FloOR NUMAER 2 - - •.2bb98E-04
R:OTATION FO' THE F LODA. NUHP.EP 3 - -.75696E-05
VECTOR OF HIE FORCfS ANO HOt1ENTS IN THE lOCH COQROINHE $'I'STF"4 FI]~ T4F ~RlCI~G 'iUM!JER 4
SUI Il O.
SUI 21 O.
SUI 31 O.
SV( Il -.47003E+03
SV( 21 .14075E+02
sv ( 3l .1 7Q32:€ .03
,( Il O.
M( 21 O.
M( 31 O.
CONDENSED STIFFNESS MATR IX (IN THE GlOBAl COOR.DINATE SYSTE!'(I FO" r", "TOT I\IUCINGS ( xr cr - 4 l
IUTKI l, Il- .b20Q2E+07 HATK ( l, 21- -.14189E+08 MATK ( 1, 31- .lob4o~+n~ "1ATIC ( l. ~ l- .~423qE+07 l'IAIK ( " 51- -.5'464E+07MAli< ( l, b l- .41b37E+07 /'\ATI< ( " 71- -.27424E+I0 MA.TKI l, Bl- .62162E+10 MA TIC ( l, Ql- -.4712H+I0 KA TK ( 2, 11- -.1"189E+08l''IATK( 2, 21- o 3QOl<H+08 f1ATI< ( 2, 31- -. 401Q2E +09 MATK ( ~, 4)- -.,554b4F.+f'\'" MA TI( ( 2, ,,- .15254E+08 ,IìIIATK ( 2, bl- -.15947E+08HATK( 2, 71- • b27blE + lO HATK ( 2, Bl- -.112blE+1l HATK( 2, 91- .1BOltbE+11 '"U,TI( ( 3, 11- .10640e+08 I1ATKI 3, 21- -.4019ZE+QeHATI( ( 3. 31- .1095lE +OB NA.TI« 3, ~1- .41b31E+07 MAlK ( 3, 51- -ol5q47~·Oq "lA. 11« 3, b l- .27720E+oe MA1K ( 3, TI- -.4112lE+IOl'IAIK ( 3, e i- .18046E+11 MATK( 3, cn. -.313b5f+11 MATK { 4, 11- • 2423q~+01 "'4.1'< ( 4, ~) . -.55H>4E+Ol MATK ( 4, 31- .41637E+07HATK ( ~, 41- .62494E+07 MATK( 4, 51- -.14Z58E+oa MATK ( ~, bl- .10671E+M '14TI( I 4, 71- .H117E+I0 P1ATK ( ~, Bl- -.93846E+10"AH ( 4, Ql- • 702QQE. + lO P1ATi« 5, U- -.55464[+01 "'ATK ( 5, 21- .15Z~41E +!,U3 'UTi< ( 5• 31- -.1'9HE+08 I1ATK « 5. 4)- -.14258E+08P1ATK( 5, 51- .39204E+08 MA.TK ( 5, 6). -.409P.2F+08 H4Ti< ( 5, TI- -.93846E"+lo') ~4 TI( t 5, 81- .2'j~05E+ll l''IATK( 5, Ql- -.Z6917E+11H"'TK ( b, 11- .41637E+07 MATK ( b, 21- -.1~:jI"41E+06 'UTi< ( b, 31- • Z7720E +OQ, 1'1.6Ti( ( b, <l- .1')677E+08 "ATK( " 51- -.40962E+08"ATt< ( b, e r- .7t33ze+oB "tK( b, 71- • 70Z9QE + lO '1ATI( ( •• 81- -.Z6Q77E+ll "!4TI( ( 6, Ql- .46Q'OE+1l "ATK ( T, 11- -.Z71t1ItE+1OMATi< ( 7, 21- .62762E+10 NATK ( 7, 3) • -.47121E+lO MnK ( 7, 41- .411l7C+l') '1HI«( 7, 51- -.93646E+I0 /'lA TK( T, bl- .70299E+I0MA TI< ( T, 71- .49690E+13 M4.TK( 7, 8 l- -.1l142E+14 MATK ( T, Ql- .62303E+13 '14TI( ( B. 11- .62761E+10 I1ATK ( B, Z l- -.17261E+llHUK( B, 31- .18046[+11 MATK( B, 4). -.93846[+10 MATK( S, 5)- .25B05F.+11 "IATIC ( 8, bl- -.~6q17E+ll I1ATK ( B. TI- -.111"'2E+1'"l'I,,TI<( B, Bl- .30523E+14 MATI< ( 8, 91 .• -.31741E+14 IUTK( Q, 11- -.47121'=+1:> Mo\TK{ Q, ~) . .1~046E+11 ~ATK ( .. 31- -.31365E+11H/t.TK( Q, 41- o 7019QE+10 MATi< ( Q, 5) •. -.26977E+11 IUTK( Q, bl- • 4b9S0f.' H 1 "UTI({ Q, 71- .8.2303E+13 rUTK( Q. Bl- -.3114lE+lltf1ATK( Q. Ql- .551gl,f+14 MATK(
VECTOP. IJF THE FLDO" DISPLACEME~TS IN THE GLOBAL COORDINA TE S YSTE:'"(TRAN5lATION$ OF THf GlOB/Il COORDINA TE $Y$TEK OPIGIN ANO FLDOR -'l:nTo\TIO"lq
TRAN$lAT ION IN "t-DJRECTION FO' TH' FlOOR NUl18ER l - -.15P3F-Ol
TR,Ml$LATION IN X-DIRECTION Fa' TH' FlOOR NU/'tBER 2 - -. 7421 ~F-02
TiU,NSLAT ION IN X-OIPECTION Fa' TH' FLOOR NUIBER 3 - -.1174QF.-r)2
TRANSLATION IN Y-OI"ECTION Fa, TH' FLOOP NuM8ER. -.115,:\F-Ol
TRANSLATIfJN IN Y-DIJHCTInN Fa' TH' FlOflP NIJH3ER 2 - -.10QflQF-Ol
IRANSl Al ION '" Y-iHPECTrrN Fa, THE FLOOR NUH9ER -.3941 ';F-O'
POH TION FO' THE HClOR Nt'HI)H l -.51575E-04
P1T/lTlON FO' THE FLOQR NII,..BER ? - -.2~~~BE-04
P.QTATION FOR TH' FLOiJ" NUI'IRER 3 -.751')9f:JE-Q5
