- Email: [email protected]
TAP-86: A three-dimensional frame analysis program for tall buildings
Compurrs & Strucrures Vol. Printed in Great Britain.
32, No.
5, pp. 9%962,
1989 0
CO45-7949/89 13.00 + 0.00 1989 Maxwell Pcrgamon Macmillan plc
TAP-86: A THREE-DIMENSIONAL FRAME ANALYSIS PROGRAM FOR TALL BUILDINGS CA0 LI
The Ninth Design and Research Institute, Shanghai, China (Received 4 May 1988) Abstract--TAP-86 Program has three unique features: (1) the use of thin-wall element with opening for analyzing the torsion and warping effect of shear walls under lateral loads; (2) the use of three-dimensional analysis method for calculating the deformation and member forces of tall building structures under lateral and vertical loads; (3) the use of three kinds of dynamic analysis of structures, the plane vibration method for symmetric structures, the one direction seismic wave input method and the two direction seismic waves simultaneously input method for unsymmetric structures where the torsion-couple vibration effect should be taken into account. TAP-86 Program can be used to analyze such tall building systems as space frames, shear walls and tubular structures under static and seismic loads, and to calculate the member forces and reinforcements of beams, columns and shear walls. The program is written in FORTRAN 77 and runs on IBM-4341 and IBM-3031 computers.
1. INTRODUCTION
In designing high-rise buildings, if the building structure is asymmetric, if there is eccentric loading or if there exist many non-planar shear walls in the structure, it is necessary to use a three-dimensional structure analysis method in order to get more accurate results. It may be very difficult to use plane structure analysis to solve this sort of problem. TAP-86 Program is a three-dimensional frame analysis program for tall buildings. There are only two sorts of basic member elements in the program, i.e. a space rectangular beam element and a thin-wall column element. But the program is effective and powerful for structural engineers to use easily in analyzing high-rise building structures under lateral and vertical loads. On the other hand, there are three kinds of dynamic analysis of structures in the program, i.e. the plane vibration method for symmetric structures, the one direction seismic wave input method and the two direction seismic waves simultaneously input method for unsymmetric structures where the torsion-couple vibration effect should be taken into account. In this way, structural engineers can choose different dynamic analysis methods according to the regularity of structures. TAP-86 Program also calculates the reinforcements of beams, columns and shear walls according to the structural responses under static and dynamic loads. 2. STATIC ANALYSIS
OF STRUCTURES
(1) Beam element-the horizontal floor member connecting with columns at two ends. (2) Connecting beam element-the horizontal floor member connecting with shear walls at one or two ends. (3) Column element-the vertical member in a structure. (4) Thin-wall column element-the vertical member comprised of several plane shear walls. The stiffness matrix of space rectangular beams can be used as that of beam elements, column elements and connecting beam elements. The difference between beam and column elements is only the direction of coordinate axes. Usually, the number of degrees of freedom of a space beam is twelve, including bending, shear and torsion. The vectors of member force {f} and deformation {d} of a space beam can be expressed as the following eqns, respectively, {~}=[Nx,,NY,,Nz,,~x,,~Y,,~~,,Nx~,NY,, Nz2,
{d)
MY,,
= b,, Ul, WI, ox,,
JWT OY,, oz,,
u2,02, w2,ox,,
OY2, Oz21T.
The subscripts 1 and 2 represent the left and right ends of a beam element. The stiffness matrix of a space beam can be obtained according to the relationship between {J} and {d}, i.e.
2.1. Basic member elements Every kind of structural component in a high-rise building can be defined as following four kinds of member elements:
Mx,,
0-j=
WIW.
The stiffness matrix [Kb] can be found in many reference texts. 953
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A shear wall between two consecutive floors can be represented by a thin-wall column element. For example, the commonly used E-, H- or U-shaped open section arrangement of the core walls can be treated as thin-wall columns in this way. Usually, the thinwall column element would warp under twist. The torsion angle’s derivative Oz’ = dOz/dr caused by section warping is called warping angle. After introducing warping angle Oz’ and warping moment B to the vectors of member forces and deformations, a thin-wall column element has fourteen degrees of freedom as shown in the following eqns {f}=[Nx,,Ny,,Nz,,Mx,,My,,Mz,,B,,Nx,, Ny2, Nz,, (4
= b,, oy,
Mx2, My2, Mz,,
01, ~1, Ox,, oz,,
B21=
LI
The stiffness matrix [Kc] of a thin-wall column element is similar to that of a space beam element. The difference between the two sorts of elements is that [Kb] is a 12 x 12 matrix and [Kc] is a 14 x 14 matrix. 2.2. Transformation matrices for elements In analyzing high-rise building structures, a global coordinate system should be set up in order to represent the locations of every frame element. The transformation from a local coordinate system to a global coordinate system should be. carried out before the overall stiffness matrix of a structure can be obtained. If the angle between the axes of two coordinate systems is A, as shown in Fig. 1, the coordinate transformation matrix [ZJ is
OY,, Oz,,, ~2,029 w2,Ox2,
Oz,.]? where
[Tll=
cos A
sinA
0
-sin A 0
cosA
0
0
1
0
The subscripts 1 and 2 represent the upper and bottom ends of a thin-wall column element. According to Vlasov’s thin-walled beam theory, the relationship between torque, warping moment and torsion angle, warping angle can be shown as the following matrix,
Chkl
- 1
ICHkl - SHkl/l I- CHkl SHkllk
- I
0 cos A -sinA
sin A
0
cosA
0
0
0
1
If there are rigid arms at the ends of a member element, the transformation of rigid arms should be carried out. The beam element has the rigid arms as shown in Fig. 2. The transformation matrix of rigid
- kSHk1 1 - CHkl kSHkl 1 - CHkl
CHkl
- 1
SHkl/k - 1 I- CHkl ICHkl - SHkljk
where zz = GIz/(2 - 2CHkl+ klSHkl), k = ,/(GIz/Elw), Iz is the torsion constant, Iw is the warping constant, I is the length of a thin walled beam.
arms for beam elements is
X
0 Fig. 1.
A three-dimensional frame analysis program
j/,
955
I_dk,
0
I
2
X
Length
Fig. 2.
where
Here, w 1, w2 are the warping coordinates of thin-wall column elements on nodes 1 and 2 respectively; A, B points are the shear centers of two thin-wall column elements. The column element has rigid arms di, ah, t# and dk in two coordinate planes shown in Fig. 4. The transformation matrix of rigid arms for column elements is
-dj
1 1
1
Hl;
0
WI = _ 0 -I-iii [ 1
[H2] = [Hl] =
When a beam connects with a shear wall, it has a rigid arm whose length is equal to the distance between the shear center of the thin-wall column element and the node at which the beam is supported, as shown in Fig. 3. The transformation matrix of rigid arms for connecting beam elements is
[H2] =
I 1 01
where 1
YA - Yl Xl -XA
1 1
(Hl] =
Yl-YA 1
XA -Xl
-wl XA -Xl YA - Yl
1 0
1 1
1
YB - Y2 X2-XB
1 1 [H2] =
YB-Y2 1
XB-X2 1
-w2 XB-x2 YB - Y2
0 1
*
1 drill 1
-&1
1
956
CA0
Ll
-I
x,
*
W
Thin-wall
column
X
Fig. 3.
If the shear centers of two consecutive thin-wall column elements are not concordant with each other, the upper and lower thin-wall column elements would be connected by a rigid arm between their shear centers, shown in Fig. 5. In this way, the transformation matrix of rigid arm for thin-wall column elements is
where - YP
1
XP
1 1
[QU=
1 1
0
1 1 Here, Xp, Yp are the coordinates floor. where
of a point p on a
[H2] = [I], I
2.3. Member forces
-9
According to the transformation matrices for all member elements, their stiffness matrices in the global coordinate system are:
ex
1 [HI] =
1 ey 1
-en 1
0
beam elements,
1 1 Due to the high in-plane rigidity of the tloor, all structural components connected to the same slab are forced to move as a rigid floor in the horizontal plane, i.e. the displacement parameters u,, u, Oz, at all nodes are no longer independent, but can be set to some related values. This fact could be taken in structural analysis to reduce subs~n~ally the number of degrees of freedom associated with the overall stiffness matrix of structures. Therefore, transformation matrix of rigid floor for all elements is
[Q]=
!$!_;-&s, [ 1
Kbli = ~Q1I~~~1~~~1~~1~T3~Ql~ connecting
beam elements,
Wli = tQl~~lI~lt~~l~3i7~~l~Ql; column elements,
Kbli = ~Ql~T3~~I~~~l~~l~TlIQl; thin-wall column elements,
Nli = ~Ql~~l~~l~~~l~T7~~l~Ql. Therefore, based on the number of degrees of freedom for each member element, the overall
0
Fig. 4.
Fig. 5.
A three-dimensional frame analysis program Y
l-l
P*
X? t
structural global coordinate system respectively. Therefore, the eigenvalue eqn of a high-rise building structure is the same as a multi-degree of freedom system, i.e.
T
Y=
o-
X
4
where
Fig. 6.
structural follows,
stiffness matrix [KK] can be obtained
as
[KK] = c [K]i. , In this way, the displacement analysis is
where
eqn for structural
(D} is the structural displacement vector; {P} is the structural loading vector.
Due to the adoption of the rigid floor constraint, there are three common displacement parameters U, v, Or on each floor and three or four independent parameters w, Ox, Oy, Oz’ on each node. Varying the structural loading vectors according to dead, imposed, wind and earthquake loads respectively, the member forces can be obtained by solving the structural displacement eqn as the following eqns beam elements, if} = connecting
951
[~~lWlPl~Ql~4;
[S] is the stiffness matrix of the structure; [M] is the mass matrix of the structure, [M]=diag[ml,m2,. . .,mi,. . . ,mn] mi is the mass of the ith floor of the structure; {U} is the characteristic vector of the structure, (U}=[ul,u2 ,..., ui,..., unlT.
Solving the structural eigenvalue eqn, the response of the structure under seismic action can be analyzed by using the response spectrum superposition method based on the building code of China (TJl l-78). The earthquake force on the ith floor of the structure is Pij = C x aj x rj x Vii x Wi, where
C is the structural effect factor; uj is the seismic effect factor corresponding to the period Tj of the jth mode of vibration; Uij is the ith floor’s characteristic displacement corresponding to the ith natural mode of vibration; rj is the modal contribution factor of the jth mode of vibration,
beam elements, U-1 =
WlP’ll[fJl[QlV~~
Uij x Wi
'N' C
column elements,
Ui’j x Wi
i-l
thin-wall column elements, U-1 =
i rj =
~~clP’ll[~l[Ql~4.
These member forces are used to calculate the reinforcements of beams, columns and shear walls according to the concrete structure code of China (TJlO-74). 3. DYNAMIC ANALYSIS OF !3TRUCWRES 3.1. Plane vibration method A high-rise building structure can be treated as a lumped-mass model to represent the actual distributed mass structure. Because the simplification of the actual three-dimensional response of structures is made by assuming the design horizontal acceleration components to act nonconcurrently in the direction of each principal plane of buildings, a dynamic analysis process of structures could consist of two independent processes corresponding to each principle plane along X-direction and Y-direction of the
If there exists an eccentricity ei of the ith floor due to the non-coincidence of the center of mass and of rigidity, the torque caused by horizontal earthquake forces can be calculated as follows Mij=Pijxei. After solving the displacement equation of the structure under the earthquake loads {Pj} and {Mj}, every member force of the structure on the jth mode of vibration can be obtained. Then the modal combination of member forces can be obtained by using the method of the square root of the sum of the square of the modal maxima (SSRS method), i.e. I
s=
Jpj,i
where Sj is the structural response of the jth mode of vibration, j = 16 in TAP-86 Program. 3.2. One direction seismic wwe input method Actually, the ground acceleration has horizontal components along two perpendicular directions and
CA0 LI
958
a vertical component. Usually, the rotational components of earthquake ground motion are negligible. Because most building structures are more sensitive to horizontal or lateral distortions, it has been the practice in most instances to consider only the structural response of the horizontal components of ground motion. The effects of the vertical component of ground motion have generally not been considered significant enough to merit special attention. However, due to the horizontal components of ground motion, torsional effects in structures do occur, as a result of ground compliance and the non-coincidence of the centers of mass and rigidity. Based on the study of dynamic torsional effects in multistorey structures where this effect is more pronounced, it is necessary to use a three-dimensional dynamic analysis method to analyze the seismic response of high-rise building structures. When considering the torsion-coupled vibration of structures, there are three degrees of freedom (two horizontal displacement and one rotation) at each floor, and the total number of degrees of freedom for a structure of N floors is 3N. Therefore, the stiffness matrix of a structure could be obtained as follows,
The vector matrix [V] of the structure can be obtained by solving the eigenvector equation
where
[VI = [{Vl), {W,. **9 {vj), *. . , {Un)l and {Vj}=[xjl,xj2,..
..,xjtl,
yjl,
yj2 ,**....,
yin,
Oj l,Oj2, . . . . , Ojn]‘. The ith floor’s earthquake forces of thejth mode of vibration (Fig. 8) can also be obtained by using the response spectrum superposition method, i.e. Pijx = C x aj x rj x Xij x Wi Pgy = C x aj x rj x Yij x Wi Mij=CxajxrjxOijxJi, where C, aj are the same factors as the previous formula, N
The element Sij in [S] (i = 1,2,. . . ,3N, 3N) represents the amount of horizontal j=l,2,..., action or torque on the jth floor to produce the unit displacement on the ith floor of a structure. If the acceleration iig of one direction seismic wave and input angle B have been decided in advance (Fig. 7), the dynamic equation of the structure is
WI{ U} + PI{ 01= -Pfl
WI&,
where . . ,xn,
{V} =[Xl,X2,.
Yl, Y2,. . .) Yn,
N
sinB~mixXij+cosB~mixYij rj=
i-l
N N ~mixX?j+~mixYi2j+~JixOitj
i-l
i-l
i-1
N i-1
In TAP-86 Program, j = l-9. The structural response of every mode of vibration can be obtained by solving the displacement equation of the structure under the earthquake loads {Pjx}, {Pjy } and {Mj}. In order to consider the effects of torsion-coupled vibration, the modal combination of member forces can adopt the complete quadrate combination method (CQC method), i.e.
0 1,02, . . . ) On]‘; [M]=diag[ml,m2
,...,
mn, ml,m2 ,...,
mn,
S =
c 1 L-7
SJ
PkJ Sk,
Jl, J2,. . . , Jn]. mi, Ji is the mass and moment of inertia of the ith floor of the structure;
iig
where Sj, Sk are the structural responses under the jth and the kth mode of earthquake forces; and Pkj is the modal couple factor;
[Bb] = {{sin B}n x 1, {cos B}n x 1, {O}}. yt
0
8 ki Fig. 7.
Y
.
P,X -
4,
0 X
&X
F?,x I
Fig. 8.
A
8 x h*(l + r)r3’*
safety. In this way, the most disadvantageous input angle of seismic wave is not needed to be considered when the dynamic response of a structure is being analyzed. This method is more suitable to designing structures whose shapes are very irregular.
Pkj = (1 - r*)* + 4 x !I*(1 + r2)r h = 0.05,
959
three-dimensional frame analysis program
r = Tk/Tj,
Tk, Tj are the periods of the kth and the jth mode of vibration of the structure. 3.3. Two direction seismic waves simultaneously input
4. DESIGN AND APPLICATION OF THE PROGRAM
method
4.1. Design of the program
The direction of the ground acceleration is usually difficult to decide in advance, and the components of earthquake acting on the structure would vary with time along X-direction and Y-direction. If considering the effects of three components of earthquake acceleration, two horizontal components ug(t) and ug(t) as well as one rotation Og(t), the dynamic equation of a structure is
According to the static and dynamic analysis method of structures stated previously, a threedimensional frame analysis program can be written in FORTRAN 77 which consists of the following main function blocks.
I Input of structural
data
I
I
j”‘““’ Calculation
where
of section properties
t
[N]=diag[ml,m2 J1,32 ,.....,
,...,
mn, ml,m2,
Calculation of mass centres and weiahts of floors
.,m,
Jn]
VW = r{&!}, @g>,
ml’
1
PI 101 -[Ycl 1x01= 101 [Ol WC1. PI [ PI PI Xci, Yci are the coordinates
1-1
Forming overall stiffness matrix
Forming vertical loading vectors
of mass center of the ith
floor of the structure. Assuming that the two direction earthquake forces could be calculated by adopting the same response spectrum data, and the effect of rotational component of ground motion is negligible at the same time, the earthquake forces of each floor can be calculated as follows Pijx = C x aj x rj x Xij x Wi
and wind
of d namic roperties I Calculation and earthauake VP orces o the structure ’
I
I
Solving the displacement equation of the structure under every sort of loading vectors
I I
I
reinforcements of beams, COlumnS,
Pijy = C x aj x rj x Yij x Wi 1 End
Mij=CxajxrjxOijxJi
where C, aj are the same factors as the previous formula, N
rj=
y
1
Scheme 1.
N
CmixXij+CmixYij i-1 i-l
,$ (mi x Xiv + mi x Yifi + Ji x Oi2j)
The advantage of using two direction seismic waves simultaneously input method is its simplicity and
4.2. Application A high-rise building built in Shanghai has 34 stories and is 111.9 m high. The building is a Y shape frame-tube structure consisting of three frame arms and a hexagonal inner tube. The planar shape of shear walls in the inner tube is very irregular. The plane configuration of the structure is shown in Fig. 9. TAP-86 Program was used to do static and dynamic analysis of the structure.
960
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LI
Fig. 9.
The comparison of the three kinds of dynamic analysis methods of the structure are shown in Fig. 12. The conclusion is that the horizontal displacements of the structure under earthquake forces would get maximum values when adopting two direction seismic wave simultaneously input method, and get minimum values when using one direction seismic wave input method.
The natural modes of plane vibration of the structure are shown in Fig. 10. The results ofdynamic analysis of the structure by using plane vibration method are shown in Table 1. The natural modes of spatial vibration of the structure are shown in Fig. 11. The results of dynamic analysis of the structure by using one direction seismic wave input method and two direction seismic waves simultaneously input method are shown in Table 2 and Table 3, respectively.
Table 1. The ueriods of plane vibration and displacements of the structure Direction
Mode
1
2
3
x
Period (XC) Ux (cm) UY (cm) Oz (e - 4 rad)
2.088 8.813 0.488 1.061
0.4857 -0.767 0.0049 0.007
0.1405 0.0918 -0.0048 -0.082
Y
Period (set) Ux (cm) UY (cm) Oz (e - 4 rad)
2.124 0.046 8.733 1.254
0.4961 0.0011 -0.808 0.245
0.2251 -0.0091 0.9294 - 0.079
Fig. 10. The natural modes of plane vibration. (a) Modes along Xdirection.
(b) Modes along Y-direction.
A three-dimensional
frame analysis program
(of
961
fbf T
( / \
L
-1
-,
Fig. 11. The natural modes of spatial vibration. (a) The projection on X-O-Z plane. (b) The projection on Y-O-Z plane. (c) The projection on rotation. Table 2. The period of spatial vibration and displacements of the structure under one direction seismic wave input Direction
(4
B-90"
(XI
(b) B=O (Y)
1
Mode
2
3
4
5
Period (SW) f.Jx (Qn) 0 (cm) Oz (e -4rad)
2.2582 0.7308 2.3362 1.231
2*1054 7.1775 -2.856 1.484 -
ux WI UY (cm) Or (e - 4 radl
2.1892 7.0085 3.687
- 2.6754 0.6641 0.1099 1.0628 0.8587 0.1888 -0.553 -6.307 -4.92
6
I
8
9
1.0671 0.6625 0.6194 0.5056 0.4738 0.4455 0.3936 0.5638 0.1237 -0.0021 -0.0908 -0.4475 -0.0452 -0.0652 0.7283 0.2123 -0.0633 0.3421 -0.1545 0.0213 -0.0616 5.345 -0.554 0.855 0.006 0.556 0.766 0.024 -0.004 -0.120 1.621
0.114 -0.419 -0.01
0.2427 -0.019 0.0853 0.0098 -0.306 0.385
-0.037 -0.035 0.012
Table 3. The displacements of the structure under two direction seismic wave simultaneously input Mode
1
2
3
4
S
tJx (cm) WV(cmf Or (e - 4 rad)
2.9184 9.3427 4.914
9.8558 -3.915 2.037
- 1.227 - 1.585 11.645
-0.233 -0.401 1.047
-0.006 -0.184 2.477
Di#acament (cm) Pig. 12. The horizontal displacements of the structure under earthquake loads. (a) Displacements along X-direction. (b) Displacements along Y-direction. GY-l-Plane vibration method. GY-2-One direction seismic wave input method. GY-~-TWO direction seismic waves input method.
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5. CONCLUSION
TAP-86 Program is a three-dimensional frame analysis program especially suitable to high-rise building structures, and easier to use in this subject than general structural analysis programs. The periods of vibration, displacements and the distribution of shear forces of a structure under static and dynamic loads can be obtained accurately by
LI
using TAP-86 Program. TAP-86 Program has been used in more than ten design projects in China since 1986 ’ Acknowledgements-The research project of developing TAP-86 Program began in 1985 and ended in 1987. My colleague, Mr Xu Rong, did a great deal of work on the project. We also got a lot of help from the Mathematics Department of East-China Normal University.
32, No.
5, pp. 9%962,
1989 0
CO45-7949/89 13.00 + 0.00 1989 Maxwell Pcrgamon Macmillan plc
TAP-86: A THREE-DIMENSIONAL FRAME ANALYSIS PROGRAM FOR TALL BUILDINGS CA0 LI
The Ninth Design and Research Institute, Shanghai, China (Received 4 May 1988) Abstract--TAP-86 Program has three unique features: (1) the use of thin-wall element with opening for analyzing the torsion and warping effect of shear walls under lateral loads; (2) the use of three-dimensional analysis method for calculating the deformation and member forces of tall building structures under lateral and vertical loads; (3) the use of three kinds of dynamic analysis of structures, the plane vibration method for symmetric structures, the one direction seismic wave input method and the two direction seismic waves simultaneously input method for unsymmetric structures where the torsion-couple vibration effect should be taken into account. TAP-86 Program can be used to analyze such tall building systems as space frames, shear walls and tubular structures under static and seismic loads, and to calculate the member forces and reinforcements of beams, columns and shear walls. The program is written in FORTRAN 77 and runs on IBM-4341 and IBM-3031 computers.
1. INTRODUCTION
In designing high-rise buildings, if the building structure is asymmetric, if there is eccentric loading or if there exist many non-planar shear walls in the structure, it is necessary to use a three-dimensional structure analysis method in order to get more accurate results. It may be very difficult to use plane structure analysis to solve this sort of problem. TAP-86 Program is a three-dimensional frame analysis program for tall buildings. There are only two sorts of basic member elements in the program, i.e. a space rectangular beam element and a thin-wall column element. But the program is effective and powerful for structural engineers to use easily in analyzing high-rise building structures under lateral and vertical loads. On the other hand, there are three kinds of dynamic analysis of structures in the program, i.e. the plane vibration method for symmetric structures, the one direction seismic wave input method and the two direction seismic waves simultaneously input method for unsymmetric structures where the torsion-couple vibration effect should be taken into account. In this way, structural engineers can choose different dynamic analysis methods according to the regularity of structures. TAP-86 Program also calculates the reinforcements of beams, columns and shear walls according to the structural responses under static and dynamic loads. 2. STATIC ANALYSIS
OF STRUCTURES
(1) Beam element-the horizontal floor member connecting with columns at two ends. (2) Connecting beam element-the horizontal floor member connecting with shear walls at one or two ends. (3) Column element-the vertical member in a structure. (4) Thin-wall column element-the vertical member comprised of several plane shear walls. The stiffness matrix of space rectangular beams can be used as that of beam elements, column elements and connecting beam elements. The difference between beam and column elements is only the direction of coordinate axes. Usually, the number of degrees of freedom of a space beam is twelve, including bending, shear and torsion. The vectors of member force {f} and deformation {d} of a space beam can be expressed as the following eqns, respectively, {~}=[Nx,,NY,,Nz,,~x,,~Y,,~~,,Nx~,NY,, Nz2,
{d)
MY,,
= b,, Ul, WI, ox,,
JWT OY,, oz,,
u2,02, w2,ox,,
OY2, Oz21T.
The subscripts 1 and 2 represent the left and right ends of a beam element. The stiffness matrix of a space beam can be obtained according to the relationship between {J} and {d}, i.e.
2.1. Basic member elements Every kind of structural component in a high-rise building can be defined as following four kinds of member elements:
Mx,,
0-j=
WIW.
The stiffness matrix [Kb] can be found in many reference texts. 953
954
CA0
A shear wall between two consecutive floors can be represented by a thin-wall column element. For example, the commonly used E-, H- or U-shaped open section arrangement of the core walls can be treated as thin-wall columns in this way. Usually, the thinwall column element would warp under twist. The torsion angle’s derivative Oz’ = dOz/dr caused by section warping is called warping angle. After introducing warping angle Oz’ and warping moment B to the vectors of member forces and deformations, a thin-wall column element has fourteen degrees of freedom as shown in the following eqns {f}=[Nx,,Ny,,Nz,,Mx,,My,,Mz,,B,,Nx,, Ny2, Nz,, (4
= b,, oy,
Mx2, My2, Mz,,
01, ~1, Ox,, oz,,
B21=
LI
The stiffness matrix [Kc] of a thin-wall column element is similar to that of a space beam element. The difference between the two sorts of elements is that [Kb] is a 12 x 12 matrix and [Kc] is a 14 x 14 matrix. 2.2. Transformation matrices for elements In analyzing high-rise building structures, a global coordinate system should be set up in order to represent the locations of every frame element. The transformation from a local coordinate system to a global coordinate system should be. carried out before the overall stiffness matrix of a structure can be obtained. If the angle between the axes of two coordinate systems is A, as shown in Fig. 1, the coordinate transformation matrix [ZJ is
OY,, Oz,,, ~2,029 w2,Ox2,
Oz,.]? where
[Tll=
cos A
sinA
0
-sin A 0
cosA
0
0
1
0
The subscripts 1 and 2 represent the upper and bottom ends of a thin-wall column element. According to Vlasov’s thin-walled beam theory, the relationship between torque, warping moment and torsion angle, warping angle can be shown as the following matrix,
Chkl
- 1
ICHkl - SHkl/l I- CHkl SHkllk
- I
0 cos A -sinA
sin A
0
cosA
0
0
0
1
If there are rigid arms at the ends of a member element, the transformation of rigid arms should be carried out. The beam element has the rigid arms as shown in Fig. 2. The transformation matrix of rigid
- kSHk1 1 - CHkl kSHkl 1 - CHkl
CHkl
- 1
SHkl/k - 1 I- CHkl ICHkl - SHkljk
where zz = GIz/(2 - 2CHkl+ klSHkl), k = ,/(GIz/Elw), Iz is the torsion constant, Iw is the warping constant, I is the length of a thin walled beam.
arms for beam elements is
X
0 Fig. 1.
A three-dimensional frame analysis program
j/,
955
I_dk,
0
I
2
X
Length
Fig. 2.
where
Here, w 1, w2 are the warping coordinates of thin-wall column elements on nodes 1 and 2 respectively; A, B points are the shear centers of two thin-wall column elements. The column element has rigid arms di, ah, t# and dk in two coordinate planes shown in Fig. 4. The transformation matrix of rigid arms for column elements is
-dj
1 1
1
Hl;
0
WI = _ 0 -I-iii [ 1
[H2] = [Hl] =
When a beam connects with a shear wall, it has a rigid arm whose length is equal to the distance between the shear center of the thin-wall column element and the node at which the beam is supported, as shown in Fig. 3. The transformation matrix of rigid arms for connecting beam elements is
[H2] =
I 1 01
where 1
YA - Yl Xl -XA
1 1
(Hl] =
Yl-YA 1
XA -Xl
-wl XA -Xl YA - Yl
1 0
1 1
1
YB - Y2 X2-XB
1 1 [H2] =
YB-Y2 1
XB-X2 1
-w2 XB-x2 YB - Y2
0 1
*
1 drill 1
-&1
1
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Ll
-I
x,
*
W
Thin-wall
column
X
Fig. 3.
If the shear centers of two consecutive thin-wall column elements are not concordant with each other, the upper and lower thin-wall column elements would be connected by a rigid arm between their shear centers, shown in Fig. 5. In this way, the transformation matrix of rigid arm for thin-wall column elements is
where - YP
1
XP
1 1
[QU=
1 1
0
1 1 Here, Xp, Yp are the coordinates floor. where
of a point p on a
[H2] = [I], I
2.3. Member forces
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According to the transformation matrices for all member elements, their stiffness matrices in the global coordinate system are:
ex
1 [HI] =
1 ey 1
-en 1
0
beam elements,
1 1 Due to the high in-plane rigidity of the tloor, all structural components connected to the same slab are forced to move as a rigid floor in the horizontal plane, i.e. the displacement parameters u,, u, Oz, at all nodes are no longer independent, but can be set to some related values. This fact could be taken in structural analysis to reduce subs~n~ally the number of degrees of freedom associated with the overall stiffness matrix of structures. Therefore, transformation matrix of rigid floor for all elements is
[Q]=
!$!_;-&s, [ 1
Kbli = ~Q1I~~~1~~~1~~1~T3~Ql~ connecting
beam elements,
Wli = tQl~~lI~lt~~l~3i7~~l~Ql; column elements,
Kbli = ~Ql~T3~~I~~~l~~l~TlIQl; thin-wall column elements,
Nli = ~Ql~~l~~l~~~l~T7~~l~Ql. Therefore, based on the number of degrees of freedom for each member element, the overall
0
Fig. 4.
Fig. 5.
A three-dimensional frame analysis program Y
l-l
P*
X? t
structural global coordinate system respectively. Therefore, the eigenvalue eqn of a high-rise building structure is the same as a multi-degree of freedom system, i.e.
T
Y=
o-
X
4
where
Fig. 6.
structural follows,
stiffness matrix [KK] can be obtained
as
[KK] = c [K]i. , In this way, the displacement analysis is
where
eqn for structural
(D} is the structural displacement vector; {P} is the structural loading vector.
Due to the adoption of the rigid floor constraint, there are three common displacement parameters U, v, Or on each floor and three or four independent parameters w, Ox, Oy, Oz’ on each node. Varying the structural loading vectors according to dead, imposed, wind and earthquake loads respectively, the member forces can be obtained by solving the structural displacement eqn as the following eqns beam elements, if} = connecting
951
[~~lWlPl~Ql~4;
[S] is the stiffness matrix of the structure; [M] is the mass matrix of the structure, [M]=diag[ml,m2,. . .,mi,. . . ,mn] mi is the mass of the ith floor of the structure; {U} is the characteristic vector of the structure, (U}=[ul,u2 ,..., ui,..., unlT.
Solving the structural eigenvalue eqn, the response of the structure under seismic action can be analyzed by using the response spectrum superposition method based on the building code of China (TJl l-78). The earthquake force on the ith floor of the structure is Pij = C x aj x rj x Vii x Wi, where
C is the structural effect factor; uj is the seismic effect factor corresponding to the period Tj of the jth mode of vibration; Uij is the ith floor’s characteristic displacement corresponding to the ith natural mode of vibration; rj is the modal contribution factor of the jth mode of vibration,
beam elements, U-1 =
WlP’ll[fJl[QlV~~
Uij x Wi
'N' C
column elements,
Ui’j x Wi
i-l
thin-wall column elements, U-1 =
i rj =
~~clP’ll[~l[Ql~4.
These member forces are used to calculate the reinforcements of beams, columns and shear walls according to the concrete structure code of China (TJlO-74). 3. DYNAMIC ANALYSIS OF !3TRUCWRES 3.1. Plane vibration method A high-rise building structure can be treated as a lumped-mass model to represent the actual distributed mass structure. Because the simplification of the actual three-dimensional response of structures is made by assuming the design horizontal acceleration components to act nonconcurrently in the direction of each principal plane of buildings, a dynamic analysis process of structures could consist of two independent processes corresponding to each principle plane along X-direction and Y-direction of the
If there exists an eccentricity ei of the ith floor due to the non-coincidence of the center of mass and of rigidity, the torque caused by horizontal earthquake forces can be calculated as follows Mij=Pijxei. After solving the displacement equation of the structure under the earthquake loads {Pj} and {Mj}, every member force of the structure on the jth mode of vibration can be obtained. Then the modal combination of member forces can be obtained by using the method of the square root of the sum of the square of the modal maxima (SSRS method), i.e. I
s=
Jpj,i
where Sj is the structural response of the jth mode of vibration, j = 16 in TAP-86 Program. 3.2. One direction seismic wwe input method Actually, the ground acceleration has horizontal components along two perpendicular directions and
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a vertical component. Usually, the rotational components of earthquake ground motion are negligible. Because most building structures are more sensitive to horizontal or lateral distortions, it has been the practice in most instances to consider only the structural response of the horizontal components of ground motion. The effects of the vertical component of ground motion have generally not been considered significant enough to merit special attention. However, due to the horizontal components of ground motion, torsional effects in structures do occur, as a result of ground compliance and the non-coincidence of the centers of mass and rigidity. Based on the study of dynamic torsional effects in multistorey structures where this effect is more pronounced, it is necessary to use a three-dimensional dynamic analysis method to analyze the seismic response of high-rise building structures. When considering the torsion-coupled vibration of structures, there are three degrees of freedom (two horizontal displacement and one rotation) at each floor, and the total number of degrees of freedom for a structure of N floors is 3N. Therefore, the stiffness matrix of a structure could be obtained as follows,
The vector matrix [V] of the structure can be obtained by solving the eigenvector equation
where
[VI = [{Vl), {W,. **9 {vj), *. . , {Un)l and {Vj}=[xjl,xj2,..
..,xjtl,
yjl,
yj2 ,**....,
yin,
Oj l,Oj2, . . . . , Ojn]‘. The ith floor’s earthquake forces of thejth mode of vibration (Fig. 8) can also be obtained by using the response spectrum superposition method, i.e. Pijx = C x aj x rj x Xij x Wi Pgy = C x aj x rj x Yij x Wi Mij=CxajxrjxOijxJi, where C, aj are the same factors as the previous formula, N
The element Sij in [S] (i = 1,2,. . . ,3N, 3N) represents the amount of horizontal j=l,2,..., action or torque on the jth floor to produce the unit displacement on the ith floor of a structure. If the acceleration iig of one direction seismic wave and input angle B have been decided in advance (Fig. 7), the dynamic equation of the structure is
WI{ U} + PI{ 01= -Pfl
WI&,
where . . ,xn,
{V} =[Xl,X2,.
Yl, Y2,. . .) Yn,
N
sinB~mixXij+cosB~mixYij rj=
i-l
N N ~mixX?j+~mixYi2j+~JixOitj
i-l
i-l
i-1
N i-1
In TAP-86 Program, j = l-9. The structural response of every mode of vibration can be obtained by solving the displacement equation of the structure under the earthquake loads {Pjx}, {Pjy } and {Mj}. In order to consider the effects of torsion-coupled vibration, the modal combination of member forces can adopt the complete quadrate combination method (CQC method), i.e.
0 1,02, . . . ) On]‘; [M]=diag[ml,m2
,...,
mn, ml,m2 ,...,
mn,
S =
c 1 L-7
SJ
PkJ Sk,
Jl, J2,. . . , Jn]. mi, Ji is the mass and moment of inertia of the ith floor of the structure;
iig
where Sj, Sk are the structural responses under the jth and the kth mode of earthquake forces; and Pkj is the modal couple factor;
[Bb] = {{sin B}n x 1, {cos B}n x 1, {O}}. yt
0
8 ki Fig. 7.
Y
.
P,X -
4,
0 X
&X
F?,x I
Fig. 8.
A
8 x h*(l + r)r3’*
safety. In this way, the most disadvantageous input angle of seismic wave is not needed to be considered when the dynamic response of a structure is being analyzed. This method is more suitable to designing structures whose shapes are very irregular.
Pkj = (1 - r*)* + 4 x !I*(1 + r2)r h = 0.05,
959
three-dimensional frame analysis program
r = Tk/Tj,
Tk, Tj are the periods of the kth and the jth mode of vibration of the structure. 3.3. Two direction seismic waves simultaneously input
4. DESIGN AND APPLICATION OF THE PROGRAM
method
4.1. Design of the program
The direction of the ground acceleration is usually difficult to decide in advance, and the components of earthquake acting on the structure would vary with time along X-direction and Y-direction. If considering the effects of three components of earthquake acceleration, two horizontal components ug(t) and ug(t) as well as one rotation Og(t), the dynamic equation of a structure is
According to the static and dynamic analysis method of structures stated previously, a threedimensional frame analysis program can be written in FORTRAN 77 which consists of the following main function blocks.
I Input of structural
data
I
I
j”‘““’ Calculation
where
of section properties
t
[N]=diag[ml,m2 J1,32 ,.....,
,...,
mn, ml,m2,
Calculation of mass centres and weiahts of floors
.,m,
Jn]
VW = r{&!}, @g>,
ml’
1
PI 101 -[Ycl 1x01= 101 [Ol WC1. PI [ PI PI Xci, Yci are the coordinates
1-1
Forming overall stiffness matrix
Forming vertical loading vectors
of mass center of the ith
floor of the structure. Assuming that the two direction earthquake forces could be calculated by adopting the same response spectrum data, and the effect of rotational component of ground motion is negligible at the same time, the earthquake forces of each floor can be calculated as follows Pijx = C x aj x rj x Xij x Wi
and wind
of d namic roperties I Calculation and earthauake VP orces o the structure ’
I
I
Solving the displacement equation of the structure under every sort of loading vectors
I I
I
reinforcements of beams, COlumnS,
Pijy = C x aj x rj x Yij x Wi 1 End
Mij=CxajxrjxOijxJi
where C, aj are the same factors as the previous formula, N
rj=
y
1
Scheme 1.
N
CmixXij+CmixYij i-1 i-l
,$ (mi x Xiv + mi x Yifi + Ji x Oi2j)
The advantage of using two direction seismic waves simultaneously input method is its simplicity and
4.2. Application A high-rise building built in Shanghai has 34 stories and is 111.9 m high. The building is a Y shape frame-tube structure consisting of three frame arms and a hexagonal inner tube. The planar shape of shear walls in the inner tube is very irregular. The plane configuration of the structure is shown in Fig. 9. TAP-86 Program was used to do static and dynamic analysis of the structure.
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Fig. 9.
The comparison of the three kinds of dynamic analysis methods of the structure are shown in Fig. 12. The conclusion is that the horizontal displacements of the structure under earthquake forces would get maximum values when adopting two direction seismic wave simultaneously input method, and get minimum values when using one direction seismic wave input method.
The natural modes of plane vibration of the structure are shown in Fig. 10. The results ofdynamic analysis of the structure by using plane vibration method are shown in Table 1. The natural modes of spatial vibration of the structure are shown in Fig. 11. The results of dynamic analysis of the structure by using one direction seismic wave input method and two direction seismic waves simultaneously input method are shown in Table 2 and Table 3, respectively.
Table 1. The ueriods of plane vibration and displacements of the structure Direction
Mode
1
2
3
x
Period (XC) Ux (cm) UY (cm) Oz (e - 4 rad)
2.088 8.813 0.488 1.061
0.4857 -0.767 0.0049 0.007
0.1405 0.0918 -0.0048 -0.082
Y
Period (set) Ux (cm) UY (cm) Oz (e - 4 rad)
2.124 0.046 8.733 1.254
0.4961 0.0011 -0.808 0.245
0.2251 -0.0091 0.9294 - 0.079
Fig. 10. The natural modes of plane vibration. (a) Modes along Xdirection.
(b) Modes along Y-direction.
A three-dimensional
frame analysis program
(of
961
fbf T
( / \
L
-1
-,
Fig. 11. The natural modes of spatial vibration. (a) The projection on X-O-Z plane. (b) The projection on Y-O-Z plane. (c) The projection on rotation. Table 2. The period of spatial vibration and displacements of the structure under one direction seismic wave input Direction
(4
B-90"
(XI
(b) B=O (Y)
1
Mode
2
3
4
5
Period (SW) f.Jx (Qn) 0 (cm) Oz (e -4rad)
2.2582 0.7308 2.3362 1.231
2*1054 7.1775 -2.856 1.484 -
ux WI UY (cm) Or (e - 4 radl
2.1892 7.0085 3.687
- 2.6754 0.6641 0.1099 1.0628 0.8587 0.1888 -0.553 -6.307 -4.92
6
I
8
9
1.0671 0.6625 0.6194 0.5056 0.4738 0.4455 0.3936 0.5638 0.1237 -0.0021 -0.0908 -0.4475 -0.0452 -0.0652 0.7283 0.2123 -0.0633 0.3421 -0.1545 0.0213 -0.0616 5.345 -0.554 0.855 0.006 0.556 0.766 0.024 -0.004 -0.120 1.621
0.114 -0.419 -0.01
0.2427 -0.019 0.0853 0.0098 -0.306 0.385
-0.037 -0.035 0.012
Table 3. The displacements of the structure under two direction seismic wave simultaneously input Mode
1
2
3
4
S
tJx (cm) WV(cmf Or (e - 4 rad)
2.9184 9.3427 4.914
9.8558 -3.915 2.037
- 1.227 - 1.585 11.645
-0.233 -0.401 1.047
-0.006 -0.184 2.477
Di#acament (cm) Pig. 12. The horizontal displacements of the structure under earthquake loads. (a) Displacements along X-direction. (b) Displacements along Y-direction. GY-l-Plane vibration method. GY-2-One direction seismic wave input method. GY-~-TWO direction seismic waves input method.
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5. CONCLUSION
TAP-86 Program is a three-dimensional frame analysis program especially suitable to high-rise building structures, and easier to use in this subject than general structural analysis programs. The periods of vibration, displacements and the distribution of shear forces of a structure under static and dynamic loads can be obtained accurately by
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using TAP-86 Program. TAP-86 Program has been used in more than ten design projects in China since 1986 ’ Acknowledgements-The research project of developing TAP-86 Program began in 1985 and ended in 1987. My colleague, Mr Xu Rong, did a great deal of work on the project. We also got a lot of help from the Mathematics Department of East-China Normal University.