COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Semi-Analytical Analysis of Super Tall Building Bundled-Tube Structures
Yaoqing Gong *, Ke Li
Civil Engineering School of Henan Polytechnic University, Henan Jiaozuo, 454000 China
Email: [email protected]
Abstract A new semi-analytical method is developed for the analysis of interactions between the subgrade and the
foundation and the superstructure of the super tall building bundled-tube structure by three-dimensional model. That
is, the superstructure and its foundation of the super tall building bundled-tube structure are simplified equivalently
and continuously to a combination of stiffened-thin-wall tubes on semi-infinite elastic subgrade; then discretization is
made by some nodal lines, the unknown functions defined on the lines are used as primary unknowns, and
interpolating functions are implemented between the lines; thirdly the principle of minimum potential energy is
applied so as to change the analysis of a tall building structure into the solution of the boundary problem of a group of
ordinary differential equations, which can be solved by the precise and powerful Ordinary Differential Equation
Solver—COLSYS. The interactions between the subgrade and the foundation and the superstructure of a super tall
building bundled-tube structure due to static loadings are analyzed by the method based on the model. The numerical
results show that the analytical model is reasonable and feasible. Therefore, a practicable method for the global
analysis of the super tall building bundled-tube structure is obtained, and some valuable conclusions are also acquired
through analyzing the computing results.
Key words: super tall building, bundled-tube structure, semi-analytical method, three-dimensional model, stiffenedthin-wall tube, ordinary differential equation solver
INTRODUCTION
As a result of rapid development of human’s social activity, International Metropolitan cities’ requirement for tall
buildings is gradually toward the developing direction of larger volume, higher height, and more diversified
architectural images. Thusly, the various new structural systems of tall buildings have been yielded [1,2,3] to meet the
requirement. Especially in China, after nearly 20 year’s practice of tall building construction, the structural systems
such as the tube structure, the tube-in-tube structure, the framed tube and the like, which are used to resist lateral forces
of tall buildings, have been commonly adopted[4]. However, these types of structural systems will not be able to
satisfy the requirement of lateral rigidity when the height or the whole size of the tall building becomes big to some
kind of degrees. In this circumstance, the bundled-tube structure system can be a preferable choice since it has strong
load bearing capacity, great lateral stiffness. In addition, it remains the good ductility and excellent aseismatic
performance. So it is a sort of very prospective structural system for super tall buildings.
In the viewpoint of architecture, the bundled-tube structure provides more useful space and make flexible layout.
Because of these merits, a skyscraper with the bundled-tube structure allows the building to achieve a very high
altitude so as to reach above 100 layers [3]. For example, Sears building, in Chicago of American, adopted the
bundled-tube structure as its structural system, which then became the typical representative with innovative
significance since its reliability and economical efficiency obtained the best balance. Thusly, it becomes one of good
model in the history of super tall buildings.
However, the research reports and papers on the analysis of the bundled-tube structure are very hard to look up at
present because the bundled-tube structure belongs to the type of the large-scale complex structure. That is, its height,
flexibility and the diversity of structural system are quite different from those of the conventional structure. So its
stressed performance, mechanical analysis and structural design are much more complex comparing to the common
structural system.
⎯ 467 ⎯
In order to change the situation and to seek a fast, effective, reasonable and simplified analytical method for the stage
of preliminary design or analysis of global performance, a new semi-analytical method based on ODE (Ordinary
Differential Equation) Solver for the analysis of the interactions between the subgrade and the foundation and the
superstructure has been developed in the paper. The superstructure and the foundation of the super tall building with
bundled-tube structure are simplified equivalently and continuously to a three-dimensional model, which is a
combination of stiffened-thin-wall tubes on semi-infinite elastic subgrade. And the static analysis of a super tall
building bundled-tube structure is achieved with the three-dimensional model by the method. The numerical results
show that the simplified computing model is reasonable and the semi-analytical method is effective and powerful.
Therefore, a practicable method for the global analysis of the super tall building bundled-tube structure has been
established. Some valuable conclusions are obtained through analyzing the computing results as well.
ESTABLISHMENT OF COMPUTING MODEL
The new ideas of the paper are embodied on its physical modeling, which is a three-dimensional assemblage of the
subgrade, the foundation and the superstructure of a super tall building with bundled-tube structure.
To the subgrade, it is idealized as a semi-infinite elastic body, i.e., the bottom and the wall of the foundation pit can be
recognized as different equivalent semi-infinite elastic bodies according to the actual conditions. Furthermore, the
rigidities of the elastic bodies pertinent to various deformations of the foundation have been derived as analytical
equations by principle of energy equivalence [5], with which the reactions between the foundation and the subgrade
can be quantified expediently.
To the foundation, it is recognized as a part of the superstructure. In fact, the foundation is the extension of the
superstructure toward the underground. The only difference is the size, since in most cases the foundation must be big
enough to make the soil beneath it stable.
To the superstructure, generally, a bundled-tube structure is composed of several tubes connected by common frames
to form an external structure, and one inner tube connected with the external structure by floorboards so as to compose
the whole bundled-tube structure. Thusly, the mechanical performance of the structural system under the action of
external loadings will be similar with that of an assemblage of several closed stiffened-thin-wall tubes. In practice, the
displacement field of the entire bundled-tube structure due to static loadings is dominated by lateral displacements of
its longitudinal axis and warping of its cross sections.
In order to implement an effective semi-analytical analysis to the structural system by three-dimensional model, it is
necessary to transform the superstructure equivalently into a continuous combination of several closed
stiffened-thin-wall tubes with different stiffness by means of equivalent principle of rigidity, as shown in Fig. 1.
Therefore, the frames connecting the external tubes are equivalently simplified to a combination of continuous
thin-wall tubes; the external tubes themselves are idealized equivalently to stiffened ribs of the continuous thin-wall
tubes. The whole external structure is thusly idealized into a stiffened-thin-wall tube. Then, combine the continuous
thin-wall external tube stiffened by the ribs with the inner tube to form a tube-in-tube structure with stiffened ribs on a
semi-infinite elastic subgrade. The tubes work in concert with their ribs and the elastic subgrade consistently.
Figure 1: Sketch of stiffened-thin-wall tubes
Figure 2: Stress state of tube wall
DESCRIPTION OF DISPLACEMENT FIELD
In order to describe the displacement field of the structural system, two assumptions must be made as follows:
(1) Rigid floor slab assumption, i.e., the stiffness of the floor slab in its own plane is infinite large, on the contrary, that
out of the plane is neglected.
(2) The magnitude of circumferential normal stress of the cross section of the tubes is negligible compared with that of
longitudinal stress of the tube wall. This means that the stress state of the tube wall consists of normal stress σ(s, z )
⎯ 468 ⎯
in the longitudinal direction and shearing stress τ(s, z ) in the latitudinal directions, which are the functions of
coordinates ‘s’, along periphery direction, and ‘z’, along the longitudinal direction, as show in Fig. 2.
Above assumptions indicate that the circumferential normal stress along the central line direction of the thin-wall tube
can be neglected. The longitudinal normal stress and the latitudinal shearing stress are the main actions on the tube
wall, meanwhile the warping normal stress caused by shearing lag has been taken into consideration [6,7].
Based on above assumptions, a discretization can be made by the nodal lines (Fig. 1), which are parallel to the axis of
the entire structure. Then take the unknown functions of the longitudinal displacements of the nodal lines on the inner
tube and the external tube, and the unknown functions of transverse displacements of the centroidal line of the cross
sections of the whole structure as fundamental unknown functions. After that, adopt interpolation functions between
nodal lines. Thusly, the displacement fields of the whole structure can be expressed as
k
⎧⎧
⎫⎪
⎪
⎪
⎪
⎪
ϕi (sin )(win (z ))i = [ϕ(sin )]j {win (z )}j ⎪⎪
⎪
∑
⎪
⎪⎪
⎪⎪
⎪ i =1
u(s, z ) = ⎪
⎨⎨
⎬
k
⎪⎪
⎪
⎪⎪
ϕi (sex )(wex (z ))i = [ϕ(sex )]j {wex (z )}j ⎪⎪
⎪⎪
∑
⎪⎪
⎪
⎪⎪
⎩ i =1
⎩
⎭⎪
(1)
{v 0 (z )} = {([v 0x (z ) v0y (z ) θ(z )]T )j }
(2)
In which, u(s, z ) is the longitudinal displacement function of the structure, it is a function set composed of two groups of
functions of inner tube and external tube. {v 0 (z )} is the transverse displacement function of the structure, it is a set of
function vectors. Subscript ‘in’ and ‘ex’ represent, respectively, the inner tube and the external tube. ‘ sin ’ and ‘ sex ’
represent, respectively, the curvilinear coordinate (cross-section central line direction) of inner tube and external tube. ‘z’
represents longitudinal coordinate (axial direction) of the tubes. k is the number of the intersections of nodal lines with ‘ sin ’
and ‘ sex ’. j = 1, 2, ", n is the number of segments of the nodal lines in the longitudinal direction.
Displacement fields of the stiffened ribs work in concert with those of the thin-wall external tube, that is, the transverse
displacements of the stiffened ribs are the same as those of centroidal line of the cross-section of the structure, the axial
displacements of the stiffened ribs are identical with the longitudinal displacements corresponding to the nodal lines
located at the same positions.
DEVELOPMENT OF GOVERNING EQUATIONS
According to the above displacement fields, the total potential energy of entire structural system can be expressed as
Π = U t + U z + U tb + U tg + U zg + U p
(3)
In which, U t , U z , U tb , U tg , U zg are the elastic strain energies of the thin-wall tubes, the stiffened ribs, stored in the
elastic body around the foundation pit, stored in the elastic body under the bottoms of the thin-wall tubes, and stored in
the elastic body under the bottoms of the stiffened ribs, respectively. U p is the external loading potential energy.
Subscript ‘i’ and ‘e’ represent, respectively, inner tube and external tube. They can be written as
⎧
⎛ n ⎛
⎪
⎧
⎫
⎪ ⎡ ∂u
⎪ ⎞⎟ ⎞⎟⎟
∂u ∂vt 2 ⎥⎤
⎪
⎜ ⎜ Hi 1 ⎪
∂θ 2 ⎪
⎪
⎜
2
⎢
⎪
⎪
⎜
⎜⎜∑ ⎜ ∫
⎪
E ( ) + G(
) ⎥ bds + GJ d ( ) ⎬ dz ⎟⎟⎟ ⎟⎟
+
⎨∫
⎢
v
⎪
⎜
s
0
2 ⎪⎪ ⎢ ∂z
⎜ i =1 ⎜
∂s
∂z ⎥
∂z ⎪⎪ ⎠⎟⎟ ⎟⎟⎟
⎪
⎪
⎦
⎪ ⎣
⎪ i ⎠in
⎩
⎭
⎪⎜⎝ ⎝
Ut = ⎨
⎪
⎛ ⎛
⎧ ⎡ ∂u
⎫ ⎞⎟ ⎞⎟
⎪
⎪
⎪
∂u ∂vt 2 ⎥⎤
⎪ ⎜⎜ n ⎜⎜ H i 1 ⎪
∂θ 2 ⎪
⎟
2
⎢
⎪
⎪
⎪
E ( ) + G(
) ⎥ bds + GJ d ( ) ⎬ dz ⎟⎟⎟ ⎟⎟
+ ⎜⎜∑ ⎜⎜ ∫
+
⎨∫
⎪
⎢
v
s
⎟ ⎟⎟
⎪ ⎜ i =1 ⎜ 0 2 ⎪
∂z
∂s
∂z ⎥
∂z ⎪
⎪
⎪
⎪
⎜
⎦
⎪ ⎢⎣
⎪ ⎠⎟i ⎠⎟ex
⎩
⎭
⎪
⎪
⎩ ⎝ ⎝
(4)
⎪
∂ 2v0x 2
∂ 2v0y 2
∂w 2 ⎫⎪
Hi ⎧
∂θ 2
1 4
⎪
U z = ∑ ∑ ∫ ⎨(EI y )k ( 2 ) + (EI x )k ( 2 ) + (GJ g )k ( ) + (EA)k ( )k ⎪⎬dz
0 ⎪
∂z
∂z
∂z
∂z ⎪⎪
i =1 2 k =1
⎪⎩⎪
⎭⎪
(5)
n
U tb =
1 H1
dz ∫
C r (K tH vt2 + K nH vn2 + K tH (ρθ )2 )ds
v
∫
s
0
2
⎯ 469 ⎯
(6)
⎞
⎪⎧⎪⎜⎛ 1 ⎡
K zDu 2 + K tDvt2 + K tDvn2 + K tD (ρθ )2 ) bds ⎤⎥ ⎟⎟
(
⎢
v
⎪⎜
∫
⎜
⎦ z =0 ⎠in
⎪⎝ 2 ⎣ s
U tg = ⎪⎨
⎪⎪ ⎛ 1 ⎡
⎞
K zDu 2 + K tDvt2 + K tDvn2 + K tD (ρθ )2 ) bds ⎤⎥ ⎟⎟
(
⎪⎪+ ⎜⎜⎝ ⎢ ∫
v
⎦ z =0 ⎠ex
⎪⎩ 2 ⎣ s
U zg =
1 4 ⎡
K zDwk2Ak + K tD Ak v 02x + K tD Ak v 02y + K tD (J g )k θ 2 + (K tD I y )k (v 0′x )2 + (K tD I x )k (v 0′y )2 ⎥⎦⎤
∑
⎢
⎣
2 k =1
z =0
⎧⎪⎛ n ⎛ Hi ⎧
N
⎫
⎪
⎪⎪ ⎞⎟ ⎞⎟⎟
⎪⎜
⎪⎪⎜⎜∑ ⎜⎜ ∫ ⎪⎨∫
⎟
+
p
ubds
p
u
∑
zl l ⎬ dz ⎟ ⎟
v z
⎪⎭
⎪⎪⎜⎝ i =1 ⎜⎝ 0 ⎪⎩
l =1
⎪ s
⎪ ⎠⎟i ⎠⎟⎟in
U p = −⎨
⎪ ⎛ n ⎛ H⎧
N
⎫ ⎞⎞
⎪⎪⎪+ ⎜⎜∑ ⎜⎜ i ⎪⎨⎪ p ubds + p v + p v + m θ + ∑ p u ⎪⎬⎪ dz ⎟⎟ ⎟⎟⎟
z
x
x
y
y
z
zl
l
0
0
v
∫
∫
⎜
⎪⎪ ⎝⎜ i =1 ⎜⎝ 0 ⎪⎩⎪ s
⎪⎭⎪ ⎠⎟⎟ ⎠⎟⎟
l =1
i ex
⎪⎩
(7)
(8)
(9)
Where, E is the elastic modulus of the material of the stiffened-thin-wall tube to resist axial deformation; G is the
elastic modulus of the material of the stiffened-thin-wall tube to resist shearing deformation. GJ d is the torsion
stiffness of cross-section of the thin-wall tube; GJ g is the torsion stiffness of cross-section of the stiffened rib. ρ is
the distance between infinitesimal body and center of the cross-section of the stiffened-thin-wall tube; b is the
thickness of the stiffened-thin-wall tube; H 1 is the depth of the foundation. C r is the interfacing coefficient between
foundation and subgrade. K tD and K zD are the equivalent tangent and normal stiffness of the soil at the bottom of the
foundation, respectively; K tH and K nH are the equivalent tangent and normal stiffness of the soil at the foundation-pit
wall, respectively. Pz and Pzl are the distributed loading and the concentrated force in z direction, respectively; P x
and P y are the linear distributed forces in x and y direction of the local coordinate, respectively; mz is the linear
distributed torque in θ direction of the local coordinate along the axis of the local coordinate of stiffened-thin-wall
tubes; vt and vn are the tangent and normal displacements of the wall of an infinitesimal body obtained from
stiffened-thin-wall tubes in local coordinate, respectively; θ is the rotation angle of an infinitesimal segment of
stiffened-thin-wall tubes in local coordinate. Their corresponding expressions are:
vt (z ) = [ηt ][T ]T {v0 (z )} , vn (z ) = [ηn ][T ]T {v 0 (z )} , θ (z ) = [I θ ][T ]T {v 0 (z )}
[ηt ] = [cos α sin α ρt ] , [ηn ] = [− sin α cos α ρn ] , [I θ ] = [0 0 1]
⎡ cos β
− sin β
0⎤⎥
⎢
⎢
⎥
[T ] = ⎢ sin β
cos β
0⎥
⎢
⎥
⎢ sin β x 0 − cos β y 0 cos β x 0 + sin β y 0 1 ⎥
⎢⎣
⎥⎦
The meanings of other symbols can be referred to the reference [5].
By principle of minimum potential energy,
δΠ = 0
(10)
The governing differential equations and their corresponding boundary conditions could be obtained as follows
⎧⎪(E [A]{w ′′(z )} − G[B ]{w(z )} − G[C g ]{v 0′ (z )} + {Pz } + {Pzl })ex = {0}
⎫⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪(E [A]{w ′′(z )} − G[B ]{w(z )} − G[C g ]{v0′ (z )} + {Pz } + {Pzl })in = {0}
⎪⎪
⎪⎪
⎪⎪
T
T
⎨GD3θ ′′(z ) + (G[C g ]3 {w ′(z )})ex + (G[C g ]3 {w ′(z )})in − C r (EK )33 θ(z ) = 0
⎬
⎪⎪
⎪⎪
T
T
(4)
⎪
′′
′
′
4
(
)
(
)
[
C
]
{
w
(
z
)}
G
[
C
]
{
w
(
z
)}
C
(
EK
)
v
(
z
)
0
EI
v
z
GD
v
z
G
−
−
−
+
=
( g1
)ex ( g 1
)in r
⎪⎪
⎪⎪⎪
11 0x
y 0x
1 0x
⎪⎪
⎪
(4)
v ′′ (z ) − (G[C ]T {w ′(z )}) − (G[C ]T {w ′(z )}) + C (EK ) v (z ) = 0⎪
⎪
4
(
)
EI
v
z
GD
−
x 0y
g 2
g 2
r
2 0y
22 0y
⎪⎩⎪
⎪⎭⎪⎪1
ex
in
⎯ 470 ⎯
(11)
⎧
⎫
⎪
⎪
(E [A]{w ′′(z )} − G[B ]{w(z )} − G[C g ]{v 0′ (z )} + {Pz } + {Pzl })ex = {0}
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
′′
′
(E [A]{w (z )} − G[B ]{w(z )} − G[C g ]{v 0 (z )} + {Pz } + {Pzl })in = {0}
⎪
⎪
⎪
⎪
⎪
⎪
T
T
⎪
⎪
⎨GD3θ ′′(z ) + (G[C g ]3 {w ′(z )})ex + (G[C g ]3 {w ′(z )})in + M z = 0
⎬
⎪
⎪
⎪
⎪
(4)
T
T
⎪
4EI y v 0x (z ) − GD1v 0′′x (z ) − (G[C g ]1 {w ′(z )})ex − (G[C g ]1 {w ′(z )})in − Px = 0⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
T
T
(4)
⎪4EI x v 0y (z ) − GD2v 0′′y (z ) − (G[C g ]2 {w ′(z )}) − (G[C g ]2 {w ′(z )}) − Py = 0⎪
⎪
⎪
⎪
ex
in
⎪
⎪
⎩
⎭j
(12)
⎧⎪(E [A]{w ′(0)} − K zD [A]{w(0)})ex = {0}
⎪⎪
⎪⎪
⎪⎪(E [A]{w ′(0)} − K zD [A]{w(0)})in = {0}
⎪⎪
⎪⎪GD3θ ′(0) + (G[C g ]T3 {w(0)})ex + (G[C g ]T3 {w(0)})in − (DFK )33 θ(0) = 0
⎪⎪
⎪⎨4EI v ′′ (0) = 4K I v ′ (0)
y 0x
tD y 0x
⎪⎪
⎪⎪4EI v ′′′(0) − GD v ′ (0) − (G[C ]T {w(0)}) − (G[C ]T {w(0)}) + (DFK ) v (0) = 0
y 0x
g 1
g 1
1 0x
11 0x
ex
in
⎪⎪
⎪⎪4EI v ′′ (0) = 4K I v ′ (0)
x 0y
tD x 0y
⎪⎪
⎪⎪
T
T
⎪⎪⎩4EI x v 0′′′y (0) − GD2v 0′y (0) − (G[C g ]2 {w(0)})ex − (G[C g ]2 {w(0)})in + (DFK )22 v 0y (0) = 0
(13)
⎧
⎪
(E [A]{w ′(H )})ex = {0}
⎪
⎪
⎪
⎪
(E [A]{w ′(H )})in = {0}
⎪
⎪
⎪
⎪
θ ′(H ) + (G[C ]T {w(H )}) + (G[C ]T {w(H )}) − M = 0
⎪
GD
g 3
g 3
3
0
⎪
ex
in
⎪
⎪
⎪
⎨4EI yv 0′′x (H ) = 0
⎪
⎪
⎪
4EI yv 0′′′x (H ) − GD1v 0′x (H ) − (G[C g ]T1 {w(H )})ex − (G[C g ]T1 {w(H )})in + Px = 0
⎪
⎪
⎪
⎪
⎪
4EI x v 0′′y (H ) = 0
⎪
⎪
⎪
⎪⎪4EI v ′′′(H ) − GD v ′ (H ) − (G[C ]T {w(H )}) − (G[C ]T {w(H )}) + P = 0
x 0y
2 0y
g 2
g 2
y
ex
in
⎪
⎩
(14)
⎧⎪ {wex (z )}j = {wex (z )}j +1, {win (z )}j = {win (z )}j +1, (θ(z ))j = (θ(z ))j +1
⎪⎪
⎪⎪ (v (z )) = (v (z )) , (v ′ (z )) = (v ′ (z )) , (v (z )) = (v (z )) , (v ′ (z )) = (v ′ (z ))
j
0x
j +1
0x
j
0x
j +1
0y
j
0y
j +1
0y
j
0y
j +1
⎪⎪ 0x
⎪⎪
⎪⎪{(E [A ]{w ′(z )})ex }j = {(E [A]{w ′(z )})ex }j +1, {(E [A]{w ′(z )})in }j = {(E [A]{w ′(z )})in }j +1
⎪⎪
⎪⎪ GD3θ ′(z ) + (G[C g ]T3 {w(z )}) + (G[C g ]T3 {w(z )})
ex
in j
⎪⎪
⎪⎪
T
T
⎪⎪ = GD3θ ′(z ) + (G[C g ]3 {w(z )})ex + (G[C g ]3 {w(z )})in j +1
⎪⎪
⎪⎪ (4EI v ′′ (z )) = (4EI v ′′ (z ))
y 0x
y 0x
j
j +1
⎨⎪⎪
⎪⎪
v ′ (z ) − (G[C ]T {w(z )}) − (G[C ]T {w(z )})
⎪⎪ 4EI yv 0′′′x (z ) − GD
1 0x
g 1
g 1
in j
ex
⎪⎪
⎪⎪
T
T
⎪⎪ = 4EI yv0′′′x (z ) − GD1v 0′x (z ) − (G[C g ]1 {w(z )})ex − (G[C g ]1 {w(z )})in j +1
⎪⎪
⎪⎪ (4EI x v 0′′y (z )) = (4EI x v 0′′y (z ))
j
j +1
⎪⎪
⎪⎪
T
T
⎪⎪ 4EI x v 0′′′y (z ) − GD2v 0′y (z ) − (G[C g ]2 {w(z )})ex − (G[C g ]2 {w(z )})in j
⎪⎪
⎪⎪ = 4EI x v0′′′y (z ) − GD2v 0′y (z ) − (G[C g ]T2 {w(z )}) − (G[C g ]T2 {w(z )})
ex
in j +1
⎪⎩
⎪⎪
(15)
(
)
(
)
(
)
(
)
(
)
(
)
⎯ 471 ⎯
In which, subscript ‘in’ and ‘ex’ represent respectively the inner tube and the external tube. J g is the moment of inertia
relative to the centroid of the stiffened rib. [C g ]T1 , [C g ]T2 and [C g ]T3 represent respectively the first line, second line
and third line of the matrix [C g ]T . [A] = [A] + [Az ] , [Az ] is the matrix of composed merely by the elements of the
rib’s area whose position is determined by the nodal number of the ribs. The expressions of the other symbols appeared
in above equations are as follows:
[A] =
∫ [ϕ ] [ϕ ]bds , [C
T
s
g
] = [C ][T ]T ， [C ] =
∫ [ϕ ′] [η ]bds ， {P } = ∫ [ϕ(s )]
T
s
T
t
z
s
pzbds ，
M
{Pzl }j = ∑ [ϕj (sl )]T pzl ，GD3 = (4GJ g + GD33 )ex + (GD33 )in ，GD1 = (GD11 )ex + (GD11 )in ，
l =1
GD2 = (GD22 )ex + (GD22 )in ， [Dt ] =
∫ [η ] [η ]bds ， [D ] = [T ]([D ] + J [I
T
s
t
t
t
d
θ
]T [I θ ])[T ]T ，
[C g ]T = [T ][C ]T ， [EK ] = [T ](K tH [Dt ] + K nH [Dn ] + K tH Sa [I 0 ])[T ]T ， [Dt ] =
[Dn ] =
∫ [η
s
n
∫ [η ] [η ]ds ，
T
s
t
t
]T [ηn ]ds ， (DFK )33 = ((DK )33 )ex + ((FK )33 )in ， (DFK )11 = ((DK )11 )ex + ((FK )11 )in ，
(DFK )22 = ((DK )22 )ex + ((FK )22 )in ， [DK ] = [FK ] + K tD [Dθ ] ， Sa =
∫ ρ bds ， S
s
[FK ] = [T ](K tD [Dt ] + K tD [Dn ] + K tDSa [I 0 ])[T ]T ， [Dn ] =
∫ [η
⎡ cos β
0⎤⎥
− sin β
⎢
⎢
⎥
[T ] = ⎢ sin β
cos β
0⎥ ,
⎢
⎥
⎢ sin β x 0 − cos β y 0 cos β x 0 + sin β y 0 1 ⎥
⎢⎣
⎥⎦
⎡0
⎤
⎢
⎥
⎢
⎥
[I 0 ] = ⎢ 0 ⎥
⎢
⎥
⎢
⎥
1
⎢⎣
⎥⎦
⎡ 4
⎢ ∑ Ak
⎢ k =1
⎢
⎢
[Dθ ] = ⎢
⎢
⎢
⎢
⎢
⎢
⎣
4
∑A
k
k =1
s
n
2
t
a
=
∫ ρ ds ，
s
2
t
]T [ηn ]bds , [I 0 ] = [I θ ]T [I θ ] ,
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
4
⎥
⎥
(
)
J
∑
g k⎥
k =1
⎦
Equations (11) and (12) respectively represent governing ordinary differential equations of the foundation and the
superstructure, they are actually the equations of static equilibrium of the infinitesimal segment of the bundled-tube
structure, the first two equations are the balance equations of the vertical force of the bundled-tube structure and the
latter three equations are those of the horizontal force of the entire infinitesimal segment. Equations (13) and (14) are
boundary conditions which are in fact the force balance conditions at the foundation’s bottom and the superstructure’s
top, respectively. Equations (15) are connecting conditions between the segments of the subgrade and the foundation
and the superstructure; they are actually the compatibility conditions of displacement and the equilibrium conditions at
the joints.
Foregoing ordinary differential equations can be solved by the virtue of Ordinary Differential Equation solver, termed
ODEs, a general purpose program developed to solve various ODE problems, such as COLSYS [8].
NUMERICAL EXAMPLE AND ANALYSIS OF COMPUTING RESULTS
1. Example The cross section of the bundled-tube structure of reinforced concrete is shown in Fig. 3. The average
equivalent thickness of the equivalent thin-wall external tube and the equivalent thin-wall inner tube are 0.45m and
0.55m, respectively. The depth of the foundation is 16m, and the height of the tubes is 200m. The elastic modulus of
the material of the stiffened-thin-wall tube to resist axial and shearing deformation are E = 3.25 × 107 KPa ,
⎯ 472 ⎯
Figure 3: Sketch of cross section of bundled-tube structure
G = 1.22 × 107 KPa . The equivalent normal and tangent rigidities of the soil at the foundation bottom
are K zD = rd 20.0 × 106 KPa/m , K tD = rd 15.7 × 106 KPa/m , respectively; and the equivalent normal and
tangent rigidities of the soil at the wall of the foundation pit are K nH = rd 18.0 × 106 KPa/m ,
K tH = rd 13.5 × 106 KPa/m , respectively. rd is a coefficient depending on the site circumstance [9,10,11] ( in this
example, rd = 1.0 ). C r is an interfacing coefficient between foundation and subgrade ( in this case, C r = 1.0 ).
Assume the gravity loading is 50 KN/m 3 , the even horizontal loadings of the two directions are 5 KPa , and the even
torque is 10 KN ⋅ m/m . Table 1 to Table 6 are some of the computing results of the example.
Table 1 Warping on the top of the building and maximum stress of the joint between foundation and
superstructure due to the gravity and transverse loadings
Maximum stress of the joint between
foundation and superstructure (MPa)
Warping on the top of the building (mm)
Maximum warping
Warping difference
External tube
4.44
1.2
Inner tube
27.99
0.26
Normal stress
Position
−15.47
Corner point of
inner tube
Table 2 Maximum settlement and stress of the subgrade due to gravity and transverse loadings
Settlement of subgrade (mm)
Maximum normal stress of subgrade (MPa)
Maximum settlement
Settlement difference
Normal stress
Position
0.825
0.02
−16.50
Corner of inner tube
Table 3 Maximum shearing force and torque of the foundation and the joint between the foundation and
the superstructure due to gravity and transverse loadings
Maximum shearing force and torque
Maximum shearing force and torque of the joint
between foundation and superstructure
of foundation
Qx (KN)
Qy (KN)
M z (KN ⋅ m)
Qx (KN)
Qy (KN)
M z (KN ⋅ m)
143467
143467
−10.57
−54000
−54000
-2000
⎯ 473 ⎯
Table 4 Sidesway and maximum shearing force of superstructure
Sidesway due to transverse loadings (mm)
Δ
Maximum shearing force and torque of
cross sections of superstructure
Maximum sidesway
between layers
Sidesway on the top
Δ/H
δ
δ /h
force and torque
Position
x
3.61
1/55402
0.057
1/52631
Qx
−54000 The joint with foundation
y
3.61
1/55402
0.057
1/52631
Qy
−54000
The joint with
foundation
Mz
−2000
The joint with
foundation
θ
0.002 degree
Table 5 Maximum sidesway and settlement with variation of stiffness of subgrade and foundation
Coefficients varied with
rigidities of subgrade
rd = 0.01
Coefficients( RP1 )varied
with rigidities of foundation
rd = 0.1
rd = 1.0
RP1 = 3.4
rd = 10.0
RP1 = 3.4
RP1 = 13.5
RP1 = 15.5
Maximum sidesway (mm)
9.47
4.25
3.61
3.47
3.47
3.47
Maximum settlement (mm)
81.69
8.22
0.83
0.082
0.088
0.097
Table 6 Variation of axial stress of the tube a with variation of stiffness of subgrade
Height (m)
rd = 0.01
rd = 0.1
rd = 1.0
rd = 10.0
144.00
−558.95
−558.17
−557.63
−557.39
112.00
−943.87
−941.83
−940.25
−939.60
80.00
−1341.37
−1336.17
−1331.59
−1329.80
48.00
−1748.91
−1735.13
−1721.11
−1715.98
−1.60
−3218.45
−3122.07
−3001.46
−2960.86
−6.40
−3328.32
−3209.36
−3057.67
−3007.32
−9.60
−3402.48
−3266.59
−3091.60
−3034.49
−12.80
−3477.65
−3323.16
−3122.56
−3058.38
2. Analysis of the computing results Through analyzing the computing results, some conclusions can be
obtained as follows:
(1) Maximum normal stress of the subgrade appears in the corner point of the foundation’s bottom, and that of the
superstructure takes place in the corner point of its joint with the foundation as well. It is observed that shearing lag has
considerable effect on the stress distribution, as show in Table 1 and Table 2.
(2) Maximum shearing force of the superstructure always occurs on its bottom section where the foundation is
connected, but the position of maximum shearing force of the foundation becomes lower and lower with the increase of
its rigidity, as show in Table 3 and Table 4.
(3) The sidesway on the top of the bundled-tube structure is very small under the action of the transverse loadings,
which indicates that the structural system has advanced lateral and spatial rigidity, as show in Table 4.
⎯ 474 ⎯
(4) The relative rigidities of the subgrade and the foundation have the remarkable influence on the sidesway of the
superstructure and the settlement of the ground. Yet the influence will be very small enough to omit when the rigidity
of the subgrade approaches or is higher than that of the foundation. In the circumstance, the subgrade can be idealized
as the rigid body. Furthermore, when the rigidities of both subgrade and foundation are many times bigger than that of
the superstructure, we may consider them as the rigid subgrade and the rigid foundation. That is to say, traditional
method simplifying the restraint between the superstructure and the foundation as a build-in restraint is reasonable
only when the rigidities of the subgrade and the foundation are many times greater than that of the superstructure, as
show in Table 5.
(5) The influence field of stress in the superstructure due to the variation of rigidity of the subgrade only limits to the
vicinity of the foundation, which has again proven the accuracy of assumption of St. Venant [12], as show in Table 6.
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⎯ 475 ⎯