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Download Luận văn Numerical analysis of externally prestressed concrete beams
In recent years, the beams prestressed by means of external cables have attracted the
engineer’s attention. Especially, the use of external prestressing is gaining popularity in
bridge constructions because ofits simplicity and cost-effectiveness. A large number of
bridges with monolithic or precast segmental block have been built in the United States,
European countries and Japan by using the external prestressing technique. The external
prestressing, moreover, is applied not only to new structures, but also to existing structures,
which need to be repaired or strengthened. Although various advantages of external
prestressing have been reported elsewhere, there still remain certain problems concerning
the behavior of externally prestressed concrete beams at ultimate that must be examined in
great detail
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may be differentiated with respect to x and gives:
-41-
q
GAdx
dQ
GAdx
vd s ωω
=−=2
2
(3.29)
where
dx
dQq −=
By summation of Eq.(3.27) and Eq.(3.29) gives a general equation to determine deflection
with considering shearing force effect in Eq (3.30).
⎟⎠
⎞⎜⎝
⎛
+= q
GA
EIM
EIdx
vd y ϖ12
2
(3.30)
Fig.3.3 shows the effect of shear deformation on the total deflection of the beam. When
beam is subjected to loading, the total displacement is equal to the sum of flexural
displacement and shear displacement and expressed by an equation νy=νb+νs. However, when
the span-to-depth ratio is big, the effect of shear deformation on the total displacement is
extremely small, and in practice most analyses are usually taken by ignoring of shear
deformation. But, the span-to-depth ratio is small, especially for the deep beams, it should be
taken into account, because the shear deformation is approximately 10~15% of the total
deflection at ultimate.
3.2.3 Shear stiffness model
Based on Timoshenko’s theory, shear deformation as well as shear strain depends on value
of the shear stiffness GA. However, in the elastic zone, effect of shear deformation (before
formation of Cr-ack) on the total deformation is extremely small. In this case, the shear
stiffness can be approximately calculated from the well-known relationship:
)(
AE
GA c
ν+
=
12
(3.31)
Fig.3.3 Effect of shear deformation
Lo
ad
Deflection
Vb Vs
GA is infinite
GA is variable
Total displacement
Vy=Vb+Vs
Vy
Lo
ad
-42-
where Ec is Young’s modulus of concrete;ν is Poisson coefficient; A is area of cross section.
In the beams that are subjected to large shear forces and are web reinforced accordingly,
diagonal Cr-acks must be expected during the service condition. These Cr-acks can increase the
shear deformation of the beam, considerably. Shear distortions occurring in the web may be
approximated by using the analogous truss model, in which vertical stirrup and 45o diagonal
concrete struts are assumed to form web member (see Fig.3.4). The elongation of the stirrups
is Δs, and the shortening of the compression strut is Δc. Applying Williot’s principle, the shear
distortion can be found as:
csRsv ΔΔΔΔΔ 2+=+= (3.32)
where
vs
s
s AE
SV
=Δ and
wc
s
c bE
V
.
22
=Δ and substituting into Eq.(3.32), the shear distortion is given
as:
⎟⎟⎠
⎞
⎜⎜⎝
⎛
+=+=
c
s
v
w
ws
s
wc
s
vs
s
v E
E
A
Sb
bE
V
bE
V
AE
SV 422
2Δ (3.33)
where Es is the Young’s modulus of stirrup; S is the stirrup space and bw is the width of the
web. Therefore, the shear distortion per unit length of the beam becomes:
⎟⎟⎠
⎞
⎜⎜⎝
⎛
+=⎟⎟⎠
⎞
⎜⎜⎝
⎛
+=
Δ
= n
dbE
V
E
E
A
Sb
dbE
V
d vws
s
c
s
v
w
ws
sv 41
4
ρ
γ (3.34)
where ρv=Av/bwS is ratio of stirrup per unit of space of stirrup and n=Es/Ec is modulus ratio.
Therefore, the shear stiffness of beam with 45o diagonal Cr-acks, in accordance with truss
action is the value of Vs when γ=1, and is thus given by:
Fig.3.4 Truss model for shear stiffness model
Δ S
CΔ
Δ R ΔV
ΔC
Δ S
Δ R
VS
VS
45o
d
d
s
d
-43-
⎟⎟⎠
⎞
⎜⎜⎝
⎛
+
=
v
v
ws n
dbEGA
ρ
ρ
41
(3.35)
Similar expression can be derived for other inclination of compression struts α and stirrupβ.
It may be easy shown for the general case that the stirrup stress will be:
ββαρ 2sin)cot(cot += v
s
s
Vf (3.36)
where the stirrup length is d/sinβ. Therefore, the shear stiffness can be defined as expression:
dbE
n
GA ws
v
v
βρα
βαβαρ
44
244
sinsin
)cot(cotsinsin
+
+
= (3.37)
where )sin( βρ SbA wvv = for the general case.
3.3 PROPOSED EQUATION OF CABLE STRAIN
Despite the extensive literature on the topic of beam prestressed with external cables, the
problem has been approached mainly in the elastic range. The solved case, for which a clearly
stated analytical model is available both in linear and nonlinear range, is the case of the
simply supported beams with rectilinear cables, for which the writing of compatibility
equations is trivial and balance conditions are particularly simple. The treatment for more
complex situations led to formulations referring to very specific cases and the solving method
often adopted involved procedures. In particular, the coupling between an entire beam
deformation and the cable local strain does not make it convenient to write the balance
equations in terms of the equilibrium of beam cross section.
Therefore, the global deformation compatibility between the concrete beam and the
prestressing cable is necessary in order to establish the member-analysis procedure for the
evaluation of externally prestressed concrete beams. In order to provide a satisfactory answer
to these questions, a solution for computing the increase of strain in an external cable has been
developed to analyze accurately the response of the beams prestressed with external cables,
and will be presented in this section. It is important to note that the initial development of the
current methodology for the analysis of externally prestressed concrete beams was carried out
at Concrete Structures Laboratory of Nagoya University45~48). Although this methodology was
applied for the analysis of simply supported beams with external cables and symmetrical
loading conditions, it is, however, very important and useful materials for the further
-44-
development. It is truly said that without the previous researcher’s successes, the current
developed methodology may not be possible to be going on.
3.3.1 Previous development of equation for cable strain
Numerous investigations of analytical models for the beams prestressed with unbonded
cables or external cables were carried out in the past. Here, only the most relevant models to
the present research are briefly presented.
Before the formulation of cable strain for externally prestressed concrete beams, it is very
important to note that we cannot review the computing method for the cable strain without
evoking of the initial formulation for prestressed concrete beams with partially bonded cables.
For the calculation of cable strain of the concrete beams prestressed with partially bonded
cables, Tanabe, T., et al.49) were earlier authors, who proposed the following equation:
cs
l
cscsss dxl
k εΔεΔεΔεΔ +⎟⎠
⎞⎜⎝
⎛
−= ∫01 (3.38)
where ks is the slipping coefficient; Δεs Δεcs are the increments of cable strain and concrete
strain at the cable level, respectively; l is the total length of cable between the extreme ends.
From Eq.(3.38), it can be seen that there are two extreme cases, namely, perfectly bonded
and completely free slip. For the case of perfect bond, the slipping coefficient should be equal
to zero (ks=0), i.e., the cable strain Δεs is equal to the concrete strain Δεcs at the cable level.
While, for the case of completely free slip, the slipping coefficient should be equal to 1.0
(ks=1.0), i.e., the total deformation of cable is equal to the total deformation of concrete beam
at the cable level, and the cable strain is constant over the whole its length between the
extreme end anchorages.
To apply Eq.(3.38), an intensive investigation of experimental and numerical studies on the
behavior of prestressed concrete frame with partially bonded cables was carried out by
Umehara, H., et al.50~52) at Concrete Laboratory of Nagoya Institute of Technology. It is
shown that the behavior of partially bonded PC beams can be satisfactorily predicted, and
analytical results are in good agreement with the experimental data. The accuracy of the
proposed equation for the cable strain is again verified by comparing the predicted results
with the experimental observations. Umehara also carried out the numerical investigations on
the effect of the bond condition, and the predicted results of these investigations is plotted in
Fig.3.5.
-45-
A more application of Eq.(3.38) for the analysis of partially bonded PC plate was carried
out by Qutait, A.R., et al.53). The authors stated that strain energy of cable varies according to
the extent of bondage between the concrete and the cable. When perfect bond and no sliding
are occurred, the change in the cable strain due to the applied load would exactly follow the
change of strain in the concrete fiber at the same level. On the other hand, when the bond
between the concrete and the cable is artificially reduced to zero as in the case of perfectly
unbonded members, the magnitude of cable strain will be the same throughout its entire
length...
Download miễn phí Luận văn Numerical analysis of externally prestressed concrete beams
In recent years, the beams prestressed by means of external cables have attracted the
engineer’s attention. Especially, the use of external prestressing is gaining popularity in
bridge constructions because ofits simplicity and cost-effectiveness. A large number of
bridges with monolithic or precast segmental block have been built in the United States,
European countries and Japan by using the external prestressing technique. The external
prestressing, moreover, is applied not only to new structures, but also to existing structures,
which need to be repaired or strengthened. Although various advantages of external
prestressing have been reported elsewhere, there still remain certain problems concerning
the behavior of externally prestressed concrete beams at ultimate that must be examined in
great detail
Để tải bản DOC Đầy Đủ xin Trả lời bài viết này, Mods sẽ gửi Link download cho bạn sớm nhất qua hòm tin nhắn.
Ai cần download tài liệu gì mà không tìm thấy ở đây, thì đăng yêu cầu down tại đây nhé:
Nhận download tài liệu miễn phí
Tóm tắt nội dung:
shear force Q in continuous function whichmay be differentiated with respect to x and gives:
-41-
q
GAdx
dQ
GAdx
vd s ωω
=−=2
2
(3.29)
where
dx
dQq −=
By summation of Eq.(3.27) and Eq.(3.29) gives a general equation to determine deflection
with considering shearing force effect in Eq (3.30).
⎟⎠
⎞⎜⎝
⎛
+= q
GA
EIM
EIdx
vd y ϖ12
2
(3.30)
Fig.3.3 shows the effect of shear deformation on the total deflection of the beam. When
beam is subjected to loading, the total displacement is equal to the sum of flexural
displacement and shear displacement and expressed by an equation νy=νb+νs. However, when
the span-to-depth ratio is big, the effect of shear deformation on the total displacement is
extremely small, and in practice most analyses are usually taken by ignoring of shear
deformation. But, the span-to-depth ratio is small, especially for the deep beams, it should be
taken into account, because the shear deformation is approximately 10~15% of the total
deflection at ultimate.
3.2.3 Shear stiffness model
Based on Timoshenko’s theory, shear deformation as well as shear strain depends on value
of the shear stiffness GA. However, in the elastic zone, effect of shear deformation (before
formation of Cr-ack) on the total deformation is extremely small. In this case, the shear
stiffness can be approximately calculated from the well-known relationship:
)(
AE
GA c
ν+
=
12
(3.31)
Fig.3.3 Effect of shear deformation
Lo
ad
Deflection
Vb Vs
GA is infinite
GA is variable
Total displacement
Vy=Vb+Vs
Vy
Lo
ad
-42-
where Ec is Young’s modulus of concrete;ν is Poisson coefficient; A is area of cross section.
In the beams that are subjected to large shear forces and are web reinforced accordingly,
diagonal Cr-acks must be expected during the service condition. These Cr-acks can increase the
shear deformation of the beam, considerably. Shear distortions occurring in the web may be
approximated by using the analogous truss model, in which vertical stirrup and 45o diagonal
concrete struts are assumed to form web member (see Fig.3.4). The elongation of the stirrups
is Δs, and the shortening of the compression strut is Δc. Applying Williot’s principle, the shear
distortion can be found as:
csRsv ΔΔΔΔΔ 2+=+= (3.32)
where
vs
s
s AE
SV
=Δ and
wc
s
c bE
V
.
22
=Δ and substituting into Eq.(3.32), the shear distortion is given
as:
⎟⎟⎠
⎞
⎜⎜⎝
⎛
+=+=
c
s
v
w
ws
s
wc
s
vs
s
v E
E
A
Sb
bE
V
bE
V
AE
SV 422
2Δ (3.33)
where Es is the Young’s modulus of stirrup; S is the stirrup space and bw is the width of the
web. Therefore, the shear distortion per unit length of the beam becomes:
⎟⎟⎠
⎞
⎜⎜⎝
⎛
+=⎟⎟⎠
⎞
⎜⎜⎝
⎛
+=
Δ
= n
dbE
V
E
E
A
Sb
dbE
V
d vws
s
c
s
v
w
ws
sv 41
4
ρ
γ (3.34)
where ρv=Av/bwS is ratio of stirrup per unit of space of stirrup and n=Es/Ec is modulus ratio.
Therefore, the shear stiffness of beam with 45o diagonal Cr-acks, in accordance with truss
action is the value of Vs when γ=1, and is thus given by:
Fig.3.4 Truss model for shear stiffness model
Δ S
CΔ
Δ R ΔV
ΔC
Δ S
Δ R
VS
VS
45o
d
d
s
d
-43-
⎟⎟⎠
⎞
⎜⎜⎝
⎛
+
=
v
v
ws n
dbEGA
ρ
ρ
41
(3.35)
Similar expression can be derived for other inclination of compression struts α and stirrupβ.
It may be easy shown for the general case that the stirrup stress will be:
ββαρ 2sin)cot(cot += v
s
s
Vf (3.36)
where the stirrup length is d/sinβ. Therefore, the shear stiffness can be defined as expression:
dbE
n
GA ws
v
v
βρα
βαβαρ
44
244
sinsin
)cot(cotsinsin
+
+
= (3.37)
where )sin( βρ SbA wvv = for the general case.
3.3 PROPOSED EQUATION OF CABLE STRAIN
Despite the extensive literature on the topic of beam prestressed with external cables, the
problem has been approached mainly in the elastic range. The solved case, for which a clearly
stated analytical model is available both in linear and nonlinear range, is the case of the
simply supported beams with rectilinear cables, for which the writing of compatibility
equations is trivial and balance conditions are particularly simple. The treatment for more
complex situations led to formulations referring to very specific cases and the solving method
often adopted involved procedures. In particular, the coupling between an entire beam
deformation and the cable local strain does not make it convenient to write the balance
equations in terms of the equilibrium of beam cross section.
Therefore, the global deformation compatibility between the concrete beam and the
prestressing cable is necessary in order to establish the member-analysis procedure for the
evaluation of externally prestressed concrete beams. In order to provide a satisfactory answer
to these questions, a solution for computing the increase of strain in an external cable has been
developed to analyze accurately the response of the beams prestressed with external cables,
and will be presented in this section. It is important to note that the initial development of the
current methodology for the analysis of externally prestressed concrete beams was carried out
at Concrete Structures Laboratory of Nagoya University45~48). Although this methodology was
applied for the analysis of simply supported beams with external cables and symmetrical
loading conditions, it is, however, very important and useful materials for the further
-44-
development. It is truly said that without the previous researcher’s successes, the current
developed methodology may not be possible to be going on.
3.3.1 Previous development of equation for cable strain
Numerous investigations of analytical models for the beams prestressed with unbonded
cables or external cables were carried out in the past. Here, only the most relevant models to
the present research are briefly presented.
Before the formulation of cable strain for externally prestressed concrete beams, it is very
important to note that we cannot review the computing method for the cable strain without
evoking of the initial formulation for prestressed concrete beams with partially bonded cables.
For the calculation of cable strain of the concrete beams prestressed with partially bonded
cables, Tanabe, T., et al.49) were earlier authors, who proposed the following equation:
cs
l
cscsss dxl
k εΔεΔεΔεΔ +⎟⎠
⎞⎜⎝
⎛
−= ∫01 (3.38)
where ks is the slipping coefficient; Δεs Δεcs are the increments of cable strain and concrete
strain at the cable level, respectively; l is the total length of cable between the extreme ends.
From Eq.(3.38), it can be seen that there are two extreme cases, namely, perfectly bonded
and completely free slip. For the case of perfect bond, the slipping coefficient should be equal
to zero (ks=0), i.e., the cable strain Δεs is equal to the concrete strain Δεcs at the cable level.
While, for the case of completely free slip, the slipping coefficient should be equal to 1.0
(ks=1.0), i.e., the total deformation of cable is equal to the total deformation of concrete beam
at the cable level, and the cable strain is constant over the whole its length between the
extreme end anchorages.
To apply Eq.(3.38), an intensive investigation of experimental and numerical studies on the
behavior of prestressed concrete frame with partially bonded cables was carried out by
Umehara, H., et al.50~52) at Concrete Laboratory of Nagoya Institute of Technology. It is
shown that the behavior of partially bonded PC beams can be satisfactorily predicted, and
analytical results are in good agreement with the experimental data. The accuracy of the
proposed equation for the cable strain is again verified by comparing the predicted results
with the experimental observations. Umehara also carried out the numerical investigations on
the effect of the bond condition, and the predicted results of these investigations is plotted in
Fig.3.5.
-45-
A more application of Eq.(3.38) for the analysis of partially bonded PC plate was carried
out by Qutait, A.R., et al.53). The authors stated that strain energy of cable varies according to
the extent of bondage between the concrete and the cable. When perfect bond and no sliding
are occurred, the change in the cable strain due to the applied load would exactly follow the
change of strain in the concrete fiber at the same level. On the other hand, when the bond
between the concrete and the cable is artificially reduced to zero as in the case of perfectly
unbonded members, the magnitude of cable strain will be the same throughout its entire
length...