Civil Engineering Reference
In-Depth Information
(1) Internal force analysis of retrofitted beam
The prestressed internal force, which is caused by lower-supported prestressed reinforce-
ments in retrofitted beam, is shown in Fig. 3.16. The effective prestressed internal force in
beam section
L
1
is:
⎫
⎬
M
px
1
=
σ
p
1
·
A
p
(
h
a
θ
−
X
sin
θ
)
V
p
1
=
σ
p
1
·
A
p
sin
θ
(3
.
17)
⎭
N
p
1
=
σ
p
1
·
A
p
cos
θ
The effective prestressed internal force caused by reinforcements in beam segment
L
2
between two supporting points is:
⎫
⎬
M
p
2
=
σ
p
2
·
A
p
(
h
b
+
a
p
)
V
p
=0
N
p
2
=
σ
p
2
·
(3
.
18)
⎭
A
p
where
A
p
is the total sectional area of prestressed reinforcements;
σ
p
1
,
σ
p
2
is the effective
prestress of the prestressed reinforcement in beam segments
L
1
and
L
2
, respectively, which
is equal to the value that controls stress
σ
con
minus the loss of prestress force
σ
l
in each
beam section (see detail case of
σ
con
and
σ
l
in the following text); X is the distance be-
tween anchoring point and calculation section;
θ
is the angle between oblique tensile bar
and longitudinal axis;
a
p
is the distance between composite force of horizontal prestressed
reinforcements and lower edge of section;
h
a
is the distance between anchoring point and
longitudinal axis of original beam;
h
b
is the distance between longitudinal axis of original
beam and lower edge of section.
The value of
N
p
2
is a little less than the value of
N
p
1
because of friction. When construction
is finished, the value of internal force of section equals the difference between the internal
force (
M
0
,V
0
) caused by external loads and the internal force (
M
p
,V
p
) caused by prestress,
which is
⎫
⎬
M
=
M
0
−
M
p
V
=
V
0
−
V
p
(3
.
19)
⎭
N
=
N
p
(2) Calculation of inverted camber and deflection of retrofitted beam
Prestress produces an invert arch in a retrofitted beam, so prestress retrofitting cannot
only effectively strengthen beam, but also reduce deflection. When calculating retrofitted
beam's deflection, the deflection
f
1
before stretching, the inverted camber
f
p
caused by
prestress and the deflection
f
2
caused by later load after retrofitting should be considered,
respectively, and then be superimposed together:
f
=
f
1
−
f
p
+
f
2
(3
.
20)
a. Calculation of
f
1
.
Undischarged load acts on the beam before stretching, which results in the deflection
f
1
. At this stage, the beam stiffness enhances a little with the increase of discharged load.
However, due to the original beam long-term deformation, the stiffness under long-term
load should be adopted when calculating deflection caused by undischarged load. Before the
beam is retrofitted, beam stiffness, which is changing in a certain range, has related to the
ratio of reinforcement and undischarged load and so on. For convenience, this stiffness is
suggested to be:
B
1
=(0
.
35
∼
0
.
5)
E
c
I
c
(3
.
21)
where
E
c
and
I
c
are elastic modulus of original concrete and inertia moment of transformed
section, respectively.
