Civil Engineering Reference
In-Depth Information
For simply supported beam,
f
1
can be approximately calculated in the following formula:
48
α
ML
2
5
f
1
=
(3
.
22)
B
1
where
α
is the influence coecient of discharged load. The original beam has been unloaded,
but the deflection cannot be recovered entirely because of concrete's plastic property and
creep deformation etc., so
α
can usually be adopted as 1.1.
b. Calculation of
f
p
.
At the initial phase of stretching, owing to the existence of cracks at the bottom of beam,
inverse stiffness is small and inverse deflection develops rapidly. With prestress increasing,
cracks tend to close, stiffness enlarges and inverse deflection grows slowly. For simplification,
stiffness is better considered as a constant. Considering that too large calculating value of
inverted camber will influence members' safety, when calculating inverted camber using
structural mechanics method, beam stiffness is suggested as the following formula:
B
p
=0
.
75
E
c
I
c
(3
.
23)
c. Calculation of
f
2
.
When retrofitting is finished, beam produces deflection
f
2
under the later load. To calcu-
late
f
2
, stiffness can be accounted as follows:
The beam with prestressed reinforcements in the external, retrofitted structure becomes
a composite structure in fact, in which tensile bar is prestressed reinforcement, while the
original beam works as an arch. The deflection can be calculated with structural mechanics.
For simplification, the method above-mentioned can also be used to calculate this deflection,
and retrofitted beam stiffness is suggested to be calculated by the following formula:
B
2
=(0
.
7
∼
0
.
8)
E
c
I
c
(3
.
24)
3. Bearing capacity calculation of retrofitted beam
(1) Bearing capacity calculation of normal section
As to retrofitted beam with prestressed reinforcements after being retrofitted, prestressed
reinforcements contact with original beam only at anchoring point and supporting point.
When the beam deflects with the increasing of loads, original reinforcement in beam elon-
gates with the increasing of curvature of original beam; however, deformation of prestressed
reinforcements is not the same as that of original reinforcements, and relates only to the
beam deflection at supporting point and anchoring point. Certain research indicates that
stress increment ratio in prestressed reinforcements is far less than that in original reinforce-
ment (only 18%
35%). Because of this deformation incongruity, equivalent load method is
used to calculate sectional bearing capacity of this kind of retrofitting beam.
Equivalent load means that action by prestress in original beam can be replaced by cor-
responding load, and the internal forces (moment, shear force and axial force) of original
beam are equal. Equivalent load method is to apply prestress as equivalent load on original
beam in calculating bearing capacity of retrofitted beam, and then verify bearing capacity
of original beam according to its size and distributed steel condition.
The stress of prestressed reinforcements is the sum of the stress when stretching is fin-
ished and the stress increment that is caused by later load. However, this kind of stress
increment is actually very small, and is almost the same value (about 5.35 MPa) per in-
creasing unit load, which exerts little influence on the stress of prestressed reinforcements.
When high strength prestressed reinforcements are adopted, areas of reinforcements become
smaller, so this influence to the total prestress internal force becomes even less. For con-
venient calculation, this stress increment can be neglected in design, and internal force of
∼
