Structural Concrete - Hassoun

19.10 End-Block Stresses 777
The compressive force, C, on the section is equal to the tension force, T:
C = T = A ps f se
In terms of stresses,
C
= A ps f se
= f
A c A
c1
c
where f c1
is an assumed uniform stress on the section.
For a preliminary design, a triangular stress distribution is assumed with maximum allowable
compressive stress, f ca , on one extreme fiber; therefore, the average stress is 0.5 f ca = f c1
.
The allowable compressive stress in concrete is f ca = 0.45 f c. ′ Thus, the required concrete area, A c ,
can be estimated from the force, T, as follows:
A c = T
f c1
A c =
= A ps f se
f c1
T
0.5 f ca
=
= A ps f se
0.5 f ca
= A ps f se
0.225 f ′ c
M T
(0.65h)(0.5 f ca ) =
M T
0.33 f ca
= M T
0.15 f ′ c
(19.62)
(19.63)
This analysis is based on the design for service loads and not for the factored loads. The eccentricity,
e, is measured from the centroid of the section to the centroid of the prestressing steel and can be
estimated approximately as follows:
e = K b + M D
(19.64)
F i
where K b is the lower Kern limit and M D is the moment due to the service dead load.
19.10 END-BLOCK STRESSES
19.10.1 Pretensioned Members
Much as a specific development length is required in every bar of a reinforced concrete beam, the
prestressing force in a prestressed concrete beam must be transferred to the concrete by embedment
or end anchorage or a combination thereof. In pretensioned members, the distance over which
the effective prestressing force is transferred to the concrete is called the transfer length, l t .After
transfer, the stress in the tendons at the extreme end of the member is equal to 0, whereas the stress
at a distance l t from the end is equal to the effective prestress, f pe . The transfer length, l t , depends
on the size and type of the tendon, surface condition, concrete strength, f c, ′ stress, and method of
force transfer. A practical estimation of l t ranges between 50 and 100 times the tendon diameter.
For strands, a practical value of l t is equal to 50 tendon diameters, whereas for single wires, l t is
equal to 100 wire diameters.
In order that the tension in the prestressing steel develop full ultimate flexural strength, a bond
length is required. The purpose is to prevent general slip before the failure of the beam at its full
design strength. The development length, l d , is equal to the bond length plus the transfer length,
l t . Based on established tests, the ACI Code, Section 25.4.8, gives the following expression for
computing the development length of seven-wire pretensioning strands:
l d (in.) =
(
fse
3000
)
d b −
( fps − f se
1000
)
d b (19.65)