Page 267 - Handbook of Civil Engineering Calculations, Second Edition

2.52 REINFORCED AND PRESTRESSED CONCRETE ENGINEERING AND DESIGN

the lateral restraint, the tendons are deformed to a wedge shape across a relatively short
distance at each end of the member. It is within this distance, termed the transmission
length, that the steel becomes bonded to the concrete and the two materials exert their pre-
stressing forces on each other. However, unless greater precision is warranted, it is as-
sumed for simplicity that the prestressing forces act at the end sections.
The tendons may be placed either in a straight line or in a series of straight-line seg-
ments, being deflected at designated points by means of holding devices. In the latter
case, prestressing forces between steel and concrete occur both at the ends and at these
deflection points.
In posttensioning, the procedure usually consists of encasing the tendons in metal or
rubber hoses, placing these in the forms, and then pouring the concrete. When the con-
crete has hardened, the tendons are tensioned and anchored to the ends of the concrete
beam by means of devices called end anchorages. If the hoses are to remain in the mem-
ber, the void within the hose is filled with grout. Posttensioning has two important advan-
tages compared with pretensioning: It may be performed at the job site, and it permits the
use of parabolic tendons.
The term at transfer refers to the instant at which the prestressing forces between steel
and concrete are developed. (In posttensioning, where the tendons are anchored to the
concrete one at a time, in reality these forces are developed in steps.) Assume for simplic-
ity that the tendons are straight and that the resultant prestressing force in these tendons
lies below the centroidal axis of the concrete section. At transfer, the member cambers
(deflects upward), remaining in contact with the casting bed only at the ends. Thus, the
concrete beam is compelled to resist the prestressing force and to support its own weight
simultaneously.
At transfer, the prestressing force in the steel diminishes because the concrete con-
tracts under the imposed load. The prestressing force continues to diminish as time elaps-
es as a result of the relaxation of the steel and the shrinkage and plastic flow of the con-
crete subsequent to transfer. To be effective, prestressed-concrete construction therefore
requires the use of high-tensile steel in order that the reduction in prestressing force may
be small in relation to the initial force. In all instances, we assume that the ratio of final to
initial prestressing force is 0.85. Moreover, to simplify the stress calculations, we also as-
sume that the full initial prestressing force exists at transfer and that the entire reduction
in this force occurs during some finite interval following transfer.
Therefore, two loading states must be considered in the design: the initial state, in
which the concrete sustains the initial prestressing force and the beam weight; and the fi-
nal state, in which the concrete sustains the final prestressing force, the beam weight, and
all superimposed loads. Consequently, the design of a prestressed-concrete beam differs
from that of a conventional type in that designers must consider two stresses at each
point, the initial stress and the final stress, and these must fall between the allowable com-
pressive and tensile stresses. A beam is said to be in balanced design if the critical initial
and final stresses coincide precisely with the allowable stresses.
The term prestress designates the stress induced by the initial prestressing force. The
terms prestress shear and prestress moment refer to the vertical shear and bending mo-
ment, respectively, that the initial prestressing force induces in the concrete at a given
section.
The eccentricity of the prestressing force is the distance from the action line of this re-
sultant force to the centroidal axis of the section. Assume that the tendons are subjected to
a uniform prestress. The locus of the centroid of the steel area is termed the trajectory of
the steel or of the prestressing force.
The sign convention is as follows: The eccentricity is positive if the action line of the
prestressing force lies below the centroidal axis. The trajectory has a positive slope if it