F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

Chapter 10
Prestressed concrete continuous beams
10.1 Primary and secondary moments
In prestressed concrete, one important difference between simple beams
and continuous beams is that, in the latter, prestressing generally induces
support reactions. Consider the simple beam in Fig. 10.1-1(a). If for the
time being, we do not consider the effects of the dead and imposed loads,
then whatever the magnitude of the prestressing force and the tendon
profile, there will be no reactions at the supports A and B.* Of course, the
prestressing force produces a moment -Pees, where the negative sign
is used because Pees is a hogging moment for positive values of es
(Fig. 10.1-1(b)). This moment is called the primary moment, M 1 • For the
tendon profile shown here, the primary moment causes the beam to deflect
upwards (Fig. 10.1-1( c)). Suppose the upward deflection at a point C is
ac. If the beam is restrained against deflection at C by an additional
support (Fig. 10.1-1( d)), then the support C must exert a reaction Rc on
the beam. Rc also induces reactions RA and R8 , so that the three reactions
form an equilibrium set of forces; these support reactions cause a
secondary moment, M 2, to act on the beam (Fig. 10.1-1(e)) such that
the downward deflection at C due to M 2 is numerically equal to ac. The
algebraic sum of the primary moment and the secondary moment is called
the resulting moment (Fig. 10.1-1( e)):
M3 =
resulting
moment
Mt +
primary
moment
Mz
secondary
moment
(10.1-1)
where, it should be pointed out, the so-called secondary moment may
sometimes be of larger magnitude than the primary moment.
At any section of the continuous beam, the effect of the prestressing
force Pe and the resulting moment M3 is equivalent to that of a force Pe
acting at an eccentricity eP, where
ep = -M3/Pe (10.1-2)
(If for any reason, such as friction, Pe cannot be regarded as constant along
• It is assumed that provisions are made to permit horizontal displacement at supports.