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)

Analysis of prestressed continuous beams: elastic theory 383
a2. p; a3. p; ... a;, p ... an_ 1. p respectively
Also, for the beam simply supported at 1 and n, let a;i be the upward
deflection at support position i due to a unit upward force at position j.
Since in fact there is no deflection at the typical support i, the following
compatibility condition is satisfied:
a; 2R 2 + a;3R 3 + . . . + a;;R; + ... a;, n _ 1Rn _ 1 + a;, p = 0
(10.2-1)
where R; is the upward reaction at support i of the continuous beam caused
by the primary moment; the coefficients a;j and a;, P are deflection
coefficients of the simple beam and are hence readily determined.
Equation (10.2-1) is satisfied for all values of i from i = 2 to i = n - L
Therefore, we have n - 2 simultaneous equations to solve for the n - 2
support reactions, using standard computer routines if necessary. This
method of analysis is of general applicability. However, for practical
purposes, a more convenient method is available. Before describing this
method, brief reference will be made to some well-known relations in
elementary structural mechanics. Referring to the short length of beam in
Fig. 10.2-2, the shear force V, the bending moment M and the load q are
related by
V = dM/dx
q = -dV/dx = -d 2 M/d.x 2
(10.2-2)
(10.2-3)
J: q dx = J: - d V = VA - VB
[~]A- [d:L
(10.2-4)
The following general statements may be made:
(a) The load between two sections of a beam is equal to the change in
slope of the bending moment diagram between the points (see eqn
10.2-4).
(b) If the curvature of the bending moment diagram is constant between
the two sections, then the load is uniformly distributed (see eqn
10.2-3). In practice where the tendon profile is circular or parabolic,
or is approximately circular or parabolic, the load due to prestressing
is taken as uniformly distributed.
(c) If the bending moment diagram for a certain length of the beam
q
M IUUHWM+dM
t(l l)l
1--dx--l
V V+dV
Fig.10.2-2