Shear strength prediction—plastic method - Reinforced Concrete Deep Beams - page 197

where the contributions from the web concrete and from the reinforcement
are calculated by Eqns (8.1) and (8.2), respectively.
The work equation W E = W l gives the upper bound solution:

(8.21)

The lowest upper bound is determined by minimising with respect to the
variable angle a . A minimum is found for dV / d ( a +¦)=0, which gives:

(8.22)

Inserting into Eqn (8.21) and introducing f c * =v f c, f

= T y / bhf and cot ¦= a/h we

find:

for f£ v/2. The validity range arises from the condition a +¦ ‡p /2, together

with Eqn (8.22).
For a +¦ £p /2 we have dV/d ( a +¦) £ 0 which means that the lowest upper

bound is obtained with /2. This is the case also if a contribution to the

rate of internal work is assigned to compressed reinforcement cf. Eqn (8.2).
Thus we get:

a +¦ =p

for f‡ v/2. The situation a +¦= p /2 corresponds to a relative displacement rate
which is perpendicular to the beam axis ( Figure 8.10 ), in which case the
longitudinal reinforcement does not yield, (i.e. the beam is over-reinforced).
The upper bound solution is seen to be identical with the lower bound
solution, Eqn (8.11). This means that the flexural capacity, Eqn (8.16), is the
exact plastic solution if we have t = t l given by Eqn (8.18), and s = s l given by
Eqn (8.19). For t > t l and/or s > s l (and unchanged shear span 1), the lowest
upper bound will exceed the highest lower bound.
Figure 8.11 shows an alternative, flexural mechanism, consisting of a
clockwise rotation ? of the beam end about a point O at the distance y below
and the distance x outside the inside edge of the load platen. The rate of
external work done by the load is: W E = V ( a + s /2 + x ) h . The rate of internal
work dissipated in the mechanism is:

The work equation W E = W l gives the upper bound solution:

(8.23)