Civil Engineering Reference
In-Depth Information
Figure 8.12
Failure mechanism with hyperbolic yield line.
The rate of external work is found by considering the displacement rate
of the support reaction relative to the mid section of the beam:
W
E
=vß
+¦) -
a
/2-
s
/2)
The rate of internal work is found from Eqn (8.1):
h
(-(
h
/2-c) tan(
a
The reinforcement does not contribute to the rate of internal work because
the relative displacement rate is perpendicular to the reinforcement at the
intersection with the yield line.
The work equation
W
E
=
W
I
gives the upper bound solution:
(8.24)
The lowest upper bound solution is determined by minimising with respect
to the variable angle
a
. The condition
dV/d
a
=0 gives:
Inserting into Eqn (8.24) and introducing and cot¦=
a/h
we recover
Eqn (8.15). Thus also in this case is the upper bound solution identical with
the lower bound.
