1.
STRENGTH DESIGN METHOD
Let therefore,
Let and (steel yields at failure)
2.
If the steel does not yield at failure (larger steel ratios)
From the above quadratic equation value of c can be determined
When c and are known the moment capacity can be calculated by
At balance failure the concrete strain reaches and steel
stress is (i.e. )
3.
Then and
Design actions (forces/moments)
It can be noted above that the actual geometric shape of the
concrete compressive stress distribution varies considerably
and that, in fact, one need not know this shape exactly,
provided one does know two things:
1. the magnitude C of the resultant of the concrete
compressive stresses and
2. the location of this resultant.
Information on these two quantities was obtained from the
results of experiment al research and expressed in the two
parameters and .
4.
In the ACI 318 methodology, the actual stress distribution is
replaced by an equivalent one of simple rectangular outline. The
intensity of this equivalent constant stress and its depth
are easily calculated from the two conditions that (1) the
total compression force C and (2) its location, i.e., distance from
the top fiber, must be the same in the equivalent rectangular as in
the actual stress distribution.
From above fig. the first condition gives:
from which, , with and . The second
condition simply requires that in the equivalent rectangular stress
block, the force C be located at the same distance from the
top fiber as in the actual distribution. It follows that .
,
and
5.
Balanced Condition:
Under-reinforced Beams:
(such that steel yields before balanced condition is reached)
ACI provisions for under-reinforced beams:
The ACI code ensures use of under reinforced beams by further
encouraging use of higher reduction factors for larger values of yield
strains in the reinforcement ( at )
The corresponding steel ration is given by:
6.
Simple relations for analysis of singly reinforced beam
, where
For practice design purposes the relationships can be written as:
By imposing the strength reduction factor
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