STRENGTH DESIGN METHOD

Let therefore,

Let and (steel yields at failure)

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. )

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 .

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

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:

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

Design Aids: