By straight line proportion (figure 93(b)) we can calculate the strain in
compression
steel when the extreme concrete fiber has a compressive strain of
0.003.
ε_{s}′ = ( x  d′ ) (0.003) / x =
(4.997  2.5) (0.003) / 4.997 = 0.0015
yield strain of steel, ε_{y} = f_{y}
/E_{s} = 60 / 29000 = 0.00207
ε_{s}′ is
less than ε_{y} ,
this means that compression steel has not yielded before crushing of concrete.
hence the assumption is not confirmed and therefore the calculated value of x is
not valid.
In such cases when compression steel has not yielded we have to find
the depth of neutral axis by considering the equilibrium of forces
acting on the section;
Force in compression steel will be calculated on the basis of
actual stress f_{s}′ in compression rebar
at the time of crushing of concrete.
f_{s}′ = ε_{s}′
* E_{s} ={( x  d′ ) (0.003) / x}*29000
C_{s} = (f_{s}′  0.85 f_{c}′)
A_{s}′ =[{( x  2.5 ) (0.003) / x}*29000
 0.85 * 5] *1.2
C_{c} = 0.85 f_{c}′ ba
= 0.85 *5 *14 *(0.8*x)
T = A_{s} f_{y}
= 5.08*60 = 304.8 kip
Apply equilibrium of froces at the section;
C_{c} + C_{s} = T;
substituting the values of C_{c}, C_{s}, T in the above equation and simplifying
we get;
47.6 x^{2}
 205.5 x
 261 = 0;
the above quadratic equation is solved to get the positive
value of x;
therefore x
=
5.34 in.
a
= β_{1} x
= 0.8 * 5.34 = 4.27 in.
