RCC Singly Reinforced Beams-working stress method

Hi, For design of the structures, we have different methods but here I am going to discuss here with you the working stress method of design of the singly reinforced beams.
Singly reinforced rectangular beams are the beams which are reinforced either at the top or at the bottom. Reinforcement is provided at top in case of the cantilever beams otherwise we provide the reinforcement generally at the bottom.

There are few assumptions in this method and some of the most important are:
All the tension is taken up by the reinforcement provided, so the tensile force of the concrete in the tensile section is ignored.

Critical Neutral Axis(Xc):  Location of the Critical neutral axis is found out using the geometrical relations between the stresses in the compression side and the tensile sides, which are assumed to vary linearly with the depth of the beam. Stresses are zero at the critical neutral axis and compressive on  one side and tensile on the other varying linearly. So there are two similar triangles formed and one can find out the location of the neautral axis using the relations between the various sides of the similar triangles.

Xc= mc/(t+mc) * d

here, Xc= Critical Neutral axis
m= modular ratio = 280/(3*compressive stress in concrete in bending)
c= maximum compressive stress in concrete
d= effective depth of the beam section

Actual Neutral axis(Xa):
Actual neutral axis of the section may be found by equating the moment of the areas on the two sides of the neutral axis about the neutral axis. On the compressive side the area of the concrete is considered  and on the tensile side the equivalent area of the steel reinforcement is taken and its moment is taken about the neutral axis. By equating the two we can get the actual neutral axis.

Balanced Section:
When Xa=Xc, it is a balanced section    (c'=c, ta=t)

Over-reinforced section:
When  Xa>Xc, it is a over- reinforced section   (c'= c, ta< t)

Under-reinforced section:
when Xa<Xc                                            (c'<c, ta=t)                                          

where,  (Xa= Actual neutral axis, Xc=Critical Neutral Axis
 c'- actual compressive stess in concrete, c- maximum compressive stress in concrete, ta= actual tensile stress in steel, t= maximum tensile stress in steel)
Thanks for your visit!