To understand the term doubly reinforced section, one should have knowledge about design of beams and moment of resistance.
The main consideration of beam is moment of resistance. We design the beams such that it will not fail due to bending moment acting on it. We provide the depth and area of beam according to the bending moment acting on it. Bending moment acting on it is resisted by developing the tension and compression forces in beam. These forces results in moment with liver arm as distance between them.
As shown in the above figure the compression force C and tension force T will produce moment with Z as liver arm.
Some times, to produce the moment of resistance equal to bending moment acting on it, liver arm has to be increased which leads to the increase in the depth of the beam.But increase in depth leads to decrease in free height between floor and beam, which causes uncomfortable to roam in room. In these case we can increase the width of the beam which can compensate the moment as compressive force = 0.36*f ck*b*x.
But due architectural consideration, to get aesthetic look both depth and width are restricted.
In that case we provide the reinforcement on the top as compressive reinforcement and in bottom as tension reinforcement. In this way we are increasing the moment of resistance by increasing the compressive force without increasing in width and depth.