## Doubly Reinforced Beam

- Beams reinforced with steel in compression and tension zones are called doubly reinforced beams. This type of beam will be found necessary when due to headroom consideration or architectural consideration the depth of the beam is restricted.
- The beam with its limited depth, if reinforced on the tension side only, may not have enough moment of resistance, to resist the bending moment.
- By increasing the quantity of steel in the tension zone, the moment of resistance cannot be increased indefinitely. Usually, the moment of resistance can be increased by not more than 25% over the balanced moment of resistance, by making the beam over-reinforced on the tension side.
- Hence, in order to further increase the moment of resistance of a beam section of unlimited dimensions, a doubly reinforced beam is provided.

### Depth of Neutral Axis

0.36f_{ck}bx_{u} + Asc (f_{sc }– f_{cc}) = 0.87f_{y}A_{st}

### Moment of Resistance

M_{u} = 0.36f_{ck}bx_{u }(d-0.42x_{u}) + Asc (f_{sc} -f_{cc}) (d-d’)

If d’/dis between 0.05-0.2, fcc is taken equal to 0.45f_{ck}.

## Design Procedure

Limiting Moment of Resistance using Singly Reinforced Beam Formula,

M_{u, lim} = 0.87f_{y}A_{st1} [d-0.42x_{u, limit}]

M_{u, lim }= 0.36f_{ck}x_{u, limit }b (d-0.42x_{u, limit})

Or when Section is Balanced,

A_{st1} = (0.36f_{ck}bx_{u, limit}) / 0.87f_{y}

Where, d = Effective Depth

X_{u, limit }= Limiting Depth of Neutral Axis

If Factored or Given Moment is Greater than the Limiting Moment of Resistance (M_{u} > M_{u, lim}), then doubly reinforced beam is required,

M_{2} = M_{u} – M_{u, lim}

Area of Steel in Compression,

A_{sc} = (M_{u} – M_{u, lim}) / f_{sc}(d-d’)

For the additional Moment, Area of steel Required

A_{st2} = A_{sc} f_{sc }/ 0.87f_{y}

Total Tension Steel Required,

A_{st} = A_{st1} + A_{st2}

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