Design of Doubly Reinforced Beams

Strain Compatibility in Doubly Reinforced Beams

Tension steel in one layer or more.Compression steel in one layer.Tensile steel assumed yielding but in analysis problems, this may be untrue Compression steel may yield or not.Strain expressions are derived from strain compatibility of plane sections (use of similar triangles).

= Area, depth and strain at tension steel centroid

= Depth and strain at bottom tension layer (max. depth)

= Depth and strain at minimum depth tension layer

= Area, depth and strain at compression steel

Strain Compatibility in Doubly Reinforced Beams

Strain expressions are derived using similar triangles :

Centroid tension

Bottom tension

Min.depth tension

Compression

Compression steel stress

Nominal Moment 

Assume tension steel has yielded

Assume compression steel has yielded

Compute stress block depth a and neutral axis depth c

Compute compression and tension steel strains

If compression and tension steel have yielded, OK continue,perform strain checks and compute nominal moment

If compression steel has not yielded, express unknown stress block depth and compression steel stress in terms of neutral axis depth c and solve resulting quadratic equation. Perform strain checks and compute nominal moment.

If tension steel has not yielded, use adapted method

(Compression steel has not yielded)

Analysis  (Compression steel has not yielded)

Multiplying all terms by c and assembling leads to a quadratic equation with respect to the neutral axis depth c :

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