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Balanced, under, over sections with strain diagrams

Short answer

Balanced: x_u = x_u,max, concrete strain 0.0035 and steel at yield strain, simultaneous failure. Under-reinforced: x_u < x_u,max, steel strain exceeds yield, ductile failure. Over-reinforced: x_u > x_u,max, steel strain below yield, concrete crushes suddenly. Only under-reinforced or balanced sections are permitted in design.

Differentiate among the balanced, under reinforced and over reinforced section in a rectangular reinforced concrete section in limit state method with corresponding strain diagram.

PointBalancedUnder-reinforcedOver-reinforced
Neutral axisxu=xu,maxx_u=x_{u,max}xu<xu,maxx_u<x_{u,max}xu>xu,maxx_u>x_{u,max}
Steel areaAst=Ast,limA_{st}=A_{st,lim}Ast<Ast,limA_{st}<A_{st,lim}Ast>Ast,limA_{st}>A_{st,lim}
Concrete strain0.00350.00350.0035
Steel strainεy=0.87fyEs+0.002\varepsilon_y=\dfrac{0.87f_y}{E_s}+0.002>εy>\varepsilon_y<εy<\varepsilon_y
Steel stress0.87fy0.87f_y0.87fy0.87f_y<0.87fy<0.87f_y
First to failBoth togetherSteel yieldsConcrete crushes
FailureSimultaneousDuctile, with warningSudden, brittle
Moment0.36fckbxu,max(d−0.42xu,max)0.36f_{ck}bx_{u,max}(d-0.42x_{u,max})0.87fyAstd(1−Astfybdfck)0.87f_yA_{st}d(1-\dfrac{A_{st}f_y}{bdf_{ck}})Limited to Mu,limM_{u,lim}

Strain diagrams (all with concrete strain 0.0035 at the top):

 Under-reinforced   Balanced        Over-reinforced
 0.0035             0.0035          0.0035
 |\                 |\              |\
 | \                | \             |  \
 |  \ xu<xumax      |  \ xu=xumax   |    \ xu>xumax
 +----  NA          +----  NA      +-------  NA
 |   \              |   \           |      \
 |    \ es>ey       |    \ es=ey    |       \ es<ey

The depth of the neutral axis is found from the strain diagram: xu,maxd=0.00350.0035+εy\dfrac{x_{u,max}}{d}=\dfrac{0.0035}{0.0035+\varepsilon_y}, which gives 0.53, 0.48 and 0.46 for Fe250, Fe415 and Fe500. Only under-reinforced (or balanced) sections are accepted in design.

✓ Answer checked by Claude Opus 5.5 (high effort)

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