Limit of eccentricity for solid rectangular section

>>Along breath

Let consider a column is subjected to compressive load ‘P’ having cross section dimensions breath ‘b’ and thickness ‘d’ as shown in fig. 4.a.

Let,

σ₀= Direct stress;

σ_b = Bending stress;

P = Applied load (compressive or tensile);

A = Cross-sectional area
A= b x d;

M = Bending Moment acting on column
M= P.e_yy;

e_yy = Eccentricity w.r.t. Y-axis;

I_yy = Moment of inertia of column section along Y-axis
$I_{yy}=\frac{db^{3}}{12}$ ;

y_max = Max. centroidal distance;
$y_{max}=\frac{b}{2}$

Z = Section modulus;
$Z=\frac{\left( \frac{db^3}{12}\right)}{\left(\frac{b}{2}\right)}=\frac{db^3}{12}\times\frac{2}{b}=\frac{db^2}{6}$

Now, for no tension condition;
$\sigma_{0}=\sigma_{b}$
$\frac{P}{A}=\frac{M}{Z}=\frac{P\cdot e_{yy}}{\frac{db^{2}}{6}}$
$e_{yy}=\frac{P}{A} \times \frac {db^{2}}{6} \times \frac {1}{P}$
$e_{yy}=\frac {db^{2}}{6A} = \frac {db^{2}}{6 \cdot bd}=\frac{b}{6}$
This eccentricity when load ‘P’ on right side of Y-axis. Similarly eccentricity on left side of Y-axis i.e. b/6. Therefore total width of eccentricity at middle portion of section becomes
$=\frac{b}{6}+\frac{b}{6}=\frac{b}{3}$

>>Along thickness

As eccentricity calculates in breath same way we can find eccentricity along thickness. It becomes
$e_{xx}=\frac{d}{6}$
Total width of eccentricity at middle portion of section
$=\frac{d}{3}$
Kernel or Core Section: By jointing limiting eccentricity we get a section that section is called as ‘Kernel or Core Section’.
For rectangular section width of Kernel Section is b/3 along breadth and d/3 along thickness or depth as shown in fig.4.b.