Condition for No Tension in the Section:

Middle Third Rule

In Figure (b) f₀ < f_b and thus, stress changes sign, being partly tensile & partly compressive across the section. In masonry & concrete structures, the development of tensile stress in the section is not desirable, as they are weak in tension. This restricts the eccentricity e to a firm value which shall be investigated now for different sections.

In order that the stress might not change sign from compressive to tensile, we have

 f₀  ≥  f_b

i.e.    P/A ≥ (Pe /I )× (d/2)

i.e.     P/A ≥      Pe d/ (2 AK ²)

 or        e ≤  2k ² /d

here, k = radius of gyration of section with regard to N.A. & d is the depth of the section.

Therefore, for no tension in the section, the eccentrically should not exceed

 Now, Let us take a rectangular section and discover out the limiting value of e. For a rectangular section of width b & depth d,

I =  (1 / 12) bd ³     and   A = bd

k ²  =  I / A = d²/12

Putting in Eq. (7.5), we obtain

e ≤       2d ²/ d × 12 ≤ d/6

∴          I _max     = d/ 6

The value of eccentricity might be on either side of the geometrical axis. Therefore, the stress shall be of the same sign during the section if the load line is in the middle third of the section.

In particular case of rectangular section, the maximum intensities of extreme stresses are given by following

f  = P/ A ±  Pl / Z xx   =  (P/ bd ) ± 6 pe/bd ²

=  (P/ bd)[1 ± (6e/d) ]

Core or Kernel of a Section

If the line of action of the stress is on neither of the centre lines of the section, the bending is unsymmetrical. Though, there is certain area within the line of action of the force P must cut the cross-section if the stress is not to become tensile. This area we call it as 'core' or 'kernel' of the section. Let us determine this for a rectangular section.