State of stress at the junction of flange:

A beam of I-section as illustrated in Figure (a) is subjected to a bending moment of

84.928 kN m and shear force of 106.1 kN. Verify the state of stress at the junction of flange & web.

(a)                                                            (b)

 Figure

Solution

The neutral axis of the section shall be the horizontal axis through the centroid of the section.

The bending stress at any layer 'y' above the neutral axis is specified as,

s_x  =    - My / I

where, I is refer to the moment of inertia of the section and M is refer to the bending moment on the section.

The shear stress at any particular layer is specified as

where,is the moment of the area of that location of the section above or below the layer regarding the neutral axis, b is the breadth of the layer and Q, the shear force at the section.

For the cross-section specified, the moment of inertia is specified as 2.1232 x 10⁸ mm⁴. Total depth of beam = 400 mm.

∴ Neutral axis is at a depth of 200 mm from top.

The distance of the layer at junction of web & flange through the neutral axis is 180 mm.

Determination of Bending Stress

s_x   = - My /I =  - 84.928 x 10⁶ x180 /2.1232 x 10⁸  = - 72 N/mm²

Determination of Shear Stress

In this case   shall be given by the moment of the flange area regarding the neutral axis.

= 120 x 20 x (180 + (20 /2)) = 4.56 x 10⁵  mm³

∴          t_xy        = (106.1 x 10³x4.56 x 10⁵)/(2.1232 x 10⁸ x 10 )=       22.8 N/mm²

Therefore, we obtain,

s₁  = 6.613 N/mm2

s₂  = - 78.613 N/mm2

t_max  = 42.613 N/mm2

You might obverse here that the junction among flange and web shall become the critical zone, while the shear force at the section is considerable. Though, experience shall tell you, where to look for critical zones in different cases.