Question 23.1: A cantilever beam (Fig. 23.1) carries concentrated loads as ...

We note that stiffeners HKD and JK are required at the point of application of the 4,000 N load to resist its vertical and horizontal components. A further transverse stiffener GJC is positioned at the unloaded end J of the stiffener JK, since stress concentrations are produced if a stiffener ends in the center of a web panel. We note also that the web panels are effective only in shear, so that the shear flow is constant throughout a particular web panel; the assumed directions of the shear flows are shown in Fig. 23.1. It is instructive at this stage to examine the physical role of the different structural components in supporting the applied loads. Generally, stiffeners are assumed to withstand axial forces only, so that the horizontal component of the load at K is equilibrated locally by the axial load in the stiffener JK and not by the bending of stiffener HKD. By the same argument, the vertical component of the load at K is resisted by the axial load in the stiffener HKD. These axial stiffener loads are equilibrated in turn by the resultants of the shear flows q_{1} and q_{2} in the web panels CDKJ and JKHG. Thus, we see that the web panels resist the shear component of the externally applied load and at the same time transmit the bending and axial load of the externally applied load to the beam flanges; subsequently, the flange loads are reacted at the support points A and E. Consider the free body diagrams of the stiffeners JK and HKD shown in Fig. 23.2. From the equilibrium of stiffener JK, we have

\left(q_{1}-q_{2}\right) \times 250=4,000 \sin 60^{\circ}=3,464.1 N  (i)

and, from the equilibrium of stiffener HKD,

200 q_{1}+100 q_{2}=4,000 \cos 60^{\circ}=2,000 N  (ii)

Solving Eqs. (i) and (ii), we obtain

The vertical shear force in the panel BCGF is equilibrated by the vertical resultant of the shear flow q_{3}. Thus,

from which

Alternatively, q_{3} may be found by considering the equilibrium of the stiffener CJG. From Fig. 23.3,

or

from which

The shear flow q_{4} in the panel ABFE may be found using either of the above methods. Thus, considering the vertical shear force in the panel,

from which

Alternatively, from the equilibrium of stiffener BF,

from which

The flange and stiffener load distributions are calculated in the same way and are obtained from the algebraic summation of the shear flows along their lengths. For example, the axial load P_{ A } at A in the flange ABCD is given by

or

Similarly,

that is,

The complete load distribution in each flange is shown in Fig. 23.4. The stiffener load distributions are calculated in the same way and are shown in Fig. 23.5. The distribution of flange load in the bays ABFE and BCGF could be obtained by considering the bending and axial loads on the beam at any section. For example, at the section AE, we can replace the actual loading system by a bending moment

and an axial load acting midway between the flanges (irrespective of whether or not the flange areas are symmetrical about this point) of

Thus,

and

This approach cannot be used in the bay CDHG except at the section CJG, since the axial load in the stiffener JK introduces an additional unknown.