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' Sx,l?xy

Ixx^yy

Similar expressions are obtained for a closed section beam from Eq. (9.80).

Example 9.15

Calculate the deflection of the free end of a cantilever 2000 mm long having a channel section identical to that in Example 9.13 and supporting a vertical, upward load of 4.8 kN acting through the shear centre of the section. The effective direct stress carrying thickness of the skin is zero while its actual thickness if 1 mm. Young's modulus E and the shear modulus G are 70 000 N/mm2 and 30 000 N/mm2 respectively.

The section is doubly symmetrical (i.e. the direct stress carrying area) and supports a vertical load producing a vertical deflection. Thus we apply a unit load through the shear centre of the section at the tip of the cantilever and in the same direction as the applied load. Since the load is applied through the shear centre there is no twisting of the section and the total deflection is given, from Eqs (9.86), (9.88), (9.89) and (9.90), by

JO Elxx Jo \ J section Gt J

and z is measured from the built-in end of the cantilever. The actual shear flow distribution has been calculated in Example 9.13. In this case the q\ shear flows may be deduced from the actual distribution shown in Fig. 9.52, i.e.

qx = <7„/4.8 x 103 Evaluating the bending deflection, we have f20oo 4 8 x io3(2000 — z)2dz

JO 70000 X 48 x 106 The shear deflection As is given by

3.81 mm

0 0

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