Page 362 - Aircraft Stuctures for Engineering Student

9.10 Deflection of open and closed section beams 343

y2 dA = Ixx and SA xy dA = Iry , we have

+(MX.lIYY - My,lIxy)(K,OIyy - My?oIx,)Ixx
+[(Kv*lIrx - Mx,lIxy)(Mx,oIyy - My:~I.vy)
+(Mx:Jyy - ~y,lIxywy~oIxx - ~x.oLy)lITyl dz (9.85)

For a section having either the x or y axis as an axis of symmetry, Ixy = 0 and Eq.
(9.85) reduces to

(9.86)

The derivation of an expression for the shear deflection of thin-walled sections by
the unit load method is achieved in a similar manner. By comparison with Eq. (9.84)
we deduce that the deflection As, due to shear of a thin-walled open or closed section
beam of thickness t, is given by
)
As = JL ( jxction 71YOtdS dz (9.87)

where T~ is the shear stress at an arbitrary point s around the section produced by a
unit load applied at the point and in the direction As, and ^lo is the shear strain at the
arbitrary point corresponding to the actual loading system. The integral in paren-
theses is taken over all the walls of the beam. In fact, both the applied and unit
shear loads must act through the shear centre of the cross-section, otherwise
additional torsional displacements occur. Where shear loads act at other points
these must be replaced by shear loads at the shear centre plus a torque. The thickness
t is the actual skin thickness and may vary around the cross-section but is assumed to
be constant along the length of the beam. Rewriting Eq. (9.87) in terms of shear flows
q1 and qo. we obtain

(9.88)
as = JL (Isection Grds ) dz
qo4'
where again the suffixes refer to the actual and unit loading systems. In the cases of
both open and closed section beams the general expressions for shear flow are long
and are best evaluated before substituting in Eq. (9.88). For an open section beam
comprising booms and walls of direct stress carrying thickness tD we have, from
Eq. (9.75)

(9.89)