## Ixl

where A2 = Gt(2B + A)/dEAB. Examination of Eq. (11.38) shows that q changes sign when cosh Az = (sinh XL)/XL, the solution of which gives a value of z less than L, i.e.

Finally, the shear flow distributions are obtained from Eqs (11.39), thus _-0Pm_ PAX sinh X(L — z)

Again we see that each expression for direct stress, Eqs (11.47), (11.49) and (11.51). comprises a term which gives the solution from elementary theory together with a correction for the shear lag effect. The shear flows c/| and q2 are self-equilibrating, as can be seen from Eqs (11.52) and (11.53), and are entirely produced by the shear lag effect (q\ and q2 must be self-equilibrating since no shear loads are applied).

### 11.5 Constraint of open section beams

Instances of open section beams occurring in isolation are infrequent in aircraft structures. The majority of wing structures do, however, contain cut-outs for undercarriages, inspection panels and the like, so that at these sections the wing is virtually an open section beam. We saw in Chapter 10 that one method of analysis for such cases is to regard the applied torque as being resisted by the differential bending of the front and rear spars in the cut-out bay. An alternative approach is to consider the cut-out bay as an open section beam built-in at each end and subjected to a torque. We shall now investigate the method of analysis of such beams.

If such a beam is axially unconstrained and loaded by a pure torque T the rate of twist is constant along the beam and is given by

We also showed in Section 9.6 that the shear stress varies linearly across the thickness of the beam wall and is zero at the middle plane (Fig. 11.22). It follows that although the beam and the middle plane warp (we are concerned here with primary warping), there is no shear distortion of the middle plane. The mechanics of this warping are more easily understood by reference to the thin-walled I-section beam of Fig. 11.23(a). A plan view of the beam (Fig. 11.23(b)) reveals that the middle plane of each flange remains rectangular, although twisted, after torsion. We now observe the effect of applying a restraint to one end of the beam. The flanges are no longer free to warp and will bend in their own planes into the shape shown in plan in

Fig. 11.22 Shear stress distribution across the wall of an open section beam subjected to torsion

Middle plane

Middle plane

Fig. 11.22 Shear stress distribution across the wall of an open section beam subjected to torsion

Fig. 11.23 (a) Torsion of l-section beam; (b) plan view of beam showing undistorted shape of flanges.

Fig. 11.23 (a) Torsion of l-section beam; (b) plan view of beam showing undistorted shape of flanges.

Fig. 11.24. Obviously the beam still twists along its length but the rate of twist is no longer constant and the resistance to torsion is provided by the St. Venant shear stresses (unrestrained warping) plus the resistance of the flanges to bending. The total torque may therefore be written T = Tj + Tr, where Tj = GJ dd/dz from the unconstrained torsion of open sections but in which d6/dz is not constant, and Tr is obtained from a consideration of the bending of the flanges. It will be instructive to derive an expression for Tr for the I-section beam of Fig. 11.25 before we turn our attention to the case of a beam of arbitrary section.

Suppose that at any section z the angle of twist of the I-beam is 6. Then the lateral displacement u of the lower flange is

Fig. 11.24 Bending effect of axial constraint on flanges of l-section beam subjected to torsion,

The integral in this equation is evaluated by substituting pR = (d/d?)(2^4R) and integrating by parts. Thus

At each open edge of the beam qT, and therefore Jo 2^4Rfds, is zero so that the integral reduces to — Jc AA%_tds, giving

d3_e dz3

where rR = Jc 4y4R/ds, the torsion-bending constant, and is purely a function of the geometry of the cross-section. The total torque T, which is the sum of the St. Venant torque and the Wagner torsion bending torque, is then written d9 d30

(Note: Compare Eq. (11.59) with the expression derived for the I-section beam.)

In the expression for rR the thickness t is actually the direct stress carrying thickness tD of the beam wall so that rR, for a beam with n booms, may be generally written

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