Page 376 - Aircraft Stuctures for Engineering Student

Problems 357

Fig. P.9.24
Am. de/& = 29.3 x rad/mm, w3 = -w4 = -0.19 mm,

wz = - ~1 = -0.056m

P.9.25 A uniform beam with the doubly symmetrical cross-section shown in Fig.
P.9.25, has horizontal and vertical walls made of different materials which have shear
moduli G, and Gb respectively. If for any material the ratio mass density/shear
modulus is constant find the ratio of the wall thicknesses tu and tb, so that for a
given torsional stiffness and given dimensions a, b the beam has minimum weight
per unit span. Assume the Bredt-Batho theory of torsion is valid.
If this thickness requirement is satisfied find the a/b ratio (previously regarded as
fixed), which gives minimum weight for given torsional stiffness.
Ans. tb/ta = Gu/Gb, b/a = 1.

Fig. P.9.25

P.9.26 Figure P.9.26 shows the cross-section of a thin-walled beam in the form of
a channel with lipped flanges. The lips are of constant thickness 1.27 mm while the
flanges increase linearly in thickness from 1.27mm where they meet the lips to
2.54mm at their junctions with the web. The web has a constant thickness of
2.54 mm. The shear modulus G is 26 700 N/mmz throughout.
The beam has an enforced axis of twist RR' and is supported in such a way that
warping occurs freely but is zero at the mid-point of the web. If the beam carries a
torque of 100Nm, calculate the maximum shear stress according to the St. Venant