Aircraft structures for engineering students - part 7 ppt

352 Open and closed, thin-walled beams

Calculate and sketch the distribution of shear flow due to a vertical shear force S,

acting through the shear centre S and note the principal values. Show also that the

distance & of the shear centre from the nose of the section is tS = 1/2( 1 + a/b).

Am. q2 = q4 = 3bSY/2h(b + a), q3 = 3SY/2h. Parabolic distributions.

P.9.15 Show that the position of the shear centre S with respect to the intersection

of the web and lower flange of the thin-walled section shown in Fig. P.9.15, is given

by

5's = -45a/97, 7s = 46a/97

Fig. P.9.15

P.9.16 Figure P.9.16 shows the regular hexagonal cross-section of a thin-walled

beam of sides a and constant wall thickness t. The beam is subjected to a transverse

shear force S, its line of action being along a side of the hexagon, as shown.

Find the rate of twist of the beam in terms oft, a, S and the shear modulus G. Plot

the shear flow distribution around the section, with values in terms of S and a.

Fig. P.9.16

Problems 353

Ans. dO/dz = 0.192S/Gta2 (clockwise)

q1 = -OXS/a,

q4 = q6 = 0.13S/a,

q2 = qs = -0.47S/a,

q5 = 0.18S/a

q3 = 47 = -0.17S/a

Parabolic distributions, q positive clockwise.

P.9.17 Figure P.9.17 shows the cross-section of a single cell, thin-walled beam

with a horizontal axis of symmetry. The direct stresses are carried by the booms B1

to B4, while the walls are effective only in carrying shear stresses. Assuming that

the basic theory of bending is applicable, calculate the position of the shear centre

S. The shear modulus G is the same for all walls.

Cell area = 135000mm2. Boom areas: B1 = B4 = 450mm , B2 = B3 = 550mm . 2 2

Wall Length (mm) Thickness (mm)

12, 34

23

41

500

580

200

0.8

1 .o

1.2

Ans. 197.2mm from vertical through booms 2 and 3.

100 mm

100 mm

- ------ --

1.0 mm 100 mm

0.8 mm

500 mm

Fig. P.9.17

P.9.18 A thin-walled closed section beam of constant wall thickness t has the

cross-section shown in Fig. P.9.18.

Fig. P.9.18

354 Open and closed, thin-walled beams

Assuming that the direct stresses are distributed according to the basic theory of

bending, calculate and sketch the shear flow distribution for a vertical shear force

S,, applied tangentially to the curved part of the beam.

Ans. qol = S,,( 1.61 cos 8 - 0.80)/r

q12 = Sy(0.57SS - 1.14rs - 0.33)/r

P.9.19 A uniform thin-walled beam of constant wall thickness t has a cross￾section in the shape of an isosceles triangle and is loaded with a vertical shear force

Sy applied at the apex. Assuming that the distribution of shear stress is according

to the basic theory of bending, calculate the distribution of shear flow over the

cross-section.

Illustrate your answer with a suitable sketch, marking in carefully with arrows the

direction of the shear flows and noting the principal values.

Ans. q12 = SY(33/d - h - 3d)/h(h + 2d)

q23 = S,,(-6$ + 6h~2 - h2)/h2(h + 2d)

3

Fig. P.9.19

P.9.20 Find the position of the shear centre of the rectangular four boom beam

section shown in Fig. P.9.20. The booms carry only direct stresses but the skin is

fully effective in carrying both shear and direct stress. The area of each boom is

lO0mm2.

Ans. 142.5 mm from side 23.

3

I 14 I- 240 mm I

Fig. P.9.20

Problems 355

250 mm I

P.9.21 A uniform, thin-walled, cantilever beam of closed rectangular cross￾section has the dimensions shown in Fig. P.9.21. The shear modulus G of the top

and bottom covers of the beam is 18 000 N/mm2 while that of the vertical webs is

26 000 N/m' .

The beam is subjected to a uniformly distributed torque of 20 Nm/mm along its

length. Calculate the maximum shear stress according to the Bredt-Batho theory

of torsion. Calculate also, and sketch, the distribution of twist along the length of

the cantilever assuming that axial constraint effects are negligible.

Am. T~~ = 83.3N/mm2, 0 = 8.14 x lop9

t

-- 2.1 mm 2.1 mm

1.2mm

1

11 11.2 mm

Fig. P.9.21

P.9.22 A single cell, thin-walled beam with the double trapezoidal cross-section

shown in Fig. P.9.22, is subjected to a constant torque T = 90 500 N m and is con￾strained to twist about an axis through the point R. Assuming that the shear stresses

are distributed according to the Bredt-Batho theory of torsion, calculate the distribu￾tion of warping around the cross-section.

Illustrate your answer clearly by means of a sketch and insert the principal values of

the warping displacements.

The shear modulus G = 27 500 N/mm2 and is constant throughout.

AFZS. Wi = -Wg = -0.53m, W2 = -W5 = O.O5mm, W3 = -W4 = 0.38m.

Linear distribution.

356 Open and closed, thin-walled beams

1.25 mm 3 1.25 mm r￾1.25 mm

500mm d+ 890 mm

Fig. P.9.22

P.9.23 A uniform thin-walled beam is circular in cross-section and has a constant

thickness of 2.5 mm. The beam is 2000 mm long, carrying end torques of 450 N m and,

in the same sense, a distributed torque loading of 1 .O N m/mm. The loads are reacted

by equal couples R at sections 500 mm distant from each end (Fig. P.9.23).

Calculate the maximum shear stress in the beam and sketch the distribution of twist

along its length. Take G = 30 000 N/mm2 and neglect axial constraint effects.

Am. r,, = 24.2N/mm2, 8 = -0.85 .x 10-82rad, 0 < z < 500mm,

8 = 1.7 x 10-8(1450~ - z2/2) - 12.33 x rad, 500 < z < 1OOOmm

Fig. P.9.23

P.9.24 A uniform closed section beam, of the thin-walled section shown in Fig.

P.9.24, is subjected to a twisting couple of 4500Nm. The beam is constrained to

twist about a longitudinal axis through the centre C of the semicircular arc 12. For

the curved wall 12 the thickness is 2 mm and the shear modulus is 22 000 N/mm2.

For the plane walls 23, 34 and 41, the corresponding figures are 1.6mm and

27 500 N/mm2. (Note: Gt = constant.)

Calculate the rate of twist in radians/mm. Give a sketch illustrating the distribution

of warping displacement in the cross-section and quote values at points 1 and 4.

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