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5.8. Thin-walled cellular sections

The Bredt-Batho theory developed in the previous section may be applied to the solution of problems involving cellular sections of the type shown in Fig. 5.7.

Fig. 5.7. Thin-walled cellular section.

Fig. 5.7. Thin-walled cellular section.

Assume the length RSMN is of constant thickness 11 and subjected therefore to a constant shear stress t\ . Similarly, NOPR is of thickness t2 and stress t2 with NR of thickness i3 and stress Z3.

Considering the equilibrium of complementary shear stresses on a longitudinal section at N, it follows that nh = r2t2 + r3t3 (5.17)

Alternatively, this equation may be obtained considering the arrows shown to be directions of shear flow q(= zt). At N the flow qi along MN divides into q2 along NO and <73 along NR, i.e. <?i = <72 + <73

The total torque for the section is then found as the sum of the torques on the two cells by application of eqn. (5.14) to the two cells and adding the result, i.e. T = 2q\A\ + 2q2A2

Also, since the angle of twist will be common to both cells, applying eqn. (5.16) to each cell gives

6_ L f tisi + r3s3\ _ L_ f r2s2 - r3s3 \ ~ 2G\ At 2G\ A2 j where si, s2 and S3 are the median line perimeters RSMN, NOPR and NR respectively.

The negative sign appears in the final term because the shear flow along NR for this cell opposes that in the remainder of the perimeter.

5.9. Torsion of thin-walled stiffened sections

The stiffness of any section has been shown above to be given by its value of GJ or GJeq ■ Consider, therefore, the rectangular polymer extrusion of simple symmetrical cellular constructions shown in Fig. 5.8(a). The shear flow in each cell is indicated.

But because of symmetry q\ must equal q3 q2 = 0;

i.e., for a symmetrical cellular thin-walled member there is no shear carried by the central web and therefore as far as stiffness of the section is concerned the web can be ignored.

Qi B q3

Qi B q3

Fig. 5.8(a). Polymer cellular section with symmetrical cells, (b) Polymer cell with central web removed but reinforced by steel I section.

Stiffness of complete section, from eqn. (5.16)

4A2t

where A and s are the area and perimeter of the complete section.

Now since G of the polymer is likely to be small, the stiffness of the section, and its resistance to applied torque, will be low. It can be reinforced by metallic insertions such as that of the I section shown in Fig. 5.8(b). For the I section, from eqn. (5.8)

GJe = GZk2db3

and the value represents the increase in stiffness presented by the compound section.

Stress conditions for limiting twist per unit lengths are then given by: For the tube

and for the I section

Hk\d

Gb^kj d

Usually (but not always) this would be considerably greater than that for the polymer tube, making the tube the controlling design factor.

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