To prove the validity of (2.1), let us multiply (6.6) by G, differentiate over z, and apply formula (6.5). As a result we obtain

In the first sum of (6.7), we substitute m þ 1 by m, in so doing its low limit becomes equal to unit, while the upper limit is equal to n.

Let us separate out the term at m ¼ n in the newly obtained sum and the term at m ¼ 0 in the second sum of (6.7), and add together all remained sums. As a result, we obtain

If we include the second term standing in the right-hand side of (6.8) into the sum, and express the value ð 1ÞnZ;ðnÞ; then we are led to relationship (2.1).

6.3The Main Kinematic and Dynamic Characteristics of the Wave Surface

spatial rectangular Cartesian coordinates, x- and y- are the main axes of the cross section of the beam (Fig. 3.1), ½ui;ðkÞ& ¼ ½okui=otk&; t is the time, vi ¼ ui;ð1Þ; ki; ki; and si are the components of the unit vectors of the tangential to the centroid axis, and directed along the main axes, respectively, and Latin indices take on the values 1, 2, 3.

where q is the density of the beam’s material.

Eliminating the value ½rkk& from (6.13) and (6.15), we find the velocity of the quasi-longitudinal wave propagating in the thin-walled beam of open section

Relationship (6.13) with due account for (6.16) takes the form

Now multiplying (6.9) by kikj and considering that on the quasi-longitudinal wave G ¼ G1; we obtain

Substituting (6.18) in (6.11), we are led to the following equations valid on the quasi-longitudinal wave:

Note that in the three-dimensional medium only one value, i.e. ½uk; k&; is nonzero on the quasi-longitudinal wave, while in the one-dimensional medium, where the ‘beam-wave’ propagates, on the quasi-longitudinal wave there are two nonvanishing values, namely, ½ux; x& and ½uy; y&; resulting to the fact that, as distinct to the statics of the thin-walled beams of open section where the contour of the cross-section remains to be rigid during the process of its deformation, in the dynamic problems this contour experiences the deformation on the quasi-longitudinal wave.

It is interesting to emphasize that relationships (6.19), which are valid only on the front of the quasi-longitudinal wave propagating in the thin-walled beam of open profile, coincide by their form with the well-known formulas for simple beams of solid cross-section being under simple tension-compression, i.e.

ux; x ¼ uy; y ¼ muz; z;

which are valid for all cross sections of the simple beam with the longitudinal z-axis.

86 6 Peculiarities of Transient Wave Propagation

Multiplying (6.10) by kikj and by kisj and (6.14) by ki and by si; respectively,

where ½vx& ¼ ½vi&ki and ½vy& ¼ ½vi&si.

Eliminating the values ½rkx& and ½rky& from (6.20) and (6.21), we find the

Thus, we have just prove that the dynamic theory of thin-walled beams of open profile proposed in this book admits only two transient waves, quasi-longitudinal and quasi-transverse, propagating with the velocities (6.16) and (6.22), in so doing the velocity of the quasi-longitudinal wave is equal to that of longitudinal wave in a thin elastic rod of solid cross section, while the velocity of the quasi-transverse wave coincides with the velocity of the shear wave propagating in a threedimensional elastic medium.

Reference

1. T.Y. Thomas, Plastic flow and fracture in solids, (Academic Press, New York, 1961)