Substituting the found arbitrary constants (4.51) in the ray series (4.44)–(4.49), we obtain the final expressions for the desired fields. Thus, for example, knowing the values a2 and a3 (4.51), it is possible to determine aðtÞ (4.40) and a(t) (4.17), and therefore to obtain the typical time-dependence of the contact force (4.9)

where P ¼ Pjt¼t ¼ ka 3=2:

Equating to zero the expression for the contact force (4.52), we obtain the approximate formula for the duration of contact of the impacting rod with the thinwalled beam of open section.

Note that the solution for a particular case of a straight thin-walled beam of open profile could be obtained by putting Rt ¼ R ! 1 and R0 ¼ r0; as it follows from (4.10), in Eqs. 4.42–4.52.

4.2.3 Numerical Example

As an example, let us consider the impact of a steel rod with a rounded end upon a steel arch with a constant radius of curvature and zero torsion, the cross

the increase in the initial axial compression results in the increase of both the maximal magnitude of the contact force and the duration of contact.

The curve of the erkk-dependence of the dimensionless initial velocity of impact

e0 ¼ V0G 1 resulting in the local damage of the thin-walled open-section beam

V 0 ;

in the place of contact is shown in Fig. 4.4 at the given magnitude of the dimensionless yield limit ery ¼ ryðq0G20Þ 1: From Fig. 4.4 it is evident that with the increase in the initial axial compression the initial velocity of impact, which may lead to the local damage of the structure, decreases.

Fig. 4.3 The dimensionless time-dependence of the dimensionless contact force

Fig. 4.4 The erkk- dependence of the dimensionless initial velocity of impact in the case when the contact stress is equal to the yield limit