10. Dynamic load action

10.1. Dynamic load

Up to now we studied the action on the construction of the static load. It is known that the static loads change the final values from zero very slowly so the acceleration the construction elements receive under that is scornfully small. However, very often the loads have a dynamic character because they change a large velocity in time. The action of these loads is accompanied by the construction vibrations and their separate elements.

The stresses arising under the detail vibrations can be in many times more by its values than the stresses from the action of the static loads.

The design of the construction details under the dynamic load is more complicated than the design under the static load. Difficulty, on the one hand, consists in more complicated methods determining the internal forces and stresses setting up from the action of the dynamic loads and on the second hand – in more complicated methods determining the mechanical material properties under the dynamic load.

For example, under the action of the impact load (i.e. the load has a very small duration) many materials work as brittle, under the action of the static loads they become plastic; under the action of constant repeated variable load the material strength goes down sharply.

The general design method under the dynamic load is based on famous D^’Alembert^’s principle from theoretical mechanics. According to this principle any moving body can be considered as one being in the instantaneous equilibrium state if acting on it the external forces the force of inertia is added which is equal to the production of the body mass and its acceleration and directed to the side which is opposite the acceleration. That is why in cases when the forces of inertia are known, the section method can be applied without any limits and to determine the internal forces the equilibrium equations can be used.

In that case when determining the forces of inertia is difficult, for example under the impact, to determe the dynamic stresses and deformations the energy preservation law is used.

10.2. Calculating stresses under the uniformly accelerated motion

In many cases the acceleration with which the machine details displace is known. In this case the dynamic stresses are calculated without any difficulty. Consider some examples.

Example 1. The load of the weight G is lifted up with the acceleration (Fig. 10.1). Determine the stress in the rope ignoring its weight.

Fig. 10.1. Fig. 10.2.

Solution. Apply to the load the force of inertia which is equal to directed down. Use the method of sections. Do the cut n-n and remove the upper rope part. Denote by the stress in the rope and as soon as the stresses in axial tension distribute uniformly over the section we can accept that where is the unknown dynamic stress in the rope.

Projecting all forces including the forces of inertia on the vertical axis we get

(10.1)

(10.2)

where is the stress under the static load action; is the dynamic coefficient.

Thus, in many cases the dynamic stress can be given by the static stress and the dynamic coefficient. It is very convenient as the dynamic coefficient must be often determined by an experimental way.

Example 2. The bar of the weight q of the length 1 m is lifted by using two threads fastened to its ends (Fig. 10.2). The motion is translational at the acceleration a. Determine the stress in the bar.

Solution. Apply to each element of the bar with the length equal to the unit the force of inertia See, that this problem is equivalent to the problem of the simple beam loaded by the uniformly distributed load of the intensity

The maximum bending moment will be at the section in the middle of the beam:

(10.3)

where is the bending moment from the static uniformly distributed load of the intensity q.  is the dynamic coefficient.

The maximum dynamic stress is determined by the usual bend formula

(10.4)