Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

In the loaded system, dynamic loading produces stresses and strains, the magnitude
and distribution of which depend not only on the usual parameters (the dimensions of
the member, the loading applied, the elastic modulus of the member, etc.), but also on the
velocity of the strain waves that propagate through the solid material. This latter consideration,
although very important when loads with extremely high velocities are applied, may
often be neglected when the velocity of the impact load is relatively low. The loading may
be considered a low-velocity impact when the loading time permits the material to act in the
same manner as it does under static load—that is, when the relations between stress and
strain and between load and deflection are essentially the same as those already developed
for static loading. For low-velocity impacts, the time of application of the load is greater
than several times the natural period of the loaded member. If the time of application of the
load is short compared with the natural period of vibration of the member, the load is usually
said to be a high-velocity impact.
Energy methods can be used to obtain solutions to many problems involving impacts
in mechanics of materials, and such methods enable us to develop some insight into the
significant differences between static and dynamic loading.
Investigation of Impact Loading with Simple
Block-and-Spring models
Freely Falling Weight: As an illustration of an elastic system subjected to an impact load,
consider the simple block-and-spring system shown in Figure 17.17. A block having mass
m = W/g is initially positioned at a height h above a spring. Here, W represents the weight of
the block and g the gravitational acceleration constant. At this initial position, the block’s
velocity is v = 0; hence, its kinetic energy is also zero. When the block is released from rest,
it falls a distance h, where it first contacts the spring. The block continues to move downward,
compressing the spring until its velocity is momentarily halted (i.e., v = 0). At this instant, the
spring has been compressed an amount D max , and the kinetic energy of the block is once again
zero. If the mass of the spring is neglected and the spring responds elastically, then the principle
of conservation of energy requires that the potential energy of the block at its initial
position must be transformed into stored energy in the spring at its fully compressed position.
In other words, the work done by gravity as the block moves downward a distance h + D max
is equal to the work required to compress the spring by D max . The maximum force F max developed
in the spring is related to D max by F max = k D max , where k is the spring stiffness (expressed
in units of force per unit of deflection). Therefore, from conservation of energy, and
assuming no dissipation of energy at impact, the external work done by the weight W of the
block as it moves downward must equal the internal work of the spring as it stores energy:
2
Wh ( +D ) = ( kD ) D = kD (a)
max
1
2 max max
1
2 max
k
m
729
IMPACT LOAdINg
h
Δ max
m
FIGURE 17.17 Freely falling
weight and spring.
At the instant a moving body is
stopped, its energy (both
potential and kinetic) has been
transformed into internal energy
in the resisting system.
Note that the factor one-half appears in Equation (a) because the force in the spring
gradually increases from zero to its maximum value. Equation (a) can be rewritten as
2
2W
2W
Dmax
− Dmax
−
k k h = 0
This quadratic equation can be solved for D max , and the positive root is
2
W W W
D max = + ⎛ 2 h
k ⎝ ⎜ ⎞
⎟ + ⎛ k ⎠ ⎝ ⎜ ⎞
⎟
k ⎠
(b)