# STADYN Wave Equation Program 9: Addition of Coefficient of Restitution to Cushion and Interface Properties

In our last STADYN post we discussed the addition of $\alpha$ factors to take into account adhesion phenomena with cohesive soils.  In this post the addition of a more mundane but nevertheless important parameter for impact pile hammer systems is done: consideration of plastic losses in hammer and pile cushions, and interfaces as well.

Most impact pile hammers use some kind of hammer cushion; additionally, concrete piles are almost always driven with pile cushion at the pile head.  Cushions of both kinds are subject to significant plastic deformation and generation of heat.  There are several possibilities of modelling these elements in a simulation such as STADYN.

The first is to use velocity-dependent (viscous) damping to simulate the dissipation of energy.  STADYN in its current form has no velocity-dependent parameters; to add these would involve some major changes in the code, and in any case the testing of cushion material does not produce a result that would indicate such a property.

The second is to use an elastic-purely plastic approach similar to the one used in the soils.  The problem with this is that it would “flat-top” the impulse to the pile, and there is no evidence that the cushion material fails in this way.

The third is to use a “coefficient of restitution” approach, where the rebound of the cushion takes place at a different stiffness than the compression.  This is illustrated in two variants below.

The conventional model dates back to Smith, and is still used in GRLWEAP.  The ZWAVE model is described by Warrington (1988).  In both cases the energy lost in the cushion is represented by the shaded area.

For STADYN the conventional model was adopted.  Implementing this took a little more care in a finite element code than in finite-difference codes like WEAP and GRLWEAP but it was done.  To accomplish this, it was necessary to compute the force in the cushion incrementally, as with plasticity the response is now path-dependent.  When the cushion rebounded (i.e., the distance between the cushion faces increased from one step to the next) the rebounding stiffness is used.  In this way multiple rebounds can be modelled properly.

Since the inverse methods do not model the hammer, the Mondello and Killingsworth case is not considered here.  This leaves the other two cases, and these can be summarised very briefly.

The Finno (1989) case had a blow count increase from 15.8 to 17.0 blows/30 cm. For the SE Asia case, the blow count increased from 11.8 to 13.5 blows/30 cm.  Additionally for the latter case comparisons with the pile head force and ram velocity vs. time tracks were produced.

The pile head force until peak was identical, and then decreased more rapidly afterwards. There was an additional “kick” at 2L/c not present in the previous run.

The ram (point) velocity is the same until rebound, and then the ram is essentially stationary with the coefficient of restitution until 2L/c, after which the ram velocity in the two cases is very close.  The sawtooth effect is mostly due to the “ringing” of the ram, i.e., a stress wave going up and down the ram.

While it is evident that the method of energy transfer is different with the addition of the coefficient of restitution, the actual effect of plasticity on the blow count is not great.  This is probably due to two factors: most of the energy transfer takes place during compression of the hammer cushion, and both hammers are using micarta and aluminium, which has a relatively high coefficient of restitution (0.8).  Nevertheless cushion losses are greater in materials such as plywood, which is used with concrete piles.  It is to this type of pile that STADYN’s development now turns.

## Author:Don Warrington

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