View Full Version : Adjustment of specimen stiffness based on system stiffness

Costi, John (rgh)
05-03-2004, 02:23 PM
Dear members

We do a considerable amount of testing (soft tissue and bone) using an
Instron servohydraulic testing machine that often involve the custom design
and manufacture of jigs and fixtures to allow clamping of specimens as well
as ensuring forces and moments are applied in as best a physiological manner
as possible. As you may appreciate, in some cases the stiffness of the test
rig (N/mm or Nm/degree) is substantially less than the stiffness of the
Instron. In these cases we quantify the stiffness of the test rig using
rigid steel blocks in place of the specimen to obtain a "system stiffness".
We then perform our tests on the specimen to obtain a "specimen stiffness".
The stiffness is calculated using linear regression over the most linear
portion of the final cycle of the loading curve (quite often Rsquared values
of 0.98 or better are achieved). The specimen stiffness therefore represents
components of the system stiffness as well as the true specimen stiffness
itself. Usually we are happy to use the specimen stiffness as calculated and
feel this is justified as studies are conducted using the same test rig
having the same system stiffness allowing direct comparisons.

However, in some cases, modifications may be made to test rigs thereby
changing its system stiffness and any subsequent specimen tests would
produce a stiffness that may be different due to the change in system
stiffness. Once again this is acceptable so long as we do not wish to
compare the specimen stiffnesses to specimens tested using the original test
rig. If we wish to compare specimen stiffnesses to earlier tests then the
differences in system stiffness between specimens must be taken into
account. This is particularly important if the system stiffness is not at
least an order of magnitude larger than the specimen stiffness.

My question is: has anyone encountered this before and if so what steps were
taken to adjust the specimen stiffnesses so that they were comparable?

I believe that the stiffness may be adjusted on the assumption that the
region where the stiffness is calculated is linear, which seems to be the
case in the majority of the testing we do, although it is clear that the
overall specimen behaviour is nonlinear and viscoelastic. The method I have
used to adjust the stiffness is quite simple and is outlined as follows
(apologies for the messy terminology in advance!):

Let: S represent the stiffness of the specimen tested in the rig
SRig represent the system stiffness of the rig using steel
SAdj represent the desired adjusted stiffness of the specimen
after taking into account the system stiffness

At the same force (F), the adjusted displacement (d_Adj) can be calculated
as the displacement of the specimen minus the displacement of the system
(d_S - d_SRig). It follows then that SAdj can be calculated between the
linear range of forces (F2 and F1) as:

SAdj = (F2 - F1) / (d_Adj2 - d_Adj1)

SAdj = delta_F / (d_Adj2 - d_Adj1)

And substituting for d_Adj we have:

SAdj = delta_F / [(d_S2 - d_SRig2) - (d_S1 - d_SRig1)]


SAdj = delta_F / [(d_S2 - d_S1) - (d_SRig2 - d_SRig1)]


SAdj = delta_F / (delta_d_S - delta_d_SRig)

Assuming that the stiffnesses were calculated over the same force range
delta_F we have:

delta_d_S = delta_F / S and:

delta_d_SRig = delta_F / SRig

We know delta_F, S and SRig therefore we can substitute and simplify to

SAdj = delta_F / [(delta_F / S) - (delta_F / SRig)]

I have verified this method using the above formula and also by using
experimental data and adjusting the displacements at the same force and then
using linear regression to calculate the adjusted stiffness and found that
both methods produce the same adjusted stiffness.

I'd appreciate any comments or suggestions with using this method and will
post a summary of replies in due course.

Kind regards

John Costi
Biomechanical Engineer, Final Year PhD Candidate & COES (SF-36 software)
Department of Orthopaedics
Division of Surgery
Repatriation General Hospital
Daws Road, Daw Park
Phone: +61 8 8275 1126
Fax: +61 8 8374 1998
Email: John.Costi@rgh.sa.gov.au


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