Higher order equations for circular plates
Background
This master thesis project considers the derivation of higher order equations for isotropic circular
plates using both analytical and numerical methods (e.g. Mathematica, Matlab). Basic structural
elements such as rods, beams, plates and shells are of huge importance in mechanical, civil, aerospace
and nuclear engineering. Among these, plates and shells are perhaps of most importance when
studying systems in three dimensions. The majority of works on plates presented in the literature
consider the most simple plate geometries (rectangular and circular) where it is straightforward to
compare results between different theories.
A plate may be modeled using three dimensional theory, but it is well known that the solutions to
these equations are very hard to obtain. Instead, several approximate theories have been developed
making use of the fact that the plate thickness is (usually) much smaller than the lengths in the plate
plane, e.g. the Kirchhoff and the Mindlin plate equations. Such theories have proved to be useful for a
variety of structural problems, especially in statics for slender structures and for dynamics at relatively
low frequencies. However, these classical plate theories are known to be inaccurate when the plate
thickness is not considered small, or at higher frequencies (also for thin plates). Even for slender plates
at low frequency, the classical theories do not model such things as the stress distribution within the
plate properly.
Project description
These matters concerning shortcomings among the traditional theories are to be addressed in this
project using higher order equations for circular plates. Here a series expansion method is to be used
that is straightforward to implement. Using low order truncation, approximate plate equations are
obtained that in many cases are superior to classical plate theories (based on our experience for
rectangular plates). Using high order truncation, high accuracy results are obtained that converge to
the three dimensional theory and thus act as benchmark solutions (based on our experience for
rectangular plates). Some preliminary analyses for circular plates have recently been done, which will
be beneficial for the student.
Expertice
The Division of Dynamics has for more than a decade been studying higher order equations for
various structures such as rods, beams, plates and shells. Among these, various work on rectangular
plates have been studied (isotropic, anisotropic, porous, micromechanic) but for some reason not for
circular plates. Hence, this project will fill that gap (for isotropic plates) where circular plates without
holes and with holes (annular plates) are to be studied. Our experiences from the earlier highly related
projects will certainly be beneficial to this project.
Outcome
It is my intention that this work will be part of a scientific publication. Most likely, the results from
this thesis could be presented at an international conference (that was the case for a much related
master thesis a few years ago, see link below).
http://studentarbeten.chalmers.se/search/index.xsql?start=0&doSearch=true&query=abadikhah&su
bmit01=S%C3%B6k
Longitudinal displacement in a rod, radius a and length L, with clamped end boundaries for the
lowest eigenfrequency. Exact theory (solid), our lowest order theory (dashed), and various classical
theories for different radii at x=L/4.
Number of students: 1
Student background: MPAME, MPSEB
Supervisor and contact: Peter Folkow, peter.folkow@chalmers.se
Examiner: To be decided