The basic objective of this course is to understand analysis
of thin-walled structures. To a great
extent, the structures used in aerospace and ocean structures are
thin-walled. Assuming the thickness to be small helps in reducing the complexity
of the analysis as a two–dimensional analysis will be sufficient. That is, instead of using six stress
components to completely express the state of stress at a point, one needs only
three. Similarly, instead of six strain components, one requires only
three. An appropriate stress/strain
matrix (Hooke’s law) will be sufficient.
Once we have stresses and deformations, we can check if the
structure will satisfy service requirements.
That is, if the deformations will be less than maximum allowable
deformations and if the structure is sound.
For structural integrity, we need to determine if the material will fail
under the applied stresses. A task that
the so-called failure theories perform.
We will discuss some of these theories in the class.
After briefly going over the concepts related to state of
stress, strain, and shear flow, we will study the response of beams with
asymmetric cross section. A section is
termed unsymmetric if it does not have at least one axis of symmetry. For such a section, the product moment of
inertia does not vanish. A key
observation will be that for beam with asymmetric cross-sections, we have out
of plane deflections. We will also determinate the shear stresses caused by
transverse shear and how for thin walled tubes the related concept of shear
flow is used. Recall that the presence
of shear stresses cause an angle of twist.
Therefore, for all beams, unless the transverse shear force passes
through a specific point, called the shear center, there will also be an
associated twist of the cross-section. Determination of the shear center for a
cross-section is an important exercise in the analysis of thin-walled
structures. It is rather easy for an
open section, but involves significant calculations for closed sections.
Modern-day structural analysis of complex structures, due to
the availability of fast computers, is often performed using the so-called
Finite Element Method (earlier called Matrix Analysis of Structures). In this method, a complex structure is
represented as a combination of simple elements. For most of the structures of interest, it is rather straight-forward
to develop the governing equations for these simple elements (axial bars,
truss, beams, plates and shells). This
can be done either using statics or using energy methods. The governing
equations for simple elements are then assembled (using compatibility and
equilibrium) to generate a very large set of linear (or nonlinear if the
problem is nonlinear) algebraic equations.
These equations are solved using computers to determine the response of
the structure. Basic concepts related
to the Matrix Analysis of thin-walled structures will be covered in this
course. You will be able to write a
simple computer code to perform matrix analysis of truss structures.
In industry, the finite element analysis is performed using
commercially available codes such as NASTRAN, ABAQUS, and ANSYS.