In this section, we discussed linear elasticity which is perhaps the simplest of the constitutive
models for structural properties. But that doesn’t mean that it is too simple
to be of practical use. In fact, a liner elastic material behavior
is enough in answering solving a lot of structural problems.
There’s a huge spectrum of problems that make use of linear elastic material models
but in this lesson, we have picked three specific examples.
The first one demonstrates the stress analysis of a bike frame,
the second one shows the harmonic analysis of an electronic device on a shaker table.
For the third example, we’ll do something different.
We’ll see how the microstructure of a solid influences it’s macroscopic behavior.
This will still involve linear elasticity, but we’ll see the anisotropic aspects of it.
Let’s start with our first example which is the stress analysis of a bike frame.
The bikes are a great means for both transport and recreation.
A bike frame is the main structural component of the bike as it supports
all the loads that are acting on the bike and it also holds the whole structure together.
So, it is important to make sure that the loads are distributed in the frame
such that it can support them without undergoing permanent deformation.
In this example we’ll perform stress analysis on one such bike frame.
Here’s the geometry of the bike frame, the weight of the rider is the major load that acts on it
and it is generally distributed on 3 different locations
One portion of the load acts through the handle,
one through the bike seat and the another through the bike pedals.
The whole frame is supported on two roller supports
which are at the shaft connecting the wheels.
Using these as the loads and boundary conditions one can perform the stress analysis.
In this example we assume that the bike frame is made of structural steel, so we
model it using a linear elastic material model and here are the material properties.
Now, we apply the total weight of the rider
at these three locations and then apply the roller supports at these two locations.
Since there are no net accelerations in the part due to these loads,
the inertial effects are negligible. So, we can perform a static stress analysis.
Here’s the stress contour for this model using a static analysis.
We can see the way the stresses are distributed in the model.
From the legend, we see that the maximum stress developed in the bike frame is about 30.6 MPa.
The purpose of this simulation is to see how the stresses are distributed in the bike frame
and if it can take this load without undergoing plastic deformation.
The structural steel material is known to yield at a stress around 275 MPa which is
way more than the maximum stress developed in this frame.
So, we may conclude that the current design of the bike frame passes this design criteria.
Notice that even if we decided tin change the design or if we increase the applied load to such
an extent that the frame starts to yield, it is not necessary to use a nonlinear model.
We only need to know whether the maximum stress is more than the yield
strength to decide on the design. If the maximum stress exceeds this limit,
then the design already failed, and we need not perform the simulation using a plasticity model
as it won’t add anything more to our analysis as far as our application is concerned.
A similar stress analysis can be performed for
several structural members such as trusses in bridges and building, or aircraft fuselage where
it is very important to have a design that does not yield.
The results from such an analysis can also be used as inputs for auxiliary
studies such as fatigue analysis or the safety factor studies.
Now, let’s look at our second example which is an electronics device that is
mounted on a shaker table. The electronics devices usually have many small components
that are soldered to a printed circuit board.
These days we can see such systems mounted on many environments
that subjected to large vibrations such as on rockets, aircrafts or commercial cars.
Under such vibrations the devices may be subjected to a lot of stresses which can
result in failure of solder joints and result on component failure.
This may be avoided by distributing the mass of the components on the board in
such a way that the joint of no component is stressed beyond its failure limit.
Let’s simulate one such device that is mounted on a shaker table. All
the material used in this model are linear elastic materials.
We’ll subject the shaker to a harmonic load to mimic the vibrations,
so we use a harmonic analysis. To do this, we first perform a model analysis
followed by a harmonic analysis using a mode super position method.
This way we need not perform a full transient analysis which is way more expensive.
As an output from this analysis, we get the frequency responses,
in this case we look at the frequency response of equivalent stress.
Here’s the curve that we get for this particular example.
What we see here is the max. stress developed in the assembly
when it is vibrating at different frequencies.
From the chart we see that the max. stress in the model is developed at a frequency of 1160 Hz
and if we look at the stress contour developed in the part at this frequency,
we can see where the max. stresses are developed in the full assembly.
This analysis tells us which components are most susceptible to failures,
and like I mentioned earlier all the materials are linear elastic.
If one is interested in a more accurate assessment of
the component failure, then the analyst may still extract the reaction forces
at the component and then use this as input for a component level model.
So, this is another example that shows how useful a linear elastic material is used
in performing structural analysis of mechanical systems.
Now, let’s look at the final example. In this example we’ll not focus on a mechanical design,
rather we’ll focus on the micro-structure of a solid
and how it influences the macroscopic properties of the solid.
We have discussed earlier that all the material properties are related
to the micro-structure of the solid. If the micro-structure has any directional structure
it’ll introduce anisotropic properties.
To understand how this pans out between the scales,
let’s look at three different structures and see how it affects the macroscopic behavior.
Let’s start with a body centered crystal structure which is commonly seen in metals.
This micro-structure has two different materials called as the base and the particle.
Each of these materials has different material properties and they’re shown here.
This unit is called as a representative volume element
which is extracted from a solid at a bigger scale.
Each unit has two different materials that are structured in a certain way, when many such units
come together, they result in the overall mechanical behavior of the solid.
In this case, when we calculate the effective material property of the solid, here are the
Young’s modulus, shear modulus and Poisson’s ratios in three principal directions.
As we can see from the table all the values are uniform in all the directions. So, this
material can be isotropic in nature even though it is made of two different materials.
Now, let’s look at the next representative volume unit, which again has two different materials,
wood matrix and wood fiber that have different material properties. In this case,
we see a clear directionality where the wood fibers are present in one direction.
Let’s perform a similar calculation for this unit and calculate the effective material properties.
Here are the Young’s modulus, shear modulus and Poisson’s ratio in three
principal directions. As we can see from the values,
they’re different in all three directions which makes this material anisotropic.
Finally, let’s look at one last unit where the structure is seen in woven composites.
In this case, the material is braided and immersed in a matrix and together
they provide the structural support to the material.
In this case we have a matrix and the yarn, and each has very different material properties.
Let’s calculate the effective material properties for this unit.
Here are the values for the young’s and shear modulus and Poisson’s ratio.