Introduction to Large Deformation Earlier we talked about how large
deformation changes the shape of a structure and introduces nonlinearity in a system’s response.
But we haven’t talked about when we consider large deformation in our analysis.
The answer to this question is very elaborate so we’ll dedicate the current lesson to discuss it.
When we use the large deformation formulation in an analysis,
it considers the change in shape of the structure and the corresponding stresses developed in it.
It’s important to note that stiffness of a mechanical depends on both the material
and the geometry of the system. For instance, a bar of stainless steel
has a very stiff response compared to a spring made of the same material.
In this case, the helical geometry of the spring makes the structure more flexible.
So, large deformation considers the changes in the
shape and introduces geometric nonlinearities in the model.
There are 4 important types of geometric nonlinearities, namely:
1. Large strain 2. Large rotation
3. Stress stiffening 4. And spin softening
All these nonlinearities are introduced when large deformation formulation is used
but depending on the application one or more of them are predominant.
We’ll look at few examples to demonstrate how each of these nonlinearities plays a role
in analyzing mechanical systems. Let’s start with large strain,
to understand how large strain affects a system let’s look at a fishing rod for example.
We hold onto one of the fishing rod and tie the bait to a rope which is hanging at the other end.
When a fish takes the bait and gets caught to the hook, we start winding the rope to bring
the fish out of water. At this stage,
the weight of the fish is transferred as load to the free end of the fishing rod
and at this stage it pretty much acts a slender cantilever beam.
So, the weight of the fish is creating a moment acting on the free end of the rod.
If you notice, the fishing rod bends due to this applied moment and in that process
span of the cantilever beam changes and in turn reduces the magnitude of moment.
So, you can clearly see how the change in shape is changing the amount of load acting
on the structure. Due to this adjustment, the displacement increment changes automatically.
During this whole process the weight of the fish is not changing,
well, at least once it’s out of the water. So, the displacement at the very beginning
may be changing linearly but as it deforms further the varying moment
changes the displacement behavior and ends up with a nonlinear force-displacement response.
If we do not consider this changing moment, we will end up with an incorrect prediction
of deflection and may end up with a bad design for fishing rods.
Now, let’s look at the next geometric nonlinearity which is large rotation.
This is an interesting nonlinearity which must be considered even if there’s a rigid rotation.
Let’s take a drive shaft for instance, it is used to transmit torque or rotation from one component
to another. It is commonly used in automobiles in transferring power from engine to the wheels.
During its operation it may experience torque that shears it about its own axis.
When this happens, the points move in a circular trajectory on its surface
and this new updated coordinate are nonlinear functions,
trigonometric functions to be more precise, of the angle of shear.
If we assume small rotation, mathematically, we may approximate Sin(theta)=theta and Cos(theta)=0.
By doing so we can linearize the governing equations and therefore use linear solver.
But the error introduced by this approximation increases as the angle of shear increases. As
a result, at larger rotations, the shaft appears to grow radially which is physically incorrect.
Such errors can be avoided by using large deformation formulation
in case of large rotations, even if the structure is not deforming as much.
So, if in an application, a structural component is known to undergo large rotations
using large deformation is highly recommended.
Now, coming to the third type of nonlinearity which is stress stiffening. This is another
nonlinearity which can drastically change the stiffness of a system.
A classic example that demonstrates the effect of stress stiffening is string instruments such
as a guitar. A crucial step that every guitarist takes before playing is to tune the instrument.
This is done by turning a few knobs that are attached to the guitar strings.
By doing so, they’re stretching the strings and introducing stress in them.
As a result, the strings become stiffer and their frequencies change. The presence of
stress in the string increases its stiffness and in turn changes the way the strings vibrate.
The stress stiffening is not just limited to string instruments,
in any bolted assembly when we tighten the bolts it introduces some stress in the structure.
As a result, the stiffness of that structure increases.
In fact, not considering the stress stiffening can result in inaccurate results.
A pre-stress can be added to a linear system as well but using large deformation formulation
will automatically account for its affect. But as an engineer it is still important to judge
whether a system has any pre-stress to begin with before analyzing its structural response.
Finally, let’s talk about the fourth nonlinearity which is spin softening.
All the rotating machinery such as turbines and jet engines have heavy
components that are rotating at high speeds and as a result, they experience centrifugal forces
that are acting away from their center. Since these components are fixed at their center,
they tend to deform radially due to these centrifugal forces.
As they deform radially outward their radius of rotation changes
which in turn results in increases the centrifugal force and hence the radial deformation too.
As a result, the spinning components are prone to deforming more than the same components that
are at rest. This phenomenon is nothing but the spin softening which is captured
only when we are considering the change in shape of the structure during deformation.
So, using large deformation formulation will consider this phenomenon.
This is very crucial in studying the vibrational properties of heavy rotating machinery
such as wind turbines and jet engines.
So, in this lesson we have learned how large deformation introduces four different kinds of
nonlinearities into the model response and their importance in accurate engineering calculations.
While we have discussed the physical aspects of large deformation,
we haven’t yet talked about the mathematical treatment of this formulation.
This is again a very important topic to learn so we’ll save it for a separate discussion.