In the previous sections, we saw that tightening the bolts produces compressive stress on the
mating components even before any operational loads are applied.
These stresses affect both the static and the dynamic behavior of the assembly.
Applications such as seals and gaskets that are static in nature, show how the prestress
due to bolt preload creates contact pressure to achieve proper closure before operational
loads are applied.
For dynamic systems such as mounting vibrating equipment or tires in vehicles, not having
enough bolt preload can result in unpleasant vibrations that can be both annoying and dangerous.
This is because the initial stress developed in the assembly due to bolt preload changes
the overall stiffness of the assembly.
For example, when the applied bolt preload is insufficient, the overall stiffness of
the bolted assembly reduces, and this in turn changes its natural frequencies.
But what’s natural frequency and how is it affected by prestress?
Let’s take a step back and see what the natural frequency of an assembly is.
Consider a spring-mass system, with one end of the spring fixed and a mass attached to
its other end.
If the mass is displaced from its mean position and then let go, the system starts to oscillate
with a certain constant frequency which is nothing but the natural frequency of the system.
The natural frequency for this spring-mass system is given by the square-root of the
spring stiffness K, divided by the mass M. Therefore, natural frequency of any system
depends on both its stiffness and its mass.
To understand the effect of prestress on natural frequency, let’s look at the example of
guitar tuning.
When we tune a guitar, we effectively stretch or loosen the strings.
By doing so, we alter the stress in them which changes its stiffness.
Therefore, it affects its natural frequency.
A tensile prestress generally increases the natural frequency while a compressive prestress
reduces the natural frequency.
Bolt preload plays a similar role on bolted joints.
Applying more or less bolt preload changes the initial stress state of the assembly and
this may change its natural frequency.
This can be realized by studying the governing equations of motion for such systems.
Here are the equations of motion for any general dynamic system that is under influence of
an external load.
Its overall response is captured by three solution terms, namely, displacements, velocities
and accelerations.
These responses are affected by three properties of the system which are: the mass matrix,
the damping matrix and the stiffness matrix.
Each of these matrices shows how the respective properties are distributed throughout the
system.
In case of undamped systems, the damping matrix reduces to zero and therefore we can eliminate
that term.
What we’re left with are the equations of motion for an undamped system.
In cases where the inertial effects are negligible, which is generally the case in bolted structures
during the preloading stage, the mass term can be neglected which leaves us with a simplified
version of this equation.
This equation represents the response of a static system and this is the equation that
we solve for in calculating stresses, pressures and reaction forces in a preloaded bolted
joint.
As we mentioned earlier, the presence of stress changes the stiffness of a system.
So, when we analyze a preloaded bolted joint, we solve for the stresses developed in the
part in first load step, lock the bolt preload, and then apply external loads.
When we do this, the change in stiffness due to preload in step one is calculated in the
form of a stress stiffening matrix, and when we apply external loads in step 2 the modified
system’s response due to stress stiffening is considered.
In fact, even when we study dynamic characteristics of such system, stress stiffening matrix is
first calculated and added to the initial stiffness of the system.
This modified stiffness is used in calculating the natural frequencies and that’s how the
effect of prestress on dynamic characteristics is captured.
Let’s revisit the example of guitar tuning and calculate its natural frequencies before
and after tuning.
Before tuning, here are the first ten natural frequencies of the system.
Now, if we introduce some stress in the strings by stretching them during tuning, here are
the same 10 frequencies calculated after introducing prestress.
It’s very clear that the strings are vibrating at a much larger frequency after tuning.
The same concept extends to preload in bolts.
Tightening the bolts increases preload and introduces stress in the system.
As the stiffness of the system increases, it does not vibrate that easily.
This can be realized by calculating the natural frequencies of a bolted structure before and
after applying preload.
But before we get that far, there’s one more topic to discuss pertaining to solver
settings that must be considered.
It’s whether to use large deformation.
Large deformation is a type of formulation that is used in situations that involve, well,
large deformation.
Particularly, when the change in shape of the system is noticeable, then it’s a good
idea to turn this formulation ON.
Using large deformation in solving the governing equations of motions takes into account four
types of nonlinearities, namely
Large strain, large rotation, stress stiffening and spin softening.
While all these are important factors, it’s worth noting that in case of bolted joints,
large strain and stress stiffening are prominent.
So, we’ll focus on them.
The stress stiffening here is same as what we discussed earlier.
So, turning on large deformation considers both the change in shape and the stress stiffening.
We have seen why stress stiffening is important, but is change in shape important too?
Let’s look at the example of a fishing pole to learn and see how change in shape matters!
The fishing pole is a tall slender structure and it acts like a cantilever beam under the
influence of fish weight.
If the weight of the fish is W and the length of the fishing pole is l then the deflection
of the pole at the tip is given by this relation.
When the length of the pole remains unchanged, we can see that this relationship reduces
to a linear function between the tip deflection and the weight of the fish.
But the length of the pole does not remain constant if there’s even slightest deflection.
At very small deflections the change in length is negligible and this relationship is valid
but as the deflection increases the change in length is noticeable and therefore this
equation is no longer valid.
This shows how the results deviate from linear behavior at large deformations and why it
is important to consider them while analyzing bolted structures.
Using large deformation makes the analysis nonlinear which is typically more expensive.
Also, in a multi-step analysis, when we use large deformation in the first load step the
following steps continue using it which adds to the computation time.
So, it may be beneficial to use small deformation in large models that do not deform so much.
But how do we decide whether the deformation is large or small?
There’s no rule of thumb to answer this, it’s subjective to the application.
A few instances in case of bolted joints where it’s recommended to use large deformation
are:
First, when the mating parts or the bolt have nonlinear material behavior
Secondly, when there are nonlinear contacts such as frictional contacts are used in the
model.
Third, when some of the parts undergo large deformations, such as gaskets and seals.
Or finally, when sliding of the components is of interest such as in case of shear joints.
If it’s not clear whether large deformation is needed for an application, it is recommended
to always use large deformation.
In case of design iterations, one may run two cases: one with and one without large
deformation to check the difference in results and decide whether to use large deformation
in all other simulations.
In terms of governing equations, the large deformation captures changes in stiffness due to change in shape, material nonlinearities, contact nonlinearities, spin softening and stress stiffening.