General Mechanical

Should the stiffness of this cantilever beam be different for these two conditions?

• Rameez_ul_Haq
Subscriber

So, I was just doing some Finite Element Analysis (FEA) to figure out what would be the effect on the stiffness of a cantilever beam if another beam is attached to it on its free end.

• Rameez_ul_Haq
Subscriber
• peteroznewman
Subscriber
Yes, the change in deformation is due to the addition of the block. Move the block to the other end of the cantilever and you will see a much larger change in tip deformation. Move the block to the halfway point along the length and the tip deflection will be between the other two results.
The addition of the block is changing the stiffness of the beam. While the change in local stiffness is the same in the three positions of the block, the effect on tip deflection depends on where the stiffness increase was located. The effect is largest when the stiffness increase was near the base and smallest when the stiffness increase was near the tip. The reason is the bending moment is large at the base and small at the tip, so adding stiffness at the location where the bending moment is largest has the greatest effect on tip deflection.
I should add that if you use a Kinematic Mount to connect a block to a beam, that will have zero influence on the stiffness of the beam. That means the static tip deflection will not change whether the block is mounted at the tip, the base or not mounted at all. However, since many systems are subject to gravity or vibration, the mass of the block will cause the mounting position to influence other results such as the first natural frequency or the deformation under a gravity load instead of a tip force.
• Rameez_ul_Haq
Subscriber
The reason is the bending moment is large at the base and small at the tip, so adding stiffness at the location where the bending moment is largest has the greatest effect on tip deflection.
Is this coming from the general Bernoulli beam theory? And also, not only for this system, but for any type of system if I increase the stiffness in the region where the moment is highest, that will have the greatest effect on the max displacement of the beam, is that correct? Say, a beam load in the middle with two pin supports at each end.
In you last para, you mentioned these:
I should add that if you use a Kinematic Mount to connect a block to a beam, that will have zero influence on the stiffness of the beam.
However, ........ , the mass of the block will cause the mounting position to influence other results such as the first natural frequency ....
how is that, can you please provide a visual example to elaborate what you meant by connecting the gray beam to the green one using Kinematic Mount? i would be grateful. I know what you meant is that now the gray beam can be thought of a simple Point Mass, but how can kinematic mount achieve that, I want to understand this.
Secondly, why would the location of this gray beam (or a point mass) change the natural freqeuncy, since still the total mass in the system and its overall stiffness is still the same?

• Rameez_ul_Haq
Subscriber
,would be glad to catch a reply from you on this one, sir :)
• peteroznewman
Subscriber
You could compute the stiffness of each length of the beam using Bernoulli beam theory and show that putting the high stiffness section closest to the base reduces the tip deflection the most.
I have provided many images of a Kinematic Mount. Here is one using three balls in three Vees created by a pair of cylindrical pins with magnets at the center to pull the two plates together.
While the total mass and stiffness stays the same, the distribution of mass plays a large role in the modal frequencies.
• Rameez_ul_Haq
Subscriber
thank you, sir. And how would you comment on this statement I made:
And also, not only for this system, but for any type of system if I increase the stiffness in the region where the moment is highest, that will have the greatest effect on the max displacement of the beam, is that correct? Say, a beam load in the middle with two pin supports at each end.

• peteroznewman
Subscriber
Yes, the center of a simply supported beam has the highest bending moments, so mounting at the center might reduce the static deflection due to gravity as long as the increase in stiffness compensates for the added mass.
However if you look at a Modal analysis, mounting at the center might also reduce the first natural frequency (which is bad), because the increase in mass might push the frequency down faster than the increase in stiffness pushes it up.
• Rameez_ul_Haq
Subscriber
and if instead, I mount that small vertical beam not at the center, but at the a different location (where the moment is not quite high), then the modal freq decrease rate will now be higher, did I get that right? Since the addition of mass is still the same but the increase in stiffness is now not at that level (as compare to mounting it exactly at the middle of this beam).
Or we cannot say anything since the mass distribution has also changed?
Plus, I just gave an example like this beam (i.e. roller support at one end and pin support at other) just to be sure that the addition of extra material should always be there (for a beam) where the moment is the highest, in order to achieve the max increase in stiffness (in order to reduce the max deflection). I mean this is just the general idea, right? It doesn't necessarily depend upon what loads or boundary conditions we apply to a beam.
• peteroznewman
Subscriber
Attaching a mass to a flexible body will create a local increase in stiffness, if the connection is not a kinematic mount. Adding mass and changing the local stiffness is going to change the deformation due inertial loads such as accelerations and it is going to change the first natural frequency of the structure. Build a model and run the analyses to determine the responses of interest. Compare the responses to the requirements. It might turn out that any mounting location is acceptable, or it might turn out that none of the mounting locations are acceptable without further changes to the structure.
When the structure has a first mode requirement, plotting Elemental Potential Energy (ENERGYPOTENTIAL) from a Modal analysis, provides valuable feedback on design changes. Increase the stiffness where the Potential Energy is high. This will increase the first natural frequency. Reduce weight where the Potential Energy is low. This may also increase the first natural frequency.
• Rameez_ul_Haq
Subscriber
,yes I recall you already mentioned and explained the usecase of ENERGYPOTENTIAL in this thread, last year:
Secondly, can you please comment on the last paragraph of my previous comment?
• peteroznewman
Subscriber
If you are talking about optimizing a design, you first need to describe all the requirements and the quantity that is to be minimized (or maximized).
In the case of a cantilever beam, you can minimize the mass subject to a maximum stress and maximum deflection for a constant tip force by varying the thickness from base to tip at a constant width, or varying the width profile from base to tip at a constant beam thickness. In either case, there is more material near the base where the bending moment is highest.
• Rameez_ul_Haq
Subscriber
,I was just saying that for a cantilever beam or a beam supported by pin at one end and roller at other, you have already made it clear that increasing the stiffness in the region of the maximum moment will have the maximum effect on deflection i.e. the decrease in deflection will be most significant. However, you also mentioned that this fact is coming from the bernoulli beam theory. I don't know if bernoulli beam (or even Timoshenko) assumes and limits itself to what load or boundary conditions, or in other words what kind of loads and boundaries will make these theories to be valid for application and use for beams.
If you are talking about optimizing a design, you first need to describe all the requirements and the quantity that is to be minimized (or maximized).
So assume I have a bolt connecting two solid bodies. I am modelling the bolt using solid elements, and then applying the necessary contacts to properly connect them to the bodies. I now what to minimize the max deformation of this bolt. For the optimization with regards to max stress, I am already aware of the method/process and I use it frequently to appropriately size my bolts. However, I don't size it for max deformation of this bolt, but this time in my project, I desire to do so. Now, I am not exactly sure that where to apply extra material so that it will have the max effect on the deformation for the intention of lowering it (for the same addition of mass coming from extra added material). I don't know how to progress on this problem. I mean can I just directly increase the stiffness on the solid bolt (for example, although it doesn't practically make sense, but just for the sake of discussion) in the region where moment is highest, or it all depends on what the boundary conditions, loads, contacts, etc are for this statement to be valid:
so adding stiffness at the location where the bending moment is largest has the greatest effect on tip deflection.
• peteroznewman
Subscriber
Since we are talking about using Bending Moment to decide where to increase stiffness, that implies that we are talking about loads that create bending in the beam. In other words, axial loads would not be a valid load to consider when distributing stiffness along the length.
Bolts should only see axial loads, so bolts are not a good place to apply the method of reducing bending by adding stiffness.
• Rameez_ul_Haq
Subscriber
,I mean there can be shear forces in the bolts as well, which can cause bending moment altogether in the bolt. Isn't it?
But I hope I was able to clarify my question that if boundaries, loads, contacts, materials, etc matters. Or this statement which you gave, that increasing the stiffness in a structure at the location of max bending moment will have the max impact in reducing the max deformation, was infact generic.
• peteroznewman
Subscriber
Shear forces are supported at a bolted joint by the friction between the two flanges. The bolt only provides the axial clamping force that squeezes the flanges together. If the shear force is larger than the limiting friction force, the joint needs to be redesigned using more bolts or larger bolts that have a higher axial force to generate a higher limiting friction force.
Consider two cantilever beams that have the exact same area, but one has a lower tip deflection for the same force.
It is too simple to say to "increase stiffness at the location of maximum bending moment". You can see that was done on the T-shaped cantilever. The tapered cantilever has lower deflection for the same weight of material.
• Rameez_ul_Haq
Subscriber
,I mean yeah now I understand what you mean. I was just trying to get an affirmation from you so that I can use this fact:
so adding stiffness at the location where the bending moment is largest has the greatest effect on tip deflection.
in any kind of beam or structure, without being cautious of what boundaries, loads, contacts, materials, etc does this beam possess at its ends or through its length (ofcourse, assuming that this beam is carrying the bending moment somehow). Well thats a good option to completely replace the beam with a different beam (of same mass but different cross sectional bending stiffness) if the max tip deflection needs to be reduced, but what if I cannot change the beam's cross section or shape as a whole, but what I can do is to add some extra material somewhere to avail the max benefit of this additional stiffness (so that it reduces the deformation the most).
• peteroznewman
Subscriber
If the tip deflection of a cantilever beam is too large and you can't change the beam cross-section or shape, the most weight efficient addition to the structure is a cable or strut that supports the tip from above or below. A strut in tension is better than a strut in compression because you don't have to worry about buckling. This approach has nothing to do with the location of the maximum bending moment. It is adding a new load path to ground as close to the tip as possible.