# Cantilever Beam Small Deflection Approximation

• randall016
In summary: No, I think you can just do it once and then use the calculated deflection for the rest of the application.

#### randall016

I am calculating the strain in a cantilever beam with a point load for a given deflection. The deflection is around .5mm for a beam that is just over 5mm long (width is 3.8mm and height is 0.15mm). I was told for the assumption of small deflections to be valid, deflection should be 2% of the length and in this case it would be 10%. Is this rule of 2% valid? I would assume it would be dependent on the thickness of the beam. Thanks in advance for any input.

Because the total strain and deflection measured is a composite of the the strains at each point along the beam the reliability/repeatability of load vs deflection at very low loads/deflections is primarily due to potentially localized stress relaxations due to the lack perfect homogeneity of the material properties along the length of the beam's stressed surfaces. These localized effects can be eliminated by first cycling the beam to a significant stress/strain level before applying the desired smaller load vs deflection measurements or intended loads.
At the same time, at very low loads or deflections, the accuracy of the associated load and deflection measurement instrumentation/systems and the possible external effects of operating temperature variations upon the beam material and/or instrumentation accuracy must also be taken into consideration.

randall016 said:
I am calculating the strain in a cantilever beam with a point load for a given deflection. The deflection is around .5mm for a beam that is just over 5mm long (width is 3.8mm and height is 0.15mm). I was told for the assumption of small deflections to be valid, deflection should be 2% of the length and in this case it would be 10%. Is this rule of 2% valid? I would assume it would be dependent on the thickness of the beam. Thanks in advance for any input.
I think a deflection of 2% of the length is probably more than you should look for.

In structural applications, a max. deflection of L/360 is generally the limit for most beams. This works out to about 0.28% L, rather than 2% L.

With such large deflections in such simple beam geometries, I would also be concerned that the bending stress in the beam has exceeded the yield stress of the material, and a permanent set has been created.

randall016,
(Edited post)

What is your calculated maximum stress for the beam at this deflection?

I have run the calculation in my US units for your 10% deflection and seen that the stress is about 46,000 psi, which, if correct, is acceptable if you are using spring steel or alloy. So 2% is very safe and your 10% is acceptable for these kinds of applications as opposed to civil structural standards.

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I calculated 590MPa maximum stress (85,000 PSI). I am designing a flexure made of BeCu that a strain gage is mounted on. My max deflection is known so I used euler-bernoulli beam theory to calculated the deflection for a point load (y(x=L)=PL^3/3EI. I solved for P and substituted it into the elastic flexure stress (stress=MC/I which is equal to E*strain from hookes law). After substituting and solving for strain I get strain=3*y*c/L^2 where y is the deflection and c is the distance from the neutral axis so half the thickness of the flexure. Assuming E is 131GPa for BeCu I get the 590MPa. The yield strength is around 1.5GPa so this should be within the elastic region.

I guess my question is to see if it is appropriate to use euler-bernoulli beam theory when it assumes small deflections or should I use another beam theory such as Timoshenko that would account for shear? I would assume that the strain would end up being less using Timoshenko beam theory so Euler-Bernoulli may be a more conservative estimate of the strain.

Just a note regarding your strain gage application. I originally started addressing that part of the issue but since you did not include the strain gage that in your original post I changed my mind and deleted it.
Assuming you are using foil adhesive mounted gages, I will leave all of the factors related to gage factor, gain and instrumentation accuracy to you; but, just to be safe, I do want to mention two items, of which you are most likely already aware. One is that preflexing is to a significant strain to address adhesive bonding relaxation and slip is important for these applications; and the second, particularly critical for your low strain application is accurate temperature compensation either by foil to base material thermal coefficient matching or by including a static (unloaded) gage mount in your measuring circuit.

Thank you for the comment. I was not aware of the preflexing issue. Would this need to be done every time before taking measurements or just an initial break in? Everything is already temperature compensated.

The preflexing is only required for initial "break in".

After break in you should run a repeatability cycling test to check the percent of error at your required deflection. The amount of scatter during that test will tell you whether or not the gage output and your measurement system at that deflection is repeatable and within your % of error requirements.
When I was working with foil strain gages, even after break in there was a "rule of thumb" minimum percent of strain for reliable gage readings; unfortunately, I no longer remember what that value was.
I recommend you contact your strain gage manufacturer/supplier for any technical documents or input they have regarding their gage applications.
Also, for a reference on all elements of foil strain gage application see:
http://www.omega.com/techref/pdf/StrainGage_Measurement.pdf

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How are you going to mount a strain gauge to work on such a tiny sensing area ?

## 1. What is the Cantilever Beam Small Deflection Approximation?

The Cantilever Beam Small Deflection Approximation is a simplification technique used in structural engineering to estimate the deflection of a cantilever beam under a load. It assumes that the deflection of the beam is small compared to its length, and neglects higher-order terms in the equations of motion.

## 2. When is the Cantilever Beam Small Deflection Approximation applicable?

This approximation is applicable when the deflection of the beam is less than 1/10th of its length, and the material is linearly elastic. It is commonly used in the design of simple structures such as bridges, balconies, and shelves.

## 3. How is the Cantilever Beam Small Deflection Approximation calculated?

The deflection of a cantilever beam under a load can be calculated using the following equation:
δ = (FL^3)/(3EI)

Where δ is the deflection, F is the applied load, L is the length of the beam, E is the Young's modulus of the material, and I is the moment of inertia of the cross-section of the beam.

## 4. What are the limitations of the Cantilever Beam Small Deflection Approximation?

While this approximation is useful for simple structures, it has some limitations. It does not take into account the effect of shear forces, and it assumes that the material is linearly elastic. It also becomes less accurate as the deflection becomes larger, and is not suitable for highly curved or irregularly shaped beams.

## 5. How accurate is the Cantilever Beam Small Deflection Approximation?

The accuracy of this approximation depends on the assumptions made and the conditions of the beam. In general, it provides a good estimate for small deflections and can be improved by using more refined methods, such as finite element analysis, for more complex or critical structures.