Help with the three point bending test of a composite rod

[musacci]

Hello everyone,

I am trying to evaluate the spring constant of a coaxial polymer cylindrical fiber. This fiber has a stiff core and a more soft sheath.

I know the geometry of the system and I know the Young's moduli of both components. I can find separate models for the core or for the sheath where using the area moment of inertia I can extract the spring constant. My problem is that I don't know how to extract the spring constant of the whole object.

I need this to have a theoretical value to compare it with the results of three point bending experiments in which I directly measure the spring constant and extract the bending modulus of the whole structure.

Any help would be great!

 

[desertfox]

Try this:-

http://classes.mst.edu/civeng120/lessons/composite/beam_of_two_materials/index.html

“Do not worry about your problems with mathematics, I assure you mine are far greater.” Albert Einstein

 

[musacci]

Thanks for the reply,

I had thought of that but my problem is slightly different since I'm using a rod (circular cross section) and not a beam. Also, the whole rod is surrounded by a material which has a different elastic modulus.

I was thinking of calculating the spring constant of the outer material as if it was a hollow rod. From there I can calculate what would be the equivalent diameter of a hollow rod made of my inner material but having the same spring constant as the actual outer sheath.
In this way I would end up having a single material rod with an equivalent diameter.

Would this reasoning be valid?

 

[desertfox]

One other idea would be to look at the calculations for a reinforced concrete beam, I think the theory is still the same.

“Do not worry about your problems with mathematics, I assure you mine are far greater.” Albert Einstein

 

[musacci]

I cannot seem to find the theory for a concrete beam reinforced by a single rod. Other than that it should be quite similar.

 

[MintJulep]

Solve each simultaneously with part of the load applied to each piece. Constraint is that the deflection must be equal. This give you the load portion for each.

 

[musacci]

Hi MintJulep,

thanks for your reply, however I don't understand what you mean.

The deflection in my experiments is equal since the two core and the sheath are chemically bonded together. What do you mean by "solve each simultaneously"? And how should I divide the load into parts to be applied to each piece?

 

[MintJulep]

Think about it some more.

 

[rb1957]

does the "rule of mixtures" for combining materials of different E apply ?

why does a rod invalidate the beam test, proposed in the first reply ?

do you want a bending stiffness or an axial stiffness ?

another day in paradise, or is paradise one day closer ?

 

[robyengIT]

Roark's formulas for stress and strain - 7th edition appendix C (if you google you will find the whole book free of charge)

 

[musacci]

Thank you everyone for your replies,

could anyone confirm if this approach would be correct:

I express the bending rigidity of the composite beam as the sum of the bending rigidity of the separate components:

E_compositeI_composite=E_coreI_core+E_sheathI_sheath

From this I can estimate the equivalent Young's modulus that the rod would have if it was a fully uniform material.

 

[rb1957]

that isn't the usual way we account for different E. read up on "rule of mixtures", basically convert the section to one material, keeping the centroid of the transformed part in the same place as the original. in your case, I might keep the mid-thickness radius the same and transform the thickness.

another day in paradise, or is paradise one day closer ?

 

[MintJulep]

Rule of mixtures assumes an approximately homogeneous distribution of fibers of the different elements. That is not what we have here.

Max deflection of a simply supported beam, point load at center: y=Wl³/48EI

For the core: y_core = W_corel³/48E_coreI_core

For the sheath: y_sheath = W_sheathl³/48E_sheathI_sheath

Because, as stated by musacci, "The deflection in my experiments is equal since the two core and the sheath are chemically bonded together" y_core=y_sheath=y

Therefore: W_corel³/48E_coreI_core = W_sheathl³/48E_sheathI_sheath

Also: W_total = W_core + W_sheath

Solve.

Finally: E_composite= W_totall³/48yI_composite

Since you can do this with any arbitrary length, choose l=1 and all the l³ terms go away.

 

[rb1957]

so you're saying w_core/w_sheath = EI_core/EI_sheath

I guess i'd've worked from assuming that bending strain at the interface of the two materials is the same.

I'd expect that the stiffer material (higher EI) would dominate the behaviour, and that the softer material probably doesn't change the result significantly (ie the composite beam behaves like the stiffer component); assuming that there is a significant difference in EI. this is, I think, counter to the above result. If the two materials are similar in stiffness, then maybe they'd behave like the above result (ie sharing the load).

another day in paradise, or is paradise one day closer ?

 

[MintJulep]

so you're saying wcore/wsheath = EIcore/EIsheath

Yes. The load is shared in proportion to the stiffness.

Seems perfectly consistent with what you are saying. If EI of one component goes to zero its share of the load goes to zero.

Consider the analogous case of two coil springs of same free height but different k in parallel and co-axial.

F=kD

F_inner + F_outer = F

F_inner = k_innerD_inner

F_outer = k_outerD_outer

D_inner = D_outer

 

[MintJulep]

Or maybe not.

https://books.google.com/books?id=H...e&q=load shared by concentric springs&f=false

 

[rb1957]

yes, but I'm not seeing the sharing so clearly.

if the two rods were independently reacting the load then the sharing is obvious. The softer rod will take some load to deflect the same as the stiffer one.

But the two rods are not independent, but constrained so that their bending straindeflection is the same at the interface. I'm thinking that the axial deflection in the softer rod would be larger than that of the stiffer rod, given the same bending deflection if the two were free. being glued together I see that the stiffer rod may restrain the softer one, so that the moment in the softer one may possibly be negative.

this could be an interesting FEA.

another day in paradise, or is paradise one day closer ?