Airy's stress function for a cantilever beam

[Divya Shyam Singh]

I have to calculate Airy's stress function for a cantilever beam made of two different material along its length.
The young's modulus of the first half is E1 and E2 for second half. The beam is made such that these materials are joined to each other one after other along its length.
At the interface, there will be a force balance and same displacement for both the beams.

How will I calculate the forces at the interface?

 

[Chestermiller]

I have to calculate Airy's stress function for a cantilever beam made of two different material along its length.
The young's modulus of the first half is E1 and E2 for second half. The beam is made such that these materials are joined to each other one after other along its length.
At the interface, there will be a force balance and same displacement for both the beams.

How will I calculate the forces at the interface?

The normal and shear components of the traction have to be continuous at each location across the interface.

 

[Divya Shyam Singh]

The normal and shear components of the traction have to be continuous at each location across the interface.

Thank you for your time.

I have done some stress calculation as a result of which i can see a stress discontinuity at the interface where beam of first material ends and the other starts. But in actual conditions there won't be any discontinuity in stress and any change in the stress will be smooth. But i have no idea at all about how to caculate the expression for stress or strain fields.
So how to approach this problem?

 

[Chestermiller]

Thank you for your time.

I have done some stress calculation as a result of which i can see a stress discontinuity at the interface where beam of first material ends and the other starts. But in actual conditions there won't be any discontinuity in stress and any change in the stress will be smooth. But i have no idea at all about how to caculate the expression for stress or strain fields.
So how to approach this problem?

I don't have any experience working with the Airy stress function.

 

[Nidum]

Which way round is the problem ? Do you want to :

(1) Find an expression for the actual stresses in the beam and then derive an expression for the Airy stress function ?
or
(2) Find and solve an expression for the Airy stress function and then derive an expression for the actual stresses ?

Your wording suggests (1) but let's be clear on this .

 

[Nidum]

Assuming (1)

Start by finding expressions for the shear force and bending moment at any location along the beam . Best to draw the diagrams as well .

To obtain an expression for Airy stress function at any point in the beam you will then need to derive expressions for each of σ_x , σ_y and σ_xy at any point in the beam .

nb: This is a 2D problem

 

[Divya Shyam Singh]

Assuming (1)

Start by finding expressions for the shear force and bending moment at any location along the beam . Best to draw the diagrams as well .

To obtain an expression for Airy stress function at any point in the beam you will then need to derive expressions for each of σ_x , σ_y and σ_xy at any point in the beam .

nb: This is a 2D problem

Thanks for your reply

I already did the first way but i am not getting a proper result.
I would like to add one more detail to the problem. The areas of the two beams are different. Because of this, there will be a discontinuity in stresses at the interface however the force would be same. So i feel that there will be a redistribution of stresses of some sort, I am not sure.

How can i solve this problem taking into account the redistribution of stress?

 

[Divya Shyam Singh]

(2) Find and solve an expression for the Airy stress function and then derive an expression for the actual stresses ?

Also, please do explain the method for (2)

 

[Nidum]

The areas of the two beams are different

If you have a beam of one section joined end on to a beam of a different section then there could be a very complicated stress pattern in the transition area between the two beams .

What do you actually know about the two beam sections ?

 

[Divya Shyam Singh]

If you have a beam of one section joined end on to a beam of a different section then there could be a very complicated stress pattern in the transition area between the two beams .

What do you actually know about the two beam sections ?

We know the Youngs modulus and all the dimensions of the beam. We know the force applied. We have to find airy's stress function, stresses, strains and displacement functions. If i could solve for stress, i would be able to solve for the rest of them.

 

[Nidum]

Please do a proper drawing of the connection zone between the two beams sections and post it for us to discuss .

 

[Divya Shyam Singh]

https://postimg.org/image/arh46w7lh/
This is the link to the image.

 

[JBA]

By treating the total beam as two series connected cantilever beams you should be able to get the stress, shear and deflection data you need for your final calculation.

 

[Nidum]

I've processed your sketch to make it a bit clearer .

Realistically I don't see how you can properly obtain the stress distribution at the junction of the two beams without using numerical methods .

Just to get an approximation you could assume some distance either side of the joint where the beam stresses revert to having a simple distribution and then sketch in best guess for distribution in the transition area .

There will be a stress concentration to consider in the corner of that abrupt step .

 

[Nidum]

One thing bothers me a little :

The original problem was relatively simple - do you really need to add in all this complexity to give an answer at a level consistent with the question ??

 

[Divya Shyam Singh]

One thing bothers me a little :

The original problem was relatively simple - do you really need to add in all this complexity to give an answer at a level consistent with the question ??

Yes, actually this is the actual problem i am working on. Thanks for your help so far.
I have assumed that the stress at the beam interface with larger area will be the same as the beam with smaller area. After going the distance equal to its thickness(according to St, Venant principle), the stress in the beam with larger area will become equal to the stress calculated by the basic beam equations.
There will be no change in stress distribution in the beam of the smaller area.
The reason for this assumption is that if we change stress distribution in the beam with smaller area, the moment of these stresses are not coming to be equal to the moment at that section. So i assumed that there will be no stress change in the beam with smaller area.
Is my assumption correct?

 

[JBA]

Because the shear and bending force profiles for the beam are independent of the beam sections' size and modulus, I recommend determining the magnitude of those profiles with your applied end load and then using those to calculate the bending and shear stresses for each beam segment based upon its size and its modulus. The bending moment and shear force profiles for the beam will be continuous throughout its length but there will be a discontinuity in the M/EI and stress profiles at the junction of the two beam segments based upon each segment's height and modulus.

If you can get access to a copy of "Mechanics of Materials" by E.P Popov, there is a discontinuous beam analysis example in Chapter 11 of that textbook illustrating this.

 

[Divya Shyam Singh]

Because the shear and bending force profiles for the beam are independent of the beam sections' size and modulus, I recommend determining the magnitude of those profiles with your applied end load and then using those to calculate the bending and shear stresses for each beam segment based upon its size and its modulus. The bending moment and shear force profiles for the beam will be continuous throughout its length but there will be a discontinuity in the M/EI and stress profiles at the junction of the two beam segments based upon each segment's height and modulus.

If you can get access to a copy of "Mechanics of Materials" by E.P Popov, there is a discontinuous beam analysis example in Chapter 11 of that textbook illustrating this.

Thanks a lot! :D

 

[Nidum]

@Divya Shyam Singh . Have you solved this problem now ?

 

[Divya Shyam Singh]

Yes, upto some degree. I managed to calculated the stress field at every position of the beam. However there will be some highly non linear stress field in the vicinity of the interface. I am pondering about the same, how to calculate the non linear stress near the interface...
any thoughts?

 

[Nidum]

Basic variable section beam theory does not help you solve this problem at all .

As I said in post #14 :

Realistically I don't see how you can properly obtain the stress distribution at the junction of the two beams without using numerical methods .

Just to get an approximation you could assume some distance either side of the joint where the beam stresses revert to having a simple distribution and then sketch in best guess for distribution in the transition area .

There will be a stress concentration to consider in the corner of that abrupt step .

Let us look at the approximation method first .

Start by drawing out the joint area of the two beam sections neatly and large size on graph paper . Sketch in you best guess of the stress distribution .

 

[JBA]

The change at the interface is instantaneous with the stress profile having a vertical line at the interface. It is as though there are two individual beams connected in series at that junction.