How To Verify Deep Beam Fem Results

[WR1]

I have to submit this week my assignment for finite element analaysis course. I have to make a FEM model for a deep beam with different support conditions and then verify.

 

I want to know what are different verification methods i could use, and how? I mean how can I knw the non-linear distribution of stress and strain of deep beams, like in shallow beams it was Stress= My/I.

 

Any suggestion for FEM model? I will be using LUSAS program.

 

( I dont know if the material is concrete...so dont want to use ACI theories..i just have E and other data)

[UmarMakhzumi]

I have to submit this week my assignment for finite element analaysis course. I have to make a FEM model for a deep beam with different support conditions and then verify.

 

I want to know what are different verification methods i could use, and how? I mean how can I knw the non-linear distribution of stress and strain of deep beams, like in shallow beams it was Stress= My/I.

 

Any suggestion for FEM model? I will be using LUSAS program.

 

( I dont know if the material is concrete...so dont want to use ACI theories..i just have E and other data)

I suppose your material is concret because shear faiulre for steel doesnt make sense in flexural. What you can do is:  There are different shear strength models that you can use. E.g., you can do a manual Strut and Tie calclation and compare the results with your FEA. Does your beam has any shear reinforcement or is it a deep beam without reinfo. I did a lab in my masters where we failed a beam without shear reinfo and measured strain values at different location. We verified the results using Canadian Code forumlas that allow calculating shear strenght of beam without transverse reinforcement. For your case, using strut and tie model will be a resonable approach. Give a heads up to your teacher to see what does he say.

[Badar (BAZ)]

The way you have written about your assignment, it can be done in different ways. One way has been mentioned by Umar, which involves calculation of strength of beam.

 There could be another way, the easier one. Perform linear elastic F.E analysis on your deep beam, and then compare the results- which could be deflection, bending strains and shear strains- with Themoshenko equations that take into account bending , as well as, shear deformations. You can find these equations in "Timoshenko S.P. and Gere J.M., (1971), Mechanics of materials, Van Nostrand Reinhold Company", or "Timoshenko S.P., (1957), Strength of Materials – Part I, D. Van Nostrand Co. Inc., London, England".

[WR1]

@umar: No, concrete is not mentioned as the material and the course is not about concrete, ok we know it happens in concrete but these assignments carries 25% of total marks so i dont wanna involve concrete here. I just have E, no reinforcement or shear reinf is mentioned. Any way I like baz answer, and i was studying what he said lately.

 

 

The way you have written about your assignment, it can be done in different ways. One way has been mentioned by Umar, which involves calculation of strength of beam.

 There could be another way, the easier one. Perform linear elastic F.E analysis on your deep beam, and then compare the results- which could be deflection, bending strains and shear strains- with Themoshenko equations that take into account bending , as well as, shear deformations. You can find these equations in "Timoshenko S.P. and Gere J.M., (1971), Mechanics of materials, Van Nostrand Reinhold Company", or "Timoshenko S.P., (1957), Strength of Materials – Part I, D. Van Nostrand Co. Inc., London, England".

 

For others I would also like to share the following link where this professor has developed notes on FEM based on Lagoon FEM book and I loved the way he has explained it. http://www.ce.memphis.edu/7117/notes/notes.html (See chapter 4a) (this is not my univ)

 

@baz just to tell that I am asked that assignment should not be more than 500 words and 5 support conditions and no more than 12 figures, will this validation also fit in this limit? or is there any other simple approach. Any past assignments from you?

[Badar (BAZ)]

Yes, you will easily accommodate those restrictions.

I think that is, mentioned in previous comment, the simplest, and widely accepted way of validating your results of elastic FEM.

You can also compare distribtution of principle compressive stresses, distribution principle tensile stresses, deflection due to shear deformation, due to bending deformation, plot (ratio of deflection due shear/deflection due to bending) with length/height of beam, and consequently comment on that L/H <4 criteria. Shear contribution of deflection will be more for L/H less than 4, i.e deep-beam behavior.

 

I have attended the course, but that was in context of Geotechnical engineering with emphasis on tunneling; consequently, we were assigned a tunneling problem.

[WR1]

thank u all!

[WR1]

but again, as we know that strain will be non linear in this example, but i am asked to do a linear elastic analysis? now what?

[Badar (BAZ)]

In your last comment, I  think , you are talking about distribution of stresses along cross-section of member; linear for bending and nonlinear for shear stresses(rectangular section). This distribution corresponds to a state when material is behaving elastically. Non linear distribution of shear stress, along cross section, does not mean that material is in plastic state.

[WR1]

sir baz, Im tired, you wont believe how much material I have read on google about these things, but still I couldnt understand a thing.

 

Can you pls clarify some issues?

 

 I came to know that M/EI in timoshenku beams is not double derivative of dy/dx but single derivative of d (phi) / dx, after solving this you get same stresses that is sigma = My/I. It is same like Euler-Bernoulli. It means flexural stresses are unaffected. That means they are still linear.. Then how come they are non linear? We know  they are but how to find it?

 

In Timoshenku beam, the stiffness matrix includes shear effects but that way I can use F=[k]x where u contains phi1, u1, phi2, u2.

 

and [f] = P1, m1, P2, m2

 

This way ok I can find either the displacements or the forces but does it help in validating the results? I was thinking to find the stresses by some mean and comapre with my LUSAS model.

[Badar (BAZ)]

Curvature of beam is double derivative of deflection. There are other ways to express curvature; it depends upon what you want achieve. I think you are talking about the relation of curvature that Timoshenko used for deriving equations for deflection.

 

I am not able to understand your next query. Where are these stresses linear, and where are they non-linear? I am not sure what you are referring to?

 

You don't need to derive stiffness matrix and other matrices to compare results with FEM. You need to use equation d, or h, on page 172 and 173 of his book (strength of materials, second edition, the book is in two parts). These two  equations can be used to compute deflection of beam ( one for udl and other for point load), and equations, take into account, shear, as well as, bending behavior.

[WR1]

For deflection it is easy to compare including both bending  +  shear components of deflection and then I can compare this with my FEM results. But I was asking if could calculate stresses manually and compare them with my FEM model too? or it is not possible?

[Badar (BAZ)]

You can compute principle stresses, given in the book under the article: "principle stresses in bending". These are the same relations that are used for plane stress problem.

[WR1]

You need to use equation d, or h, on page 172 and 173 of his book (strength of materials, second edition, the book is in two parts). These two  equations can be used to compute deflection of beam ( one for udl and other for point load), and equations, take into account, shear, as well as, bending behavior.

 

deflection is calculated from those equations. Done. Now If I want to calculate stress that would be

 

Axial stress = My/I

 

Shear Stress = ??

 

Then I will calculate principal stress with the formula given in Timoshenku book. But how would i calculate shear stress in deep beam? This is very complex

 

Shear stress = K² G [ -phi(x) + dy/dx ]

[Badar (BAZ)]

You will need two values to compute maximum principle stress, minimum principle stress and maximum shearing stress at a given section and at a particular point: these values are bending stress(sigma x) and shearing stress for a particular point at that section.

Sigma x is My/I and shear stress can be computed by using equation 65 of that book, i.e the  equation for computing variation of shear stress over a rectangular cross-section. You will need to pick a point to compute these values, if you have taken a mid of cross-section, then shear stress will be maximum and bending stress will be zero. Put these two values in