Plate Buckling (All Sides Built-In) - LEVY (1942) Solution

[Prathik123]

Hi All,

For a rectangular plate buckling with all sides built-in, Levy has solved this problem in his 1942 paper: "BUCKLING OF RECTANGULAR PLATES WITH BUILT-IN EDGES", S. LEVY, JOURNAL OF APPLIED MECHANICS, VOL. 9, PG. 171, 1942.

In his paper, the buckling coefficient, K is determined by solving Equation 8, which is a doubly infinite set of equations. The solved values are summarized in Fig. 2. This figure is famously found in Bruhn Fig. C5.2 or C5.3.

However, I am trying to mathematically solve Equation 8. Can someone please help me figure this out or point me in the right direction? I am trying to code this in Matlab, so if someone can give me advice via code (of any language) is also fine by me.

Any help is greatly appreciated.

 

[Stress_Eng]

I’ve looked into rectangular plates with various edge restraints and have found the Minimum Potential Energy approach very useful.

 

[Prathik123]

Hi Stress_Eng,

I get what you mean, but I am not looking to go down a new mathematical path for now.
I was hoping to just solve the equation provided by Levy. The convergence should also be quick and with today's computational power, should take just a fraction of a second.

Thanks,
Prathik

 

[SWComposites]

Prathik - what is the issue that you are having? Eqn 8 needs to be written out for a selected set of m and n values in matrix form, then take the determinant and solve for the critical load.

 

[Prathik123]

Hi SW,

My issue is I can't figure out how to set up the matrix itself.
I have never attempted to solve a simultaneous infinite series before.

Are you suggesting something like this?

Taking the first equation of Equation 8:
Below are the coefficients of km for m,n (lets say I only go up to 5):
| m=1,n=1 m=3,n=1 m=5,n=1 |
| m=1,n=3 m=3,n=3 m=5,n=3 |
| m=1,n=5 m=3,n=5 m=5,n=5 |
Then I solve for K (embedded in the coefficients) by setting the determinant of the matrix above to 0.

I repeat the above for tn.

The lower K value of the two is the desired critical K value.

Is this the correct approach?
The problem is, I have 2 equations. The above only does for the first equation.

Thanks,
Prathik

 

[Prathik123]

In Levy's paper, he stated that for the first calculations, he used t1, k1, k3, k5, and k7.

This means he used n = 1 and m = 1 to 7.

If I assume aspect ratio = 1, then the equations will be this:

(k1 + t1)/(4-K) + (3k3 + 9t1)/(100-9K) + (5k5 + 25t1)/(676-25K) + (7k7 + 49t1)/(2500-49K) = 0 ------ EQ. 8A
(k1 + t1)/(4-K) + (k3 + 3t1)/(100-9K) + (k5 + 5t1)/(676-25K) + (k7 + 7t1)/(2500-49K) = 0 ------ EQ. 8B

I now have 5 variables: t1, k1, k3, k5 and k7 but only 2 equations.
I cannot set up a square matrix, hence I can't calculate a determinant.

This is the confusion that I am having.

Thanks,
Prathik

 

[SWComposites]

Yeah, the paper is not very clear on the details, and its not at all obvious. I haven't messed with something like this for 30+ years, so am a bit fuzzy. When I get back from vacation in a couple of weeks I'll try to find time to write out all of the equations and have a go at the a/b=1 case. Suggest picking one of the cases from Table 2, and start with the first approximation column. As noted in the text, "the determinant involving t1, k1, k3, k5, and k7" which I'm assuming means t1, k1, k3, k5, and k7 are the independent variables, with n = 1 and m = 1, 3, 5, 7. I haven't yet sorted out how eqns 9-11 get used; probably need to write out the eqn 8 series and see how it looks.

 

[Prathik123]

It's no problem, SW. I totally understand. Please take your time.

If anything, please let me know.

 

[Stress_Eng]

I believe the steps are as follows
Consider a column matrix 1,2,3 … representing a n and m case. Considering m to be even and n odd, case 1 is where m=2 and n=1. Now determine k and t for case 1. Case 2, m=4 and n=3, determine k and t for case 2, and so on for say i cases. Fingers crossed!

 

[Prathik123]

I get what you are saying. The problem is, all the coefficients of the km and tn variables must be set in a square matrix and solved simultaneously (determinate = 0) to determine the K. What makes things even more interesting is Levy solved with n = 1 and managed to get rapid convergence with just m up to 7.

 

[SWComposites]

Prathik,

I think the solution is in the "(n = 1 , 3 , 5 ... )" and "(m = 1, 3, 5, ...) at the ends of eqns 8a and 8b.

So for the case of a/b=1, and t1, k1, k3, k5, and k7, with n = 1 and m = 1, 3, 5, 7, you construct one 8a eqn using n=1, and four 8b eqns using m = 1, 3, 5, 7 respectively. Which gives 5 eqns for the 5 variables. Then take the determinant of that and solve for K.

Hope it works.

Steve

 

[centondollar]

Out of curiosity, where do you have a plate that is prone to buckling and that is built-in (in reality, not just on paper)?

My suggestion is to find a deflection approximation satisfying clamped boundary conditions, retaining a few terms, plugging it into the total potential energy expression of a thin plate and solving for the eigenvalue (buckling load). Easier than what you propose, and it will likely compare favorably to the FEA output you no doubt will also produce if this problem represents a real design situation.

 

[SWComposites]

There are many aerospace structures which have plate areas between stiffeners where the plate boundary conditions are close to fixed. Aerospace structures are usually weight critical so pinned edge assumptions are often too conservative. Some solutions using energy methods can overestimate buckling loads. Also, satisfying the clamped edge boundary conditions is sometimes difficult with some energy based series solutions. I did derive a Galerkin type solution for anisotropic plates for both pinned and clamped edge conditions many years ago.

 

[Prathik123]

Thanks SW.

Let me try that.

 

[Prathik123]

Hi SW,

I just tried it and it doesn't work. I hope I am doing it right.

Assuming aspect ratio = 1, and K = 10 (for illustration purposes only), I get the following equations:

Equation 8a; n = 1, m = 1, 3, 5, 7
(-1/6)k1-(1/6)t1+(3/10)k3+(9/10)t1+(5/426)k5+(25/426)t1+(7/2010)k7+(49/2010)t1=0
Simplifying:
0.8164t1-(1/6)k1+(3/10)k3+(5/426)k5+(7/2010)k7 = 0

Equation 8b; n = 1, m = 1
-k1/6-t1/6=0

Equation 8b; n = 1, m = 3
k3/10+3t1/10=0

Equation 8b; n = 1, m = 5
k5/426+5t1/426=0

Equation 8b; n = 1, m = 7
k7/2010+7t1/2010=0

Putting the coefficients in a matrix, ordering them by t1, k1, k3, k5 and k7, gives the following:
| 0.81640 -0.16667 0.30000 0.01174 0.00348 |
|-0.16667 -0.16667 0.00000 0.00000 0.00000 |
| 0.30000 0.00000 0.10000 0.00000 0.00000 |
| 0.01174 0.00000 0.00000 0.00235 0.00000 |
| 0.00348 0.00000 0.00000 0.00000 0.00005 |

However, when I put a random K value like 12, it does not goal seek close to 10.
The determinate value does not make sense.

I have summarized these calculations in the attached Excel file.

 

[Stress_Eng]

Just wanted to say, if you do decide to try an alternative method to solving equation 8, I do have some examples of solving the deflection of rectangular plates with all edges built-in, using the minimal potential energy method. The examples were generated out of personal interest only and have not been used in industry. Although they haven't been checked, I've compared the deflection and BM stresses to those obtained from Roark, and there is good agreement. I've looked at and compared to Roark a uniform pressure distribution (1/2 waveform displacement) and a linearly tapering pressure (0 to peak in one direction). By tweaking the tapering pressure (to give -ve on one side and +ve on the opposite), a 2 buckle waveform can be produced in one direction. The strain energy method does permit the buckling load to be derived from a known waveform. I haven't looked at any higher wave forms. If you do try an alternative method, they may give you some ideas.

 

[Prathik123]

Hi Stress_Eng,

It's not that I'm not open to other methods. It's just that the trusted values are from Levy's paper (referenced by Timoshenko) for this special case of the plate (CCCC with uniaxial compression). Although most engineers simply go to Bruhn without actually realizing Bruhn is referencing from NACA 3781.

However, I don't mind looking at what you have. Have you tried comparing your method with Levy's buckling factor, K in Table 2? How close are they?

 

[Stress_Eng]

Hi Prathik123,
I haven’t tried. I’ll have a look and let you know what I find.

 

[centondollar]

If you want to verify buckling coefficients obtained from tables in literature or from hand-calculations, you may model the same plate in any FEM program and compare the results. Many reputable software vendors offer free licenses which allow plate buckling calculations, and if this is work-related, you presumably have an industry standard FE program available on your desktop.

An infinite series solution with finite terms retained is not conceptually more accurate than a weighted residual approach or Ritz approach, and the weighted residual or Ritz approaches are more easily adapted to changing boundary conditions and loading.

 

[Prathik123]

Hi cent,

I am looking for a closed-form solution to the above stated problem. Hence, why I want to solve this Levy equation. This method is also industry accepted.
Regarding what you mentioned about its convergence, I do agree, but in a physical practical sense, I think it is accurate enough.

About FEM, in general, I usually have a hand calculation / theoretical value before I validate a FEM result. I am not comfortable with going in the other direction. Although I have done many comparisons with SOL 105 for simply-supported plates via the enforced displacement method, this is not what I am looking for here.

Of course, the truth is in the test data.