Hello Everybody,
I would really appreciate any form of help for the following problem.
I want to calculate the deflection formula and the critical buckling load for a rectangular composite plate under uniaxial in plane load. The plate is clamped on the loaded ends and free on the unloaded ends(clamped-free-clamped-free).
Therefore I use the Rayleigh Ritz Method(RRM) for minimizing the formula of Potential Energy of the Plate.
For those of you who use Maple the following link contains a explained worksheet file. Alternatively I attached the file below this thread.
https://drive.google.com/folderview?id=0B6ODPZfF_qeUQXNSTzRZOUxfUzQ&usp=sharing
The governing differential equation is the following:
D11*W(xxxx)+(2*D12+4*D66)*W(xxyy)+D22*W(yyyy)+N*W(xx)=0
,where
the smallest value of N is the Critical Buckling Force
and W is the deflection in z-direction
and D are the values of the bending stiffness matrix of the material
Taking the boundary conditions into account I assumed only buckling in x direction and therefore the assumed shape function of W is:
W(x,y):=Sum(i=1...M)(c*(cos(2*i*Pi*x/a)-1)
,where c are the so called Ritz coefficients (also amplitudes of the deflection) that i don't manage to calculate.
The theory of the RRM says to develope the Potential Energy of the plate and derive it to each ritz coefficient and set these equations =0.
P=1/2*integral((D11*ddwx^2-N*dwx^2),[x=0..200,y=0..50])
For instance set the number of terms to M=2:
eq[1] :=−0.0139687c[2]+2067.20c[1]−4.93480Nc[1]=0
eq[2] :=33075.2c[2]−0.0139687c[1]−19.7392Nc[2]=0
This should result in M equations with M unknowns but in my case there is still the unknown of N(critical buckling force).
I managed to calculate the force N but not the coefficients c1...cM which I need for the shape function of W(x,y).
I already tried to calculate the coefficients by using the boundary conditions without success. Another approach could possibly be the Newton Raphson Method, but I don't know how to apply it to this problem and I think I made a mistake earlier in my calculations.
Any tipps or help would be greatly appreciated.
Thanks in advance
Sam
View attachment Plate_Buckling.mw.zip
I would really appreciate any form of help for the following problem.
I want to calculate the deflection formula and the critical buckling load for a rectangular composite plate under uniaxial in plane load. The plate is clamped on the loaded ends and free on the unloaded ends(clamped-free-clamped-free).
Therefore I use the Rayleigh Ritz Method(RRM) for minimizing the formula of Potential Energy of the Plate.
For those of you who use Maple the following link contains a explained worksheet file. Alternatively I attached the file below this thread.
https://drive.google.com/folderview?id=0B6ODPZfF_qeUQXNSTzRZOUxfUzQ&usp=sharing
The governing differential equation is the following:
D11*W(xxxx)+(2*D12+4*D66)*W(xxyy)+D22*W(yyyy)+N*W(xx)=0
,where
the smallest value of N is the Critical Buckling Force
and W is the deflection in z-direction
and D are the values of the bending stiffness matrix of the material
Taking the boundary conditions into account I assumed only buckling in x direction and therefore the assumed shape function of W is:
W(x,y):=Sum(i=1...M)(c*(cos(2*i*Pi*x/a)-1)
,where c are the so called Ritz coefficients (also amplitudes of the deflection) that i don't manage to calculate.
The theory of the RRM says to develope the Potential Energy of the plate and derive it to each ritz coefficient and set these equations =0.
P=1/2*integral((D11*ddwx^2-N*dwx^2),[x=0..200,y=0..50])
For instance set the number of terms to M=2:
eq[1] :=−0.0139687c[2]+2067.20c[1]−4.93480Nc[1]=0
eq[2] :=33075.2c[2]−0.0139687c[1]−19.7392Nc[2]=0
This should result in M equations with M unknowns but in my case there is still the unknown of N(critical buckling force).
I managed to calculate the force N but not the coefficients c1...cM which I need for the shape function of W(x,y).
I already tried to calculate the coefficients by using the boundary conditions without success. Another approach could possibly be the Newton Raphson Method, but I don't know how to apply it to this problem and I think I made a mistake earlier in my calculations.
Any tipps or help would be greatly appreciated.
Thanks in advance
Sam
View attachment Plate_Buckling.mw.zip