Dear Colleague,

the application of the boundary conditions for a beam are reported in the following article.

Please pay attention on the number of points you have.

You have 4 boundary conditions at the two ends (two at the first end and other two at the other end)

and only (N-4) domain points to use.

The total number of equations must to be equal to N.

So you must write (N-4) domain equations for indexes (3,4,....,N-2) and 4 equations for the boundary conditions for the indexes (1,2) and (N-1,N).

If you need help please let me know.

All the Best,

Francesco Tornabene

Article Critical Flow Speeds of Pipes Conveying Fluid by the General...

I tried increasing the domain also. Even then I am getting the frequency from second mode. 

Dear Colleauge,

you can find answer in the following review.

All the Best,

Francesco Tornabene

Article Strong Formulation Finite Element Method Based on Differenti...

Dear Colleague,

the application of the boundary conditions for a beam are reported in the following article.

Please pay attention on the number of points you have.

You have 4 boundary conditions at the two ends (two at the first end and other two at the other end)

and only (N-4) domain points to use.

The total number of equations must to be equal to N.

So you must write (N-4) domain equations for indexes (3,4,....,N-2) and 4 equations for the boundary conditions for the indexes (1,2) and (N-1,N).

If you need help please let me know.

All the Best,

Francesco Tornabene

Article Critical Flow Speeds of Pipes Conveying Fluid by the General...

Thank you very much  sir.

I will try again with these inputs.

Dear Colleagues,

thank you very much. I am happy to helping you.

All The Best,

Francesco Tornabene 

 I have tried many times to solve differential equations by using a polynomial

series expansions, which  transform a  differential equation into an algebraic equation.

    Sometimes I utilized transcendent functions. In every case it is necessary to look at the specific  problem.

All the best

 Prof. Silvano Tizzi

Dear Sir,

I read the paper and tried to solve accordingly, but I am not able to converge to the values given in the review  paper. I am not able to find, where I have gone wrong.

The details of my work is in the doc file I am attaching.

Please help with this.I have attached the review paper along with the question.

Dear Sir,

I read the file. In this case as we have w1=w2=w5=w6=0, the 6*6 coefficient matrix reduces to a 2*2 matrix.

Then I took the determinant of the matrix, which when solved will give the frequency value. I used this method to solve both for 6 points and 7 points, but still I am getting wrong answers.

Do you think the potential cause of error is due to this method of obtaining the frequency or Is there any problem with my weighting coefficients?

Dear All,

here you can find a simple MATLAB code to solve the Euler-Bernoulli simply-supported beam:

% Beam Problem - Simply-supported Euler-Bernoulli Beam

clear

clc

% data

L=2;

b=0.01;

h=0.01;

A=b*h;

I=1/12*b*h^3;

E=2.1e11;

ni=0.3;

rho=7800;

q0=100;

% numerical data

N=31;

m=4;

% Weighting coefficients for derivation

r=zeros(1,N);

for k=1:N

r(k)=cos(((2*k-1)*pi)/(2*N));

end

r=sort(r);

% first derivative of Chebyshev polynomial of the first kind

dCheb_1=(N*sin(N*acos(r)))./(1 - r.^2).^(1/2);

c=zeros(N,N,m);

for p=1:m

if p==1

% first order derivative weighting coefficients

for i=1:N

for j=1:N

if i==j

else

c(i,j,p)=dCheb_1(i)/((r(i)-r(j))*dCheb_1(j));

end

end

end

else

% higher order derivative weighting coefficients

for i=1:N

for j=1:N

if i==j

else

c(i,j,p)=p*(c(i,j,1)*c(i,i,p-1)-c(i,j,p-1)*((r(i)-r(j))^(-1)));

end

end

end

end

% diagonal weighting coefficients for each order derivative

for i=1:N

c(i,i,p)=-sum(c(i,:,p));

end

end

% domain length

L1=r(N)-r(1);

% coordinate transformation

x=(r-r(1))/(r(N)-r(1))*L;

% weighting coefficients transformation

for p=1:m

c(:,:,p)=(L1/L)^p*c(:,:,p);

end

% Strong formulation

Ks=E*I*c(:,:,4);

Fs=q0*b*ones(N,1);

Ms=rho*A*eye(N);

% buondary conditions

Ks(1,1)=1;

Ks(1,2:N)=zeros(1,N-1);

Ks(2:N,1)=zeros(N-1,1);

Ks(N,N)=1;

Ks(N,1:N-1)=zeros(1,N-1);

Ks(1:N-1,N)=zeros(N-1,1);

Ks(2,:)=E*I*c(1,:,2);

Ks(N-1,:)=E*I*c(N,:,2);

% buondary conditions for loads

Fs(1)=0;

Fs(2)=0;

Fs(N-1)=0;

Fs(N)=0;

% static analysis

us=Ks\Fs;

% dynamic analysis

Kdds=Ks(3:N-2,3:N-2);

Kdbs=[Ks(3:N-2,1:2) Ks(3:N-2,N-1:N)];

Kbbs=[Ks(1:2,1:2) Ks(1:2,N-1:N); Ks(N-1:N,1:2) Ks(N-1:N,N-1:N)];

Kbds=[Ks(1:2,3:N-2); Ks(N-1:N,3:N-2)];

Mddgs=Ms(3:N-2,3:N-2);

% eigenvalue problem

Hs=sqrt(eig(Mddgs\(-Kdbs*(Kbbs\Kbds)+Kdds)));

% circular frequencies

omegas=sort(Hs);

% frequencies

freqs=real(omegas/(2*pi));

% Exact solution

u_ex=q0*b*L^4/(24*E*I)*(x.^4/L^4-2*x.^3/L^3+x/L);

% Exact solution: dynamic analysis

freq_ex=zeros(N-4,1);

for i=1:N-4

freq_ex(i,1)=(((i^2)*(pi/2))/(L^2))*sqrt((E*I)/(rho*A));

end

Freqs=freqs./freq_ex;

% static deformation

figure

plot(x,u_ex,'r-',x,us,'kd')

set(gca,'Ydir','reverse')

% dynamic frequencies comparison

figure

plot(((1:N-4)-1)/(N-5),0*(Freqs-1)*100,'r-',((1:N-4)-1)/(N-5),(Freqs-1)*100,'kd-')

ylim([-10 10])

xlim([0 1])

All the Best,

Francesco Tornabene

    It would be necessary to look at the differential equation, and particularly

it would be good to know to what physical  problem this refers.

    In my experience many times I have tried to utilize  series expansions,

which can transform a differential equation into an algebraic  equation.

     Many times I found the requested solutions with limited computer time.

   Best greetings from prof. Silvano Tizzi

Dear Prof. Francesco Tornabene

Many thanks for your continous efforts.

Can you help me for implementing the boundary conditions of a cantilever beam with tip point load or a simply supported beam with a mid-span point load.

Regards

Dear Professor Attia,

if you want consider a tip load you can change the following line in the previous code:

% buondary conditions

Ks(1,1)=1;

Ks(1,2:N)=zeros(1,N-1);

Ks(2,:)=c(1,:,1);

Ks(N-1,:)=E*I*c(N,:,2);

Ks(N,:)=E*I*c(N,:,3);

% buondary conditions for loads

Fs(1)=0;

Fs(2)=0;

Fs(N-1)=0;

Fs(N)=-1000;

This case is related to a cantilever beam with a concentrated load at the second end.

In the case you want to consider a concentrated load at the middle of a beam I suggest to read the following paper about this topic. Otherwise you must divide the beam in two parts where you want to applied the concentrated load.

All the Best,

Francesco Tornabene

http://www.sciencedirect.com/science/article/pii/S0263822315004237

Many thanks Prof. F. Tornabene for your kind help.

Dear Dr. Ugurcan 

I tried to implement the determinant which you gave in the uploaded doc, however I am not able to get to the correct eigenvalues. My coefficient matrix seems to be correct. Any suggestions where I would be lacking?

Thanks

Sajal

Hello,

I can help to implement the boundary conditions of a cantilever beam with a point mass attached to the free end ?.

I am considering Euler beam theory-Bernoullii. I have understood that in considering the point mass without the effect of the rotational inertia of the system is reduced to N-3. But I can not solve the system.

If anyone modeled with Timoshenko beam theory considering the mass point also considering the rotational inertia, I would help too.

Thank you very much, I hope answers.

Hi Gilardi

Prof. Francesco has shared a code to implement the tip mass at the free end. The change is in the BC. Did you try that? I can have a look at your code and help you in pointing out the error.

Best!

Sajal

 Dear Professor Sajal Sagar Singhji,

Can you please share that MATLAB code with me. My e-mail address is [email protected]  

Thank you very much in advance.

 You should share the code here.I am working in the area of vibration problems and facing the same difficulties calculated the weighing coefficients and implementation of boundary condition is also correct but i am not getting the required convergence. I got the correct results on 50 grid points but it should be 15 points.      

Hi,

How can I apply central point load on a simply supported beam using DQM?

Any suggestion / help is highly appreciated .

Thanks

Dear Collegaue,

you can refer to the article below:

https://www.researchgate.net/publication/307558421_On_the_Mechanics_of_Laminated_Doubly-Curved_Shells_Subjected_to_Point_and_Line_Loads?ev=prf_pub

and you can refer to the very good book by X. Wang:

http://www.sciencedirect.com/science/book/9780128030813

In both cases, you will find a simple way to apply concentrated loads.

Another way is to divide the domain in different part and apply the concentrated load 

in the junction point, see for example the papers:

1) https://www.researchgate.net/publication/302314119_Transient_Dynamic_Response_of_Generally-Shaped_Arches_Based_on_a_GDQ-Time-Stepping_Method?ev=prf_pub

2) https://www.researchgate.net/publication/223912505_Analytical_and_Numerical_Results_for_Vibration_Analysis_of_Multi-Stepped_and_Multi-Damaged_Circular_Arches?ev=prf_pub

All the Best,

Francesco Tornabene

Article On the Mechanics of Laminated Doubly-Curved Shells Subjected...

Article Transient Dynamic Response of Generally-Shaped Arches Based ...

Article Analytical and Numerical Results for Vibration Analysis of M...

Thank you Prof. Francesco Tornabene for this useful information. I would appreciate, if you can provide some references were the beam problems (subjected to concentrated load) are solved by DQM.

Dear Colleague,

please read the book by X. Wang.

All the Best,

Francesco Tornabene

Thanks Francesco Tornabene for sharing the references.

Anyone have some reference for solving this problem when variables are in polar form

you can do it in matlab its so easy

Already tried so many times but unable to correct reault. for rectangular plates it works well.

see

Computational Mechanics ’95 || A New Approach for Solving Fourth-Order Equations by the Differential Quadrature Method

Atluri, S. N., Yagawa, G., Cruse, Thomas

https://booksc.xyz/book/40629490/d8b0d1