Abstract
Structures such as buildings and bridges consist of a number of components such as beams, columns, and foundations, all of which act together to ensure that the loadings that the structure carries is safely transmitted to the supporting ground below. The study of the design and deflection of the beam under load play an important role in the strength analysis of a structure. In the present paper, we have applied high-order compact finite difference scheme using MATLAB to approximate the solution of Euler–Bernoulli beam equation which determines the deflection of the beam under the load acting on the beam.
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Acknowledgements
This paper is part of the project [No.-UCS&T/R&D/PHY SC.-10/12-13/6180] funded by Uttarakhand State Council for Science and Technology, Dehradun, Uttarakhand, India.
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Appendix
Appendix
Matlab Programs:
Example 1:
clc; clear all; t=0; t=cputime; format long; n=input(‘enter the number of subintervals’); x=linspace(0,1,n+1); h=1/n; %----------------------------------------------------- f=[]; g=[]; fori=1:n+1 f=[f;(8*(pi^4)*sin(2*pi*x(i)))]; end %----------------------------------------------------- %right side function g for v fori=2:n g=[g;(f(i-1)/12)+(f(i+1)/12)+(5*f(i)/6)]; end %----------------------------------------------------- %left boundary lb=0; %right boundary rb=0; %----------------------------------------------------- %coefficient matrix A A=sparse(n-1,n-1); %diagonal elements of A fori=1:n-1 A(i,i)=-2/(h^2); end %left elements of A fori=2:n-1 A(i,i-1)=1/h^2; end %right elements of A fori=1:n-2 A(i,i+1)=1/h^2; end %----------------------------------------------------- %modified g due to left boundary g(1)=g(1)-(lb/h^2); %modified g due to right boundary g(n-1)=g(n-1)-(rb/h^2); %----------------------------------------------------- v=[]; v=A\g; v=[lb;v;rb]; %----------------------------------------------------- %right side function q for u q=[]; fori=2:n q=[q;(v(i-1)/12)+(v(i+1)/12)+(5*v(i)/6)]; end %----------------------------------------------------- %modified q due to left boundary q(1)=q(1)-0; %modified q due to right boundary q(n-1)=q(n-1)-0; %----------------------------------------------------- %calculate the value of u u=[]; u=A\q; %----------------------------------------------------- %numerical solution nums=[]; nums=[nums;0;u;0]; %----------------------------------------------------- %exact solution ex=[]; fori=1:n+1 ex=[ex;(sin(pi*(x(i)))*cos(pi*(x(i))))]; end %----------------------------------------------------- %absolute error er=[]; fori=1:n+1 er=[er;abs(nums(i)-ex(i))]; end %----------------------------------------------------- %maximum absolute error maxer=max(er) timing=cputime-t %----------------------------------------------------- % graph plot(x,nums,’b’,x,ex,’*’); plot(x,er);
Example 2:
format long; clc; n=input(‘enter the number of subintervals’); x=linspace(0,1,n+1); h=1/n; %----------------------------------------------------- f=[]; g=[]; fori=1:n+1 f=[f;((pi^4)*x(i)*sin(pi*(x(i))))
-(4*(pi^3)*cos(pi*(x(i))))]; end %----------------------------------------------------- %right side function g for v fori=2:n g=[g;(f(i-1)/12)+(f(i+1)/12)+(5*f(i)/6)]; end %----------------------------------------------------- %left boundary lb=2*pi; %right boundary rb=-2*pi; %----------------------------------------------------- %coefficient matrix A A=sparse(n-1,n-1); %diagonal elements of A fori=1:n-1 A(i,i)=-2/(h^2); end %left elements of A fori=2:n-1 A(i,i-1)=1/h^2; end %right elements of A fori=1:n-2 A(i,i+1)=1/h^2; end %----------------------------------------------------- %modified g due to left boundary g(1)=g(1)-(lb/h^2); %modified g due to right boundary g(n-1)=g(n-1)-(rb/h^2); %----------------------------------------------------- v=[]; v=A\g; v=[lb;v;rb]; %----------------------------------------------------- %right side function q for u q=[]; fori=2:n q=[q;(v(i-1)/12)+(v(i+1)/12)+(5*v(i)/6)]; end %----------------------------------------------------- %modified q due to left boundary q(1)=q(1)-0; %modified q due to right boundary q(n-1)=q(n-1)-0; %----------------------------------------------------- %calculate the value of u u=[]; u=A\q; %----------------------------------------------------- %numerical solution nums=[]; nums=[nums;0;u;0]; %----------------------------------------------------- %exact solution ex=[]; w=[]; fori=1:n+1 ex=[ex;(x(i)*sin(pi*x(i)))]; end %----------------------------------------------------- %absolute error er=[]; fori=1:n+1 er=[er;abs((nums(i)-ex(i)))]; end %----------------------------------------------------- %maximum absolute error maxer=max(er) %----------------------------------------------------- % graph plot(x,nums,’b’,x,ex,’o’); % plot(x,er);
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Pathak, M., Joshi, P. (2019). High-Order Compact Finite Difference Scheme for Euler–Bernoulli Beam Equation. In: Yadav, N., Yadav, A., Bansal, J., Deep, K., Kim, J. (eds) Harmony Search and Nature Inspired Optimization Algorithms. Advances in Intelligent Systems and Computing, vol 741. Springer, Singapore. https://doi.org/10.1007/978-981-13-0761-4_35
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