1-D Beam Code

Point (1):

For EI(x)=1, f=1, L=1, with Boundary Conditions w=0 , dwdx

=0 at x=0 and M=0,V=1 at

x=1.Given t=0.01.

Fig 1. Displacement solution for EI(x)=1, f=1, L=1

Fig 2. Displacement Derivative solution for EI(x)=1, f=1, L=1

Fig 3. Shear force Distribution for EI(x)=1, f=1, L=1

Fig 4. Bending Moment Distribution for EI(x)=1, f=1, L=1

Fig 5. Bending Stress Distribution of top surface for EI(x)=1, f=1, L=1

Fig 6. Bending Stress Distribution of Bottom surface for EI(x)=1, f=1, L=1

As the exact solution is the fourth order solution, the finite element solution does not match for one element and it is converging as the number of elements increases. The derivative of the solution is continuous and that also converging as the number of elements increases. Bending Moment, Bending Stresses on the top and the bottom surfaces are linear for one element and converging as the number of elements increases.

Point(2):

For EI(x)=1, f=x, L=1, with Boundary Conditions w=0 , dwdx

=0 at x=0 and M=0,V=0 at

x=1. Given t=0.01.

Fig 7. Displacement solution for EI(x)=1, f=x, L=1

Fig 8. Displacement Derivative solution for EI(x)=1, f=x, L=1

Fig 9. Shear force Distribution for EI(x)=1, f=x, L=1

Fig 10. Bending Moment Distribution for EI(x)=1, f=x, L=1

Fig 11. Bending Stress Distribution of top surface for EI(x)=1, f=x, L=1

Fig 12. Bending Stress Distribution of Bottom surface for EI(x)=1, f=x, L=1

As the exact solution is the fifth order solution, the finite element solution does not match for one element and it is converging as the number of elements increases. The derivative of the solution is continuous and that also converging as the number of elements increases. Bending Moment, Bending Stresses on the top and the bottom surfaces are linear for one element and converging as the number of elements increases.