Use one dimensional finite element method to solve the 8elem

Use one dimensional finite element method to solve the 8-element problem. Show you analysis approach step-by-step in details on (1) generating each element stiffness matrix in your model; (2) assembling and reducing the global stiffness matrix and load vector by applying boundary conditions; and (3) solving the problem by following Element and Node IDs as defined in the figure. What are the displacements at Nodes 3 and 4, and stress in Element 2? (Please take advantage of the symmetry of structure to simplify the problem.)

Solution

1) Use symmetry and reduce the proble to half by considering the centre line between element 4 and 5

we are left with element 1 , 2 , 3 , and 4. The identical problem exists for the rhs ( mirror image)

Using 1-d fem, the stiffness matrix for each element is

k = AE/L {1 -1} first row

      AE/L{-1 1 } second row

Do this for each element 1,2,3,4 using the given E,A,L

2) Now to assemble, one needs to construct the global matrix for the 4 elements

Using the coordinates 1,2,3,4,5 as in the figure ( 1 at the wall 5 at the RHS where the mirror starts)

Assemble the matrix entries for element 1 using the table entries for 1 and 4 ( ie the matrix entries of the individual stiffness matrix of element 1 go into places (1,1), (1,4), (4,1), (4,4) in that order in the global matrix

Similalry for the other matrices, their entries are assembles in the corresponding locations in the overall 5X5 matrix, eg for element 2, where the memebr goes from 2 to 3 using the positions (2,2 ), (2,3),(3,2),(3,3)

and so on

Assemble the global matrix by adding togaether the values in the same locations in the global matrix ( ie overlapping entries).

Calling the global matrix as B, the force equation is then set up as

Bq =F

Where q is a 5x1 column vector containg the displacements at locations 1 trhu 5, F is a columns vector (5X1) having the applied forces at the same locations.

Now we are given: displacements at left wall are zero so q1=q2=0

P1=P2=P3=P5=0, P4 =100

Because of the zero entries in the q vector, the 1st 2 rows and columns of the B matrix get eliminated, leaving one with a 3X3 matrix involving the entries from B obtained by removing the 1st wo rows and columns.

(Note now by symmetry q5 =0 since this point does not move as identical forces are opposing here, so 5th row and column can also be removed)

This system of 2 equations can easily be solved by hand.

The equations are B33*q3 + B34*q4   =0

B43*q3 + B44*q4 =100

Solve for q3,q4

Stress in each element is found from stress = E * strain

Where strain is the [-1,1]/L

so stress in q2 would be E/L[-1,1]*{q2,q3}

(Sorry I cant give the anser . It needs you to enter the numbers and do it yourself)