User:Eml5526.s11.team6.deshpande/hwk3

=Problem 3.2= For the spring system shown in the figure below in given data.

Find
A. Number the elements and nodes.

B. Assemble the global stiffness and force matrix.

C. Partition the system and solve for the nodal displacements.

D. Compute the reaction forces.

Given
Spring system with closed ends as shown in the figure



Solution
A: The elements and nodes are numbered as follows: Element numbers are indicated in parentheses and node numbers are indicated at the bottom of the figure.



B: The elements of the stiffness matrices are:

Where global numbers corresponding to the element nodes are indicated above each column and to the right of each row.

By direct assembly, we get the global stiffness matrix as follows

The displacement and force matrices are as follows

Combined force matrix can be written as

C: Now the global system of equations is

As first two displacements are prescribed, partition after two rows and two columns. We get,

Where,

From eq.(3.2.10), we can write

From equations (3.2.12), (3.2.13), (3.2.14), (3.2.15), we get,

and

Solving above two equations, we get

D:Calculation of the reaction forces

From eq.(3.2.10), we can also write

From equations (3.2.11), (3.2.13), (3.2.14), (3.2.19), we get,

and

Solving above two equations, we get

=6=

= Problem 6.7 Rules of Matrix algebra=

Solution
Hence, Therefore, Now, Hence, from eq. (6.7.6) and eq. (6.7.11)

Now, Using eq. (6.7.8), we can write Verified using | Wolfram Alpha

Hence, from eq.(6.7.14) and eq.(6.7.15) we can say that

=Problem 6.8=

Solution
Nodes and weights for the 5-point Gauss–Legendre formula according to | NIST Handbook are as follows:

For the table given in the problem statement, the calculated value of each term is as follows

In Fish and Belytschko's book on page 89 The position of Gauss points and corresponding weights are as follows

Hence, from the above tables we can conclude that the table given in problem statement (from [[media:fe1.s11.mtg36.djvu|Mtg 36 (a,x)]]) is reasonably accurate when compared to the above tables from NIST handbook and Fish and Belytschko's book on page 89.