Assembly of Stiffness Matrix and Force Vector

After the evaluation of element stiffness matrix and element force vector for all the elements, these quantities need to be " assembled " to get the global stiffness matrix and global force vector. As stated at the end of section 6.2, this procedure has two steps:

1. Expansion of the element stiffness matrix and element force vector to the full size.
   
2. Addition of the expanded matrices and vectors over all the elements. At this stage, the second term of the expression for (equation 6.8) also needs to be added.

Let us first discuss the first step. Note that equations (6.25) and (6.26) are the expressions for the element stiffness matrix and the element force vector while equations (6.21) and (6.22) are the expressions for their expanded versions and . When we compare equations (6.25) with (6.21), we observe that (1,1) component of occupies the position of the expanded matrix . This is because is the global number of the local node 1 of the element k. Thus, the first step involves:

• Choose the component , ,
  
• Find the global number of the local nodes and of the element . Let they be and respectively.
  
• Then the component occupies the location in -th row and s -th column of the expanded matrix . Thus, the component goes to the location in the expanded matrix.
  
• Repeat the steps (i)-(iii) for the other values of and . The remaining components of are made zero.

The first step can be expressed mathematically by introducing a matrix , called as the connectivity matrix , which relates the local and global numbering systems. The number of rows in the connectivity matrix is equal to the number of elements and the number of columns is equal to the number of nodes per element. Thus, the row index of denotes the element number and the column index of represents the local node number. The elements of are the corresponding global node numbers. Thus, for the mesh of Fig. 6.1, the connectivity matrix becomes

(6.34)

The first row of the connectivity matrix contains the global numbers of the first and second local nodes of element 1. The global numbers corresponding to the first and second local nodes of element 2 are written in the second row. Continuing in this way, the global numbers of the first and second local nodes of element appear in the -th row. The last row contains the global numbers associated with the first and second local nodes of the last, i.e. -th element. The expression (6.34), in the index notation, can be expressed as

(6.35)

It means the global number of the local node of the element is obtained as the value of the component of the connectivity matrix in -th row and -th column. As an example, consider the case of = 3 and = 2. The expression (6.35) gives . This means 4 is the global number of the second local node of the element 3. This can be verified from Fig. 6.1.

Now, the first step of the assembly procedure can be expresses as follows. The expanded matrix is obtained from the element stiffness matrix by the relation:

(6.36)

Similarly, to obtain the expanded vector from the element force vector , we use the relation:

(6.37)

Thus, we use the following procedure:

1. Choose the component
2. Find the global number of the local node from the connectivity matrix. Let it be .
3. Then, the component goes to the location of the expanded matrix.
4. Repeat the steps (i)-(iii) for the other values of . The remaining components of are made zero.

The second-step is straight-forward. After obtaining the expanded versions of the element stiffness matrix and the element force vector for all the elements, they are added as follows:

,	(6.38)
	(6.39)

The matrix corresponds to the second term of equation (6.8). Note that, the only basis function which is nonzero at is . Further, it's value at is 1. Thus

Therefore, the vector {P} can be written as

(6.41)