Expressions for Element Stiffness Matrix and Force Vector

Now, we shall determine the row and column numbers of the matrix equation (5.14) to which the -th element makes the contribution. For this purpose, consider equation (5.13):

(6.6)

where the expressions for the stiffness matrix and the force vector are given by equations (5.11) and (5.12):

(6.7)

and

(6.8)

The above expression can be written as

(6.9)

(6.10)

and

(6.11)

(6.12)

Note th at and represent the contributions of the -th element to the stiffness matrix and the force vector . The second term of equation (6.11) is a contribution from the last element only. This term needs to be treated separately and therefore not included in the expression for and the present discussion. Using the expressions (6.10) and (6.12), the contribution to equation (6.6) of the -th element can be written as

(6.13)

Note that the values of for which is nonzero over the -th element are = and = . Therefore, the only nonzero rows of and are = and = . Thus, the only nonzero equations of the set (6.13) are = and = . These equations are as follows:

,	(6.14)
	(6.15)

Since, the only values of for which is nonzero over the interval are = and . Therefore, the above equations become:

,	(6.16)
.	(6.17)

In matrix form, this can be written as

,

(6.18)

where

	(6.19)

and

(6.20)

In elemental notation, the above quantities can be written as

	(6.21)

and

(6.22)

Note that, the elements of and are zero when or and or . Thus,

(6.23)

and

(6.24)

Therefore, these quantities are often expressed as

,

(6.25)

and

(6.26)

The matrix (6.25) is called as the element stiffness matrix whereas the vector (6.26) is called as the element force vector . Note that the notation for the elemental quantities uses small letters : small k for the stiffness matrix and small f for the force vector. The notation for the corresponding full matrix or full vector uses capital letters : capital K for the stiffness matrix and capital F for the force vector.

The expressions (6.25) and (6.26) help in the systematic evaluation of the global stiffness matrix and global force vector of equation (6.6). The procedure can be described as follows:

1. First, the stiffness matrix and the force vector of each element are evaluated using the expressions (6.25) and (6.26). This can be done using a DO loop. Thus, for , the elements stiffness matrix and element force vector are evaluated using

(6.27)

and

(6.28)

2. Next, these quantities are expanded to the full size and . This is done using the relationship between the local and global node numbering systems.
   
3. Finally, the expanded matrices and vectors of all the elements are added to get the global stiffness matrix and global force vector . At this stage, the second term of the expression for (equation 6.8) also needs to be added.

The last 2 steps constitute what is called as the global assembly procedure. The details of the procedure are discussed in section 6.4. Before that, section 6.3 discusses an example on element calculation.