On substituting equation(10.84) into the first set of equation (10.83), we get

which can be can be written as

where is the condensed stiffness matrix. Now internal DOFs could be eliminated from the shape assumed functions also. On substituting a , b , and c obtained from equation (10.67)into equation , we get

which could be simplified as

with

Similarly, on substituting a , b and c obtained from equation(10.84) in equation (10.86), we get

with

 

10.3.3 Weak Formulation of the Timoshenko Beam Element for the Dynamic Case

For dynamic case the shape functions derived in the previous section will still be valid and could take the following form