Time-dependent internal forces in uncracked structures: analysis by the displacement method - Concrete Structures: Stresses and Deformations - page 148

Figure 5.1 Example of a coordinate system (b) employed for the analysis of a plane frame
(a) by the displacement method.

this con guration are determined at the n coordinates. This process is

repeated for unit values of displacement at each of the coordinates, respect-
ively. Thus a set of n × n sti

fi

ff ness coe cients is calculated, which forms the

sti ness matrix [ S ] n × n of the structure; a general element S ij is the force

required at coordinate i due to a unit displacement at coordinate j . The values
of the actions [ A u ] are also determined due to unit values of the displace-
ments; any column j of the matrix [ A u ] is composed of the values of the
actions at the desired locations due to D j

ff

= 1.

Step 4 The displacement { D } in the actual (unrestrained) structure is
obtained by solving the equilibrium equation:

[ S ] { D } = − { F } (5.1)

The equilibrium Equation (5.1) indicates that the displacements { D } must
be of such a magnitude that the arti fi cial restraining forces { F } are

eliminated.

Step 5 Finally, the required values { A } of the actions in the actual structure