Advanced Methods of Structural Analysis

286 8 The Displacement Method

Discussion:

1.The active load acts at the level of the crossbar and the bending moments in the
primary system due to this load do not arise. However, this is not to say that all
free terms of canonical equations are zero. Introduced constraints prevent angular
and linear displacements of the frame. Sofree terms present the reactive moment
and force in the introduced constraints. The reactive momentR1P D 0 , while
the reactive force of introduced constraintR2P¤ 0.
2.Even if the primary system has a nonzero bending moment diagram within the
crossbar (diagramMN 1 ), the resulting bending moment diagram along the cross-
bar is zero. This happens becauseZ 1 D 0 ,MN 2 D 0 ,andMP^0 D 0.
3.If the flexural stiffness of the crossbar is increasedntimes, then unit reaction
r 11 becomesr 11 D 2 EIn, while all other coefficients and free terms remain
the same. So equations (b) lead to the same primary unknowns and the resulting
bending moment diagram remains same. This happens because for given design
diagram the bending deformations for crossbar are absent.
4.The lengthlof the span has no effect on the bending moment diagram.
5.If the frame, shown in Fig.8.7a, is modified (the number of the vertical members
beingk), then the reaction of each support isP=k.
The next example presents a detailed analysis of a frame by the displacement
method in canonical form. This frame was analyzed earlier by the force method.
Therefore, the reader can compare the different analytical approaches to the same
structure and see their advantages and disadvantages.

Example 8.2.A design diagram of the frame is presented in Fig.8.8a. The flexu-

ral stiffnesses for the vertical member and crossbar areEIand 2 EI, respectively;

their relative flexural stiffnesses are shown in circles. The frame is loaded by force

PD 8 kN and uniformly distributed loadqD 2 kN=m. Construct the bending mo-

ment diagram.

Solution.It is easy to check the number of independent linear displacements

nd D 1 (the hinged scheme is not shown) and the total number of unknowns of

the displacement method is 2. The primarysystem is obtained by introducing two

additional constraints, labeled 1 and 2 (Fig.8.8b). Constraint 1 prevents only angu-

lar displacement of the rigid joint and constraint 2 prevents only linear displacement

of the crossbar.

The flexural stiffness per unit length for each element of the structure is as

follows:

i 1 - 3 D

1 EI

5

D0:2EIIi 6 - 8 D

2 EI

10

D0:2EIIi 4 - 5 D

1 EI

3

D0:333EI;

where the subscript of each parameteriindicates an element of the frame (Fig.8.8c).