Advanced Methods of Structural Analysis

372 11 Matrix Stiffness Method

The beam in Fig.11.3a has one unknown of the displacement method, i.e., the

degree of kinematical indeterminacy equals one. So the degree of freedomnD 1.

The frame in Fig.11.3b has three unknowns of the displacement method (two angu-

lar displacementsZ 1 andZ 2 of the rigid joints andone linear displacementZ 3 of

the cross bar). So the number of degrees of freedomnD 3.

a Z 1 b Z^1 Z 2 Z^3

Fig. 11.3 Elastic curves and concept of degrees of freedom

Thus the term “degrees of freedom” implies possibility of angular, and linear

displacements of the joints are caused bydeformationof the structure. It can be seen

that for kinematical analysis of a structure (Chap. 1 ) the term “degrees of freedom”

had other meaning; in this case this concept implies independent displacements of

the system which contains theabsolutely rigiddiscs.

11.2 Ancillary Diagrams....................................................

To present a structure in such form which can be accepted by a modern computer,

the entire design diagram should be expanded by three ancillary diagrams. They

are joint-load (J-L), displacement-load (Z-P), and internal forces-deformation (S-e)

diagrams.

11.2.1 Joint-Load (J-L)Diagram...................................

This diagram presents the transformation of the arbitrary load into theequivalent

joint loads. For construction of joint-load diagram, the followings are necessary:

1.Identify the possible angular and linear displacements of the joints
2.Construct the bending moment diagramMP^0 in the primary system of the dis-
placement method (first state) due to external exposures (loads, settlements of
supports, temperature change)
3.Present the moments and forces as thejoint loadin direction of the possible
displacements (second state).
Continuous beam in Fig.11.4has one unknownangulardisplacement at support 1.
We need to convert the given load to the equivalentmomentat the support 1. The
first state is the bending moment diagramMP^0 in primary system. The fixed-end