Advanced Methods of Structural Analysis

13.4 Stability of Continuous Beams and Frames 471

where dimensionless parameter

̨D

EI

kl

:

Equation (c) had been derived earlier (Table13.1, case 5).

Discussion:

1.It may appear that we used only one boundary condition. However, it is not
true. There are two boundary conditionsused: the secondboundary condition
isM.l/D 0 and this condition allows write expression forRD

k' 0
l
2.IfkD1. ̨D0/then the stability equation becomes tannlDnl. This case
corresponds to clamped-pinned beam (Table13.1, case 2). Ifk D 0 (simply-
supported beam) then the stability equation becomes tannl D 0 (Table13.1,
case 3).
3.The initial parameter method may be effectively applied for beams with over-
hang, intermediate hinges, sliding, and elastic supports.

13.4 Stability of Continuous Beams and Frames...........................

This section is devoted tothe stability analysis of compressed continuous beams

and frames. We assume that beams are subjected to axial forces only. Frames are

subjected to external compressed loads, which are applied in the joints of a frame. If

several different compressed loadsPiacts on a structure, we assume that all loads

can be presented in terms ofoneloadP. It means that growth of all loads up to

critical condition of a structure occurs in such way, so that relationships between all

loads remain constant, i.e., the loading is simple.

The both classical methods can be applied for stability analysis of continuous

beams and frames. However, for such structures, the displacement method often oc-

curs more effective than the force method.The primary system of the displacement

method must be constructed as usual, i.e., by introducing additional constraints,

which prevent angular and linear displacements of the joints. The primary unknowns

are linear and angular displacements of the joints. However, calculation of the unit

reactions has specific features.

13.4.1 Unit Reactions of the Beam-Columns.......................

A primary system of the displacement method contains theone-span standard

(pinned-clamped, clamped-clamped, etc.) members. These members are subjected

not only to the settlements (angular or linear) of the introduced constraints, but also

to the axial compressed loadP. For stability analysis, this is a fundamental feature.