Advanced Methods of Structural Analysis

216 7 The Force Method

determinate beam. Analysis of this beam (calculation of all reactions, con-

struction of the internal force diagram, and elastic curve) creates no difficulties

at all.

The version 2 of the primary unknown and corresponding compatibility equation

is shown also in Table7.1. The primary systems 1 and 2 are not unique. Using a

rational primary system can significantly simplify the analysis of a structure.

The following procedure may be recommended for analysis of statically indeter-

minate structures by the superposition principle:

1.Determine the degree of statical indeterminacy
2.Choose the redundant unknowns; their number equals to degree of statical
indeterminacy
3.Construct the statically determinate structure (primary structure) by eliminating
all redundant constraints
4.Replace the eliminated constraints by primary unknowns. These unknowns
present reactions of eliminated constraints
5.Form the compatibility equations; their number is equal to degree of statical in-
determinacy. Each compatibility equation should be presented in terms of given
loads and primary unknowns
6.Solve the system of equations with respect to primary unknowns
7.Since reactions of the redundant constraints are determined, then the computation
of all remaining reactions and analysis of the structure may be performed as for
the statically determinate structure

Example 7.1.Determine the reactions of the beam shown in Fig.7.6and construct

the bending moment diagram. Bending stiffnessEIis constant.

Actual

state

q

l

q

Reactions of

A B supports A B

RA RB

MB

HB

Fig. 7.6 Design diagram of the beam and reactions of supports

Solution.The structure has four unknown reactions (the vertical reactionsRA,RB,

horizontal reactionHB, and support momentMB/, so the structure is the first degree

of statical indeterminacy. A detailed solution for two versions of primary systems is

presented in Table7.2.