Advanced Methods of Structural Analysis

7.5 Settlements of Supports 247

Note that unlike the analysis of structures subjected to loads, the termMs^0 in (7.15)

is absent.

Kinematical verification of the final bending moment diagram may be performed

by the following expression

XZ MMN†

EI

dsC

X

isD0; (7.16)

whereMN†is a summary unit bending moment diagram.

Procedure for analysis of redundant structures subjected to the settlement of sup-

ports is as follows:

1.Provide the kinematical analysis, determine the degree of redundancy, choose
the primary system of the force method, and formulate the canonical equations
(7.13)
2.Construct the unit bendingmoment diagrams and calculate the unit displace-
ments
3.Calculate the free terms of canonical equations
4.Solve the canonical equation with respect to primary unknownsXi
5.Construct the internal force diagrams
6.Calculate the reactions of supports and provide their verifications

Let us consider a two-span uniform beam with equal spans. The middle support 1 is

shifted byasshowninFig.7.17a. It is necessary to calculate the bending moment

at support 1.

The degree of static indeterminacy of the structure equals one. The primary sys-

tem is the set of two simply supported beams (Fig.7.17b). Bending moment diagram

due to unit primary unknown is shown in Fig.7.17c.

l

Δ

l

l

0 1 2

Δ

M 1

1

X 1 =1

b

c

a

deX 1 =1

Δ

1/l 1/l

Δ

l

l

Δ = − 2 Δ

1 s

Δ

Δ

0 l 2

X 1 =1

Fig. 7.17 (a–c) Two-span beam. Design diagram, primary system and bending moment diagram

due to X 1 D 1 .(d,e) Two approaches for computation of free term of canonical equation