Advanced Methods of Structural Analysis

246 7 The Force Method

7.5 Settlements of Supports................................................

If any statically indeterminate structure is subjected to settlement of supports, then
internal forces arise in the members of thestructure. Analysis of such structures
may be effectively performed by the force method in canonical form. The primary
system and primary unknowns should be adopted as in the case of the fixed loads.
Let us consider any statically indeterminate structure withnredundant con-
straints. Some of the supports have linear and/or angular displacementsdi. Canoni-
cal equations are

ı 11 X 1 Cı 12 X 2 CCı1nXnC1sD 0

ı 21 X 1 Cı 22 X 2 CCı2nXnC2sD 0

 (7.13)

ın1X 1 Cın2X 2 CCınnXnCnsD0;

where free termsks.kD1;2;:::;n/represent displacements of the primary sys-
tem in the direction of primary unknownsXkdue tosettlementsof the supports. For
calculation of these terms, we need to use the theorem of reciprocal unit displace-
ments and reactions (Rayleigh second theorem).
Let the supportihas the unit displacementıi. The displacement at any point
kmay be calculated using the above-mentioned theorem, i.e.,ıkiDrik.So,the
displacement in directionkdue to unit displacement at directionimay be calculated
as reaction at supporticaused by unit load at directionk.
If the supportihas nonunity displacementdi, then the displacementkat any
pointkmay be calculated using formulakiDRNikdi,whereRNikpresents the
reaction at supportidue to unit load at directionk. In fact, it means that both parts
of formulaıkiDrikare multiplied bydi. In case of several displacementsdiof
the supports, the free terms of canonical equation of the force method are calculated
using the following expressions (indexs– “settlements” – atksis omitted):

kD

X

i

RNikdi: (7.14)

In this formulaRNikis reaction of the constraint in the direction of a given displace-
mentdidue to unit primary unknownXk D 1 .Inotherwords,RNi1andRNi2are
found as reactions in the primary system due to primary unknownsX 1 D 1 and
X 2 D 1 , respectively; these reactions are determined in supports, which are sub-
jected to displacement. After calculation of the primary unknownsXiconstruction
of internal forces is performed as usual.
The final bending moment diagram can be constructed by formula

MsDMN 1 X 1 CMN 2 X 2 CCMNnXn: (7.15)