Advanced Methods of Structural Analysis

7.5 Settlements of Supports 249

Primary system

X 1

X 2

1

5

4

3

6 8

6

3

a^4

b

φ

Δ1s

Δ2s

a

b

φ φ

10m

5m

3m

b

C

B

1 2

1

a

A

abDesign diagram

c d 21

d 11

d 22

10 d 12

10

X 2 =1

X 1 =1

3

8

M 2

M 1

Ra 2 = 0

Rj 2 = 8

Rb 1 = 0 Rj 1 =^10 Rb 2 = 1

Ra 1 = 1

X 2 =1

(^6) X 1 =1
18
3
13
10
MkNm MΣ
(factor 10–3EI)
2.1228
1.1190
1.0038
4.5418
1.119
2.1228
1.0038
Factor 10–3EI
de
Fig. 7.18 (a,b) Design diagram, primary system and corresponding displacements. (c) Bending
moment diagrams in unit conditions and reactions at the supportA.(d) Resulting bending moment
diagram; (e) Summary unit bending moment diagram
The free terms1sand2spresent the displacementsin the direction of primary
unknownsX 1 andX 2 in a primary system due to settlement of support (Fig.7.18b).
According to (7.14), we have:
1sD
X
RNi1diD.RNa1aCRNb1bCRN'1'/;
2sD
X
RNi2diD.RNa2aCRNb2bCRN'2'/; (7.3)
wherea,b,and'are the vertical, horizontal, andangular displacements of support
A. The unit reactionsRNrelated to support, which is subjected to the given settle-
ment. Each reaction has two indexes. The first index (a,b,or') corresponds to
given direction of the settlement of the support. The second index (1 and 2) means
the primary unknown (X 1 orX 2 ) which leads to appearance of unit reactions. Thus,
RNa1;RNa2are reactions in direction ofa-displacement due to primary unknowns
X 1 D 1 andX 2 D 1
RNb1;RNb2are reactions in direction ofb-displacement due to primary unknowns
X 1 D 1 andX 2 D 1