Advanced Methods of Structural Analysis

248 7 The Force Method

The canonical equation and primary unknown are ı 11 X 1 C 1s D 0 ,
X 1 D1s=ı 11.
The unit displacement

ı 11 D

MN 1 MN 1

EI

D 2 

1

2

l 1 

2

3



1

EI

D

2l

3 EI

:

The free term of canonical equation1sis a displacement in direction of the primary
unknownX 1 (i.e., themutualangle of rotation at support 1) caused by the given
settlements of support). In this simple case, the1scan be calculated by procedure
shown in Fig.7.17d. If support 1 is shifted, then slope at supports 0 and 2 are=l,
therefore the mutual angle of rotation1sD



lC



l


D 2 l.Negativesign
means that each momentXproduce the negative work on the angular displacement
=ldue to the settlement of support.
The free term can be calculated using a general algorithm (7.14) as follows: let
us applyX 1 D 1 and compute thereaction in direction of given displacement.
As a result, two reactions each1= larises at the support 1, as shown in Fig.7.14e.
Expression (7.14) leads to the same result. Indeed,1sD2.1=l/.
Canonical equation becomes.2l =3EI/X 1 2.=l/D 0. The primary un-
known, i.e., the bending moment at support 1 equalsX 1 DM 1 D 3



EI=l^2


.
A positive sign shows that extended fibers are located below the longitudinal axis of
the beam. Note that bending moments at supports of uniform continuous beams with
equal spans caused by vertical displacements of one of its supports are presented in
Ta b l eA.18.

Conclusion: In case of settlements supports, the distribution of internal forces de-
pends not only on relative stiffness, but on absolute stiffnessEIas well.

Example 7.3.The design diagram of the redundant frame is the same as in Example
7.2. No external load is applied to the frame, but the frame is subjected to settlement
of fixed supportAas presented in Fig.7.18a. Construct the internal force diagrams
and calculate reactions of supports. Assume that the vertical, horizontal, and angular
settlements areaD 2 cm,bD 1 cm, and'D0:01radD 3403000 , respectively.

Solution.Let the primary system is chosen as in Example 7.2, so the primary un-
knowns are reactionsX 1 andX 2 (Fig.7.18b).
Canonical equations of the force method are

ı 11 X 1 Cı 12 X 2 C1sD0;

ı 21 X 1 Cı 22 X 2 C2sD0; (a)

whereıikare unit displacements, which havebeen obtained in Example 7.2. They
are equal to

ı 11 D

666:67

EI

;ı 12 Dı 21 D

275

EI

;ı 22 D

170:67

EI

: (b)