Advanced Methods of Structural Analysis

490 13 Stability of Elastic Systems

13.5.2 Complex Arched Structure: Arch with Elastic Supports

In practical engineering, the stiffness coefficientkof the elastic supports is not
given; however, in special cases it can be determined from an analysis of adjacent
parts of the arch. Let us consider a structure shown in Fig.13.21a. The central part
of the structure presents the circular arch; supports of the arch are rigid joints of the
frames. The arch is subjected to uniformly distributed hydrostatic loadq. Assume
thatRD 20 m and the central angle2 ̨D 60 ı. The stiffness of all members of the
structure isEI.

6m 10m

8m

6m

a=30°

EI

q

R=20m

A B

a

M=k=? R^1 P M=k=?

i 1 =EI/ 6

i 2 =EI/ 8

1

r 11 Z 1 =1

4 i 2

2 i 2

3 i 1

bc d e

Fig. 13.21 (a) Design diagram of the structure. (b–e) Calculation of the stiffnesskof the elastic
supports of the arch

Since the left and right frames are deformable structures, then the each jointA
andBhas some angle of rotation, so the archABshould be considered as arch with
elastic supports with rotational stiffnessk. For this case of circular arch with given
type of load, the stability equation according (13.22) becomes:

tann ̨D

n

cot ̨C



n^2 ^1



EI

kR

: (13.28)

Rotational stiffness coefficientkis a coupleM, which arises at elastic support of
the arch if this support rotates through the angle'D 1. Since the jointsAandBare
rigid, so the angle of rotation for frame and arch are same. Therefore, for calculation
of the stiffnesskwe have to calculate the coupleM, which should be applied at the
rigid jointAof the frame in order to rotate this joint by angle'D 1.
The frame subjected to unknown momentMDkis shown in Fig.13.21b. For
solving of this problem, wecan use the displacement method.
Primary system of the displacement method is shown in Fig.13.21c. Canonical
equation is
r 11 Z 1 CR1PD0: