Advanced Methods of Structural Analysis

13.5 Stability of Arches 485

then (13.13) may be presented in polar coordinates in the following form

d^2 w

d^2

CwD

M

EI

R^2 ; (13.13a)

which is called the Boussinesq’s equation (1883).

R

2 a k

k

q

q

j

N

m

M 0 =k j

M 0 =k j

M 0 =k j

M 0

sin a

sin qM

m = 0

q a

j

j

w(q)

w(q)

M 0

N=q R

N=q R

N

dq

ds

ab

c

d

Fig. 13.20 (a) Design diagram of the circular arch with elastic supports; (b) Reactions and anti-
symmetrical buckling forms; (c) Distribution of bending moments caused by two reactive moments
M 0 ;(d) Free-body diagram for portion of the arch (loadqis not shown)

It is easy to show that the axial compressive force of the arch caused by uniformly
distributed hydrostatic loadqisNDqR. Indeed, the total load within the portion
ds(Fig.13.20b) equalsqdsDqRd, and all load which acts on the left half-arch is
perceived by the left support, so the horizontal and vertical components of reaction
Nare

HD

Z ̨

0

qRsindDqRcos ̨ and VD

Z ̨

0

qRcosdDqRsin ̨;

so

ND

p

H^2 CV^2 DqR:

The slope'at the elastic support and corresponding reactive momentM 0 are related
asM 0 Dk'. Distribution of the bending moments caused by two antisymmetrical
angular displacements'of elastic supports (or reactive momentsM 0 )ispresented
in Fig.13.20c. Bending moment at section with central anglecaused by only
reactive momentsM 0 equalsm. The total bending moment at any section, which
is characterized by displacementw(free-body diagram is shown in Fig.13.20d),
equals

MDqRw

sin

sin ̨

k': (13.14)

For two-hinged arch, the second term, which takes into account moment due to
elastic supports, should be omitted.