Advanced Methods of Structural Analysis

484 13 Stability of Elastic Systems

constant radiusRof curvature (circular arch), which is subjected to uniformly dis-
tributed pressure normal to the axis of the arch. In this case, only axial compressed
forces arise before the buckling arch. Thus this problem, as all previous problems in
this chapter, has a general feature – a structure before buckling is subjected to only
compressed load.
Figure13.19a, b presents statically indeterminate arches with pinned and fixed
supports as well as three-hinged arch. The loss of stability of an arch may occur
in two simplest forms. They are symmetrical form (a), when elastic curve is sym-
metrical with respect to vertical axis of symmetry and, otherwise, antisymmetrical
form (b).

Arch with pinned ends

jC = 0

dC ≠ 0

jC ≠ 0

dC = 0

jC ≠ 0

dC = 0

jC ≠ 0

dC = 0

A B

C

jC = 0

dC ≠ 0 dC ≠^0

C

Arch with fixed ends

A B

Three-hinged arch

A B

C

jA≠ 0

jA=0

jA≠ 0

jA≠^0 jA≠^0

jA=0

a

b

Fig. 13.19 Arches with different boundary conditions. (a) symmetrical and (b) antisymmetrical
buckling forms

The symmetrical and antisymmetrical forms mean that both supports rotate in the
opposite directions and the same directions, respectively. As it shown by analytical
analysis and experiments, the smallest critical load for hingeless and two-hinged
arches corresponds to antisymmetrical buckling form.

13.5.1 Circular Arches Under Hydrostatic Load...................

Assume that symmetrical circular arch of radiusRhas the elastic-fixed supports;
their rotational stiffness coefficient isk[kN m/rad]. The central angle of the arch is
2 ̨; the flexural rigidity isEI, and the intensity of the radial uniformly distributed
load isq(Fig.13.20a).
For stability analysis of the arch, we will use the following differential equation

d^2 w

ds^2

C

w

R^2

D

M

EI

; (13.13)

wherewis a displacement point of the arch in radial direction (Fig.13.20b), andM
is bending moment which is produced in the cross sections of the arch when it loss
a stability.
Let dspresents the arc, which corresponds to central angle d.Since

dw

ds

D

dw

d

d

ds

D

1

R

dw

d

and

d^2 w

ds^2

D

1

R^2

d^2 w

d^2

;