Advanced Methods of Structural Analysis

488 13 Stability of Elastic Systems

Solution of this transcendental equation for given ̨and dimensionless parameter

k 0 is the critical parametern. According to (13.15a) the critical load

qcrD



n^2  1

EI

R^3

: (13.23)

If the central angle2 ̨D 60 ı, then the roots of equation (13.22) for differentk 0 are

presented the Table13.3:

Table 13.3 Critical parameternfor circular arch with elastic clamped

supports,2 ̨D 60 ı

k 0 0.0 1.0 10 100 1000 105

n 6.0000 6.2955 7.5294 8.4628 8.6051 8.621

Limiting cases. The general stability equation (13.22) allows us to consider some

specific arches.

1.Two-hinged arch. In this case the stiffnesskD 0 and stability equation (13.22)

is tann ̨D 0. The minimum root of this equation isn ̨D ,sonD

̨

and

corresponding critical load equals

qcr minD



2

̨^2

 1



EI

R^3

: (13.24)

Critical load for ̨D=2(half-circular arch with pinned supports) equals

qcr minD 3

EI

R^3

:

2.Arch with fixed supports. In this case the stiffnessk D1and stability equa-
tion (13.22) becomes
tann ̨Dntan ̨ (13.25)
In case ̨D=2(half-circular arch) this equation can be presented in the form

cot

n

2

D0;so

n

2

D

2

;

3

2

;

SolutionnD 1 is trivial because this solution, according to expression (13.15a),

corresponds toqD 0. Thus, minimum root isnD 3. Thus for a half-circular arch

with clamped supports the critical load equals

qcr minD 8

EI

R^3

:

Rootsnof stability equation (13.25) for different angle ̨are presented in the

Ta b l e13.4.