Advanced Methods of Structural Analysis

13.5 Stability of Arches 489

Table 13.4Critical parameternfor circular arch with fixed

supports

̨ 30 ı 45 ı 60 ı 90 ı

N 8:621 5:782 4:375 3:000

It is worth to present the critical load for three-hinged symmetrical arch under
hydrostatic load. The criticalload for antisymmetric buckling form coincides with
critical load for two-hinged arch. In case of symmetrical buckling form, the critical
load should be calculated by using the formula

qcrD



4 u^2

̨^2

 1



EI

R^3

; (13.26)

where parameteruis a root of transcendental equation

tanuu

u^3

D 4

.tan ̨ ̨/

̨^3

: (13.27)

Roots of this equation are presented in the Table13.5.

Table 13.5 Circular three-hinged arch. Critical parameterufor

symmetrical buckling form

̨ 30 ı 45 ı 60 ı 90 ı

U 1.3872 1.4172 1.4584 =2D1:5708

For all above cases, the critical load may be calculated by the formula

qcrDK

EI

R^3

;

where parameterKis presented in the Table13.6.

Table 13.6 ParameterKfor critical hydrostatic load of circular

arches with different boundaryconditions

Types of arch ̨D 15 ı 30 ı 45 ı 60 ı 75 ı 90 ı

Hingeless 294 73:3 32:4 18:1 11:5 8

Two-hinged 143 35 15 8 4:76 3

Three-hinged 108 27:6 12 6:75 4:32 3

(symmetrical form)

Ta b l e13.6indicates that for three-hinged arch the lowest critical hydrostatic load
corresponds to loss of stability in symmetrical form. For arches with elastic sup-
ports, factorKsatisfies condition

K 1 KK 2 ;

whereK 1 andK 2 are related to hingeless and two-hinged arches.