In this note we show that for a pinned half-sine arch under end couples snap-through buckling will occur unsymmetrically if the initial height of the shallow arch is greater than $6.5466r,$ where r is the radius of gyration of the cross section. The closed-form expression for the critical couple can be obtained analytically.

1.
Fung, Y. C., and Kaplan, A., 1952, “Buckling of Low Arches or Curved Beams of Small Curvature,” NACA Technical Note 2840.
2.
Simitses, G. J., 1986, Elastic Stability of Structures, R. E. Krieger, Malabar, FL, Chap. 7.
3.
Simitses, G. J., 1990, Dynamic Stability of Suddenly Loaded Structures, Springer, New York, Chap. 7.
4.
Hsu
,
C. S.
,
1967
, “
The Effects of Various Parameters on the Dynamic Stability of a Shallow Arch
,”
ASME J. Appl. Mech.
,
34
, pp.
349
358
.