Instability of imperfect non-uniform shallow arch under uniform radial pressure for pinned and fixed boundary conditions

The instability of imperfect non-uniform circular shallow arch under radial pressure for both pinned-pinned and fixed-fixed boundary conditions is discussed in the article. The imperfection includes the symmetric initial displacement inducing no stress and the non-uniformity which represented by three piecewise constant-stiffness segments. Analytical theoretical formulation is derived by the least potential energy principle. Through mathematical analysis, it's shown rigorously that the effect of imperfection still doesn't vanish as the mode number of imperfection approaching to infinity. The stiffer end case and stiffer center case are discussed concerning equal potential energy load and snap-through criterion. Distinct snap-through behavior is observed for stiffer end case with fixed boundary condition by a proposed index plot. Moreover the two limiting cases including rigid end case and rigid center case are investigated by employing the augmented potential energy with Lagrangian multipliers considering the imperfection effect. Snap-through criteria are obtained in a closed form for rigid center case. And for the rigid end case with pinned BC, when the center segment's size is tending to zero, an asymptotical solution is derived and by this solution, the importance of multiplicity of roots of some equations governing the snap-through behavior is illustrated clearly. Besides, the equal potential energy load of two limiting cases is analyzed in detail with emphasis on imperfection effect. This paper aims to extend the understanding of imperfect non-uniform circular shallow arch under radial pressure.