This paper studies the time estimation of a cracked pipe structure from the crack initiation until the collapse. The crack propagation behaves in the plastic behavior of an indeterminate pipe under the forced displacement due to the vertical earthquake. The analytical foundation of this study is the combination technique of limit analysis and elastic-plastic fracture mechanics to study the relation between JIC and dJ/da so that the formation of the plastic hinge can be understood. The numerical results are plotted as the associated figures that can show the crack growth.