Clustered failure times often arise from studies with stratified sampling designs where it is desired to reduce both cost and sampling error. Semiparametric accelerated failure time (AFT) models have not been used as frequently as Cox relative risk models in such settings due to lack of efficient and reliable computing routines for inferences. The challenge roots in the nonsmoothness of the rank-based estimating functions, and for clustered data, the asymptotic properties of the estimator from the weighted version have not been available. The recently proposed induced smoothing approach, which provides fast and accurate rank-based inferences for AFT models, is generalized to incorporate weights to accommodate stratified sampling designs. The estimator from the induced smoothing weighted estimating equations are shown to be consistent and have the same asymptotic distribution as that from the nonsmooth version, which has not been developed before. The variance of the estimator is estimated by computationally efficient sandwich estimators aided by a multiplier bootstrap. The proposed method is assessed in extensive simulation studies where the estimators appear to provide valid and efficient inferences. A stratified case-cohort design with clustered times to tooth extraction in a dental study illustrates the usefulness of the method.