In this paper we present a framework for designing algorithms in shared memory of GPUs without incurring memory bank conflicts. Using our framework we develop the first comparison-based shared memory sorting algorithm that incurs no bank conflicts. It can be used as a subroutine for GPU sorting algorithms to replace current use of sorting networks in shared memory. Using our bank conflict free shared memory sorting subroutine as a black box, we design BCFMergesort, an algorithm for merging sorted streams of data that are larger than shared memory. Our algorithm performs all accesses to global memory in coalesced manner and incurs no bank conflicts during the merge.