Traceability is one of the basic tenets of all safety standards and a key prerequisite for software safety certification. In the current state of practice, there is often a significant traceability gap between safety requirements and software design. Poor traceability, in addition to being a non-compliance issue on its own, makes it difficult to determine whether the design fulfills the safety requirements, mainly because the design aspects related to safety cannot be clearly identified. The goal of this article is to develop a framework for specifying and automatically extracting design aspects relevant to safety requirements. This goal is realized through the combination of two components: (1) A methodology for establishing traceability between safety requirements and design, and (2) an algorithm that can extract for any given safety requirement a minimized fragment (slice) of the design that is sound, and yet easy to understand and inspect. We ground our framework on System Modeling Language (SysML). The framework includes a traceability information model, a methodology to establish traceability, and mechanisms for model slicing based on the recorded traceability information. The framework is implemented in a tool, named SafeSlice. We prove that our slicing algorithm is sound for temporal safety properties, and argue about the completeness of slices based on our practical experience. We report on the lessons learned from applying our approach to two case studies, one benchmark and one industrial case. Both studies indicate that our approach substantially reduces the amount of information that needs to be inspected for ensuring that a given (behavioral) safety requirement is met by the design.