This paper examines decision making under radical uncertainty in engineering design, that is, engineering decision making in those situations where it is not possible to know the outcomes and/or construct the utility functions and probabilities needed to support rational-human decision making. In these situations, despite being faced with radical uncertainty, engineers do (and must) proceed forward in a linear, clear, and predictable manner. Yet, they may not proceed in a manner that is well described by current engineering design frameworks. Examining the role of decision making in business and other social enterprises, Tuckett and Nikolic  have proposed conviction narrative theory (CNT) to describe how rational decision-makers confronted with situations in which insufficient information is available to support traditional decision-making tools use narrative and intuition to reach convincing and actionable decisions. This paper proposes that, in a manner similar to what is described in CNT, narrative and engineering judgment play a critical role in engineering design situations dominated by radical uncertainty. To that end, this paper integrates the traditional rational-human view of decision making as expressed by Hazelrigg in the well-known Decision-Based Design (DBD) framework and CNT as proposed by Tuckett and Nikolic. In the resulting rational, narrative-based design framework, narrative structures are used to describe and develop design alternatives and provide the ideas, beliefs, and preferences needed by the DBD framework. The resulting preferred design is expressed as a narrative and tested using engineering judgement. Specifically, the goal of the design process is expressed as a high-level guiding narrative that fosters the development of design narratives (design alternatives), and ultimately results in a convincing narrative that describes the preferred design. The high-level guiding narrative outlines the event(s), entity(s), preferences, and beliefs needed to support the design. The design narratives are narrative fragments that are nested within the high-level narrative and include the proposed action (idea), the specific challenges that the design faces, and the possible (but not yet verified) outcomes. The convincing narrative is the validated, preferred option that results from the DBD analysis and optimization process and is reviewed using engineering judgement. Following development of the rational, narrative-based design framework, the value of the framework is discussed within the context of practical engineering design.