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This paper presents a study of $R$-quadratic Finsler spaces and a new class of Finsler metrics called $bar{D}$-metrics. The non-Riemannian curvatures of $R$-quadratic Finsler spaces and their special case, the $R$-quadratic generalized $(alpha, beta)$-metrics, are analyzed to gain insights into their behavior. The paper then introduces the $bar{D}$-metrics, which are shown to be a proper subset of the class of $GDW$-metrics and contain the class of Douglas metrics. This paper contributes to the understanding of $R$-quadratic Finsler spaces and their properties, and presents a novel class of Finsler metrics with potential applications in the field.