The structural assessment of bridges for load capacity is inherently uncertain due to the imprecision and incompleteness of information about material properties, geometric dimensions, dead loads, live load models, and the degradation of structural resistance over time. Traditional probabilistic reliability methodssuch as Monte Carlo simulation and first-order reliability methods (FORM)require precise knowledge of probability distributions for all uncertain quantities, an assumption that is rarely satisfied in practice, particularly for older bridges, in-service inspection data of limited scope, or infrastructure networks in data-poor developing-nation contexts. This paper presents a rigorous framework for uncertainty quantification in bridge load capacity using interval mathematics and its extension to fuzzy set theory and affine arithmetic, which are suited to epistemic (knowledge-based) uncertainty characterisation where insufficient data exist to specify probability distributions. The interval model is developed for a simply supported composite steel-concrete bridge beam under combined dead load, live load per HL-93 and Class A vehicle models, and wind loading. The uncertain parameterselastic modulus [E], yield strength [f_y], slab thickness [h_s], effective span [L], and live load model factor [_Q]are represented as interval numbers or triangular fuzzy numbers (TFNs), and the load capacity rating factor RF is computed as a sharp interval using dependency-aware affine arithmetic to avoid the wrapping effect of naive interval arithmetic. An interval-valued reliability index _int is derived by bounding the first-order failure probability within a probability box (p-box) framework. The methodology is demonstrated on a case study of a 40 m span reinforced concrete slab