Bolted steel-bamboo scrimber-steel (SBSS) connections are commonly employed in composite bamboo structures, but their equivalent stiffness in the elastic phase is difficult to determine. This study develops a novel theoretical formula for estimating the equivalent stiffness in the elastic phase of single-bolted SBSS shear connections considering various failure modes. Specifically, the method focuses on calculating the contact stiffness between the bolt and the pre-drilled hole wall, as well as the shear load applied to the bolt end by the steel plate. The method uses the double-shear yield mode, which is included by European Yield Modes (EYM), and the beam on elastic foundation theory (BEFT), and considers the bamboo scrimber embedment stiffness, bolt diameter and steel plate strength class. The method is tested by conducting experiments on single-bolted SBSS connections with varying load and fiber direction (LFD) angles, bolt diameters, steel plate thicknesses and strength grades. The results show that the connection stiffness increases with bolt diameter and steel plate strength grade, decreases with LFD angle, and is not affected by steel plate thickness. The parameter mu, which is important in the theoretical equations, is also calibrated based on the experimental data. The proposed method is validated by comparing it with existing test results of single-bolted SBSS connections. This paper provides a simple and reliable model for predicting the equivalent stiffness in the elastic phase of bolted SBSS connections in bamboo scrimber structures and offers useful insights for their design.