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In-vivo tissue stiffness identification can be useful in pulmonary fibrosis diagnostics and minimally invasive tumor identification, among many other applications. In this work, we propose a palpation-based method for tissue stiffness estimation that uses a sensorized beam buckled onto the surface of a tissue. Fiber Bragg Gratings (FBGs) are used in our sensor as a shape-estimation modality to get real-time beam shape, even while the device is not visually monitored. A mechanical model is developed to predict the behavior of a buckling beam and is validated using finite element analysis and bench-top testing with phantom tissue samples (made of PDMS and PA-Gel). Bench-top estimations were conducted and the results were compared with the actual stiffness values. Mean RMSE and standard deviation (from the actual stiffnesses) values of 413.86 KPa and 313.82 KPa were obtained. Estimations for softer samples were relatively closer to the actual values. Ultimately, we used the stiffness sensor within a mock concentric tube robot as a demonstration of \textit{in-vivo} sensor feasibility. Bench-top trials with and without the robot demonstrate the effectiveness of this unique sensing modality in \textit{in-vivo} applications., Comment: 6 pages (excluding the references), 5 figures