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We study experimentally the distribution of local stresses in a granular material confined inside a vertical cylinder. We use an image correlation technique to measure the displacement field of the container induced by the forces exerted by the grains on the inner wall. We describe an optimization procedure based on the linear theory of elastic shells to deduce the distribution of these forces from the measured displacement field. They correspond to the stress field of the granular material close to the container's inner wall. We first confirm the validity of Janssen's description for various experiments, including the influence of the bead diameter and the effect of an additional mass on top of the granular column. We then apply this method to determine the stress field during the gravity-driven discharge of a silo through an aperture.