The plane elastic bending rod serves as a model of biological and artificial vibrissae. One task of vibrissae is to scan surfaces in order to detect their textures. We sketch the theory of static bending rod problems with a view to this field of applications. The solutions of the respective autonomous boundary value problems are given analytically. Numerical simulations are done in some worked out examples. Being aware of the fact that (1) in papers, object detecting by means of touch is seemingly exclusively investigated under the assumption of ideal contact (no stiction or friction), and (2) our present omission of this assumption is based upon simplifying model properties, the conclusions give some corresponding discussions and proposals for improvements and necessary work in future.