Twining plants achieve vertical growth by revolving around supports of different sizes on which they exert a pressure. These well-known observations raise many intriguing questions that can be addressed within the framework of elastic filamentary structures. Here mechanical aspects of this problem are investigated by modeling the stem close to the apex as a growing elastic rod with intrinsic curvature and twist. The analysis reveals that vertical growth is achieved thanks to the presence of discrete contact points as well as regions with continuous contact. Further the experimentally observed fact that the plant stem is under tension is shown to be directly related to the positive contact pressure. Finally the maximal radius of the pole around which a twiner can climb is identified.