Tendon driven continuum robots are often considered to navigate through-and operate in cluttered environments. While their compliance allows them to conform safely to obstacles, it leads them also to buckle under tendon actuation. In this work, we perform for the first time an extensive elastic stability analysis of these robots for arbitrary planar designs. The buckling phenomena are investigated and analyzed using bifurcation diagrams, complementing the current state of the art and adding new knowledge about robots composed of n spacer disks. We show the existence of multiple robot configurations with different shapes, achievable with the same actuation inputs. A global stability criterion is also established which links the critical tendon force, until which the robot is stable, to the design parameters. Finally, the buckling phenomena are used to actively soften the robot for a better compromise between compliance and payload. An open loop control strategy is proposed, which can theoretically decrease the stiffness to zero while maintaining the same robot shape. Experimentally, the robot is made 4 times more compliant than it is nominally using tendon actuation only.