The paper deals with the stiffness analysis and stability study of a new type of tensegrity mechanism based on dual-triangle structures, which actuated by adjusting elastic connections between the triangle edges. For a single segment of such mechanism, the torque-deflection relation was obtained as a function of control inputs and geometric parameters. It was proved that a single section of the mechanism can has either a single or three equilibrium configurations that can be both stable and unstable. Corresponding conditions of stability were found allowing user to choose control inputs ensuring the mechanism controllability, and the obtained results are confirmed by the simulation examples. The structure composed of two segments in serial was also analysed and an equivalent serial structure with non-linear virtual springs in the joints was proposed. It was proved that the stiffness of such structure decreases while the external loading increases, which may lead to the buckling phenomenon.