Natural frequencies of bonded and unbonded prestressed beams–prestress force effects

The paper rigorously presents the effect of the magnitude of the prestress force on the natural frequencies of prestressed beams with bonded and unbonded tendons. A nonlinear analytical model is formulated for the dynamic behavior of prestressed beams, in order to achieve this goal. The equations of motion for a prestressed beam and its associated boundary and continuity conditions are rigorously derived using the variational principle of virtual work following Hamilton's principle. The mathematical model is rigorous and general, and is valid for any kind of boundary and continuity conditions as well as any cable layout. The kinematic relations for the concrete are those of large displacements and moderate rotations, in order to take into account the compressive force effect caused by the prestress force. For the tendons, the same kinematic relation has been used including the cable profile as an imperfection. The effect of the magnitude of the prestress force on the natural frequencies is mathematically examined. Numerical examples are presented to illustrate the difference between natural frequencies determined by the proposed model and other models mentioned in the literature.