Exploding wires are widely used in many experimental set-ups and pulsed power systems such as Z-pinch, high-current switches, copper-vapour lasers and high-brightness x-ray lithography. However, many aspects of the process of wire explosion still remain unclear. If the current density is not too high, the wire may break up in the solid state. The experiments have shown that the wires break in tension due to longitudinal forces of unknown nature. Previous theoretical and numerical investigations served to provide a search for these forces and have identified the pinch effect and thermal expansion as a source of strong longitudinal vibrations. But the mechanism does not give a satisfactory explanation for the phenomenon in the wires with clamped ends. In this investigation, we use a simplified magneto-thermo-elastic model to study flexural vibrations induced by high pulsed currents in wires with clamped ends on account of their role in the disintegration process. Several aspects are studied, namely (i) the buckling instability due to simultaneous action of the thermal expansion and the magnetic force, and (ii) the flexural vibrations induced in initially bent wires. It is shown that the induced flexural vibrations are strong enough to lead to the breaking of the wire in a wide range of parameters.