P-I diagrams for linear-elastic cantilevered Timoshenko beams including higher modes of vibration

L.J. van der Meer, J.G.M. Kerstens, M.C.M. Bakker
Faculty of Architecture, Building and Planning, Eindhoven University of Technology, Eindhoven, the Netherlands

Pressure-impulse (P-I) diagrams are commonly used to assess damage of structural components subjected to blast loading. The response to blast loading is characterized by three main aspects: (a) the excitation of higher modes of vibration, (b) strain rate effects, and (c) ductility. P-I diagrams are often based on equivalent single degree of freedom (SDOF) systems. These SDOF systems can take strain rate effects and ductility into account, to a certain extent, but the influence of higher modes is neglected. All three aspects can be incorporated in finite element simulations, but at the cost of generality and comprehensibility. For a better understanding, this theoretical study derives P-I diagrams including higher modes, without strain rate effects and ductility. These P-I diagrams are derived for a continuous, linear-elastic, cantilevered Timoshenko beam, which represents the load-bearing structure of a multistory building, subjected to idealized blast loading. The response is determined analytically using mode superposition. Modal contribution factors for dynamic base shear are derived, based on earthquake engineering. For the material behaviour and boundary conditions investigated, it is concluded that P-I diagrams based on equivalent SDOF systems are not conservative for short-duration (impulsive) loading, and non-uniform spatial load distributions. Moreover, the shape of the P-I diagrams that have been derived, deviates significantly from the hyperbolic shape of P-I diagrams for SDOF systems.

Key words: Pressure-impulse diagram, Timoshenko beam, modal contribution factor, dynamic load factor, blast load, higher modes