discontinuous beam structures can be found in various applications such as adhesive bonded joints and piezoelectric energy harvesters. in adhesive bonded joint design, the single lap joint is commonly used and is constructed by adhering two overlapped materials with an adhesive. it is important to evaluate the static and dynamic behaviours of a single lap joint to mitigate the chance of failure under normal operating conditions. piezoelectric energy harvesters use piezo-ceramic materials, which are glued to a host structure with a thin layer of adhesive. studying the dynamic behaviours of piezoelectric energy harvesters is key to achieving optimal power output. this research presents the use of numerical analysis approaches to investigate the static and dynamic behaviours of discontinuous beam structures, with a focus on applications which require single physics (i.e. static and dynamic behaviours of a single lap joint), and multi-physics (i.e. dynamic behaviours of a piezoelectric energy harvester). throughout this research, the primary numerical approach used is the finite element method. an alternative approach is the transfer matrix method, to determine the dynamic behaviours of a single lap joint. these numerical approaches are compared with analytical methods and experimental testing, to validate their use. in this research, an experimental apparatus was developed for testing purposes. overall, the results from the numerical approaches used closely match those from the analytical methods and experimental testing for all applications. this work investigates key relationships and factors that influence the behaviours of the single lap joint and piezoelectric energy harvesters. the effect of the adhesive layer thickness and the overlap region length on a single lap joint are studied. increasing the adhesive layer thickness reduces the adhesive stress and the natural frequencies of a single lap joint. the effects of the electrical load resistance and a proof mass on the performance of a piezoelectric energy harvester are discovered. it is found that increasing the electrical load resistance of the piezoelectric energy harvester causes an increase in voltage across the resistor, and the impedance matched resistance, yielding the maximum power output. the proof mass reduces the fundamental frequency, increases the transmissibility function magnitude at this frequency, increases efficiency, and increases the impedance matched resistance of the piezoelectric energy harvester.