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This paper describes analytical and numerical methods to analyze the steady state periodic response of an oscillator with symmetric elastic and inertia nonlinearity. A new implementation of the homotopy perturbation method (HPM) and an ancient Chinese method called the max-min approach are presented to obtain an approximate solution. The major concern is to assess the accuracy of these approximate methods in predicting the system response within a certain range of system parameters by examining their ability to establish an actual (numerical) solution. Therefore, the analytical results are compared with the numerical results to illustrate the effectiveness and convenience of the proposed methods.

An improved He's energy balance method (EBM) for solving non-linear oscillatory differential equation using a new trial function is presented. The problem considered represents the governing equations of the non-linear, large-amplitude free vibrations of a slender cantilever beam with a rotationally flexible root and carrying a lumped mass at an intermediate position along its span. Based on the simple EBM, the variational integral of the non-linear conservative system is established, and the Fourier series expansion is employed to address the governing algebraic equations. An alternate procedure for a particular value of the initial condition is then used to estimate the constants. This semi-analytical representation gives excellent approximations to the exact solutions for the whole range of the oscillation amplitude, reducing the respective error of angular frequency in comparison with the simple EBM. Two illustrative examples are considered in order to elucidate the methods described, and to reveal the improvements made by the modified method.

A practical approach to explain the consequences of seismic hazards for society and decision making organisations is predicting the seismic risk. In this regard, the Probabilistic Seismic Hazard Analysis (PSHA) is used extensively to investigate the probability of different seismic hazard levels at a geographical location. In addition, a further advancement is the introduction of Probabilistic Seismic Demand Analysis (PSDA) method because it provides a new insight into the Performance-Based Earthquake Engineering (PBEE) by evaluating the seismic risk specifically for a structure. To evaluate this seismic risk, the probability that the structural seismic demands may exceed a specific value is calculated under different ground motion intensities through a probabilistic approach. This approach is called the fragility analysis. This paper provides a review of recent research advancements in seismic fragility analysis. Different methods and related solutions which can be used for the fragility analysis are discussed. In addition, uncertainty quantification, as a significant feature in fragility analysis, is described and the important parameters which may influence the seismic fragility of a structure are explained. Finally, the authors offer their recommendations for improving the fragility analysis for further studies in the future.

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