Porous structures are a new class of lightweight structural forms with excellent potential in various engineering applications. Functionally graded (FG) porous structures introduce non-uniformly distributed internal pores with preferred porosity distributions, resulting in advanced and novel structural properties compared to the conventional porous structures with uniform or random porosities, which, however, haven’t been studied systematically. To address this issue, this thesis presents a detailed static and dynamic analysis concerning the buckling, postbuckling, free vibration, forced vibration, and nonlinear vibration responses of FG porous beams and plates. Different non-uniform distributions of internal pores are considered, including varying types of symmetric and asymmetric porosities. The typical mechanical properties of open-cell metal foams and closed-cell cellular solids under Gaussian Random Field scheme are employed to determine the material properties of the proposed structures. The theoretical formulations of beams and plates are constructed by using Timoshenko beam theory and the first-order shear deformation plate theory, respectively. Governing equations are derived based on the minimum total potential energy principle, Lagrange equation method, and Hamilton’s principle for different problems, then solved using Ritz method for beams and Chebyshev-Ritz method for plates. The equation of motion in the forced vibration analysis is discretised with Newmark-β method in the time domain and Ritz method in the space domain. The postbuckling and nonlinear vibration behaviors are investigated by applying von Kármán type nonlinearity and a direct iterative algorithm. The comparisons between the present results and those in open literature or commercial software outputs are adopted as a validation of the proposed formulations. The influences of different porosity distributions, porosity coefficients, boundary conditions, beam slenderness ratios and plate aspect ratios are discussed in detail to reveal their effects on the critical buckling loads, postbuckling equilibrium paths, bending deflections and stresses, fundamental natural frequencies, forced vibration deflections and nonlinear vibration frequency ratios of porous beams and plates. The energy absorption capacity of FG porous structures is examined with an explicit dynamic finite element study focused on the in-plane crushing of closed-cell metal foams under different loading modes, including a constant-velocity impacting and an initial-velocity impacting. Two non-uniformly symmetric, two non-uniformly asymmetric and one uniform distributions of internal pores along the impact direction are constructed with Voronoi tessellation. The deformation of cell walls is evaluated using a plastic kinematic material model. Selected settings are implemented to ensure the validity of this simulation which is verified by comparing the numerical results to the experimental outcomes. The effects of varying parameters on the energy absorptions, deformation patterns, and stress-strain curves of FG porous structures are presented in both tabular and graphical forms. The nanofiller reinforcement on multilayer porous structures by using graphene platelets (GPLs) is also investigated in this thesis. The internal pores and GPL nanofillers are uniformly dispersed within each layer but both the porosity coefficients and GPL volume contents change from layer to layer, resulting in the position-dependent elastic moduli, mass density and Poisson’s ratio varying along the thickness direction. The effective material properties of the nanocomposites are computed based on the Halpin-Tsai micromechanics model for Young’s modulus and the rule of mixture for mass density and Poisson’s ratio. Special attention is given to the reinforcement effects of different GPL weight fractions, dispersion patterns and platelet geometry ratios on the linear and nonlinear mechanical behaviors of porous nanocomposite beams and plates. This thesis is focused on the static and dynamic analysis of FG porous structures. With a detailed investigation on the novel structural performance induced by the FG porosity distributions, this study sheds important insights into the porosity designs to achieve the advanced buckling, bending, vibration resistance, and the improved energy absorption capacities under different loading conditions, promoting the adoption of this new structural form in various engineering areas.