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Using the state space method, a novel analytical method is proposed to investigate the dynamic behavior of multi-cracked reinforced concrete (RC) beams strengthened with fiber-reinforced polymer (FRP) plates, while taking damping effects into account. The flexural cracks in RC beams are modeled as rotational springs, with their stiffness dependent on the crack depths. A numerically stable technique and a frequency-scanning strategy are employed to solve the homogeneous state equations associated with mode function vectors and the frequency equation, respectively. This allows for the determination of natural frequencies and corresponding modal shapes of FRP-strengthened multi-cracked RC beams under generalized boundary conditions. The orthogonality relation of vibration modes is established based on the state space formulae and the concept of symplectic inner product. Analytical solutions for the dynamic responses of FRP-strengthened multi-cracked RC beams subjected to arbitrary dynamic loads are derived using the orthogonality relation and the mode superposition method. Numerical examples are provided to predict the dynamic responses of strengthened cracked beams under a dynamic uniformly distributed step load and a concentrated triangular impulse load. The effectiveness of the proposed analytical method is validated through comparisons with finite element simulations and experimental results. Furthermore, parametric studies are performed to analyze the effects of damping ratio, crack depth, crack position and the number of cracks on the dynamic responses of these strengthened cracked beams. The results demonstrate that the strengthening strategy using externally bonded FRP plates is effective, while the effects of cracks and the first-order damping ratio on the dynamic responses are significant and cannot be ignored. Additionally, the proposed method can efficiently and accurately analyze the dynamic behavior of FRP-strengthened RC beams with arbitrary distributions of cracks and dynamic loads.

The smoothed finite element method (S-FEM) has been found to be an effective solution method for solid mechanics problems. This paper proposes an effective approach to compute the lower bound solution of free vibration and the upper bound solution of the forced vibration of solid structures, by making use of the important softening effects of the node-based smoothed finite element method (NS-FEM). Through the gradient smoothing technique, the strain-displacement matrix is obtained in the smoothing domain based on the element mesh nodes. Subsequently, the stiffness matrix is computed in a manner consistent with the standard finite element method (FEM). Here, the practical Lanczos algorithm and the modal superposition technique are employed to obtain the frequencies, modes, and transient responses of a given homogeneous structure. For three-dimensional (3D) solid structures, the automatically generated four-node tetrahedron (T4) element meshes are utilized. The results obtained from the NS-FEM are compared with the standard FEM in terms of accuracy, convergence and computational efficiency.

This paper deals with the evaluation of time response analyses of typical aerospace metallic structures. Attention is focussed on detailed stress state distributions over time by using the Carrera Unified Formulation (CUF) for modeling thin-walled reinforced shell structures. In detail, the already established component-wise (CW) approach is extended to dynamic time response by mode superposition and Newmark direct integration scheme. CW is a CUF-based modeling technique which allows to model multi-component structures by using the same refined finite element for each structural component, e.g. stringers, panels, ribs. Component coupling is realized by imposing displacement continuity without the need of mathematical artifices in the CW approach, so the stress state is consistent in the entire structural domain. The numerical results discussed include thin-walled open and closed section beams, wing boxes and a benchmark wing subjected to gust loading. They show that the proposed modeling technique is effective. In particular, as CW provides reach modal bases, mode superposition can be significantly efficient, even in the case of complex stress states.

Cantilever plate structures are widely used in civil and aerospace engineering. Here, a semi-analytical method is proposed to calculate the dynamic responses of cantilever plates subjected to moving forces. The Rayleigh–Ritz method is used to obtain the semi-analytical modal frequencies and shapes of a thin, isotropic, and rectangular cantilever plate using the assumed mode shapes that fulfill the boundary conditions of the plate. The modal superposition method is used to decouple the motion equations of the cantilever plate to obtain a series of modal equations. Then, the generalized forces are transformed into a Fourier series in terms of discrete harmonic forces. The dynamic responses of the cantilever plate are obtained by superimposing the analytical responses of a number of single-degree-of-freedom modal systems under discrete harmonic forces. The proposed semi-analytical method is verified through comparison with the numerical method. Then, the vibration of the cantilever plate under the action of moving forces is investigated based on the semi-analytical results. It is found that the contribution of the high-order modes to the dynamic responses of the plate cannot be ignored. In addition, the wavelengths of the mode shapes not only affect the magnitude of the modal forces but also the dominant frequency of the modal forces. Resonant responses of the plate are produced by the moving forces when the load interval equals the wavelength of the mode shape of a high-order mode and the exciting frequency of the moving forces equals the natural frequency of this mode.