Journal of Engineering Mechanics

A semianalytical method for the transient response of functionally graded material (FGM) rectangular plates under various boundary conditions, by the use of state space method, differential quadrature method, and numerical inversion method of Laplace transform, is proposed in this study. The FGM rectangular plate is simply supported at two opposite edges with any classical boundary conditions on the other two opposite edges, such as simply supported, clamped, and free conditions. Comparisons show the results generated by the present method agree well with those predicted by the finite element method (FEM), no matter which variation law of material properties, functionally graded (FG) index, geometry, load, and boundary conditions are employed. Convergence studies for different numbers of sampling points along the length direction and different layer numbers along the thickness direction show that the present method has a fast convergence rate. In addition, the effects of FG index, FG direction, and thickness on the plates response are investigated. Besides the semianalytical method based on state space method (SSM) and for the purpose of validation, another analytical method is developed based on the Kirchhoff thin plate theory, the Laplace transform, and its inversion. Explicit solutions for thin FGM simply supported plates under different transient loads are derived. Comparison shows that the results generated by the two analytical methods agree with each other excellently for thin plates. Both of the two analytical methods can be used to generate benchmark results for the response of FGM rectangular thin plates subjected to transient loads. However, studies show that as the thickness increases, the results obtained by the SSM and FEM agree with each other well, whereas the Kirchhoff method generates more errors.

To evaluate the predictive capability of the proper orthogonal decomposition (POD) in approximating dynamics of wave impact, the method is applied to the special case of sloshing problems that are widely found in engineering applications and actively pursued using numerical simulations. Because pressure is of major concern in hydrodynamic impact on structures, the POD is applied to pressure data obtained from short--duration numerical simulations of the sloshing phenomena. It is found that the number of POD modes necessary for accurate impact pressure prediction varies from $\sim {10}^1$ to $\sim {10}^2$, depending on the fill level, sloshing period, and amplitude. The POD predictions with the first few ($<10$) modes are sufficient for “mild” sloshing scenarios but tend to underestimate the jet-induced impact pressure at the top of the tank. It is also found that the first few modes exhibit a highly symmetric pattern. This pattern is quickly lost in the higher modes in violent sloshing scenarios, whereas it persists up to the first 10 modes for mild cases.

A three-dimensional (3D) method of analysis is presented for determining the free-vibration frequencies of complete hollow spherical shells of revolution with variable thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based on the 3D dynamic equations of elasticity. Displacement components $u_r$, $u_{\theta }$, and $u_z$ in the radial, circumferential, and axial directions, respectively, are taken to be periodic in $\theta$ and in time, and algebraic polynomials in the $r$- and $z$-directions. Potential (strain) and kinetic energies of the complete hollow spheres are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper-bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the complete hollow spheres. Comparisons are also made between the frequencies from the present 3D method, a 2D thin-shell theory, and two other 3D analyses.

A combined analytical and experimental investigation of solitary waves interacting with a submerged impermeable breakwater is presented in this study. To the leading order of approximation, the integral forms of the analytical solutions of reflected and transmitted velocity potentials and wave elevations are derived according to the Fourier integral approach. The unknown coefficients are determined from the mixed-breakwater boundary conditions using the method of least squares. Experimental data in terms of wave elevations in the reflected and transmitted regions under various wave and breakwater conditions were recorded in a wave tank for validating the analytical solutions. Generally, the analytically predicted incident and transmitted wave profiles are found to agree reasonably well with the experimental measurements, although the analytical model in the cases with greater breakwater height overestimates the peak of the reflected waves. The variations of wave transmission–related and wave reflection–related solution coefficients versus the breakwater height are analyzed. The effect of breakwater height on the peak of reflected and transmitted waves is also examined. It is found that with an increase in the breakwater height the transmission coefficient decreases. Also, more wave energy is transmitted through the breakwater than that contributed to the reflected waves, even in the cases with high ratio of breakwater height to water depth.

Asphalt concrete exhibits rate- and time-dependent behavior, and therefore viscoelastic response analysis is necessary to model the pavement structure behavior and explain the distress mechanisms. The newly developed Mechanistic–Empirical Pavement Design Guide emphasizes the importance of accurate determination of response at the pavement surface to efficiently model the top-down cracking and implement the sublayering scheme. However, it is shown in this study that the integrand of the Laplace-transformed step-response function at the pavement surface exhibits complicated oscillating behavior and slow convergence, making it difficult to achieve accurate viscoelastic solutions. This study proposes a procedure to effectively solve this problem. The proposed procedure uses the Lucas algorithm to reduce the complex oscillations to regular oscillations by separating the integrand into high- and low-frequency components and then uses integration, summation, and extrapolation methods to accelerate the coverage. The results obtained from the proposed procedure are extensively verified against the boundary conditions and finite-element results. The verification results show that the proposed procedure can accurately determine the surface viscoelastic response and provide an effective tool for pavement analysis and design.

In nature, the ground is usually composed of different soils. To simplify the soil structure, we only consider the case of parallel soil layers with a small inclined angle to the horizontal. When a ground surface without vegetative cover is subjected to a rainfall event, overland flow will happen eventually. Therefore, a mathematical model is presented herein to study the integrated surface and subsurface flows over multilayered soils with and without rainfall. The upper layer is a homogeneous water flow over the ground, and the lower layer is a pore-water flow through permeable parallel multilayered soils with infinite thickness. Both water flows are considered as laminar flows. The flow profiles, vertical velocity distribution, and shear-stress distribution are solved analytically by introducing adequate parameters. Furthermore, when a uniform rainfall event is under consideration, the fourth-order Runge–Kutta technique is used to solve the flow profiles. In addition, the phreatic surface is also depicted for the cases of larger slopes; this has never been discussed in the literature.

Response of lateral pile groups is modeled using the more accurate (than any other numerical modeling) $p-y$ curves–based load transfer model. It is essentially underpinned by limiting force per unit length $p_m{p}_u$, modulus of subgrade reaction $p_{m}k$, and $p$-multiplier $p_m$ (to cater for pile-pile interaction, $p_m=1$ for single piles). With the model, new closed-form solutions are developed incorporating the cap-rotational stiffness $k_r$. The solutions are presented in nondimensional charts for free-head ($k_r=0$) through fixed-head ($k_r>10$ times the pile bending stiffness). The study reveals that the existing $p_m$ (bearing no link to the stiffness $k_r$) is inconsistent with $p_m=0.25$ for capped piles (at limiting state of elastic solutions). This casts doubt about the accuracy of available solutions, and a compatible stiffness $k_r$ and $p_m$ is required. The compatible normalized stiffness $k_{nr}$ is equal to 0.275–0.333 ($n=0.7$) and 0.333–0.564 ($n=1.7$) for the associated $p_m$ at the design level of (ground-level) bending moment specified in the JGJ code. Use of the solutions is elaborated for a typical offshore pile group against measured response, which largely substantiates the deduced stiffness $k_{nr}$. The coupled $k_r$ and $p_m$ revealed are fundamental to design of the capped piles using any methods. The new solutions using the $k_{nr}$ and $p_m$ values should be employed to conduct pertinent design.