The thin-walled tubes used in the front rails of the vehicle's body in white play a major role in absorbing the energy of the collision. In this work, the energy absorption characteristics of the functionally graded foam-filled thin-walled tubes are studied by using quasi-static finite element analysis. The simulation of all specimens has been conducted by utilizing finite element software ABAQUS/Explicit TM . A high-precision constitutive equation is used to obtain mechanical properties of foam in different densities. Uniform density foam-filled and functionally graded foam-filled tubes with various gradient functions and configurations are simulated, and force-displacement and energy-displacement diagrams are extracted and compared to each other. The simulation carried out on the layered samples showed that the 13-layered sample is an appropriate approximation for the continuous variations of density along the axial direction. According to the results, linear variations have had better energy absorption characteristics compared to the samples with gradient exponent of m ¼ 2 and m ¼ 5 for an ascending density variation. In this study, the innovative density variation patterns as high-low-high and low-high-low are also simulated in which the high-low-high sample has better energy absorption than other specimens. It is shown for 13-layered samples, the specific energy absorption in the high-low-high design is 3 and 6.6% higher than the ascending pattern (m ¼ 1) and the low-high-low samples, respectively.

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PurposeThe authors examined the numerical natural frequency analysis of a 2D functionally graded (FG) truncated thick hollow cone using 3D elasticity theory.Design/methodology/approachThe material properties of the 2D-FGM (two dimensional-functionally graded materials) cone are graded along the radial and axial axes of the cone using a power–law distribution. The eigenvalue problem was solved using finite element analysis (FEA) employing graded hexahedral elements, and the verification of the finite element approach was assessed by comparing the current solution to earlier experimental studies.FindingsThe effects of semivertex angle, material distribution and the cone configuration on the natural frequencies have been analyzed. For various semivertex angles, thickness, length and power law exponents, many results in the form of natural frequencies and mode shapes are presented for the 2D-FGM cone. As a result, the effects of the given parameters were addressed, and the results were compared, demonstrating the direct efficiency of raising the power–law exponents and cone thickness on the rise of natural frequencies.Originality/valueFor the first time, the numerical natural frequency analysis of a 2D-FG truncated thick hollow truncated cone based on 3D equations of elasticity has been investigated. The material properties of the truncated cone have been distributed along two directions, which has not been considered before in any research for the truncated thick cone. The reason for using these innovative volume fraction functions is the lack of accurate coverage by functions that are available in the literature (Asemi et al., 2011; Babaei et al. 2021).

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