Dynamic response of foam-filled rectangular tubes subjected to low-velocity impact

Abstract

The objective of this work is to investigate dynamic response of fully clamped slender foam-filled rectangular tubes subjected to low-velocity impact. Analytical and numerical analysis are presented to predict the dynamic response of the fully clamped slender foam-filled rectangular tubes struck by a low-velocity heavy mass. In the analytical solutions, the interaction of bending and stretching induced by large deflections is considered. Good agreement is achieved between the analytical solutions and numerical results. It is shown that the filled foam strength has significant effect on the impact resistance capacity of foam-filled tubes.

Keywords

Foam-filled rectangular tube low-velocity impact overall bending

1. Introduction

A tube is an important kind of member in the family of lightweight structures [1]. Also, metal foam is a kind of lightweight material with a number of advantages [2], such as higher specific stiffness, higher specific strength, energy absorption. They are widely used in engineering, such as aerospace, aircraft, ships, etc. Filling the tubes with foam can form foam-filled tubes. Compared with the tubes, foam-filled tubes have different performances and provide more choices for design of tubes [3,4].

Over the past decades, investigations were devoted to analyzing the static and dynamic bending behavior of foam-filled tubes. Foam-filled tubes have been shown to be effective at benefit relative to hollow tubes, such as energy absorption ability, and bending resistance. Santosa and Wierzbicki [5] studied the initial and post bending collapse response of empty and thin-walled prismatic filled columns through the analytical and numerical investigations. It was found that low-density aluminum foam-filled columns have a significant increase of the bending strength of the column quite significantly. Guo et al. [6] carried out three-point bending experiments to study load-carrying capacity and energy absorption of empty, aluminum foam-filled single and double square tubes. Li and Lu [7] carried out quasi-static three-point bending experiments to study the crashworthiness characteristics of empty square and circular tubes, foam-filled single square and circular tubes, and foam-filled double square and circular tubes. Zhang et al. [8] proposed a plastic yield criterion for the foam-filled rectangular tube, and obtained analytical solutions for large-deflection of fully clamped slender metal foam-filled rectangular tubes. Sampath et al. [9] numerically and experimentally investigated energy absorption of foam-filled and empty aluminum tubes under dynamic bending. The results show filling the foam inside the tube results in more force and energy required to bend the tube.

To the authors’ knowledge, little work has been published on the analytical analysis of low-velocity impact response of foam-filled tubes. To have a better understanding of the crashworthiness of foam-filled tubes, the present research aims at the dynamic response of slender foam-filled rectangular tubes subjected to low-velocity impact. The paper is organized as follows. In Section 2, analytical solutions for the low-velocity impact response of a fully clamped metal foam-filled rectangular tubes. In Sections 3 and 4, finite element (FE) calculation is carried out, and the numerical results are compared with the analytical predictions. Also, the effect of the filled foam on dynamic response of foam-filled rectangular tubes is discussed in details. Finally, the concluding remarks are presented in Section 5.

2. Analytical solutions

Consider a fully clamped slender foam-filled rectangular tube transversely struck at midspan struck by a low-velocity heavy mass G and initial velocity of striker V _I. The length, width, height of the tube are 2L, b and b ₁, respectively, as shown in Fig. 1. The metal foam is assumed to be perfectly filled and bonded into the rectangular tube with wall thicknesses h ₁ and h in the width and height directions. The tube is modelled as the rigid-perfectly plastic (r − p − p) law with yield strength σ_f, and the filled foam is assumed to obey the rigid-perfectly plastic locking (r − p − p − l) material with yield strength σ_c and densification strain. The striker is assumed to be rigid.

Fig. 1.

Sketch of a fully clamped foam-filled rectangular tube struck by a heavy mass with initial lowvelocity at midspan.

Here, the slender foam-filled rectangular tube is assumed to deform in a global manner without local denting occurring beneath the striker, the shape of the section is unchanged as shown in Fig. 2. There are three plastic hinges developed at the impact location and the clamped supports, respectively. The transverse velocity profile is assumed to be linear and given by $\begin{eqnarray}\displaystyle {\dot{W}}=\left\{\begin{array}{@{}ll@{}}{\dot{W}}_{0}(1-x/L),\quad & \quad 0\leq x\leq L\\ {\dot{W}}_{0}(1+x/L),\quad & \quad -L\leq x\leq 0\end{array}\right. & & \displaystyle\end{eqnarray}$ (1) where ${\dot{W}}_{0}$ is the velocity at the impact point and ${\dot{W}}_{0}|_{t=0}=∼V_{0}$ , in which V ₀ is the velocity of the mass-beam system after impact, as shown in Fig. 2(a).

Fig. 2.

Overall bending deformation pattern for the neutral surface of the fully clamped foam-filled rectangular tube. (a) Transverse velocity profile, and (b) free body diagram of the left part of the foam-filled tube.

According to the law of momentum conservation, balance equations of the mass-beam system before and after impact are obtained and can be expressed as $\begin{eqnarray}\displaystyle GV_{I}=GV_{0}+G_{b}\left[\displaystyle \int _{0}^{L}V_{0}\left(1-\displaystyle \frac{x}{L}\right)dx+\int _{-L}^{0}V_{0}\left(1+\displaystyle \frac{x}{L}\right)dx\right] & & \displaystyle\end{eqnarray}$ (2) in which the mass G _b per unit length G _b = 2𝜌_f(bh ₁ + b ₁ h − 2hh ₁) + 𝜌_c(b − 2h)(b ₁ − 2h ₁). Then, we have $\begin{eqnarray}\displaystyle V_{0}=\displaystyle \frac{G}{G+\left[2{\rho}_{f}\left(bh_{1}+b_{1}h-2hh_{1}\right)+{\rho}_{c}(b-2h)(b_{1}-2h_{1})\right]L}V_{I}. & & \displaystyle\end{eqnarray}$ (3)

The left-hand part of the foam-filled rectangular tube is considered due to the symmetry of the problem Employing the moment of momentum theorem with respect to point A, we have $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle \frac{d}{dt}\left\{\int _{-L}^{0}\left[2{\rho}_{f}(bh_{1}+b_{1}h-2hh_{1})+{\rho}_{c}(b-2h)(b_{1}-2h_{1})\right]{\dot{W}}_{0}\left(1+\frac{x}{L}\right)(L+x)dx\right\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\displaystyle \frac{P}{2}L-FW_{0}-M_{1}-M_{2}\end{eqnarray}$ (4) where F ≅ N for moderate deflection, M ₁ = M ₂ and W ₀ is the deflection at midspan, as shown in Fig. 2(b). Considering the equilibrium of the striker in the vertical direction yields $\begin{eqnarray}\displaystyle -P=G\ddot{W}_{0}. & & \displaystyle\end{eqnarray}$ (5)

Combination of Eqs. (4) and (5) yields $\begin{eqnarray}\displaystyle \left\{\displaystyle \frac{[2{\rho}_{f}(bh_{1}+b_{1}h-2hh_{1})+{\rho}_{c}(b-2h)(b_{1}-2h_{1})]L^{2}}{3}+\frac{GL}{2}\right\}\ddot{W}_{0}+NW_{0}+2M_{1}=0. & & \displaystyle\end{eqnarray}$ (6)

It is assumed that the axial extensions occur in the central and end hinges, and the left-hand portion (L-a) of the foam-filled rectangular tube has a total extension e, $\begin{eqnarray}\displaystyle e=e_{1}+e_{2} & & \displaystyle\end{eqnarray}$ (7) where e ₁ and e ₂ are the axial extensions concentrated at the end of the left-hand portion and the impact point, respectively, as shown in Fig. 2(a). The total elongation and the angular rotation of left-hand part are $\begin{eqnarray}\displaystyle e\approx \displaystyle \frac{W_{0}^{2}}{2L} & & \displaystyle\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle {\psi}\approx \displaystyle \frac{W_{0}}{L}. & & \displaystyle\end{eqnarray}$ (9) For simply, we introduce the following non-dimensional parameters: $\begin{eqnarray}\displaystyle W_{0}^{\ast }=\displaystyle \frac{W_{0}}{b_{1}},\quad \overline{b}_{1}=\displaystyle \frac{b_{1}}{L},\quad \overline{t}=\frac{t}{b_{1}\sqrt{{\rho}_{f}/{\sigma}_{f}}},\quad {\dot{W}}_{0}^{\ast }=\displaystyle \frac{d{\dot{W}}_{0}^{\ast }}{d\overline{t}}, & & \displaystyle \nonumber\\ \displaystyle \ddot{W}_{0}^{\ast }=\displaystyle \frac{d{\dot{W}}_{0}^{\ast }}{d\overline{t}},\quad G^{\ast }=\displaystyle \frac{G}{2G_{b}L},\quad \overline{{\rho}}=\displaystyle \frac{{\rho}_{c}}{{\rho}_{f}},\quad V_{0}^{\ast }=\frac{V_{0}}{\sqrt{{\sigma}_{f}/{\rho}_{f}}}. & & \displaystyle \nonumber\end{eqnarray}$

The yield criterion for the foam-filled rectangular section is [8], $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}ll@{}}\displaystyle |m|=-\frac{A^{2}\overline{{\sigma}}_{c}n^{2}}{\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})}+1,\quad & \displaystyle 0\leq |n|\leq \displaystyle \frac{\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}{2\overline{h}_{1}+\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}\\[12.0pt] \displaystyle |m|=\frac{-[\overline{{\sigma}}_{c}(1-2\overline{h}_{1})(1-|n|)/2-|n|\overline{h}_{1}+\overline{h}_{1}-1/2]^{2}+1/4}{\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})},\quad & \displaystyle \frac{\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}{2\overline{h}_{1}+\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}\leq |n|\leq 1\end{array}\right. & & \displaystyle\end{eqnarray}$ (10) where $m=∼\frac{M}{M_{p}}$ , $n=∼\frac{N}{N_{p}}$ , $A=∼\frac{2\overline{h}_{1}+\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}{2\overline{{\sigma}}_{c}}$ , $\overline{{\sigma}}_{c}=∼\frac{{\sigma}_{cf}}{{\sigma}_{f}}=∼2\overline{h}+\overline{{\sigma}}(1-2\overline{h})$ , $\overline{{\sigma}}=∼\frac{{\sigma}_{c}}{{\sigma}_{f}}$ , $\overline{h}=∼\frac{h}{b}$ , $\overline{h}_{1}=∼\frac{h_{1}}{b_{1}}$ , in which the average strength of the foam-filled section ${\sigma}_{cf}=∼{\sigma}_{f}\frac{2h}{b}+{\sigma}_{c}\frac{b-2h}{b}$ , the fully plastic axial force N _p and bending moment M _p are N _p = 2σ_f bh ₁ + σ_cf b (b ₁ − 2h ₁) and $M_{p}=∼{\sigma}_{f}bh_{1}(b_{1}-h_{1})+{\sigma}_{cf}b\left(\frac{1}{2}b_{1}-h_{1}\right)^{2}$ , respectively.

According to the associated flow rule of yield criteria Eq. (10), we obtain the following relation at fully clamped end and the impact point, $\begin{eqnarray}\displaystyle \displaystyle \frac{{\dot{e}}_{1}}{\dot{{\psi}}}=\displaystyle \frac{{\dot{e}}_{2}}{\dot{{\psi}}}=\left\{\begin{array}{@{}ll@{}}b_{1}A|n|,\quad & \displaystyle 0\leq |n|\leq \frac{\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}{2\overline{h}_{1}+\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}\\[12.0pt] \displaystyle b_{1}\left[\frac{\overline{{\sigma}}_{c}}{2}(|n|-1)(1-2\overline{h}_{1})+(|n|-1)\overline{h}_{1}+\frac{1}{2}\right],\quad & \displaystyle \frac{\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}{2\overline{h}_{1}+\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}\leq \left|n\right|\leq 1.\end{array}\right. & & \displaystyle\end{eqnarray}$ (11) From Eqs. (7) to (9) and (11), relation between deflection $W_{0}^{\ast }$ and non-dimensional axial force n is $\begin{eqnarray}\displaystyle W_{0}^{\ast }=\left\{\begin{array}{@{}ll@{}}2A|n|,\quad & \displaystyle 0\leq |n|\leq \frac{\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}{2\overline{h}_{1}+\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}\\[12.0pt] (|n|-1)(\overline{{\sigma}}_{c}-2\overline{{\sigma}}_{c}\overline{h}_{1}+2\overline{h}_{1})+1,\quad & \displaystyle \frac{\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}{2\overline{h}_{1}+\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}\leq |n|\leq 1.\end{array}\right. & & \displaystyle\end{eqnarray}$ (12) When $W_{0}^{\ast }\geq 1$ and n = 1, the foam-filled rectangular tube is in a membrane state.

Substituting Eqs (10) and (12) into Eq. (6) leads to the governing differential equation $\begin{eqnarray}\displaystyle {\eta}\ddot{W}_{0}^{\ast }=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{\overline{{\sigma}}_{c}}{4[\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})]}W_{0}^{\ast 2}+1,\quad & 0\leq W_{0}^{\ast }\leq 1-2\overline{h}_{1}\\[12.0pt] \displaystyle \frac{W_{0}^{\ast 2}+2(\overline{{\sigma}}_{c}-1)(1-2\overline{h}_{1})W_{0}^{\ast }+1}{4[\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})]},\quad & 1-2\overline{h}_{1}\leq W_{0}^{\ast }\leq 1\\[12.0pt] \displaystyle \frac{2\overline{h}_{1}+\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}{2[\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})]}W_{0}^{\ast },\quad & W_{0}^{\ast }\geq 1\end{array}\right. & & \displaystyle\end{eqnarray}$ (13) where $\begin{eqnarray}\displaystyle {\eta}=-\frac{(1+3G^{\ast })[\overline{{\rho}}(1-2\overline{h})(1-2\overline{h}_{1})+2(\overline{h}+\overline{h}_{1}-2\overline{h}_{1}\overline{h})]}{6[\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})]\overline{b}_{1}^{2}}. & & \displaystyle \nonumber\end{eqnarray}$

The initial conditions are ${\dot{W}}_{0}^{\ast }|_{\overline{t}=∼0}=∼V_{0}^{\ast }$ and $W_{0}^{\ast }|_{\overline{t}=∼0}=∼0$ . Combination of Eq. (13) and the initial conditions leads to the relationship of the non-dimensional initial impact kinetic energy $U_{K}^{\ast }$ versus the non-dimensional maximum central deflection $W_{0m}^{\ast }$ $\begin{eqnarray}\displaystyle \frac{4G^{\ast }(1+3G^{\ast })}{3(1+2G^{\ast })^{2}}U_{K}^{\ast }=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{\overline{{\sigma}}_{c}}{12[\overline{{\sigma}}_{c}\left(1/2-\overline{h}_{1}\right)^{2}+\overline{h}_{1}(1-\overline{h}_{1})]}W_{0m}^{\ast 3}+W_{0m}^{\ast },\quad & 0\leq W_{0m}^{\ast }\leq 1-2\overline{h}_{1}\\[12.0pt] \displaystyle \frac{W_{0m}^{\ast 3}+3(\overline{{\sigma}}_{c}-1)(1-2\overline{h}_{1})W_{0m}^{\ast 2}+3W_{0m}^{\ast }}{12[\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})]}+K_{1},\quad & 1-2\overline{h}_{1}\leq W_{0m}^{\ast }\leq 1\\[12.0pt] \displaystyle \frac{2\overline{h}_{1}+\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}{4[\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})]}W_{0m}^{\ast 2}+K_{2},\quad & W_{0m}^{\ast }\geq 1\end{array}\right. & & \displaystyle\end{eqnarray}$ (14) where $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle U_{K}^{\ast }=\frac{G_{s}V_{I}^{2}/2}{P_{c}b_{1}},\quad P_{c}=\frac{4M_{p}}{L},\quad K_{1}=\frac{(\overline{{\sigma}}_{c}-1)(1-2\overline{h}_{1})^{3}}{12[\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})]},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \displaystyle K_{2}=\frac{(\overline{{\sigma}}_{c}-1)(1-2\overline{h}_{1})^{3}+1}{12[\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})]}.\nonumber\end{eqnarray}$

During the deformation process, the reaction force between the striker and the foam-filled rectangular tube is defined as $\begin{eqnarray}\displaystyle P_{r}=-G\ddot{W}_{0}. & & \displaystyle\end{eqnarray}$ (15) Substituting Eq. (13) into Eq. (15), we have $\begin{eqnarray}\displaystyle P_{r}^{\ast }=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{3G^{\ast }}{3G^{\ast }+1}\left\{\frac{\overline{{\sigma}}_{c}}{4[\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})]}W_{0}^{\ast 2}+1\right\},\quad & 0\leq W_{0}^{\ast }\leq 1-2\overline{h}_{1}\\[12.0pt] \displaystyle \frac{3G^{\ast }}{3G^{\ast }+1}\frac{W_{0}^{\ast 2}+2(\overline{{\sigma}}_{c}-1)(1-2\overline{h}_{1})W_{0}^{\ast }+1}{4[\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})]},\quad & 1-2\overline{h}_{1}\leq W_{0}^{\ast }\leq 1\\[12.0pt] \displaystyle \frac{3G^{\ast }}{3G^{\ast }+1}\frac{2\overline{h}_{1}+\overline{{\sigma}}_{c}(1-2\overline{h}_{1})}{2[\overline{{\sigma}}_{c}(1/2-\overline{h}_{1})^{2}+\overline{h}_{1}(1-\overline{h}_{1})]}W_{0}^{\ast },\quad & W_{0}^{\ast }\geq 1\end{array}\right. & & \displaystyle\end{eqnarray}$ (16) where $P_{r}^{\ast }=∼\frac{P_{r}}{P_{c}}$ .

3. Finite element analysis

Using the ABAQUS/Explicit code, Finite element (FE) calculations are carried out to investigate the dynamic response of fully clamped foam-filled rectangular tubes transversely struck at midspan by a heavy mass with low-velocity. The filled foam and tubes are modelled using three-dimensional eight-node and linear brick elements (Type C3D8R) with reduced integration, and four-node linear shell elements (Type S4R) with reduced integration and finite membrane strains. The striker is modeled as a rigid roller with a point mass and predefined velocity. Damping associated with the bulk viscosity in ABAQUS/Explicit is switched off by setting the bulk viscosity to be zero. Appropriate mesh refinement near the clamped end and the impact point is conducted. Half of the foam-filled rectangular tube is modelled due to the symmetry of the problem. The vertical, horizontal and rotational displacements of nodes at the clamped ends of the foam-filled tubes are set to zero. The contact between the foam-filled rectangular tube and the striker is modelled by using a contact pair surface algorithm with a frictionless contact option. The half span, width, and height of the foam-filled rectangular tubes are L = 300 mm, b = 20 mm, b ₁ = 8 mm, and wall thicknesses are h ₁ = h = 0.5 mm, i.e. b = b∕L = 1∕15, and b ₁ = b ₁∕L = 2∕75. Radius of the loading roller is R = 4 mm.

The tube obeys J ₂ flow theory of plasticity, and Deshpande-Fleck constitutive model [10] is used to model the plastic crushable behavior of metal foam implemented in ABAQUS. The tubes are made of stainless steel with yield strength σ_f = 200 MPa, elastic modulus E _f = 200 GPa, elastic Poisson’s ratio 𝜈_ef = 0.3, density 𝜌_f = 8000 kg∕m³, and linear hardening moduli E _t = 0.001E _f. The isotropic metal foam has yield strength σ_c = 10 MPa, elastic modulus E _cf = 10 GPa, relative density 𝜌 = 𝜌_c∕𝜌_f = 0.1, density 𝜌_c = 800 kg∕m³, elastic Poission’s ratio 𝜈_ec = 0.3, plastic Poission’s ratio 𝜈_p = 0. The filled foam has a long plateau-stress platform σ_c that continues up to the densification strain ϵ_D = 0.5. Metal foam rises steeply with large tangent modulus E _ct = 0. 2E _f beyond densification. It is assumed tubes and metal foam have sufficient ductility to sustain deformation without fracture.

Fig. 3.

Analytical and numerical results for non-dimensional maximum deflection $W_{0m}^{\ast }$ versus non-dimensional impact energy $U_{K}^{\ast }$ of foam-filled rectangular tubes. (a) A given mass ratio G ^∗ = 200, and (b) initial impact velocity V _I = 1 m∕s.

Fig. 4.

The effect of filled foam strength σ on (a) the non-dimensional impact force $P_{r}^{\ast }$ versus non-dimensional deflection $W_{0}^{\ast }$ , and (b) the non-dimensional maximum deflection $W_{0m}^{\ast }$ versus non-dimensional initial impact energy $U_{K}^{\ast }$ for the fully clamped foam-filled rectangular tubes subjected to low-velocity impact.

4. Results and discussion

Figure 3(a) shows the maximum deflections $W_{0m}^{\ast }$ as a function of the initial impact energy $U_{K}^{\ast }$ of the foam-filled rectangular tubes (G ^∗ = 200, σ = 0.05, h = 0.025 and h ₁ = 0.0625). In the numerical model, the local denting below the roller is too small to be considered, the shape of the section is assumed to be original. In fact, the transverse velocity profile is approximately linear in the numerical model. The analytical solutions for the maximum deflection of the tube increase nonlinearly with the impact energy of the striker, and the slope of the deflectionenergy curve gradually decreases. It can be seen that good agreement between the analytical and numerical results is achieved, and analytical predications may somewhat overestimate numerical results. Analytical and numerical results for the maximum deflection $W_{0m}^{\ast }$ of tubes (σ = 0.05, h = 0.025 and h ₁ = 0.0625) versus the initial impact energy $U_{K}^{\ast }$ of the striker are shown in Fig. 3(b). The striker velocity V _I = 1 m∕s is fixed, and the mass ratio G ^∗ is varied. One can see the good agreement between analytical solutions and numerical results.

Figures 4(a) and 4(b) show the effect of filled foam strength σ on the impact force $P_{r}^{\ast }$ versus deflection $W_{0}^{\ast }$ , and the maximum deflection $W_{0m}^{\ast }$ versus initial impact energy $U_{K}^{\ast }$ for the fully clamped foam-filled rectangular tubes subjected to low-velocity impact (G ^∗ = 200, h = 0.025 and h ₁ = 0.0625), respectively. It is seen that impact force of the slender foam-filled rectangular tubes increases with increasing filled foam strength for a given deflection. In additions, the maximum deflection $W_{0m}^{\ast }$ of the foam-filled rectangular tubes decreases with increasing σ for a given impact energy. It means that the filled foam strength has significant effect on the impact resistance capacity of foam-filled tubes.

5. Conclusions

Low-velocity impact response of fully clamped slender foam-filled rectangular tubes was investigated analytically and numerically. Employing the yield criteria for foam-filled tubes, analytical solutions were derived for large-deflection dynamic response of slender foam-filled rectangular tubes struck at midspan by a heavy mass with low velocity. Analytical solutions are in good agreement with numerical results. It is shown the filled foam strength has significant effect on the impact resistance capacity of foam-filled tubes The present analytical model can offer adequate accuracy to predict low-velocity heavy-mass impact of fully clamped slender foam-filled rectangular tubes

Footnotes

Acknowledgements

The authors gratefully acknowledge the financial supports of NSFC (11502189, 11572234).

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