Analysis of axial crushing of fiber reinforced metal tube with an eccentricity factor

The fiber reinforced metal tubes have been proved to exhibit an excellent crashworthy performance and a high ratio of strength to weight in many experiments. Some theoretical models are promoted to predict the mean crushing force, which benefits the design of these structures. Based on the solution of Wang and Lu, the theoretical analysis is presented in this paper to consider the eccentricity factor, which represents the ratio of the outward part to the total folding length. The agreement between the calculated and experimental data reveals that the analytical model is reasonable.


Introduction
Thin-walled tubes are often used as energy absorbers in moving vehicles to protect the people and structures. Many theoretical, experimental and numerical studies related to this topic have appeared in the literatures. Among different type of thin-walled structures, the circular tubes are widely employed because of their low cost and excellent energy absorption capacity. The theoretical studies on the axial crushing of thin-walled circular tubes could be tracked back to the works of Alexander [1] and Pugsley [2] around 1960s. Andrews [3] classified the deformation modes of the axial collapse of cylindrical tubes. Many other researchers made their efforts to further improve the theoretical models for the axial crushing of circular tubes. In addition to the circular metal tube, a combination of metal and composite material has been used to improve crashworthiness properties. These fiber reinforced metal tubes have been proved to exhibit an excellent crashworthy performance and a high ratio of strength to weight [4][5][6][7]. A simplified analytical model for the static crushing of externally reinforced metal tubes was presented by Hanefi [8]. In the paper, the classical Alexander's solution was modified to take into account the contribution of the metal and composite wall. Mamalis et al [9] developed the theoretical analysis of the failure mechanism of thin-walled fiberglass composite tubes under static axial compression. Wang and Lu [10] presented a theoretical model to predict the mean crushing force of arbitrarily fiber reinforced metal tubes with a ring collapse mode. This paper presented a more reasonable predictive model of axial crushing of fiber reinforced metal tubes. The crushing model took into account both the radial inward and outward folds.
In this paper, the theoretical analysis is modified to predict the mean crushing force based on the works from Wang and Lu [10]. In their works, the fold develops to the inward and outward direction, and the two displacements are assumed to be equal, as shown in figure 1(b). The experiments of axial crushing of carbon fiber reinforced steel tubes have been carried out in our laboratory. The results revealed that the inward and outward displacements were slightly different. Singace [11,12]  the axial crushing analysis considering the eccentricity factor, which was used to represent the ratio of the outward part to the total folding length. From this point of view, the eccentricity factor for the Wang and Lu's works is 0.5, which means that the length of the inward and outward part is equal. According to the Singace's works and our experiments, the eccentricity factor of 0.65 should be a more appropriate choice for the analysis, as shown in figure 1(c). The following sections will describe the analysis with the eccentricity factor of 0.65 based on the works from Wang and Lu. The theoretical prediction results are compared to the experimental data from our laboratory.

Theoretical analytical model
In the collapse of fiber reinforced metal tubes, the formation of plastic hinges leads to the developing of folds. Two folds are selected to deduce the formula, as shown in figure 1(c). Some assumptions are made to simplify the derivation. The hinges are considered as stationary plastic hinges. The metal wall obeys the Von Mises yield criterion. The work hardening of the metal is ignored. The work of the external force is dissipated by the bending at the five hinges, the circumferential stretching and compression of the tube wall between the hinges. These energy dissipation mechanisms are described in the following sections.

Energy dissipated by the bending at the hinges
With the increase of the external axial force, the bending moment of the tube wall increases. When the bending moment increases to the plastic limit bending moment M, the plastic hinges appear. For the assumption of no work hardening, the hinges continuously deform to the angle α until the adjacent folds touch together. In the five plastic hinges of two folds, hinges 2 and 4 are at outward of the tube wall, while hinges 1, 3 and 5 inward. Therefore, the hinges are restricted by fiber wall.
The plastic limit bending moment M 0 at outward convolution hinges 2 and 4 is (2) where, r is the inner radius of the tube. L is the half length of a fold.
The plastic limit bending moment M 1 at inward convolution hinges 1, 3 and 5 is where, σ czc is the axial compressive strength of the fiber reinforced layer. Considering the eccentricity factor of 0.65, during the increment d , the increment of work done by bending deformation at hinges 1, 3 and 5 is

Energy dissipated by the circumferential tensile and compressive strain of fiber reinforced layer
The circumferential compression of fiber reinforced layer between hinge 1, 3 and 5 is the same as that of the metal layer. The fiber reinforced layer in circumferential compression can be considered as a perfectly plastic material, so the strain is where, σ chc is the circumferential critical stress of the fiber reinforced layer subjected to compression.
The fiber reinforced layer between hinges 2 and 4 is under circumferential tension. The fiber layer is elastic until it fractures when the circumferential strain reaches a critical value of nearly 1.77%. Thereafter, the fiber reinforced layer does not exert any stretching resistance circumferentially. The work done in circumferentially stretching the fiber reinforced layer is concluded as follows [10]  where, σ cht and s are the critical stress and strain of the fiber reinforced layer subjected to circumferential tension, respectively.
The total work dissipated by the circumferential strain of the fiber reinforced layer is

Experimental verification
Some crushing tests on the carbon fiber reinforced steel circular tubes are performed in our laboratory to verify the solutions in this paper. The steel tubes with thickness of 1 mm and diameter of 50 mm are wrapped by the carbon fiber reinforced layers. The number of plies of the carbon fiber reinforced layers is from 1 to 5. In all samples, the fiber direction is always perpendicular to the axis of tubes.
Other parameters related to the solutions of equation (21)

Conclusions
The axial crushing of a metal tube with external fiber reinforced layers is studied in this paper with theoretical analysis method. An analytical model is modified and presented to calculate the mean crushing force based on the works of Wang and Lu [10]. In the presented model, the eccentricity factor of 0.65 is adopted to represent the ratio of the outward part to the total folding length. The theoretical results agree with the experimental data from our laboratory, which indicates the analytical model is reasonable.