Buckling Characteristics of Different Cross-Sectioned LGFR-PP Stiffeners under Axial Compression

Abstract

Long glass fiber-reinforced polypropylene (LGFR-PP) composite structures with stiffeners are important substitutes for metal parts for vehicle lightweighting; a good understanding of the buckling characteristics of LGFR-PP stiffeners would provide an important reference for engineering design. The current work is therefore intended to study the buckling characteristics of different cross-sectioned LGFR-PP stiffeners under axial compression via experimental and theoretical analysis. Firstly, LGFR-PP stiffeners with semicircular, rectangular, and trapeziform cross-sections were compressed at the axial direction using a universal testing machine to obtain the buckling process data. Then, the elasticity stability theory modified according to the experimental results was derived to estimate the buckling resistance of LGFR-PP stiffeners in different designs. The test results showed that the LGFR-PP stiffeners possessed a flexible–torsional bulking instability mode under axial compression, the LGFR-PP stiffeners with a semicircular cross-section had higher compression buckling resistance, and the rectangular and trapeziform cross-sectioned stiffeners had better rigidity. The theoretical analysis showed that the modified elasticity stability theory could generally predict the buckling resistance of LGFR-PP stiffeners under axial compression.

1. Introduction

Glass fiber-reinforced thermoplastic composites are increasingly applied in engineering design given their high strength/weight ratio, high recycling rate, low cost, and short manufacturing cycle [1,2,3,4,5,6,7,8]. Particularly in the automobile industry, glass fiber-reinforced thermoplastic composites are widely employed as substitutes for metal parts for lightweighting design [9,10], where the mechanical properties of stiffness, ultimate capacity, and failure mechanism structures are the main considerations in the design and application of composites.

In the past several decades, many studies have focused on improving the mechanical properties of structures through reinforced designs with a fiber-reinforced polymer (FRP) [11,12,13,14,15,16,17,18,19,20]. For example, Nassiraei and Rezadoost recently published a series of studies on the static capacity of tubular components in different connections (X-joints and T/Y-joints) reinforced with FRP, which provided an important reference for engineering design [11,12,13]. On the other hand, a stiffener is one of the most frequently used approaches as it can increase the distortion resistance, structural stiffness, and energy-absorbing capability with some weight addition [14]. Some studies have attempted to improve the strength and energy-absorbing capability of a polymer composite or hybrid toughened kenaf/glass epoxy composite bumper beam with reinforced stiffeners (or named as ribs) [15,16]; the results from these studies showed that the reinforced stiffeners could obviously improve the stiffness, strength, and energy-absorbing performance [15,16]. Previous studies also indicate that the geometrical (e.g., cross-section geometry and thickness) and welding parameters (e.g., bottom fillet and weld line area and position) could affect the strength of stiffeners [17,18,19]. The authors of [20] investigated the effect of the design of reinforced stiffeners on the energy-absorbing capacity of high-density expanded polypropylene, and concluded that the energy absorption of high-density expanded polypropylene could be optimized by fine-tuning the thickness and height of the reinforced stiffeners. Marco and Giovanni investigated a prototype composite rib structure and the experimental result showed that reinforced stiffeners could significantly improve the specific stiffness and strength of the laminate [9]. Despite some investigations on stiffeners, most of the above studies focus on a single type of cross-section, with few studies on long glass fiber-reinforced polypropylene (LGFR-PP) structures.

LGFR-PP is used to manufacture load-carrying structures, where the stiffener concept can be applied in LGFR-PP components to lighten their weight [21]. In semi-structural and structural components, such as battery compartments, bumper beams, etc., open-channel thin-walled composites are widely used as they are prone to buckling under compressive loading [22]. Since the buckling resistance and stiffness of LGFR-PP components could be significantly enhanced by appropriately adding stiffeners, engineers need to select the most suitable cross-section shape and size of the LGFR-PP stiffener for semi-structural and structural components under a compressive load. Moreover, the corresponding theoretical analysis on the mechanical properties and instability mode of the stiffeners would provide an important reference for the selection of LGFR-PP stiffeners. Therefore, the current study focuses on experimental investigations and theoretical analyses of the instability mode and mechanical properties of LGFR-PP stiffeners subjected to axial compression; in particular, the influences of the cross-section geometry (semicircular, rectangular, and trapeziform) are considered.

2. Experimental Study

2.1. Experimental Setup

The LGFR-PP composite used in the current work was prepared by composing long E-glass fibers with a polypropylene matrix, where glass fibers of around 40 mm in length and 13 µm in diameter were employed, which were randomly oriented in the polymer matrix. In the LGFR-PP composite, the weight of glass fibers accounted for 40%; this was chosen since this glass mass fraction could offer a peak strength to LGFR-PP composites [21]. Then, LGFR-PP stiffeners with semicircular (Type I), rectangular (Type II), and trapeziform (Type III) cross-sections were prepared via the hot-molding method, which is widely used for LGFR-PP composite production [4,10].

The geometries of the different cross-sectioned stiffeners (Type I–III) are shown in Figure 1. The thickness t of specimens is 3 mm, L (the length of specimens) is from 145 mm to 220 mm, w represents the width (98 mm), and R represents the corner radius (2 mm). The arc length (l) from point A to B is 33 mm for the semicircular and trapeziform cross-sections and 17.27 mm for the specimens in the rectangular cross-section. Then, the different cross-sectioned specimens were compressed at the axial direction using a universal testing machine (E44, MTS Systems Co., Eden Prairie, MN, USA). The loading velocity for the compression test was set as 12 mm/min loaded on the ends of the specimens. In order to guarantee reliable experimental data, five compression tests were repeated for the same designed cross-section; in total, 45 samples (3 cross-section types*3 samples of different lengths for each cross-section type*5 repetitions for each design) were tested. Then, the peak force (F_max) and the displacement (S_max) at the instant of F_max were used as the evaluation index in investigating the mechanical properties of LGFR-PP stiffeners with different cross-sections or lengths under compression. Specifically, the greater F_max reflects the greater buckling resistance of the design, and the smaller S_max means better rigidity.

2.2. Experimental Results

Figure 2 shows the average value and standard deviation of the F_max and S_max for each cross-section shape calculated from the experimental data. It is clear from the data that the F_max decreases when increasing the length of specimens (L), while the S_max increases with the enlargement of L; the specimens with the semicircular cross-section (Type I) have a higher average F_max and S_max than the specimens with the rectangular cross-section (Type II) and trapeziform cross-section (Type III) for a given L, and the average S_max for Type II specimens is similar to that of Type III when L is in the range of 145 mm to 185 mm. Figure 3 shows the typical deformation process, instability mode, failure behavior, and force–displacement relationship (simplified from the raw data) of the specimens under compression. It could be observed that the deformation was caused by a bending–torsion combined load, and the failure behavior includes matrix cracking, fiber pull-out, or fiber breakage. The force–displacement curve indicates that four typical phases (elastic–plastic–breakage–failure) could be distinguished, and the elastic–plastic deformation phase is dominant, while the breakage phase is relatively short.

3. Theoretical Study

In this section, the instability mode of LGFR-PP stiffeners subjected to axial compression is studied based on the elasticity stability theory for uniform cross-sectioned open thin-walled bars in metal materials. As shown in Figure 4, it is supposed that L is the length of a specimen, A is the area of the cross-section, O is the section centroid, x and y represent the principal inertia axes of the cross-section, and z is the centroidal axis. S represents the flexural center of the cross-section and the corresponding coordinate is (xc, yc). When the axial pressure P is increased to the critical value P_cr, the twisting or bending of the specimen may take place, which leads to a loss of stability for the specimen. Assume that the specimen is in a neutral equilibrium state, and its independent displacement components include the displacements $u$ and $\upsilon$ of the bending center S in the axes x and y direction and the torsion angle $\varphi$ of the cross-section around the bending center. The $u$ and $\upsilon$ coinciding with the forward axis are positive. The sign of $\varphi$ is determined by the right-hand screw rule. The $u$, $\upsilon$, and $\varphi$ are functions of z.

According to the experimental conditions, it is obtained that the specimens are hinged at both ends, and the cross-section of the end has freedom to warp [23,24]. When the boundary condition is $z=0$ and $z=L$, $u=\upsilon =\varphi =0$, $\frac{d^{2}u}{d{z}^2}=\frac{d^2\upsilon }{d{z}^2}=\frac{d^2\varphi }{d{z}^2}=0$, and the deformation assumption that satisfies the boundary conditions can be obtained:

$$\begin{array}{l}u=\xi \sin \frac{n\pi z}{L}\\ \upsilon =\eta \sin \frac{n\pi z}{L}\\ \varphi =\psi \sin \frac{n\pi z}{L}\end{array}(n=1,2,3,L)$$

(1)

where $\xi$, $\eta$, and $\psi$ are undetermined parameters.

The total potential energy of the open thin-walled bar under axial compression can be expressed as

$$\Pi =\frac{1}{2}{\displaystyle {\int }_0^l\left[\begin{array}{l}E{I}_y{u^{″}}^2+E{I}_x{{\upsilon }^{″}}^2+E{I}_w{{\varphi }^{″}}^2+\\ \left(G{I}_t-P{i}_0^2\right){{\varphi }^{\prime }}^2-P\left({u^{\prime }}^2+{{\upsilon }^{\prime }}^2\right)+2P{x}_0{\upsilon }^{\prime }{\varphi }^{\prime }\end{array}\right]}dz$$

(2)

where $E$ represents the elastic modulus, $G$ is the shear modulus, $I_x$ and $I_y$ are the inertia moments of the cross-section for the axis x, and $I_t$ and $I_{\omega }$ represent the torsional moment of inertia and fan-shape moment of inertia for the cross-section, respectively.

$$i_0^2=\frac{I_x+I_y}{A}+x_0^2$$

(3)

where $x_0$ is the distance between the flexural center and centroid.

By substituting Equation (1) into Equation (2) and the integral, the total potential energy $\Pi$ of the specimens tested in the current work can be calculated from Equation (4) using the parameter values shown in

Table 1. The parameter values for Equation (4).

Specimen Type		Type I	Type II	Type III
Moment of inertia (cm4)	Ix	24.51	24.31	24.25
	Iy	1.11	1.80	1.46
(xc, yc) (mm, mm)		(12.739; 0)	(23.138; 0)	(33.077; 0)
It (cm4)		25.643	26.11	25.717
Iw (cm6)		1.058	1.277	9.31
x0 (cm)		0.713	1.714	2.689
i02 (cm2)		8.022	9.498	14.625

.

$$\Pi ={\left(\frac{n\pi }{L}\right)}^2\cdot \frac{l}{4}\left\{\begin{array}{l}{\xi }^2\left[{\left(\frac{n\pi }{L}\right)}^{2}E{I}_y-P\right]+{\eta }^2\left[{\left(\frac{n\pi }{L}\right)}^{2}E{I}_x-P\right]\\ +{\psi }^2\left[{\left(\frac{n\pi }{L}\right)}^{2}E{I}_{\omega }+G{I}_t-P{i}_0^2\right]+2\eta \psi P{x}_0\end{array}\right\}$$

(4)

Based on $\frac{\partial \Pi }{\partial \xi }=0$, $\frac{\partial \Pi }{\partial \eta }=0$, and $\frac{\partial \Pi }{\partial \psi }=0$, the homogeneous linear equations are obtained as Equations (5)–(7):

$$\xi \left[{\left(\frac{n\pi }{L}\right)}^{2}E{I}_y-P\right]=0$$

(5)

$$\eta \left[{\left(\frac{n\pi }{L}\right)}^{2}E{I}_x-P\right]+\psi P{x}_0=0$$

(6)

$$\eta P{x}_0+\psi \left[{\left(\frac{n\pi }{L}\right)}^{2}E{I}_{\omega }+G{I}_t-P{i}_0^2\right]=0$$

(7)

When $n=1$, the minimum value of $P$ can be obtained. From Equation (5), Equation (8) is obtained.

$$P_1=P_{cry}=\frac{{\pi }^{2}E{I}_y}{L^2}$$

(8)

where $P_{cry}$ represents the Euler critical force of bending–buckling in the plane of symmetry. The other two roots $P_2$ and $P_3$ are calculated by letting $n=1$ and the coefficient of determination for Equations (6) and (7) being zero is expressed as

$$\left|\begin{array}{cc}{\left(\frac{\pi }{\mathrm{L}}\right)}^{2}E{I}_x-P& P{x}_0\\ P{x}_0& {\left(\frac{\pi }{\mathrm{L}}\right)}^{2}E{I}_w+G{I}_t-P{i}_0^2\end{array}\right|=0$$

(9)

Equation (10) is obtained by expanding the determinant Equation (9).

$${\left(\frac{{\pi }^{2}E{I}_x}{L^2}\right)}^2(\frac{I_{\omega }}{I_x}+\frac{G{I}_t{L}^2}{{\pi }^{2}E{I}_x})-\frac{{\pi }^{2}E{I}_x}{L^2}(\frac{I_{\omega }}{I_x}+\frac{G{I}_t{L}^2}{{\pi }^{2}E{I}_x}+i_0^2)P+(i_0^2-x_0^2)P^2=0$$

(10)

It is supposed that

$$P_x=\frac{{\pi }^{2}E{I}_x}{L^2}$$

(11)

$$S_1^2=\frac{I_{\omega }}{I_x}+\frac{G{I}_t{L}^2}{{\pi }^{2}E{I}_x}$$

(12)

By substituting Equation (11) and Equation (12) into Equation (10), Equation (13) can be obtained:

$$s_1^2{P}_x^2-(s_1^2+i_0^2)P_{x}P+(i_0^2-x_0^2)P^2=0$$

(13)

The two roots $P_2$ and $P_3$ can be obtained using Equation (13):

$$P_{2,3}=\frac{P_x}{\frac{s_1^2+i_0^2}{2s_1^2}\pm \sqrt{{\left(\frac{s_1^2+i_0^2}{2s_1^2}\right)}^2-\frac{i_0^2-x_0^2}{s_1^2}}}$$

(14)

The smaller root in Equation (14) is the bending torsional buckling critical load $P_{TF}$.

$$P_{TF}=\frac{P_x}{\frac{s_1^2+i_0^2}{2s_1^2}+\sqrt{{\left(\frac{s_1^2+i_0^2}{2s_1^2}\right)}^2-\frac{i_0^2-x_0^2}{s_1^2}}}=\frac{{\pi }^{2}E{I}_x}{l^2}\cdot \frac{1}{\frac{s_1^2+i_0^2}{2s_1^2}+\sqrt{{\left(\frac{s_1^2+i_0^2}{2s_1^2}\right)}^2-\frac{i_0^2-x_0^2}{s_1^2}}}$$

(15)

When $P_{TF}\ge P_{cry}$, the instability mode of the specimen is bending–buckling. Meanwhile, when $P_{TF}<P_{cry}$, the instability mode is flexural–torsional buckling.

The parameter values of Equation (4) for the experimental specimens were calculated as shown in Table 1. The theoretical values of $P_{cry}$ and $P_{TF}$ for the specimens with different $L$ are all $P_{cry}<P_{TF}$, and the instability mode of the specimens should be bending–buckling. The buckling resistance of the specimens is in the order of Type I > Type II > Type III, which is in the same trend as shown in the experimental results. However, the specimens’ instability mode in the theoretical analysis is quite different from that in the experimental results, where flexural–torsional bulking is the instability mode. Therefore, the theoretical value of $P_{TF}$ should be revised to coincide with the experimental results. The correction factors of the experimental specimens Type I–III with different $L$ are shown in

Table 2. The experimental and theoretical values for $F_{\max }^{}$.

Specimen Type	Length L (mm)	145	185	220
Type I	Experimental (KN)	24.89	17.91	13
	Theoretical (KN)	859.35	548.42	399.44
	Ratio	0.029	0.0327	0.0325
		The average ratio = 0.0314
Type II	Experimental (KN)	19.86	16.09	11.79
	Theoretical (KN)	694.36	480.61	357.41
	Ratio	0.029	0.033	0.033
		The average ratio = 0.0317
Type III	Experimental (KN)	17.89	12.16	9.35
	Theoretical (KN)	532.2	398.89	311.59
	Ratio	0.034	0.03	0.03
		The average ratio = 0.031

. From Table 2, it is obtained that the average value of the correction coefficient for the experimental specimens Type I–III are 0.0314, 0.0317, and 0.031, i.e., $F_{1\max }^{Ex}=0.0314F_{TF}$, $F_{2\max }^{Ex}=0.0317F_{TF}$, and $F_{3\max }^{Ex}=0.031F_{TF}$, respectively. The mean value of 0.0314, 0.0317, and 0.031 is 0.0314; using this as the correction coefficient for the theoretical analysis, the revised theoretical analysis expression can be obtained as

$$P_{th}=\frac{0.0314{\pi }^{2}E{I}_x}{l^2}\cdot \frac{1}{\frac{s_1^2+i_0^2}{2s_1^2}+\sqrt{{\left(\frac{s_1^2+i_0^2}{2s_1^2}\right)}^2-\frac{i_0^2-x_0^2}{s_1^2}}}$$

(16)

Based on the parameters in Table 1 and Equation (16), the F_max values of all specimens tested in the experimental study were calculated and compared with the average value of the experimental data in Figure 5. Generally, the estimated results were close to those of the experimental average values, with a maximum relative error within 10%.

4. Discussion

The experimental results (Figure 2a) unsurprisingly indicate the general trend that the maximum force increases with the increasing specimen length under axial compression [25]. However, the maximum displacement of buckling is relatively less sensitive to the increasing specimen length (Figure 2b) compared with the maximum force (Figure 2a). This is largely due to the low toughness of LGFR-PP [10], which limits the failure deformation–length dependency of the specimens under axial compression. It is observed from the experimental results that the cross-section geometry has a significant influence on the buckling resistance of LGFR-PP stiffeners under axial compression; in particular, the buckling resistance of the semicircular cross-sectioned stiffeners is higher than in the designs with rectangular and trapeziform cross-sections, and the trapeziform cross-section has the lowest buckling resistance. This could be explained by the fact that the semicircular cross-section has varying curvature, which is beneficial in reducing the stress concentration [26,27]. In fact, the circular tubes are also found to have a superior specific energy absorption capacity under axial quasi-static crushing than those in other cross-sections, such as triangle, square, pentagon, and hexagon shapes [28]. Regarding the rigidity (lower S_max) of the specimens from high to low, it is in the order of Type II ≈ Type III > Type I (Figure 2b), which implies that the rectangular and trapeziform cross-sections are more stable than the semicircular cross-section.

A combination of bending and torsion could be observed in the deformation of the LGFR-PP stiffeners under axial compression (Figure 3), which is similar to that of aramid fiber-reinforced polymer (AFRP) strengthened steel tubes, as found in the literature [25]. This may suggest that the LGFR-PP stiffeners are able to bear a combined load of bending and torsion. The failure behavior of the LGFR-PP stiffeners under axial compression includes matrix cracking, fiber pull-out, or fiber breakage, which is typical for LGFR-PP materials [10,21]. The force–displacement relationship of LGFR-PP stiffeners under axial compression (Figure 3) indicates that the buckling resistance of the LGFR-PP stiffeners mainly relies on elastic and plastic buckling, while the breakage and failure process is relatively sharp given the low toughness of the LGFR-PP material. This finding is similar to those of previous studies of the mechanical properties of fiber-reinforced polymer structures [10,21,25]. The above findings from the experimental study are only based on specimens using LGFR-PP with a glass mass fraction of 40% and randomly oriented fibers; the effects of the glass mass fraction and orientation have not been investigated, but they are important factors affecting the mechanical properties of LGFR-PP structures [4,21,29]. However, a change in these parameters may not affect the basic findings of the current study substantially, given the fact that all the specimens tested here were prepared using the same material and process.

The theoretical study shows that the estimated F_max values using the modified elasticity stability theory for a uniform cross-sectioned open thin-walled bar matches the experimental results well (relative error less than 10%) (Figure 5). This implies that the theoretical method demonstrated in the current study is plausible and effective for engineering design, although the “correction coefficient” approach (Table 2 and Equation (16)) is oversimplified. It should be noted that the current study only demonstrates a very basic theoretical approach to estimating the compression resistance capability of LGFR-PP stiffeners, and the influences of the non-linearity of the material being deformed, the formation and growth of micro-cracks, etc., have not been accounted for. Despite much simplification in the theoretical model, there is a relatively fixed ratio (approximately 0.03, Table 2) between the initial theoretical and experimental results, and the demonstrated theoretical method could be used as a preliminary estimation tool for engineering design.

5. Conclusions

The buckling characteristics of different cross-sectioned LGFR-PP stiffeners subjected to axial compression were investigated from experimental tests and theoretical analysis. The experimental results imply that the buckling resistance of the LGFR-PP stiffeners in the semicircular cross-section is higher than those in the designs with rectangular and trapeziform cross-sections, and the trapeziform cross-section has the lowest buckling resistance; the rectangular and trapeziform cross-sectioned stiffeners have better rigidity than those with a semicircular cross-section; the LGFR-PP stiffeners present a flexible–torsional bulking instability mode under axial compression, and the final failure behaviors include matrix cracking, fiber pull-out, and fiber breakage. The theoretical analysis results show that the elasticity stability theory for a uniform cross-sectioned open thin-walled bar modified based on a “correction coefficient” according to the test results could generally estimate the maximum buckling force of LGFR-PP stiffeners under axial compression. The findings of the current work could only provide a basic reference for the design of LGFR-PP stiffeners; the ongoing analysis focusing on the effects of the cross-section design together with the preparation techniques and materials on the mechanical characteristics of LGFR-PP structures under statistic and dynamic compression might provide further detailed information.