Buckling and layer failure of composite laminated cylinders subjected to hydrostatic pressure

1 Introduction

Composite materials have gained widespread usage in aircraft structure over the past decades owing to the significant weight-saving and other unique advantages they offer. New applications of composite materials are increasing in several fields where high specific strength and long fatigue life are the most benefits [1], [2], [3]. One common field pertains to ocean structures and underwater vehicles. Generally, submarine structures under external hydrostatic pressure have been manufactured from high-strength steels, aluminum, and titanium alloys. The buoyancy requirement for submarine structures can be easily met by applying composite materials. Additionally, the danger of being detected through sonar equipment can be reduced, whereas the resistance to corrosion can be increased because the composite materials have good sound absorption properties and are stable against the chemical reaction. For an underwater vehicle operated in deep sea, hydrostatic pressure-induced buckling tends to dominate structural performance.

Hom [4] investigated glass-reinforced plastic materials for pressure hulls of deep-submergence vehicles and proposed that glass-fiber reinforcement would probably be replaced by carbon reinforcement, which has higher properties. Ouellette et al. [5] studied the buckling of composite cylinders under external pressure and compared the experimental buckling pressures to values predicted by three different procedures: a linear buckling analysis, a proposed design code, and an existing design standard. It was found that all three procedures calculated the design pressure with values ranging from 1.7 to 11.5 times lower than the observed buckling pressure. Smith [6] contrasted the buckling behavior of pressure hulls made of fiber-reinforced polymer composites with that of high-strength steel, aluminum, and titanium alloy. He also proposed two illustrative trial designs corresponding to stiffened glass-reinforced polymer pressure hull for shallow-water operation and unstiffened carbon fiber-reinforced polymer deep-water pressure hull. Graham [7], [8] developed the analytical models in conjunction with small-scale submarine tests, which was used in the design of the large composite hull mode and also in the analysis of experimental date from the large-scale test. Liang et al. [9] optimized the design of filament-wound multilayer-sandwich submersible pressure hulls with graphite/epoxy, glass/epoxy, and boron/epoxy composite facings. The results revealed that, in the design of the wall of the submersible pressure hull, boron/epoxy is a better choice as a facing material at shallow depths, while graphite/epoxy is preferred at extreme depths. Tafreshi [10], [11] proposed a computational modeling of delamination buckling and post-buckling of laminated composite cylindrical shells subjected to external pressure or combined axial compression and external pressure. Finite element analysis verified the computational results. Several studies including those of Shen [12], Li and Lin [13] have found that there exists a circumferential stress along with an associate shear stress when the anisotropic laminated shell is subjected to lateral pressure. As Becker et al. [14], Srinivasa et al. [15] and Venkateshappa et al. [16] reported, the critical buckling load of laminated composite skew plates is definitely influenced by skew angle, aspect ratio, and stacking sequence. Moon et al. [17] studied the buckling of filament-wound composite cylinders by performing hydrostatic pressure tests. Three finite element analysis programs used for failure analysis predicted the buckling pressure with 1.9–23.9% deviations from experimental results, not considering the initial imperfections of the cylinders. Ca et al. [18] presented a reliability-based load and resistance factor design procedure for subsea composite pressure vessels subjected to external hydrostatic pressure. The longitudinal modulus, inside radius of the composite layers, unsupported length, and external pressure significantly affect the design results, whereas the transversal modulus, Poisson’s ratio, shear modulus, and winding angle have little effects. From what has been discussed in the preceding lines, carbon composites are considered promising materials [19].

This study is a preliminary research from our series of work on the application of carbon-epoxy composites to underwater vehicles. This work aims to study the stability behavior and layer failure of composite laminated cylinders under external hydrostatic pressure for underwater vehicle application. Finite element analysis was carried out by using the ANSYS software (ANSYS Inc., Pittsburgh, PA, USA). Both the linear buckling analysis and the nonlinear buckling analysis were conducted to predict the critical buckling pressure. The results from the finite element analysis were compared with the experimental results.

2 Materials and methods

A total of five carbon-epoxy composite cylinders were tested for the stability behavior under hydrostatic pressure [20]. The cylinders were marked with CTM1 to CTM5, respectively. The material properties are given in Table 1. Table 2 shows the dimensions of the composite cylinders. The composite cylinder was submerged in water, and it thus experienced axial pressure over the right end cover in addition to a lateral pressure over the cylinder surface.

3 Finite element analysis

ANSYS was used for the buckling and post-buckling analysis. Both static and stability analysis were conducted in sequence in order to obtain the critical buckling pressure. The composite cylinders were meshed using a three-dimensional structural shell element (SHELL 281), which is defined by eight nodes with 6 degrees of freedom at each node: translations in the x-, y-, and z-axes, and rotations about the x-, y-, and z-axes [21]. The left flange was fixed, and the axis direction of the right cover met the actual experimental condition. The loading condition used for the analysis was the same as that of underwater vehicles subjected to hydrostatic pressure. Figure 1 shows the finite element modeling concept. The element shape was specified to be quadrilateral with an edge length of 20 mm. The number of elements used for each finite element model is given in Table 2, which was decided by sensitivity analysis. The effect of the number of elements for finite element analysis on buckling pressure is shown in Figure 2. It can be seen that, with the total number of elements increasing, initially the buckling pressure drops sharply, followed by a slow decrease, subsequently declining slightly in the range of 1000–1500 and then tending to be constant finally. As shown in Figure 2, the total number of elements selected in Table 2 is suitable. The right end cover used the same type of elements as those for the finite element modeling of the composite cylinders, with the material properties of steel in a 13-mm-thick plate.

Figure 1:

The finite element model.

Figure 2:

Sensitivity analysis.

Although the five composite cylinders have a 316-mm nominal inner diameter and a 600-mm nominal axial length, the lengths of CTM1 and CTM2 were 564 and 569 mm, respectively. These two cylinders were shortened during the fabrication process to remove parts with serious geometric imperfection [20]. The effect of geometry parameter on the critical buckling pressure is shown in Figure 3. It can be seen that the critical buckling pressure increases with a high slope as the shell thickness increases. As for the effect of length on the critical buckling pressure, the critical buckling pressure decreases inversely as the length increases.

Figure 3:

Critical buckling pressure vs. geometry parameter of the composite cylinder. (A) Shell thickness of the composite cylinder. (B) Length of the composite cylinder.

4 First-ply failure of the angle-ply laminate

In this study, the Tsai-Wu failure criteria were used as the failure criteria to assess the ability of the angle-ply laminate pressure hull to withstand overstressing failure. Additionally, a first-ply failure strength load factor ζ was introduced to identify the characteristic of the first-ply failure of the angle-ply laminate. According to the Tsai-Wu failure criterion [22], the first-ply failure strength load factor ζ_k is defined as follows:

where σ1(k) andσ2(k) are the stresses in the longitudinal and transverse direction, respectively, and σ6(k) is the in-plane shear stress (see Ref. [9]). The expressions for the coefficients F₁₁, F₂₂, F₆₆, F₁, F₂, and F₁₂ are given by

where X_T, X_C, Y_T, and Y_C are the tensile and compressive strength of the composite materials in the longitudinal and transverse directions, and S is the in-plane shear strength. The load factor ζ^(k) is used for the failure prediction in a ply, and the Tsai-Wu failure function must be lower than 1 for the whole model. Accordingly, layer failure occurs when the load factor reaches or exceeds the value of 1.

5 Results and discussion

5.1 Buckling pressure and mode

Table 3 shows the experimental pressure used by Hur et al. [20]. The experimental buckling pressure was 0.552 MPa. The predicted buckling pressure deviated from the experimental results by 29.0%, 32.0%, and 22.6%, as determined by NASTRAN (MSC Software Company, Los Angeles, USA), MARC (MSC Software Company, Los Angeles, USA), and ACOS (School of Mechanical and Aerospace Engineering, Gyeongsang National University, Republic of Korea), respectively. It can be seen that ACOSwin relatively better predicted the buckling pressure, with a 22.6% deviation for the nonlinear analysis, and NASTRAN predicted a buckling pressure with a 29.0% deviation for the linear analysis.

Table 3

Experimental buckling pressure [20].

ID	Buckling pressure (MPa)
	Test	Linear	Nonlinear	Nonlinear
		NASTRAN (error %)	MARC (error %)	ACOSwin (error %)
CTM1	0.60	0.677 (12.8%)	0.691 (15.2%)	0.671 (11.8%)
CTM2	0.51	0.670 (31.4%)	0.684 (34.1%)	0.664 (30.2%)
CTM3	0.55	0.749 (36.2%)	0.765 (39.1%)	0.695 (26.4%)
CTM4	0.55	0.742 (34.9%)	0.759 (37.9%)	0.687 (24.9%)
CTM5	0.55	0.722 (31.3%)	0.743 (35.1%)	0.669 (21.6%)
Average	0.552	0.712 (29.0%)	0.728 (32.0%)	0.677 (22.6%)

Table 4 shows the finite element buckling pressure used by ANSYS. The deviation was 20.8% and 15.6% for the linear and nonlinear analysis, respectively. Comparing the finite analysis results to the experimental buckling pressure, it is shown that ANSYS, applying the first-order shear-deformation shell element, predicted the buckling pressure most accurately.

Table 4

Finite element buckling pressure.

ID	Buckling pressure (MPa)
	Linear	Nonlinear
	ANSYS (error %)	ANSYS (error %)
CTM1	0.632 (5.30%)	0.602 (0.30%)
CTM2	0.627 (22.9%)	0.595 (16.9%)
CTM3	0.703 (27.8%)	0.674 (22.5%)
CTM4	0.696 (26.5%)	0.669 (21.6%)
CTM5	0.678 (23.3%)	0.650 (18.2%)
Average	0.667 (20.8%)	0.638 (15.6%)

Figure 4A shows the buckling mode shapes according to the finite element method. Both the linear and the nonlinear buckling analysis show identical buckling mode shapes: four waves in the circumferential direction and one long wave in the axial direction. The experimental buckling mode shapes are similar to the predicted shapes. As shown in Figure 4B (an inside view of the cylinder), two waves are clearly observable along the circumferential direction and one wave can be seen along the axial direction. However, the other two waves are not clear in the opposite part [20].

Figure 4:

Buckling mode shape for the composite cylinder (CTM3). (A) Predicted buckling mode shapes. (B) Experimental buckling mode shape [20].

5.2 Deflection response

In the buckling of the cylinders, the load-deflection curves can differ, depending on the reference nodes; therefore, in this study, the load-deflection responses in the axial and circumferential directions were extracted from three sampling points (O, P, and Q, respectively), as shown in Figure 1. The load-deflection responses in the axial and circumferential direction are shown in Figures 5–7. The magnitudes of deflection increase slowly before the buckling and accelerate sharply near the buckling pressure in the circumferential direction, while the deflections are almost linear in the axial direction before the buckling.

Figure 5:

Typical pressure vs. deflection curves at point O (CTM3). (A) Axial direction. (B) Circumferential direction.

Figure 6:

Typical pressure vs. deflection curves at point P (CTM3). (A) Axial direction. (B) Circumferential direction.

Figure 7:

Typical pressure vs. deflection curves at point Q (CTM3). (A) Axial direction. (B) Circumferential direction.

At point O, initially, the deflection in the axial direction is negative value and the magnitudes of deflection increase slowly before the buckling, then decrease suddenly near the buckling pressure, and then the deflection becomes positive finally. At point P, the deflections are negative throughout the process and the magnitudes are larger than those at point O. At point Q, the magnitudes of deflection in the axial direction are the largest among the sampling points. As for the circumferential direction, the largest deflections occur at the middle of the cylinder among the sampling points. Figure 8 shows the static deformation under componential pressure.

Figure 8:

Static deformations under componential pressure (CTM3).

5.3 Post-buckling and failure

The post-buckling behavior was also investigated in this study. ANSYS predicted that the cylinders can carry more pressure after a slight decrease in pressure and recovery of the supporting load. The buckling mode shapes kept four waves in the circumferential direction till the final failure (see Figure 9). As the deflection developed, local failure were observed. Figures 10–12 show the failure progress at points A, B, C, D, and E after the buckling, respectively. At point A, right after the buckling, as shown in Figure 10, the load factor ζ is lower than 1 from layer nos. 1 to 24, and no failure was found, which shows that the buckling pressure is lower than the layer failure pressure. As the deformation developed to point B, the first layer failure happened corresponding to layer no. 23. At point D, layer failure occurred mainly in the outer layers (nos. 18–24) and inner layers (nos. 1–7). Point E shows the final failure of the cylinder. As shown in Figures 10–12, within the neighboring layers, the failure that occurred in the 0° layer was more serious than that in the 90° layer at the inner layers and outer layers. The load factor ζ changed almost linearly slowly at the middle layers (nos. 8–17), and the last layer failure occurred at layer no. 15.

Figure 9:

Typical pressure vs. deflection curves of the composite cylinder (CTM3).

Figure 10:

The load factor ζ of each layer at points A and B.

Figure 11:

The load factor ζ of each layer at point C.

Figure 12:

The load factor ζ of each layer at points D and E.

6 Conclusion

Both the buckling behavior and the layer failure of composite laminated cylinders under external hydrostatic pressure were investigated in this study. Without considering the initial imperfection of the cylinders, the finite element model was validated to predict the buckling pressure with an average of 20.8% (linear) and 15.6% (nonlinear) deviation from experimental results. The analyses also simulated the post-buckling and layer failure successfully. The results show that finite element analyses with shell elements can be used to evaluate the buckling load of composite laminated cylinders subjected to hydrostatic pressure.