Keywords

1 Introduction

Thin-walled cylindrical shells are extensively used as various industrial structures such as oil tanks, offshore platforms and silos, with these structures being susceptible to buckling when subjected to the hoop or axial compressive stresses. Fibre-reinforced polymer (FRP) composites are increasingly used in various structural applications, with the main reason for this being the high strength-to-weight and stiffness-to-weight ratios of FRP composites when compared to conventional structural materials. Several studies have been performed on the modelling and analysis of composite laminated cylindrical shells. The theories used in these analyses mainly extend the theories developed for isotropic shell models. Fully anisotropic laminated cylindrical shells have recently attracted the attention of researchers to achieve an optimum design of composite laminated shell structures [1,2,3,4,5,6,7].

In cylindrical shells, mechanical damages may occur due to the physical con-tact of shells with various vehicles travelling close to shell locations. Additionally, the overturning of adjacent shells can cause mechanical damage to the shells. Among various types of mechanical damage, the dent is regarded as the most crucial damage [8], which leads to local stress/strain concentration.

The cylindrical shells are subjected to external pressure when the shell is discharged; consequently, a vacuum is developed inside the shell [9]. Various researchers have investigated the buckling response of shell structures under uniform external pressure. Tsouvalis et al. [10] reported that initial imperfections had less effect on the buckling capacity of composite cylindrical shells subjected to external pressure when compared to counterpart cylindrical shells manufactured from isotropic materials. Considering the effects of nonlinear pre-buckling deformations and initial imperfections, Hui-Shen [11] proposed a boundary layer theory to predict the buckling and post-buckling response of anisotropic laminated thin-walled cylindrical shells under external pressure. Seong-Hwa Hur et al. [12] experimentally and numerically investigated the buckling and post-buckling behaviour of composite laminated cylindrical shells subjected to external hydrostatic pressure and showed the difference between numerical models and the associated experiments was limited to 15.5% if the initial imperfections were not considered. Lopatin [13, 14] proposed an analytical solution to predict the buckling capacity of composite cylindrical shells with various boundary conditions. Shen and Pan [15] reported considerable load-bearing capacity of geometrically imperfect composite cylindrical shells beyond the buckling when subjected to uniform external pressure.

There is a paucity of research concerning the effect of local geometric imperfections on the buckling behaviour of glass fibre-reinforced polymer (GFRP) cylindrical shells subjected to external pressure. This paper aims to numerically investigate the effect of local dents caused by external interferences on the buck-ling behaviour of GFRP cylindrical shells when subjected to external pressure. For this purpose, 24 finite element numerical models were developed in ABAQUS [16] then the effect of dent depth made at the 1/3rd, half, and 2/3rd of the shell on the buckling response of the cylindrical shells was investigated.

2 Finite Element Analysis

2.1 Finite Element Modeling

Four-node quadrilateral element (S4R) was used to model five layers of the composite cylindrical shells [14]. The layers of the composite shell were modelled considering the stacking sequence ([30/-30/30/-30/30]) of the composite material relative to the x-axis. The indenter was modelled as a discrete rigid surface, and a frictionless contact algorithm was used to simulate the contact between the cylindrical shell and the indenter. The dimensions of the indenter are shown in Fig. 1a [17].

Fig. 1
A 3 D view of the modeled indenter and the shell model. a. The width of a T-shaped indenter is 40 millimeters. The diameter of the U-shaped edge is 10 millimeters. b. A T-shaped indenter is fixed to the bottom of the cylindrical-shaped shell, with its related displacements.

Details of cylindrical shell and indenter models

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Convergence and mesh refinement studies were performed to ensure high accuracy in analysis, and a uniform mesh of 10 mm elements was used to model the shells. The bottom edge of the shell was constrained against vertical and radial displacements. The top edge of the shell was only restricted to radial displacements (see Fig. 1b). The shell edge was subjected to a uniformly distributed transverse load proportional to the external pressure [18]. The linear buckling analysis (Eigenvalue) and nonlinear Riks static analysis were carried out to analyse the stability of cylindrical specimens. In nonlinear buckling analysis, the load applied to the structure began from zero and gradually increased with a specific increment. Additionally, the Arc Length method was used to determine the length of the increments. Two steps are defined in the FE model to simulate the effect of dent and external pressure stages, namely:

(i) Linear general static analysis: Locally geometrical imperfection in the form of linear displacement of the indenter was applied in specific locations [19].

(ii) Riks analysis: External pressure was applied to the shell models in the form of a linear distributed load on the top edge of the cylindrical shell and uniform radial external pressure on the circumference of the shell model.

The mechanical properties of the composite material, including elastic modulus, Poisson’s ratio, and shear modulus in both longitudinal and transverse directions, were defined. To investigate the performance of composite materials, the linear elastic behaviour of undamaged materials, using Hashin theory which includes four failure modes, i.e., fibre tension and compression, matrix tension and compression, were modelled [20]. For the case of Hashin damage modelling, the beginning of shell damage is associated with the reduction in stiffness matrice of the shell.

Specifications of the Models

Twenty-four cylindrical shells were modelled, with all the shell models same geometrical specifications, i.e., a diameter of 400 mm, a height of 300 mm, and a wall thickness of 1 mm, which was composed of a five-layer composite laminate with a stacking sequence of [30/-30/30/-30/30]. The cylindrical shells were subjected to local denting load at 1/3rd, half, and 2/3rd of the shell height, where the dent depth was equal to 2, 4, 6, or 8 mm. The mechanical properties of GFRP composite are listed in Table 1. Each cylindrical shell model was assigned with a code “S-x-y-z”, where S means the cylindrical shell, x is the height of the indenter relative to the bottom edge of the shell (100, 150, or 200), y is the value of the indenter displacement (2, 4, 6, or 8 mm) and z is the orientation of the indenter relative to the axial axis of the shell (0° or 90°). For example, the S-100-2-0 refers to a shell model containing a 2 mm depth of dent at the height of 100 mm, where the dent was caused by an indenter that had no orientation relative to the shell axial axis.

Table 1 Mechanical properties of cylindrical shell material [21]
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3 Numerical Analysis and Discussions

In Fig. 2, the failure mode of S-100-4-90 and S-100-6-90 shell models are shown, where for both models the indenter location and the orientation was the same but the value of dent depth was different. Both models failed due to fibre compression mode [22], and the area of damage increased with increasing the dent depth. Additionally, for the case of S-100-6-90 model and beneath the indenter, the fibres’ damage value was approximately 80%. For a dent depth of 6 mm, placement of 6 mm for the indenter in the first step before applying of the external pressure, the fibers have been damaged by about 80% under the pressure caused by the indenter (see Fig. 3).

Fig. 2
2 contour plots of a cylindrical-shaped shell model with a T-shaped indenter at the bottom. It has 2 irregular oval-shaped failure modes in a gradient of colors. The dominant color is positive 7.846 e negative 01 at 4 millimeter depth, and positive 8.909 e negative 01 at 6 millimeter depth.

Failure of S-100-4-90 and S-100-6-90 shell models

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Fig. 3
2 contour plots of the cylindrical-shaped shell model with a T-shaped indenter at 0 and 90 degrees. It has 2 irregular oval-shaped failure modes in a gradient of colors. The dominant color is 7.220 e negative 01 at 90 degrees, and positive 8.550 e negative 01 at 0 degrees.

Fiber damage before applying external pressure

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In Table 2, the buckling capacity of the cylindrical shells with various dent depths and denting orientations are compared with the buckling capacity of counterpart cylindrical shells with no denting. The highest reduction in the buckling capacity of the shells was for a shell with a dent depth of 2 mm at the height of 100 mm (1/3rd of the shell height). Beyond 2 mm dent depth, the reduction in the buckling capacity of the shells was approximately constant. It can be deduced that a local dent in the proximity of the support had the largest effect on the buckling capacity of the shells (about 25%). Still, the depth of the local dent had a negligible impact in decreasing the buckling capacity of the shells.

Table 2 Comparison of buckling capacity of cylindrical shell models
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In Fig. 4, buckling capacities versus dent depth and indenter orientation at the mid-height of the shells (150 mm from the shell bottom) and at the 2/3rd of the shell height (200 mm from the shell bottom) are shown. The results indicated that the buckling capacity decreased by increasing the dent depth. The reduction for the case of shell models subjected to an indenter with an orientation of 90° was slightly more than the counterpart shell models with an indenter orientation of 0° (about 1%).

Fig. 4
2 stacked 3 D bar graphs compare buckling load and degree with indenter displacement. The maximum and minimum loads of around 31 kilopascals and 15 kilopascals are obtained at 90 degrees and 0 degrees, respectively.

Buckling capacity versus dent depth and indenter orientation

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4 Conclusions

This research aimed to numerically evaluate the effect of local depth caused by an indenter on the buckling behaviour of glass fabric-reinforced polymer (GFRP) cylindrical shells when subjected to external pressure. The key remarks of this research are listed below:

  • The failure mode of both models is fibre compression type, and the area of damage increased with increasing the value of dent depth.

  • For a dent depth of 8 mm and before applying the external pressure, the fibres were damaged by approximately 80% due to local stresses beneath the indenter.

  • The effect of indenter orientation on the buckling capacity of the cylindrical shell models was negligible.

  • Whilst the location of the dent depth significantly affected the buckling capacity of the cylindrical shells, the dent depth had a negligible influence on the shells’ buckling capacity.

  • The effect of dent depth on the reduction in the buckling capacity of the cylindrical shells increased from shell mid-height towards the 2/3rd of shell height.

  • By placing the indenter in the middle and near the upper edge, the buckling capacity decreases with the increasing of the amplitude imperfection.