A honeycomb sandwich structure is a typical lightweight and high-strength biomimetic structure [1
]. Based on the characteristics of a lightweight beetle forewing structure [5
], Chen et al. developed an integrated biomimetic honeycomb sandwich structure [7
], which has advantages, such as cementing free, single cast forming, and excellent mechanical properties [6
]. Currently, this plate is composed of reinforced basalt fibre epoxy resin composite material; the thickness of the honeycomb wall is approximately 2 mm [10
]. By comparison, the thickness of the honeycomb core wall of the aluminium alloy honeycomb sandwich structure is approximately 0.05 mm, which is approximately 1/40 of the former. The latter is more suitable for a large span spatial structure, which is especially sensitive to dead weight [11
]. To improve the overall stability and ultimate bearing capacity of single-layer aluminum alloy lattice shells, in this paper, the authors propose a new spatial structure system made from an aluminum alloy. The system consists of lightweight high-strength aluminum alloy honeycomb panels that are reliably connected to the single-layer aluminum alloy lattice shell by special connecting parts. This forms a new type of single-layer aluminum alloy combined lattice shell structure. In this structure, which is referred to as a new spatial combined lattice shell (NSCLS), the plates and rods support and reinforce each other [14
]. Our previous studies [15
] showed that the new spatial structure has characteristics, such as a large span, light dead weight, and high total rigidity. Because the honeycomb panels support and reinforce the structure, the overall stiffness and stable bearing capacity of the NSCLS structure are obviously higher than those of a single-layer aluminum alloy lattice shell structure with the same span. Under the same load conditions, the new system can sustain a structure with a larger span. For these reasons, it has good application prospects (Figure 1
In the study of single-layer aluminum alloy lattice shells, Hongbo Liu et al. [18
] analyzed with nonlinear finite method to clarify the stability performance of aluminum alloy single-layer latticed shell. The suggested values of rise/span ratio and initial imperfection were presented. The influencing coefficients of initial imperfection and material nonlinearity on stability bearing capacity were obtained. Sugizaki et al. [19
] tested four single-layer aluminum alloy spherical lattices using reduced-scale models. They studied the effects of three factors (high span ratio, load distribution pattern, and grid form) on the steady bearing capacity. The bearing capacity of the lattice shell with a mixed quadrilateral and triangular mesh was close to that of the one with an entirely triangular mesh. The influence of the nodal domain was taken into account in the finite element analysis [20
]. Xiaonong Guo et al. [21
] devoted to investigate the resistance of aluminum alloy gusset joint plates. It was found that the main collapse modes of AAG joint plates include the block tearing of top plates and the local buckling of bottom plates. Zechao Zhang et al. [22
] analyzed the performance of aluminum alloy single-layer latticed shell by considering the combination of dead loads, live loads, snow load, wind load, temperature effect, and other seismic dynamic loads. With the geometrical nonlinear finite element method, the stability coefficient and the weak parts of the structure were obtained when considering initial imperfections and material nonlinear. Gui Guoqing et al. [23
] used a numerical simulation to analyze the geometric nonlinear stability of an aluminum alloy lattice shell. They discussed the influence of the vector-to-span ratio, the load distribution, and the supporting condition on the stable bearing capacity of the lattice shell.
In the study of the honeycomb structure’s stability, Attard and Hunt [24
] considered the shear deformation of the panel and the core honeycomb layer. Using piecewise first-order beam theory, they analyzed the overall buckling of the honeycomb structure. Bourada M, Tounsi A et al. [11
] studied the instability of a honeycomb core with a uniform wall thickness under uni- and bidirectional in-plane compression and out-of-plane compression. Using experimental studies and numerical simulations, A. Boudjemai, R. Amri et al. [25
] studied the macroscopic deformation and plastic instability of a honeycomb structure with uniform wall thicknesses under unidirectional in-plane compression.
It is obvious that the existing studies focus on the aluminum alloy lattice shell structure or the honeycomb panel itself. Almost no studies of a combined aluminum alloy lattice shell structure, in which an aluminum alloy honeycomb panel reinforces the lattice shell have been conducted. There are some topics that should be addressed, such as the stability of these relatively lightweight and soft combined lattice shells and the difference between its unstable failure mode and that of ordinary single layer lattice shells that lack aluminum alloy. In this paper, the authors conducted experimental studies on the stability of this new structure to defects to investigate the structure’s destabilization failure mechanism and the effect of geometric imperfections on stability. The authors also verified the reliability of the finite element method. The results of this paper could provide a theoretical basis for making and revising the design codes for this new combined lattice shell structure.
3. Analysis of the Nonlinear Stability of NSCLS to Defects
3.1. Basic Assumptions
In this paper, we make the following two basic assumptions when analyzing the stability of single-layer aluminum alloy lattice shells to defects using the random defect mode method. (1) The initial error in each node follows a normal probability density function within the double mean-square deviation. That means that the random variable that was representing the initial error along the X-, Y-, and Z-axes is RX/2, where the random variable X is normally distributed. R is the maximum initial error, and the error of the random variable is within [−R, R]. (2) The random variables representing the initial error of each node are independent of each other.
3.2. Analytical Methods
First, we introduce the initial error into each node in the model to create a computational model with stochastic defects; then, a nonlinear stability analysis is performed to determine the stable bearing capacity. This value is taken as a sample in the stable bearing capacity’s sample space. Repeat the above steps several times to obtain multiple stable bearing capacities. In this way, a stable bearing capacity sample space containing N samples is created. Finally, based on the theory of probability and mathematical statistics, we perform a statistical analysis of the calculated steady bearing capacity of the samples to determine the final stable bearing capacity of the structure.
In the finite element analysis, Beam 188 element is simulated aluminium alloy bars and Shell 181 element is simulated aluminium alloy plates, while contacted elements are used between bars and plates. The boundary constraint condition of the model is fixed support. The material properties of aluminum alloy are as follows: elastic modulus E = 70 GPa, shear modulus G = 27 GPa, poisson’s ratio μ = 0.33, volume weight ρ = 2.65 × 104 N/m3.
3.3. Experimental Verification of Random Defect Mode Method
Before analyzing the nonlinear stability of the single-layer aluminum alloy lattice shells using the stochastic defect method, it is necessary to know the probability characteristics of each random variable, including its distribution and statistical parameters. This means that we need to determine the distribution of the initial geometric imperfections, and its mean, mean square deviation, and defect radius R (the absolute value of the maximum defect size), as well as other statistical parameters.
In this paper, we assume that the initial geometric imperfections are normally distributed. Due to the limitation of the nodal position deviation, we use a truncated Gaussian distribution. This is essentially a normal distribution, but only two truncation limits are set based on the normal distribution. Therefore, the input parameters for the distribution of the random variables that need to be determined are as follows: the mean μ, the standard deviation σ, the correlation coefficient ρ, and the truncation limits Xmin and Xmax. Because of the limited length of this paper, only two typical models of the six models are discussed (i.e., model A1, the lattice shell with no aluminum plates or defects, and model B2, the combined lattice shell model with defects).
3.3.1. Results for Model A1, the Defect-Free Lattice Shell Model with No Aluminum Plate
During the analysis, the authors measured the cross-sectional dimensions at the middle and at both ends of all the aluminum alloy bars. The average of the three measured cross-sectional dimensions was used for finite element modeling. In analyzing the ultimate bearing capacity of the random defect mode, as per the relevant literature, the mean was μ = 0 and the discrete coefficient was Cd = 0.423 (the average of the statistical results of the three coordinate components). The cutoff limit of the maximum amplitude of the deviation was Xt = 1.4 mm (1/1000 of the model’s span), the maximum correlation coefficient was ρmax = 0.65, and the number of random samples was N = 200.
a,b show the history and mean square deviation of the critical load samples for model A1
, respectively. The blue curve in the middle of Figure 7
b plateaus after the number of samples N exceeds 100. This indicates that the mean and mean square deviation of the critical load samples stabilize after the number of samples N exceeds 100. The red curves on the upper and lower sides are the boundaries at the 95% confidence level. The area between the curves in the graph decreases in size, which indicates that the accuracy of the history curve increases. Therefore, selecting 100 samples in practical engineering makes it possible to meet the accuracy requirements under normal conditions. The calculated results are reliable.
After using finite element analysis to analyze 200 samples with defects, we obtain a curve for the critical load samples (Figure 8
a). Using the Gaussian method (if the random variable follows a Gaussian distribution, the probability distribution function is a straight line), we plot the probability distribution of the critical load samples (Figure 8
b). The middle section of the blue curve is somewhat close to a straight line. We check that the critical load samples are normally distributed by evaluating the fit. The normal distribution assumption is acceptable at the 5% significance level.
The critical load samples are statistically analyzed and compared with the experimental results. The results are listed in Table 1
. The critical loads Pcr
in the table are determined by following the “3σ” principle (i.e., a 99.87% guarantee rate).
shows that, according to the random defect mode analysis of the lattice shell with no aluminum plate, the predicted theoretical failure pattern (Figure 9
b) is very close to the experimental failure pattern (Figure 9
3.3.2. Analysis of the Combined Lattice Shell with Defects (Model B2)
In model B2, the authors artificially set the geometric deviation of the node coordinates, i.e., we reduced the Z coordinate of the top of the model by 5.6 mm (1/250 of the combined lattice shell’s span).
As for model A1, we conducted a finite element analysis of 200 samples using stochastic defects to obtain curves for the critical load samples and graphed the probability distributions of the critical load samples using the Gaussian method.
The critical load sample were statistically analyzed and compared with the experimental results. The results are listed in Table 2
. The critical load values Pcr
in the table were determined according to the “3σ” principle (i.e., 99.87% guarantee rate).
shows that, according to the random defect mode analysis of the combined lattice shell, the predicted theoretical failure pattern (Figure 10
b) is very close to the experimental failure pattern (Figure 10
shows that the results of the finite element analysis of the aluminum alloy lattice shell model obtained using the random defect mode method are very close to the experimental values. In the finite element analysis, the mean critical load and three times the mean square deviation of the standard values are accounted for. The average error of the six models is 5.7%. This can be attributed to the small discrete coefficient and defect amplitude and to the use of a truncated normal distribution for the random defects (i.e., a truncated Gaussian distribution) in this paper. It is apparent that the testing load Pe
of the critical load is between Pmax
4. The Stability of NSCLS for Different Defect Sizes
The current Chinese “Technical specification for spatial lattice structures” [27
] stipulates that when analyzing the stability of a lattice structure, the maximum initial geometric imperfection size (i.e., the initial deviation of the surface shape) should be set to 1/300 of the lattice shell’s span. However, in this paper, the authors study a new type of single-layer aluminum alloy honeycomb combined lattice shell structure, and we find that the critical load obtained using the uniform defect mode is often not the smallest critical load. This is because the lowest-order buckling mode is frequently unable to characterize the most unfavorable distribution of defects in the structure. Therefore, the current specification of defect sizes may no longer be applicable. Therefore, it is necessary to investigate a reasonable maximum defect size for this type of structure. In this paper, we used the stochastic defect mode method to study the stable bearing capacity of a single-layer aluminum alloy lattice shell structure. To study the influence of different geometric defect sizes and plate thicknesses, the maximum defect sizes ranged from L/2000 to 16L/2000 (a total of 16 parameters), where L was the span of the combined lattice shell structure. The plate thickness was set to 0.5 mm, 1.0 mm, 1.5 mm, 2.0 mm, 2.5 mm, and 3.0 mm. The geometric dimensions and material properties that were used in the computational model were exactly the same as they were in the test model. A nonlinear finite element analysis was performed for lattice shell models with six plate thickness and 16 geometric defect sizes. Calculations were performed for 200 random samples (in total, 6 × 16 × 200 = 19,200 samples).
By processing and analyzing the nearly 20,000 samples discussed above, the authors determined the critical load Pcr
by following the “3σ” principle. Figure 11
shows the results. The thickness of the plate and the standard bearing capacity of the structure for different defect sizes are comprehensively accounted for.
shows that the single-layer aluminum alloy combined lattice shell structure is sensitive to defects. As the defect size increases, the stable bearing capacity of the structure obviously decreases. When the defect size is small, i.e., below 6L/2000, the stable bearing capacity of the structure changes approximately linearly with the defect size. The bearing capacity does not change significantly. However, when the defect size exceeds 6L/2000, the stable bearing capacity of the structure decreases nonlinearly with the defect size. The decreasing slope becomes steeper as the defect size continues increasing. Therefore, the authors recommend that when designing NSCLS in the future, the maximum size of the initial geometric imperfections should be held below 3/1000 of the span of the lattice shell.