# The Stability of New Single-Layer Combined Lattice Shell Based on Aluminum Alloy Honeycomb Panels

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Study on Stability of NSCLS

#### 2.1. The Design and Fabrication of the Test Model

_{1}, A

_{2}, and A

_{3}), and the other three were lattice shell models with reinforcement (B

_{1}, B

_{2}, and B

_{3}). Figure 2a–d show the configurations of the models. The model plane was a regular hexagon. The diameter was 1400 mm, and the vector height was 250 mm (as shown in Figure 2g,h). The models were made by assembling 90 bars, 37 nodes, and 54 triangular plates together. The bars were square aluminum alloy tubes. The side length was 20 mm, and the wall thickness was 2 mm. The bars were connected by circular disc node connectors 72 mm in diameter.

_{2}and B

_{2}). The method was to reduce the Z coordinate of the top of the model by 5.6 mm (i.e., 1/250 of the lattice shell’s span).

#### 2.2. Loading and Testing Methods

#### 2.3. Results and Discussion

#### 2.3.1. Failure Patterns of the Test Models

#### Failure Pattern of the Lattice Shell Model with No Aluminum Alloy Plate

#### The Failure Mode of the Combined Lattice Shell Model

#### 2.3.2. Load-Displacement Curve of the Test Model

_{2}) is approximately 7% lower than the capacity of those without defect (A

_{1}and A

_{3}). In the combined lattice shell model, the bearing capacity of the model with defects (B

_{2}) is 12% less than the capacity of those without defects (B

_{1}, B

_{3}).

## 3. Analysis of the Nonlinear Stability of NSCLS to Defects

#### 3.1. Basic Assumptions

#### 3.2. Analytical Methods

^{4}N/m

^{3}.

#### 3.3. Experimental Verification of Random Defect Mode Method

_{1}, the lattice shell with no aluminum plates or defects, and model B

_{2}, the combined lattice shell model with defects).

#### 3.3.1. Results for Model A_{1}, the Defect-Free Lattice Shell Model with No Aluminum Plate

_{d}= 0.423 (the average of the statistical results of the three coordinate components). The cutoff limit of the maximum amplitude of the deviation was X

_{t}= 1.4 mm (1/1000 of the model’s span), the maximum correlation coefficient was ρ

_{max}= 0.65, and the number of random samples was N = 200.

_{1}, respectively. The blue curve in the middle of Figure 7b 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.

_{cr}in the table are determined by following the “3σ” principle (i.e., a 99.87% guarantee rate).

#### 3.3.2. Analysis of the Combined Lattice Shell with Defects (Model B_{2})

_{2}, 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).

_{1}, 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.

_{cr}in the table were determined according to the “3σ” principle (i.e., 99.87% guarantee rate).

_{e}of the critical load is between P

_{max}and P

_{min}.

## 4. The Stability of NSCLS for Different Defect Sizes

_{cr}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.

## 5. Conclusions

- (1)
- By precision processing six models using a CNC machine and accounting for the initial geometric defects in the aluminum alloy lattice shell models, the authors performed a stability comparison. The results show that the overall stiffness and stable bearing capacity of the lattice shell are remarkably improved due to the aluminum alloy reinforcing plate. Regardless of whether there are geometric defects, the steady bearing capacity of the new combined lattice shells is approximately 16% higher than that of a lattice shell with the same span and no reinforcing plate. The magnitude of the increase for the lattice shell model with no defects is higher than that of the model with defects.
- (2)
- The NSCLS is a defect-sensitive structure. The influence of geometric defects on its stable bearing capacity is very obvious. The results of comparing the lattice shell models show that the sensitivities of the two types of structures are different. The bearing capacity of the defective model with no plate is approximately 7% lower than that of the model without defects. In the combined lattice shell model, the bearing capacity of the model with defects is 12% lower than that of the model without defects.
- (3)
- The finite element analysis results of applying the random defect mode method show that the theoretical failure patterns of the experimental models are basically consistent with those that were measured in the tests. The average difference between the theoretical stable bearing capacity and the experimental value is 5.7%. The theoretical load-displacement curves are also very close to the ones that were obtained in the tests. This indicates that the random defect mode method with a truncated Gaussian distribution is reasonable and reliable. It has sufficient accuracy. It can be used in structural analysis and design in practical engineering.
- (4)
- As the lowest-order buckling mode is unable to characterize the most unfavorable distribution of defects in the new structure, the critical load obtained using the uniform defect mode is frequently not the smallest critical load. Therefore, the existing specification of defect sizes can no longer be applicable. By calculating and analyzing nearly 20,000 NSCLS, we find that after the initial geometric defect value of the combined lattice shell reaches 3/1000 of the shell’s span, the stable bearing capacity decreases sharply. We recommend that the values be used as the maximum defect size for the combined lattice shell. The studies in this paper provide a theoretical basis for future design specifications for new composite lattice shell structures.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematic diagram of a large-span single-layer aluminum alloy honeycomb panel combined lattice shell: (

**a**) The overall structure; (

**b**) A local area, magnified.

**Figure 2.**Overview of the test model: (

**a**,

**b**) Lattice shell model without aluminum plate (A

_{1}, A

_{2}, and A

_{3}); (

**c**,

**d**) Combined lattice shell model (B1, B2, and B3); (

**e**) Node in the middle; (

**f**) Supporting node; (

**g**) Plan view of the model; and, (

**h**) Side view of the model.

**Figure 3.**The strain gauge layout and the testing site: (

**a**) Strain gauge locations; (

**b**) Rosette strain gage locations; and, (

**c**) The testing site.

**Figure 4.**Failure patterns of the lattice shell model without aluminum alloy plates: (

**a**) The overall failure profile; (

**b**) The amplified profile of a local destabilization area.

**Figure 5.**Failure modes of the combined lattice shell model: (

**a**) The overall failure profile; (

**b**) The amplified profile of a local destabilization area.

**Figure 6.**Load-displacement curves at the loading points of the test models: (

**a**) Curves for the lattice shell models without reinforcing plates; (

**b**) Curves for the combined lattice shell models; and, (

**c**) Curves for the six test models.

**Figure 7.**The mean and mean square deviation of the critical load samples for model A

_{1}: (

**a**) Mean; (

**b**) Mean square deviation.

**Figure 8.**History and probability distribution for the critical load of model A

_{1}: (

**a**) History; (

**b**) Probability distribution according to the Gaussian method.

**Figure 9.**Comparison of the failure patterns of the test model with no plate (A

_{1}): (

**a**) Measured failure profile; (

**b**) Theoretical failure profile.

**Figure 10.**Comparison of the failure modes of the combined lattice shell, model B

_{2}: (

**a**) Measured failure pattern; (

**b**) Theoretical failure pattern.

**Figure 11.**The critical load for different defect sizes for the lattice shell models with different plate thicknesses: (

**a**) Model with a plate thickness of 2.5 mm; (

**b**) Model with a plate thickness of 3.0 mm; and, (

**c**) Combined lattice shell models with different plate thicknesses.

**Table 1.**Comparison of the results obtained using the random defect mode analysis and the measured values for A

_{1}.

P_{e}/N | D_{max}/mm | C_{d} | P_{max}/N | P_{min}/N | P_{μ}/N | P_{σ}/N | P_{cr} |
---|---|---|---|---|---|---|---|

4879 | 1.4 | 0.423 | 5376 | 4617 | 4962 | 152 | 4506 |

_{e}—The test value of the critical load; D

_{max}—The maximum defect size; C

_{d}—The discrete coefficient; P

_{max}—The maximum of the critical load samples; P

_{min}—The minimum of the critical load samples; P

_{μ}—The mean of the critical load samples; P

_{σ}—The mean square deviation of the critical load samples; P

_{cr}—The standard value of the critical load (3 times the mean square deviation is accounted for).

**Table 2.**Comparison of the results obtained from the random defect mode analysis and the measured values for B

_{2}.

P_{e}/N | D_{max}/mm | C_{d} | P_{max}/N | P_{min}/N | P_{μ}/N | P_{σ}/N | P_{cr} |
---|---|---|---|---|---|---|---|

5058 | 1.4 | 0.423 | 6210 | 3787 | 5256 | 220 | 4596 |

**Table 3.**Comparison of the critical load of the lattice shell with no plate and the combined lattice shell.

Model Configuration | Lattice Shell with No Plate | Combined Lattice Shell | ||||
---|---|---|---|---|---|---|

Experimental Value (N) | Simulated Value (N) | Error (%) | Experimental Value (N) | Simulated Value (N) | Error (%) | |

Without defects | 4879 | 4506 | 7.6 | 5761 | 5947 | 3.2 |

4601 | 2.1 | 5556 | 6.6 | |||

With defects | 4652 | 4305 | 7.5 | 4958 | 4596 | 7.3 |

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**MDPI and ACS Style**

Zhao, C.; Zhao, Y.; Ma, J.
The Stability of New Single-Layer Combined Lattice Shell Based on Aluminum Alloy Honeycomb Panels. *Appl. Sci.* **2017**, *7*, 1150.
https://doi.org/10.3390/app7111150

**AMA Style**

Zhao C, Zhao Y, Ma J.
The Stability of New Single-Layer Combined Lattice Shell Based on Aluminum Alloy Honeycomb Panels. *Applied Sciences*. 2017; 7(11):1150.
https://doi.org/10.3390/app7111150

**Chicago/Turabian Style**

Zhao, Caiqi, Yangjian Zhao, and Jun Ma.
2017. "The Stability of New Single-Layer Combined Lattice Shell Based on Aluminum Alloy Honeycomb Panels" *Applied Sciences* 7, no. 11: 1150.
https://doi.org/10.3390/app7111150