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In this study the basic engineering principles, goals, and constraints are all combined with fuzzy methodology and applied to optimally design sandwich panel circular plate roofs for large vessels loaded statically and dynamically. These panels are made up of two stiff, strong veneer skins separated by vertical and peripheral stiffener plates. Advantages are high strength, lightweight, and sustainability. In the present approach, first the goals and constraints of the end user are identified and expressed as decision variables which are formulated using the engineering variables for materials, geometry, and function. Then same consistent fuzzy satisfaction functions are formed over the desired ranges to suit the customer's desires. The risk of extreme dynamic loadings exciting resonance is studied by natural frequency and mode analysis by FEM and analytical models. The results show the most critical locations and give guidelines for innovative remedies of the concept before detailed FEM analyses to finalize the design.

Sandwich panels with stiff light cores offer many advantages compared to solid products such as high bending and buckling load-bearing capacities and weight saving without sacrificing reliability. Especially in large industrial roofs, these properties promise benefits provided the structures are optimally designed also.

This case study is based on an industrial design task of a very large circular plate veneer sandwich roof.

The conventional analytical optimisation methods are cumbersome and not suitable for engineering design work as discussed by Gibson and Ashby [

In engineering optimisation at concept stage, most tasks are highly nonlinear, and fuzzy and design variables are few and discrete. This approach is used by Martikka and Pöllänen [

One future goal is to utilise biomimicry to copy the ingenious optimal designs in nature into technological products.

Basic mechanics is needed to understand and to utilise the sandwich design. Basic concepts for sandwich roofs are illustrated in Figures

Stress definitions. Material removal from the web can be activated at a defined area

Roof models for cylindrical vessels. (a) Model 1 with no centre support. (b) Model 2 with support. (c) Cone and sphere forms. (d) Stresses on face.

Geometry of a roof sector. Number of stiffeners is

Large circular roofs are commonly made as flat circular plates, cones, spherical, or some curved form as shown in Figure

Geometry definitions of a roof sector are shown in Figure

Design variables are classified into three main types.

Among these are loads, supports, reliability, and factors of safety. Load is external static pressure

Outer radius of roof is

Design variable options.

Variable | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

.200 | 0.22 | 0.24 | 0.26 | 0.30 | .34 | 0.36 | 0.40 | 0.45 | 0.50 | |

0.01 | 0.015 | 0.02 | .025 | 0.03 | ||||||

1 | 2 | 3 | 4 | — | ||||||

0.1 | 0.5 | 0.9 | — |

Material variables for wood and for veneer are shown in Table

Material selection data for optimisation.

E, MPa | R, MPa | Veneer, | Veneer, | ||||||
---|---|---|---|---|---|---|---|---|---|

35000 | 800 | 3500 | 80 | 10000 | 90 t, 43 c | 5000 | 50 | ||

8000 | 100 | 3500 | 80 | 500 | 4 | 5000 | 60 | ||

800 | 80 c,12 t | 3500 | 80 | 800 | 6 | 5000 | 20 | ||

2500 | 25 | 1300 | 30 | 650 | 7 | 2000 | 20 | ||

2500 | 25 | 1300 | 30 | 50 | 7 | 2000 | 20 | ||

2500 | 25 | 1300 | 30 | 650 | 7 | 2000 | 20 |

Here

The dominant stress state in the roof sandwich is the face normal stress. Bending stresses at two heights from neutral plane below and on top of the flange are

Now the boundary conditions for the roof are

Calculation of the eigenfrequencies is a difficult mathematical task. Thus some approximate modelling is useful to get fast rough answers. A sketch of a mode shape is illustrated in Figure

Sketch of an assumed lowest mode shape.

According to Soedel [

First a very simple function with two parameters is used to get reasonable estimates

Now the models by Soedel [

The physical meaning is that

In the concept stage the essential design variables are few, discrete, and their relationships are highly nonlinear. A fast enough search method is exhaustive learning search. Now all goals and constraints are formulated consistently by one flexible fuzzy function. This is illustrated in Table

Skewness parameter values.

a | b | c | d | e | |
---|---|---|---|---|---|

0.1 | 0.1 | 1 | 5 | 5 | |

5 | 0.1 | 1 | 5 | 0.1 | |

0.02 | 0.5 | 0.5 | 0.5 | 0.98 |

Principle of modelling of the general satisfaction functions.

In the design algorithm the satisfaction function is defined for each decision variable

All design variables are now discrete. A concept is formulated by making trial selections from the lists. Then decision variable and their satisfaction are calculated. Total satisfaction is calculated for each concept alternative. The one with maximal satisfaction is optimal.

The desired range for decision variable

A square plate surrogate model is used to model the buckling risk for the roof plates. A conservative model for the inelastic buckling strength is given by Blak [

The inelastic reduction factor for these edge support types is about

Material model and geometrical definitions.

The desired range for the decision variable is from small to 0.3. The bias favours small value of

Results are shown in Figure

Results of optimisation. The properties at each relative radius apply to the whole plate. Model 1 with no centre supports.

Relative radius | |

500, 30, 1 | |

243 | |

0.13 | |

0.93, 0.096 | |

0.57, 0.243 | |

0.76, 0.0052 | |

.86, 0.052, 260 | |

0.97, 0.072, 14 | |

0.38, 14000 | |

50, 13.6, 3.7 | |

60, 19, 3.15 | |

0.1, 20 |

500, 30, 3 | 500, 30, 3 | |

274 | 300 | |

0.053 | 0.056 | |

0.927, 0.06 | .97, 0.063 | |

0.5, 0.274 | 0.45, 0.3 | |

0.93, 0.044 | 1, 0.14 | |

0.54, 0.11 | 0.65, 0.09 | |

0.99, 0.18, 5.5 | .91, 0.43, 2.3 | |

0.21, 19410 | 0.21, 19410 | |

50, 10, 5 | 50, 2.6, 19 | |

60, 17, 3.5 | 60, 12, 5 | |

1.04, 20 | 2.8, 20 |

Results of optimisation. The properties at each relative radius apply to the whole plate. Model 2 with centre support.

Relative radius | |

500, 10, 3 | |

30 | |

0.49 | |

0.985, 0.054 | |

0.9982, 0.03 | |

0.757, 0.0052 | |

0.838. 0.05, 260 | |

0.85, 0.015, 65 | |

0.75, 6470 | |

50, 13, 3.12 | |

60, 4, 2.7 | |

0.1, 20 |

Two different models are considered in Figure

(a) Roof with no centre support, model 1. (b) Roof with centre support, model 2.

Total satisfaction

As shown in Figure

The cost is now defined as the cost of one sector with angle

Results are shown in Table

Model 1 with no centre support required roof total height

Model 2 with centre support required

In the final design still some detailed changes were made and also a change to slightly conical form.

Results are shown in Figures

(a) Deformed geometry and maximal principal stress range +7.5–−13.9 MPa. Minimal principal stress range 3.9–−33.4. Total deformation is at about

The first buckling mode with factor of safety 8.24. The plates with largest unsupported areas with high compressive stresses buckle first.

The first buckling mode for the whole plate model with factor of safety 8.24.

These are shown in Figures

The lowest eigen mode corresponding to eigenfrequency of 9.74 Hz. Analytical simple model gave roughly

The second eigen mode corresponding to eigenfrequency of 10.48 Hz. More complex models are shown in Figures

The second eigen mode corresponding to eigenfrequency of 13.01 Hz.

The third eigen mode corresponding to eigenfrequency of 16.89 Hz.

We have combined in this approach basic mechanics with fuzzy goal formulations and heuristics to obtain optimised sandwich panel roof of veneer. In nature, analogous ingeniously designed and biomanufactured structures are already widely and successfully used by animals and plants.

One evident advantage of the present method over the conventional ones is that more useful and reasonable optima can be found since all relevant desires like societal and ecological ones are considered simultaneously during the design. At this stage most of the total life cycle costs and sustainability are determined.

Dynamic methods are important tools to design structures with desired static and dynamic stability under extreme wind or seismic loads. The lowest frequencies by FEM were with rounding 10, 11, and 13 Hz. Analytical method gave the lowest as 11 Hz. The utility of the simple models is that they can be used in exploratory optimum concept design. The results show that a reasonable trade-off type optimum between two different constructions is obtained.

The future aim is to generalise this approach of combining basic mechanics with new goal formulations and FEM fine-tuning.

Using this design methodology designers of machines can better mimic the ingenious sustainable solutions found in nature and select the right future megatrends of technology in their design and development work.

The authors are grateful for the support to this research given by the companies Himtech Oy Engineering and Oy Scan Fibre Ltd.