Application of an Optimization Technique to Greenhouse Structural Designs

(I) The Algorithm of Design Technique and Some Results

Abstract

Greenhouses should be designed to meet requirements of construction costs, durability against loads and the possibility of providing suitable environment to plant growth. To get guides to design an optimal greenhouse from the above-mentioned three viewpoints, one of the optimization techniques, which is called the steepest descent method, is applied in this study. The direct light transmission into the greenhouse in winter is regarded as one of the most important environmental factors.
The steepest descent method gives the combination of the design variables which constitute the optimal (minimum or maximum) value in the objective function under some constraints. In this study the design variables were, for example, the shape of the house, the moment of inertia and crosssection features of each member and the ratch measure. The allowable stress or deflection of each member played a role of constraints in the course of the optimization. The objective function was the construction cost or the direct light transmission. In the steepest descent method, however, only one objective function is allowed. For this reason the following two steps were used. First the construction cost (J_s) is chosen as the objective function and is minimized. Then the inequality (1) is added to the constraints and the light transmission is maximized, where k is the coefficient of construction cost allowance and larger than 1.0, and J_sopt is the minimum construction cost.
Because in the steepest descent method all variables should be continuous and differentiable, some discontinuous variables, e.g. the number of purlins, were approximated with appropriate continuous functions. For the same reason any size and shape of steel member were assumed to exist. The crosssection features of the member are expressed with three independent variables on the basis of the empirical relations.
The optimizations were carried out under the conditions of a snow load of 30cm and a wind speed of 50m/s. The span was fixed at 9m. The latitude and the coefficient of construction cost allowance in the optimization of the light transmission were 34°N and 1.10, respectively. The following points became clear by the optimizations.
1) The steepest descent method can be applied to greenhouse design and the results show appreciable improvement both in the construction cost and in the light transmission in comparison with an example of commercial greenhouses.
2) The results of the optimization of the construction cost (cost optimization, hereafter) and the light transmission of the N-S oriented greenhouse (N-S optimization, hereafter) show a symmetric shape, whereas the optimal E-W oriented greenhouse has a ridge nearer to the south wall, so that the south roof slope becomes steeper.
3) Every optimization results in a ratch measuer of about 4m, which is wider than that of commercial greenhouses.
4) The moment of inertia of the rafters and the posts ranges from 150 to 290cm⁴. In the cases of the cost and N-S optimizations, the rafters have a larger moment of inertia than the posts. In the E-W optimization, the north rafter has an appreciably larger moment of inertia than the three others.
5) The cost optimization shows that members of the crosssection with larger depth and width are more suitable for the rafters and the posts, if the moment of inertia is the same. This holds true for the north members in the E-W optimization and for the posts in the N-S optimization. In the E-W oriented greenhouse, a member with shorter depth and larger section area and a member with shorter width are more suitable for the south rafter and for the south post, respectively.
6) In this study the strength of the glass was not taken into consideration. This resulted in very wide purlin intervals, which seem to be unrealistic, but this still proves that larger purlins with wider intervals are more suitable