Designing sustainable barn-type houses: Optimal shapes for minimal envelope and energy use

TL;DR

The paper addresses reducing energy demand and construction costs by minimizing the external envelope area of barn-type houses under either fixed volume $V$ or fixed floor area $F$. It develops a rigorous analytical framework that yields closed-form expressions for optimal dimensions—width $W$, length $L$, and height $H$—as functions of roof slope $ abla$ (alpha), together with the minimal envelope area $S_{min}$ and a dimensionless compactness measure $S/S_{min}$. It also tackles the fixed-floor-area case via a cubic equation in $W$ with a Cardano-based solution and demonstrates the methods on three real-world houses, showing varying degrees of alignment with the theoretical optimum. The work provides two open-source MATLAB-based tools for practitioners to perform optimization analyses, enabling tangible improvements in design efficiency and energy performance, and it points to future extensions incorporating dynamic energy simulations, material choices, multi-objective optimization, and broader typologies.

Abstract

Barn-type houses have become one of the most popular single-family housing typologies in Poland and across Europe due to their simplicity, functionality, and potential for energy efficiency. Despite their widespread use, systematic methods for optimizing their geometry in terms of envelope area and energy performance remain limited. This paper develops a rigorous mathematical framework for determining the optimal proportions of barn-type houses with respect to minimizing the external surface area while satisfying constraints of either fixed volume or fixed floor area. Closed-form solutions for the optimal width, length, and height are derived as explicit functions of the roof slope, together with formulas for the minimal achievable surface. A recently introduced dimensionless compactness measure is also calculated, allowing quantitative assessment of how far a given design deviates from the theoretical optimum. The methodology is applied to case studies of three existing houses, showing that while some designs deviate significantly from optimal compactness, others already closely approximate it. The results confirm that theoretical optimization can lead to meaningful reductions in construction costs and energy demand. To support practical implementation, two original freely available software tools were developed, enabling architects and engineers to perform optimization analyses.

Figures (8)

• Figure 1: Typical shape of the barn-type house.
• Figure 2: Plot of the external surface $S(r,k)$ as a function of footprint aspect ratio $r$ and slenderness aspect ratio $k$. Here $V=300 \; m^3$ and $\alpha=\pi/6=30^\circ$. Red dot is the point corresponding to the minimal surface.
• Figure 3: Plots of the optimal parameters $W_{min}$, $L_{min}$ and $H_{min}$ as functions of $\alpha$. Here $V=300 \; m^3$.
• Figure 4: Plot of the compactness measure $\frac{S}{S_{min}}$ as a function of footprint aspect ratio $r$ and slenderness aspect ratio $k$. Here $\alpha=\pi/4=45^\circ$. Red dot is the point corresponding to the global minimum equal to 1.
• Figure 5: Contour plot of the compactness measure $\frac{S}{S_{min}}$ as a function of footprint aspect ratio $r$ and slenderness aspect ratio $k$. Here $\alpha=\pi/4=45^\circ$. Red dot is the point corresponding to the global minimum equal to 1.