Hiperbolik paraboloid betonarme kabuk sistemlerin parametrik tasarımı
Parametric design of hyperbolic paraboloid reinforced concrete shell systems
- Tez No: 995056
- Danışmanlar: PROF. DR. KUTLU DARILMAZ
- Tez Türü: Yüksek Lisans
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2026
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Lisansüstü Eğitim Enstitüsü
- Ana Bilim Dalı: İnşaat Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
Hiperbolik paraboloid (hypar) kabuk yapılar, düz doğruların birleşimiyle oluşan çift eğrilikli, antiklastik yüzeylerdir. Görsel olarak karmaşık bir geometriye sahip olsalar da yapısal açıdan oldukça kararlı ve ekonomik davranış gösterirler. İnce kabuk formunda tasarlandıklarında büyük açıklıkları çok az malzeme kullanarak geçebilmeleri, onları hem mühendislik açısından hem de mimari açıdan dikkat çekici kılmaktadır. Yüzeyin çift eğriliği, yükleri büyük oranda kendi düzlemi içinde; çekme, basınç ve kesme kuvvetleriyle taşımasına olanak tanır. Böylece eğilme etkileri azalır ve yapı yüksek rijitlik kazanır. Bu araştırmanın amacı, hiperbolik paraboloid betonarme kabuk yapıların mimari ve yapısal açıdan en uygun geometrik özelliklerini parametrik çalışma ile belirlemektir. Çalışma, Félix Candela'nın Los Manantiales ve Bacardi Rum Fabrikası gibi yapılarından esinlenerek geliştirilmiş bir örnek model üzerinde yürütülmüştür. Model, keşişen iki eğrisel kenarlı hiperbolik paraboloid kabuktan oluşturulmuş olup tasarımı etkileyen temel değişkenler gözetilerek modellenmiştir. Tasarım aşamasında belirlenen değişkenler; taşıyıcı ayaklar arasındaki açıklık, kabuğun kenar yüksekliği, merkez yüksekliği ve kenar çıkmasıdır. Bu değişkenler türetilerek farklı tasarım alternatifleri üretilmiş ve en uygun formun belirlenmesinde kullanılabilecek yöntemin geliştirilmesi amaçlanmıştır. Kabuk modelin geometrisinin oluşturulma sürecinde Rhinoceros 3D programı kullanılmış olup Grasshopper eklentisi yardımıyla parametrik olarak modellenmiştir. Yapısal analizler ise SAP2000 yazılımı kullanılarak gerçekleştirilmiştir. Çalışma kapsamında; model değişkenlerinin tanımlanması, farklı modellerin üretilmesi, statik analiz için sonlu eleman ağ (mesh) yapısının oluşturulması, statik analizlerin gerçekleştirilmesi, betonarme tasarım yapılması ve sonuçların karşılaştırılması aşamaları izlenmiştir. Bu yöntem, tasarım sürecinde form ve performansın eş zamanlı olarak değerlendirilmesine olanak tanımıştır. Elde edilen bulgular, parametrik analiz araçlarının kabuk yapıların konsept tasarım aşamasında güçlü bir araştırma aracı olduğunu göstermektedir. Küçük geometrik değişiklikler bile kabuğun iç kuvvet dağılımı ve yer değiştirme davranışında önemli farklar yaratmaktadır. Sabit kalınlıkta modellenen örnek kabuk sistem, Félix Candela'nın çalışma için örnek alınan değişken kalınlığa sahip tasarımlarından farklı olsa da parametrik analiz süreci geometrik etkilerin anlaşılmasına katkı sağlamıştır. Sonuç olarak, hiperbolik paraboloid kabuklar yüksek estetik değere ve yapısal verimliliğe sahip olsalar da hassas geometrik uygulama gereksinimi, karmaşık kalıp sistemleri ve uzmanlık gerektiren işçilik sürecine ek olarak ince ve eğrilikli kesitlerde kalınlık kontrolünün zor olması bu yapı türünün günümüzde kullanımını sınırlamaktadır. Bununla birlikte, formda yapılacak küçük optimizasyonlar bile yapısal performansta önemli iyileşmeler sağlayabilmektedir. Bu çalışma, 3B modelleme ve yapısal analiz yazılımlarının erken tasarım aşamasında entegrasyonunun önemini vurgulamakta ve gelecekte değişken kalınlıklı modelleme, malzeme optimizasyonu ve gelişmiş üretim teknikleriyle bu yapıların uygulanabilirliğinin artırılabileceğini önermektedir.
Özet (Çeviri)
The hyperbolic paraboloid, often abbreviated as“hypar”represents one of the most intriguing and efficient structural forms in modern architecture and engineering. Characterized by its double curvature and anticlastic geometry, the hypar surface is generated through the intersection of two families of straight lines, which makes it a ruled surface. This seemingly complex geometry offers remarkable simplicity in construction, particularly because it can be formed using straight formwork instead of costly curved molds. The hypar's unique combination of geometric elegance and structural efficiency became a hallmark of mid-20th-century architectural design, influencing numerous pioneers who sought to merge aesthetic freedom with engineering logic. The double curvature of a hyperbolic paraboloid provides inherent structural rigidity and allows large spans to be achieved using surprisingly thin shell elements. As a surface structure, its thickness is negligible compared to its other dimensions, placing it within the broader category of thin-shell and plate structures. These forms are inherently material efficient, as their geometry enables them to transfer loads primarily through in-plane tension, compression, and shear forces, minimizing bending stresses. The result is a lightweight yet strong structure capable of spanning vast spaces with a minimal quantity of material. When properly designed, the hyperbolic paraboloid distributes loads evenly across its surface and resists deformation efficiently under uniform or asymmetrical loading conditions. The architectural and structural interest in this geometry extends beyond its mechanical advantages. Hyperbolic paraboloids create visually captivating spaces with dynamic curvature, producing light and shadow effects. This dual quality, expressive form combined with structural logic, captured the imagination of architects and engineers such as Félix Candela, Heinz Isler, Pier Luigi Nervi, Eduardo Torroja, Anton Tedesko, Bernard Lafaille, and Sergio Musmeci. Their experimental works during the twentieth century demonstrated how mathematical geometry could be transformed into poetic architectural expression. Among them, Félix Candela stands out as one of the most influential figures. His extensive experimentation with thin reinforced concrete shells produced some of the most iconic hypar structures in architectural history, including Los Manantiales in Xochimilco and the Bacardi Rum Factory in Mexico. Before Candela's groundbreaking free-edge designs, most shell structures relied on thickened edge beams or ribs for stability. Hypar shells inherently generate strong in-plane forces normal to their edges, but due to their slender cross sections, they offer little resistance in that direction. Candela's innovations lay in exploiting the geometric strength of the double curvature itself, thereby eliminating edge ribs and allowing for remarkably thin and elegant free edge vaults. Throughout his career, he explored a wide range of plan configurations, demonstrating the adaptability of the hypar form to different spatial and structural requirements. His Los Manantiales restaurant in Xochimilco remains one of the finest examples of this approach, celebrated for its aesthetic clarity and structural daring. Building upon this legacy, the present research aims to determine the optimal geometric characteristics of a reinforced concrete hypar shell in both architectural and structural terms. The study focuses on the generation of a parametric model inspired by Candela's design principles and analyzes how variations in certain parameters influence structural performance. Four primary geometric parameters were defined: support span, edge height, center height, and edge overhang. By systematically altering these parameters, a series of design alternatives were generated to identify the most efficient configuration that balances aesthetics, material economy, and structural performance. The methodology combines parametric modeling and finite element structural analysis. The parametric model was created in Rhinoceros 3D using the Grasshopper plug-in, which allowed the definition of variable geometric relationships and real time modification of the shell form. This digital approach enabled the exploration of numerous design scenarios with precise control over each parameter. The structural analyses were performed using SAP2000, a commercial software developed by Computers and Structures Inc., which allowed for accurate evaluation of stress distribution, displacement, and internal force behavior. The research workflow followed five main steps; defining the geometric parameters to form the three-dimensional model, generating multiple model variations according to parameter values, creating mesh structures for numerical analysis, performing structural analyses and design under both uniform and earthquake loading conditions, and finally comparing the results to identify the most suitable and economical configuration. Through this process, the study demonstrates the effectiveness of parametric analysis tools in bridging the gap between architectural design and structural optimization. The ability to instantly visualize and test multiple alternatives supports a more informed design process, where aesthetic and structural criteria can evolve together rather than sequentially. This approach mirrors the design philosophy of pioneers like Candela and Frei Otto, who emphasized form-finding as a way of discovering structural logic through geometry rather than imposing it afterward. In the generated models, the shell thickness was treated as a fixed parameter to isolate the influence of the four main geometric variables. While this differs from Candela's approach, who often varied shell thickness according to stress distribution, the constant-thickness assumption simplified the comparative analysis and provided a consistent basis for evaluating the geometric parameters. The findings reveal that even small adjustments in edge or center height significantly influence the internal stress patterns and deflection behavior of the shell, demonstrating the sensitivity of hypar geometry to dimensional changes. Beyond technical results, the study reinforces the architectural potential of hyperbolic paraboloids as expressive and efficient spatial forms. The fluid curvature and lightness of these shells produce visually striking interiors, where the structure itself becomes the primary architectural feature. However, despite their efficiency and aesthetic appeal, hypar roofs present practical construction challenges. Their thin reinforced concrete shells demand highly precise formwork and skilled labor. Achieving accurate curvature, maintaining uniform thickness, and controlling surface smoothness during casting all require careful execution on site. These demanding construction tolerances partly explain the decline in the use of such structures in contemporary practice, as newer spanning systems, such as steel space frames, tensile membranes, or prefabricated concrete elements, offer simpler assembly and lower labor requirements. Nevertheless, the potential for optimization remains significant. Small refinements in geometry can lead to notable improvements in mechanical performance, including reduced strain energy, lower self weight, and more uniform stress distribution. Previous studies have shown that parametric optimization can be guided by objective functions such as minimizing total deformation, controlling tensile stress concentration, or maximizing material efficiency. The integration of computational design tools into this research continues that lineage, offering a systematic framework for evaluating complex shell geometries at the conceptual stage. The outcomes of this study confirm that three-dimensional parametric modeling combined with structural analysis software provides a powerful means of enhancing both design quality and performance. Designers can visualize and modify form while immediately assessing the structural consequences, allowing for informed decisions early in the design process. This iterative feedback loop between geometry and performance promotes not only structural optimization but also the creative exploration of architectural expression. In conclusion, while the construction of hyperbolic paraboloid shells presents practical difficulties, their potential as lightweight, elegant, and materially efficient structures remains undeniable. The research demonstrates that the integration of parametric modeling and structural analysis enables a rational exploration of design space, producing forms that are both aesthetically refined and structurally sound. The study thus contributes to the ongoing dialogue between digital design tools and traditional structural logic, reaffirming the relevance of thin concrete shells in contemporary architectural practice. Future research may extend this work by incorporating variable thickness modeling, material optimization, and advanced fabrication techniques such as robotic formwork or 3D-printed molds. These developments could mitigate many of the construction challenges historically associated with hypar structures, making them more feasible in today's building industry. Additionally, further exploration of performance-based design criteria, such as environmental response, daylight optimization, and acoustic behavior, could expand the architectural versatility of these geometries. Ultimately, this study underscores that the essence of shell design lies in the synthesis of form, structure, and material. The hyperbolic paraboloid, with its elegant simplicity and geometric complexity, remains a prime example of how architectural beauty and engineering efficiency can coexist. By leveraging modern computational tools, designers can rediscover the potential of these iconic forms, adapting them to new materials, technologies, and design aspirations for the twenty-first century.
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