Comparison between Isostatic and Hyperstatic Structures

Abstract

This article presents a comparison between isostatic and hyperstatic structures, based on the analysis of beams subjected to identical loads. A theoretical description of both types of structures is provided, followed by the practical analysis of two models – a simply supported beam (isostatic) and a continuous beam with three supports (hyperstatic). Through calculations and internal force diagrams, the differences in reactions, bending moments, and structural behavior are evaluated. The study aims to deepen the understanding of these concepts among civil engineering university students.

Introduction

The classification of structures as isostatic (statically determinate) or hyperstatic (statically indeterminate) is a key concept in structural analysis. Isostatic structures can be solved using only equilibrium equations, while hyperstatic structures require additional deformation compatibility conditions due to excess constraints. This difference influences their stiffness, load redistribution capacity, and response to load changes.

Models Compared:

• Isostatic Beam (Beam A): Simply supported 6 m beam with uniform load; reactions and internal forces are straightforward to calculate.

• Hyperstatic Beam (Beam B): Continuous 6 m beam with three supports (two simple and one fixed), requiring computational tools (e.g., Ftool) to solve for forces and deformations due to redundancy.

Findings:

• The isostatic beam shows higher peak bending moments (45 kNm) concentrated at mid-span and larger deflections.

• The hyperstatic beam distributes internal forces more evenly with both positive and negative bending moments (~22.5 kNm), resulting in greater stiffness, lower deflections, and improved load redistribution.

Advantages and Disadvantages:

• Isostatic Structures are easier to analyze and construct with predictable behavior but lack redundancy, leading to higher peak stresses and vulnerability to support failure.

• Hyperstatic Structures offer enhanced stiffness, redundancy, and material efficiency but require more complex analysis, precise construction, and detailed design considerations.

Applications:

• Isostatic structures suit simpler, smaller projects (e.g., small bridges, temporary structures).

• Hyperstatic structures are preferred in large-scale, permanent constructions (e.g., multi-story buildings, large bridges) due to their safety and performance.

Key Points:

• Deformation compatibility is critical in hyperstatic analysis to ensure displacements and rotations match support conditions.

• Modern structural software greatly facilitates the analysis of hyperstatic structures by solving equilibrium and compatibility equations efficiently.

• Proper design leverages force redistribution in hyperstatic structures for optimized material use and safer, more economical construction.

Conclusion

This article aimed to compare and deepen the understanding of isostatic and hyperstatic structures—fundamental concepts in structural analysis for civil engineering students. To this end, two beams subjected to identical loads were analyzed: a simply supported beam (isostatic) and a continuous beam with three supports (hyperstatic). The analysis of the isostatic beam (Beam A) showed that it can be solved solely using equilibrium equations. The support reactions were RA = 30 kN and RC = 30 kN. The shear force diagram varied linearly from +30 kN to ?30 kN, becoming zero at mid-span. The maximum bending moment, 45 kNm, occurred at the center. The deflected shape showed a maximum downward deflection at mid-span and a counter-deflection near the double support. On the other hand, the hyperstatic beam (Beam B), due to having more unknowns than equilibrium equations, required the use of additional deformation compatibility equations and was analyzed with the aid of Ftool software. The shear force diagram showed discontinuities at the supports, indicating vertical reactions, and inclined ramps due to the distributed load. The bending moment diagram of the hyperstatic beam showed negative moments (?6.3 kNm) near the supports and a maximum positive moment (+11.3 kNm) at mid-span. The deflected shape, with its wave-like form, indicated a maximum downward deflection in the central span and counter-deflections at the ends associated with the negative moments. In summary, the comparison between the two structures revealed that, while the isostatic beam concentrated the maximum bending moment at 45 kNm, the hyperstatic beam distributed the internal forces, resulting in positive and negative moments of 22.5 kNm. This redistribution of loads in the hyperstatic structure demonstrates its greater efficiency and ability to reduce peak stresses. Although the analysis of hyperstatic structures is more complex and requires greater precision in the execution of intermediate supports, they offer higher structural efficiency and safety, being particularly advantageous for permanent structures or those subject to variable loads. In contrast, isostatic structures are simpler to analyze. This practical study reinforces the importance of a solid understanding of these fundamental concepts in the training of future civil engineers.

References

[1] Hibbeler, R. C. – Análise Estrutural [2] EN 1990: Eurocódigo - Base de Projecto Estrutural [3] EN 1992-1-1: Eurocódigo 2 – Projecto de Estruturas de Betão [4] Notas de Aula de Análise Estrutural – Universidade de Lisboa, FEUP, IST

Copyright

Copyright © 2025 Jose Fonseca. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.