Statically Indeterminate Structures: What They Are and How to Solve Them

Statically indeterminate structures play a fundamental role in structural engineering as they ensure high resistance and stability, improving the performance of complex works such as bridges, buildings, and other infrastructures. These structures are characterized by the presence of constraints in excess of those strictly necessary to maintain static equilibrium under the action of applied loads. This “redundancy” of constraints gives the system greater stiffness and resistance compared to an isostatic configuration.

However, the design and analysis of indeterminate structures require a more complex approach, as the mere equations of equilibrium are not sufficient to describe their behavior. In this article, we will explore the main characteristics of indeterminate structures and analyze the challenges related to their resolution, illustrating the most common methods to tackle such issues.

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What are statically indeterminate structures?

Indeterminate structures are structural configurations that possess a number of constraints greater than that strictly necessary to prevent any rigid body motion of the structure. In technical terms, a structure is defined as indeterminate when the degree of indeterminacy, that is, the number of constraints in excess of the available equilibrium equations, is greater than zero. This excess of constraints makes the structure not only stable and resistant but also stiffer compared to an isostatic structure, which has a number of constraints equal to that necessary for equilibrium.

However, due to this “redundancy” of constraints, in indeterminate structures the equations of equilibrium alone are not sufficient to calculate the reaction forces. To address this issue, it is necessary to abandon the assumption of rigid behavior of the structures. By considering deformability, the indeterminate unknowns can be determined by imposing not only compliance with the equilibrium conditions but also the compatibility of deformations.

A common example of an indeterminate structure is the beam fixed at both ends. In this configuration, in addition to vertical and horizontal reactions (as in a simply supported beam), there are bending moments at both ends, which result in a number of unknowns greater than the three available static equilibrium equations. This makes the structure indeterminate, or excessively constrained, and consequently more complex to analyze.

The main characteristic of indeterminate structures is their ability to distribute loads more evenly compared to isostatic structures. This means that, in the presence of external loads or partial failure conditions, internal forces can redistribute in such a way as to keep the entire structure in equilibrium, increasing the overall resistance of the project. However, this redistribution of forces makes the analysis and sizing of an indeterminate structure more complex, requiring the use of advanced methods for its resolution.

Example of an indeterminate beam

Unstable, Isostatic, and Statically Indeterminate Structures

To fully understand what statically indeterminate structures are, it is useful to compare them with the other two main categories of structures: unstable structures and isostatic structures:

• unstable structures: a structure is defined as unstable when it does not have a sufficient number of constraints to maintain static equilibrium. In other words, these structures are unstable and, under the action of loads, tend to move indefinitely. For example, a simply supported beam without additional lateral constraints could slide laterally under the action of a transverse load, making the structure unstable. Unstable structures are unable to withstand loads and, therefore, cannot be considered safe in practical applications;
• isostatic structures: isostatic structures are those in which the number of constraints is exactly equal to that required for static equilibrium. For these structures, the number of equilibrium equations (three for planar structures: sum of horizontal forces, sum of vertical forces, and sum of moments) is sufficient to determine all the constraint reactions. Isostatic structures are stable and relatively simple to analyze, as there are no excess constraints that require further considerations. However, unlike hyperstatic structures, they are unable to redistribute loads in the event of local failure, making them less resilient;
• statically indeterminate structures: as already discussed, they possess excess constraints compared to the minimum necessary to ensure static equilibrium. This “redundancy” of constraints gives these structures a greater capacity to redistribute internal loads and to resist any partial failures or local deformations, improving their safety and durability over time. However, the analysis of these structures is more complex and requires the use of specific methods to solve the equations that describe their behavior.

Here is a clear and concise comparison table between unstable, isostatic, and statically indeterminate structures:

Methods for Solving Hyperstatic Structures

The resolution of hyperstatic structures requires analytical and numerical methods that go beyond simple equilibrium equations. The two main methods used for the analysis of these systems are:

• the force method, which involves the introduction of additional unknowns represented by forces (constraint reactions), which are determined by imposing the respect of the congruence of the structure;
• the displacement method, which is based on the introduction of additional unknowns represented by displacements, which are determined by imposing the equilibrium of the structure.

These two methods allow for the determination of internal stresses and deformations of hyperstatic structures, taking into account the excess constraints and the complexity of the structural configuration.

The Force Method

The force method is one of the classical methods used to analyze hyperstatic structures. In these structures, the equilibrium equations are not sufficient to determine all the reactions and internal forces, and it is necessary to consider additional equations, defined as “congruence equations”, derived from the compatibility conditions of the deformations.

In practice, this method is based on the principle of superposition of effects and involves ideally transforming the structure into an isostatic configuration, freeing it from some of its hyperstatic constraints. These constraints are replaced by unknown forces or moments, which are then determined using a system of equations that take into account the congruence conditions.

The fundamental steps of the force method are described below:

• definition of the degree of hyperstaticity of the structure: the first step is to quantify the degree of hyperstaticity, that is, the number of constraints that exceed those necessary for the equilibrium of the structure. The degree of hyperstaticity is calculated as the difference between the total number of unknowns (constraint reactions and/or internal forces) and the number of available equilibrium equations;
• transformation into equivalent isostatic structure: proceed by eliminating a number of constraints equal to the degree of hyperstaticity, thus transforming the structure into an isostatic configuration. The removed constraints are replaced by the corresponding constraint reactions (forces or moments), which represent the unknowns of the problem. To obtain this isostatic configuration, it is also possible to degrade some constraints or introduce internal constraints that divide the structure into multiple bodies, considering among the hyperstatic unknowns also the reactions or internal forces that the original constraint or the “whole” body were able to exert;
• calculation of deformations: since the value of the hyperstatic unknowns is not known, the equilibrium equations alone are not sufficient to solve the problem. Since hyperstatic structures are considered deformable, the calculation of deformations at the locations of the suppressed constraints is performed, based on known and unknown loads, exploiting the principle of superposition of effects. This step serves, therefore, to determine the deformations induced by external forces (active loads) and those induced by hyperstatic forces;
• imposition of congruence conditions: once these deformations (functions of the hyperstatic unknowns) are defined, the congruence conditions are applied, that is, it is imposed that the displacements at the locations of the suppressed constraints are compatible with such constraints. In other words, the displacements due to the removed constraints must be consistent with the original condition of the structure. This leads to the formulation of a system of equations, called congruence equations, which allows for the determination of the unknown constraint forces;
• calculation of internal stresses: once the hyperstatic unknowns are determined, it is possible to calculate the internal stresses (such as shear, bending moment, and axial forces) in all sections of the structure. These stresses are essential to verify the resistance and stability of the entire system.

The force method is particularly effective for structures with a limited number of degrees of hyperstaticity. However, as the complexity of the structure increases, the application of this method can become more computationally demanding, necessitating the use of more advanced methods or the use of structural calculation software.

The Displacement Method

The displacement method, also known as the stiffness method, is an additional approach used for the analysis of structures where the number of equilibrium equations is insufficient to determine all the support reactions and internal forces. This method is particularly suitable for the analysis of structures characterized by a high degree of indeterminacy.

Unlike the force method, which focuses on compatibility equations where the unknowns are represented by forces or moments, the displacement method is based on equilibrium equations where the unknowns are displacements or rotations. Essentially, while the force method focuses on support forces as unknowns, the displacement method concentrates on the displacements and deformations of the structure.

The fundamental principle of the displacement method is that, in a structure, the displacements and rotations at the nodes determine the internal stresses in the structural elements (bars, beams, etc.). Once these displacements and rotations are determined, it is possible to calculate the distribution of internal forces.

The main steps for applying the displacement method are outlined below:

• identification of unknown displacements: the first step is to identify the nodes of the structure that may undergo displacements and rotations compatible with the applied constraints. These displacements and rotations, defined as degrees of freedom of the structure, are the unknowns of the problem. It is assumed that the bars are infinitely rigid from an axial point of view, but with a constant bending stiffness;
• imposition of unit displacements: for each unknown displacement, a unit value is imposed (for example, one unit of horizontal, vertical, or rotational displacement), considering one variable at a time and assuming that the other nodes remain temporarily fixed. The deformed configuration of the structure under the effect of this unit displacement is then drawn, and the internal reactions in the structural elements resulting from this displacement are evaluated;
• application of equilibrium equations: for each node, the equilibrium equations are written considering the applied external loads and the contributions of the internal forces generated by the previously calculated unit displacements;
• formulation of the stiffness matrix: once the internal reactions for all unit displacements are calculated, the global stiffness matrix of the structure is constructed. This matrix establishes the relationship between nodal displacements and nodal forces. Each element of the matrix represents the stiffness of the structure with respect to a specific degree of freedom;
• calculation of displacements: with the stiffness matrix defined, the system of linear equations is solved to obtain the nodal displacements;
• detection of internal stresses: once the nodal displacements are known, the internal stresses in the structural elements, such as axial forces, shear forces, and bending moments, are calculated using the stiffnesses of the elements and the calculated displacements.

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FAQ – “Indeterminate Structures”

What is meant by statically determinate structures?

Statically determinate structures, also known as isostatic structures, are those in which all support reactions and internal forces can be calculated using only the equations of static equilibrium. For a planar structure, there are three equilibrium equations available (sum of horizontal forces, sum of vertical forces, and sum of moments). If the number of unknowns equals three, the structure is isostatic.

What is meant by statically indeterminate structures?

Statically indeterminate structures, also known as indeterminate structures, are those in which the number of supports or unknowns exceeds the number of available equilibrium equations. In these cases, the static equations alone are insufficient to solve the system, and it is necessary to consider the deformability of the structure and use additional compatibility equations.

What is the difference between statically determinate structures and statically indeterminate structures?

The main difference concerns the relationship between the number of supports and the available equilibrium equations. Here are the key points:

• Supports:
  • Statically determinate structures: have exactly sufficient supports to ensure equilibrium.
  • Statically indeterminate structures: have excess supports beyond those strictly necessary.
• Resolution:
  • Statically determinate: solved with only the equilibrium equations.
  • Statically indeterminate: also require compatibility equations based on deformability.
• Analysis:
  • Determinate: simple, direct, quick calculations.
  • Indeterminate: more complex, requires advanced methods or structural software.
• Consideration of deformability:
  • Determinate: generally negligible.
  • Indeterminate: must always be considered for a correct analysis.
• Behavior in case of failures:
  • Determinate: less resilient, may collapse locally.
  • Indeterminate: safer due to the ability to redistribute internal loads.

In short, determinate structures are simpler to calculate, while indeterminate structures offer better performance but require more in-depth analysis.

Why is it important to analyze indeterminate structures?

The analysis of indeterminate structures is important because they offer greater safety, stability, and resistance compared to isostatic structures. However, their complexity requires advanced calculation methods to:

• determine internal forces and deformations;
• predict the behavior of the structure in case of variations in loads or supports;
• optimize the design to ensure safety and economic efficiency.

Why are indeterminate structures advantageous compared to isostatic ones?

Indeterminate structures offer greater safety and resistance, as they can redistribute loads in the event of local failures, preventing sudden collapses. Additionally, they are less sensitive to deformations and failures of supports.

What are the main methods for solving indeterminate structures?

• Force Method: hyperstatic supports are eliminated, and equivalent unknowns are introduced, then the system is solved with compatibility equations.
• Displacement Method: equations based on displacements and stiffness of the structure are set up to determine internal forces.

What are some examples of indeterminate structures?

• Beams fixed at the ends, which have more support reactions than a simply supported beam.
• Steel or reinforced concrete frames and portals, typical in buildings and bridges, which offer greater resistance to horizontal and vertical loads.
• Multi-span bridges, where continuous beams are connected at multiple supports to better redistribute loads.
• Dome and structural vaults, which must withstand complex forces due to their indeterminacy.