Unstable Structure: What It Is and How to Identify It

Structures in engineering can exhibit different levels of stability depending on how they are constrained. A structure is defined as “unstable” when the external constraints are insufficient to prevent its movement, leaving one or more degrees of freedom that allow it to move. This structural configuration is particularly dangerous, as it inevitably leads to the collapse of the entire system.

Recognizing an unstable structure is crucial for engineers and structural designers, who must be able to design safe and efficient structural systems. In this article, we will explore in detail the meaning of unstable structure and illustrate how to distinguish this category of structures from isostatic and hyperstatic structures. We will analyze the concept of degrees of freedom and explore the distinctive characteristics of each configuration, providing the basis for recognizing and correctly assessing them.

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What is an unstable structure?

An unstable structure can be generically defined as a structure that is not sufficiently constrained and, consequently, is unable to maintain balance under the action of external forces, resulting in instability and susceptibility to unwanted movements or rotations.

For example, a chair with a missing leg cannot remain stable because it lacks the necessary constraint to prevent it from tipping over. Similarly, structures are unstable when the existing constraints are insufficient to counteract all possible displacement components (both translational and rotational), leading to an unstable configuration.

An example of an unstable structure is a rod constrained at the ends with two carts, which can freely translate in the horizontal direction since there is no applied constraint capable of preventing this displacement.

Example of unstable structure

Since the concept of unstable structure is closely related to that of equilibrium, to fully understand the meaning and characteristics of this particular structural configuration, it is essential to start from the definition of static equilibrium.

The concept of static equilibrium

If we consider a structure in a given geometric configuration, it is said to be in equilibrium when each part, when considered individually, is also in equilibrium. The problem of static equilibrium of an entire system can therefore be divided into the analysis of the equilibrium of all its components.

For the purposes of determining equilibrium, each component can be treated as a rigid body. A necessary and sufficient condition for a rigid body to be in equilibrium is that two fundamental conditions are satisfied:

• equilibrium of translation: the sum of the applied forces must equal zero;
• equilibrium of rotation: the sum of the moments of the applied forces and couples must also be zero.

Mathematically, these conditions are expressed by the cardinal equations of statics. For three-dimensional systems, six equilibrium equations must be satisfied (three for translation along the x, y, z axes and three for rotation around these axes). In the two-dimensional context, that is, when the trajectories of the points of the body and the applied forces are contained in the plane, the equilibrium equations reduce to three (two for translation and one for rotation).

For structures made up of multiple components, the total number of equations necessary to ensure the static equilibrium of the entire system must be multiplied by the number of parts themselves. If a structure consists of n bodies, in the three-dimensional problem there will be 6n equilibrium equations, while in the plane problem there will be 3n equations. The equilibrium will depend on the applied external forces, constraint reactions, and internal stresses that the parts of the structure exchange with each other.

Degrees of freedom

Closely related to the concept of equilibrium is that of degrees of freedom. Degrees of freedom represent the number of independent variables necessary to completely describe the motion of a body in space or in the plane. A free body in space has six degrees of freedom (three translations and three rotations), while in the plane it has three degrees of freedom (two translations along the orthogonal axes that form the plane and one rotation around the axis perpendicular to the plane).

These degrees of freedom can be limited by the application of constraints, which can be simple, double, or triple depending on their ability to prevent one, two, or three translational or rotational movements, respectively.

With reference to the plane problem (two-dimensional), the commonly used constraints are as follows:

• • cart or simple support: simple constraint that prevents movement along the axis orthogonal to the sliding plane, allowing translations orthogonal to that axis and rotation around the constrained point. It reacts with a force perpendicular to the sliding plane;
  • pendulum or connecting rod: simple constraint equivalent to the cart and prevents movement along the axis of the connecting rod, reacting with a force orthogonal to the sliding plane;
  • double pendulum (or bipendulum): double constraint that prevents translation along the axis of the pendulums and rotation of the body, but allows translation along the direction orthogonal to that axis. It reacts with a force along the axis of the pendulums and with a couple;
  • double-double pendulum (or quadrupendulum): simple constraint that prevents rotation of the body, leaving translation free. It reacts with a couple;
  • hinge: double constraint that prevents movement in all directions of the plane, but allows rotation around the constrained point. It reacts with a force that can be decomposed into two orthogonal components;
  • fixed support: triple constraint that prevents both translations and rotation of the body. It reacts with two force components in different directions and a couple.

Types of constraint

Since, in the plane, an unconstrained structure composed of n bodies has 3n degrees of freedom (two translations and one rotation for each body), if this structure is subjected to a system of constraints chosen from those just listed, to determine its degrees of freedom it is necessary to subtract from 3n the constraint degree m, which represents the sum of the components of displacement or rotation prevented by the constraints applied to the structure (considering both external and internal constraints):

Degrees of freedom of the structure = 3n – m

where:

• n represents the number of bodies that make up the structure;
• m represents the sum of all translational or rotational displacements prevented by the constraints applied to the structure.

This relationship allows us to determine whether a structure is isostatic, hyperstatic, or unstable.

When is a structure isostatic?

A structure is considered isostatic when the number of degrees of freedom coincides with the number of constraint degrees. In the plane, this condition occurs when the following equality is satisfied:

3n = m

This means that the imposed constraints are exactly sufficient to prevent any rigid displacement of the structure. In other words, every translational or rotational movement of the bodies that make up the structure is rigidly blocked by the applied constraints. This condition ensures that the system is statically determined: the constraint reactions can be uniquely determined and there is only one solution for the conditions of static equilibrium.

When is a structure hyperstatic?

A structure is hyperstatic when the number of constraint degrees is greater than the number of degrees of freedom:

3n < m

In this case, the applied constraints are redundant with respect to the need to prevent the rigid movements of the structure. In other words, the structure has a certain number of excess constraints compared to those strictly necessary to achieve static equilibrium. This implies that the constraint reactions cannot be uniquely determined solely through the equations of static equilibrium. Therefore, the structure is statically indeterminate: multiple solutions can satisfy the conditions of static equilibrium, making it necessary to use alternative methods, such as the method of forces or stiffness, to solve the system.

When is a structure unstable?

A structure is unstable when the number of constraint degrees is less than the number of degrees of freedom:

3n > m

In this situation, the applied constraints are insufficient to prevent the rigid movements of the structure under all possible load configurations. This means that the structure can have rigid movements, or instability mechanisms, that allow parts of the structure to move freely. From a static point of view, the problem does not admit a solution because the equations of equilibrium are not sufficient to uniquely determine the constraint reactions.

Unstable structures due to poorly placed constraints

It is important to emphasize that the relationships just illustrated alone are not sufficient to define whether a structure is isostatic, hyperstatic, or unstable. It is indeed necessary to also analyze the arrangement of the constraints imposed on the structure. A structure can have a number of constraint degrees equal to or greater than the degrees of freedom and still be unstable if the constraints are not correctly positioned.
A classic example is that of two beams connected with three hinges arranged in line. Since the structure is formed by two bodies, the degrees of freedom in the unconstrained configuration are equal to six (3∙n = 3∙2 = 6). The three hinges, which prevent two components of translational displacement each, determine an overall constraint degree equal to 6. According to the relationships seen earlier, the structure should be isostatic, as the number of constraint degrees (m=6) equals the number of degrees of freedom (3n=6). However, this system is unstable, as the constraints are poorly placed, allowing unwanted movements of the structure.

Unstable structure due to poorly placed constraints

The analysis of these types of structures requires a more in-depth study than simple counting of traits, constraints, and degrees of freedom. Very often, reference is made to more advanced methods, such as the method of rotation centers, to correctly determine whether a structure is isostatic, hyperstatic, or unstable.

How to identify structural instabilities?

The presence of instability in a structure can arise from design errors, inadequate arrangement of constraints, or imperfect modeling within the calculation software. To ensure the stability and reliability of the structural model, it is essential to have tools capable of automatically signaling potentially critical situations. The structural calculation software offers specific functions for identifying instabilities, supporting the technician in the verification and correction process.
To activate the structural instability detection function:

• activate the General page of the Calculation Preferences by double-clicking on the corresponding node in the Navigator;
• press the Edit solver configuration button in the SAP Preferences section;
• in the window that opens, select the check box Instability detection on the diagonal of the structure’s stiffness matrix.

With the check box active, the calculation of stresses is not limited to classic modeling errors, but also checks for the presence of incorrect releases or connections between structural elements.

Solver configurator |EdiLus

Practical examples of unstable structures encountered

Instability on a beam between two nodes
In the first example, by performing the stress calculations without selecting the checkbox, the diagnostics report a fatal error due to a beam that is not properly supported. In particular, the beam appears suspended and not connected to the lower beams.

Beam not properly supported | EdiLus

To resolve this, simply move the beam slightly downwards by modifying the value in the Move field in the properties toolbox.
By activating the checkbox, a second fatal error emerges related to an improperly released node.
Selecting the beam shows in the properties toolbox that a release has been applied at the initial end of the node, as if it is indicated that this element, which is already free at the end, can further rotate; obviously, from the perspective of finite element modeling, this is an inadmissible condition.

Improperly released node |EdiLus

In this case, to resolve it, simply modify the constraints in the Internal Constraints section of the properties toolbox to restore the solidarity of the node in question.

Internal Constraints: solidarity | EdiLus

Instability in the modeling of stairs

Stairs represent an area particularly prone to modeling errors. In the analyzed example, three different cases of instability are presented:

• Case A – Disconnected orthogonal stairs:
  • the first flight is detached from the foundation;
  • the second, parallel to the wall, is disconnected from the wall.

Disconnected orthogonal stairs |EdiLus

To resolve, simply:

• • remove the automatic generation of the landing from the corresponding field in the Features section of the stair properties toolbox;
  • add an initial landing from the corresponding field in the Features section of the stair properties toolbox to connect to the foundation beam;

Initial landing |EdiLus

• • extend the final landings to properly connect to the orthogonal ramps.

Landing length | EdiLus

• Case B – Stair not connected to the landing:
  • the stair section rests on a CFC beam and a floor slab, but is not connected to the final landing.

To resolve this, simply add initial and final landings from the properties toolbox.

Addition of initial and final landings |EdiLus

• Case C – Stair on non-structural elements:
  • the stair rests on a floor slab and a non-structural CFC beam, and on a landing connected to a knee beam not present in the 3D model.

To resolve this, simply:

• • remove the automatic generation of the stair from the relevant field in the Features section of the stair properties toolbox;
  • insert the initial and final landings of the stair from the relevant field in the Features section of the stair properties toolbox to connect to actually structural elements.

Stair on non-structural elements – addition of initial and final landings |EdiLus

Once these modifications are made, the regeneration of the model and the subsequent calculations allow for obtaining a stable and verified structure.

Instability in the connection beams to the footings

A common mistake concerns the connection beams that do not reach the axis of the column. In these cases, the footing is modeled by the software as a simple spring and not as a finite element.

Connection beams that do not reach the axis of the column |EdiLus

Therefore, in these situations, it is necessary for the inserted connection beams to be drawn up to the axis of the column. In this case, it is enough to extend the beams to connect them to the columns to which the footing is linked.

Extension of beams until hooking them to the columns to which the footing is connected |EdiLus

Instability in the floor openings

Finally, another recurring example occurs when two parallel beams rest on non-structural CFC joists to define an opening in the floor. The software reports an instability as the CFC joists are not suitable to support such conditions.

In this situation, to correctly model the opening, one must resort to the beam object (in reinforced concrete or generic material) instead of the CFC joists.

Beam object in reinforced concrete (or other structural material) |EdiLus

Once the model is updated, the opening is correctly represented and the instability disappears.

All these examples confirm how proper modeling and the use of automatic diagnosis functions are essential tools for promptly identifying instabilities and returning a stable, coherent, and reliable structural model. For this reason, the advice is to adopt a structural calculation software that natively supports the BIM + FEM workflow: an approach that allows for speeding up checks, improving modeling accuracy, and significantly increasing overall structural safety. The tool performs a diagnosis on any instabilities in the structural modeling, signaling the structural elements that are unstable due to incorrect modeling or modification/management of constraints.

Reporting of instability | EdiLus

Read the in-depth article “Structural Design”

FAQ – Unstable Structure

What is an unstable structure?

A structure is unstable when the external constraints are not sufficient to prevent its rigid movement, leaving one or more degrees of freedom active. This leads to a condition of instability that makes the structure unable to remain in equilibrium under the action of external forces.

How can you recognize an unstable structure?

It can be recognized through the analysis of degrees of freedom and constraints. In a planar system, if 3n > m (with n = number of bodies and m = number of constraints), the structure is unstable. However, even if the equation is satisfied, incorrect arrangement of the constraints can still generate instability.

What are the differences between unstable, isostatic, and hyperstatic structures?

• Isostatic: 3n = m → balanced structure with a minimum number of constraints.
• Hyperstatic: 3n < m → structure with excessive constraints, statically indeterminate.
• Unstable: 3n > m → unstable structure, insufficient constraints.

Which structural constraints influence stability?

Constraints are classified as simple, double, or triple, and each acts by limiting one or more degrees of freedom (translations or rotations). Among the most common:

• simple support (cart);
• pendulum or connecting rod;
• hinge;
• fixed support;
• bipod and quadrupod.

Can a structure with the correct number of constraints still be unstable?

Yes. The geometric arrangement of the constraints is fundamental. Even when respecting the equation 3n = m, a poor positioning of the constraints can leave the structure free to move, as happens in the classic example of three aligned hinges.

What are the degrees of freedom in a structure?

They are the independent variables that describe the possible movements of a rigid body. In 2D there are three (two translations + one rotation), in 3D there are six. The constraints imposed in a structure serve to reduce or eliminate these degrees of freedom.

How can I identify structural instabilities during modeling?

Through automatic diagnostic tools present in FEM software, such as the checkbox for detecting instabilities on the stiffness matrix. This allows for quick identification of:

• incorrect supports;
• beams not properly supported;
• disconnected stairs;
• holes not modeled with structural beams.

What is the advantage of integrating BIM + FEM in structural design?

It allows for:

• accurately modeling architectural and structural elements;
• evaluating behavior under load through finite element analysis;
• automatically detecting instabilities and correcting them in real-time;
• increasing reliability and reducing margins of error.