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Stability of structures

Stability of structures

In mechanical engineering, the stability of systems under various forces is critical for designing durable and efficient structures. One fundamental way to explore these concepts is by analyzing systems like two rods connected at a pivot point, O, with a torsional spring of spring constant k at the pivot point. This system is similar in appearance to a scissor jack used to change tires on a car. In this case, the arms of the linkage (equivalent to the rods in this system) are entirely vertical,...

Deformation in a Circular Shaft

Deformation in a Circular Shaft

One of the distinctive characteristics of circular shafts is their ability to maintain their cross-sectional integrity under torsion. In other words, each cross-section continues to exist as a flat, unaltered entity, simply rotating like a solid, rigid slab. To understand the distribution of shearing stress within such a shaft, consider a cylindrical section inside this circular shaft. This section has a length of L and a radius of R, with one end fixed. The radius of the cylindrical section is...

Torsion of Noncircular Members

Torsion of Noncircular Members

Circular shafts undergoing torsional stress maintain their cross-sectional integrity due to their axisymmetric nature. This symmetry ensures an even distribution of stress, allowing the shaft to withstand torsion without distorting. In contrast, square bars, lacking this axial symmetry, experience significant distortion across their cross-sections when subjected to torsion, with the exception of along their diagonals and at lines connecting midpoints. A detailed examination of a cubic element...

Statically Indeterminate Problem Solving

Statically Indeterminate Problem Solving

Statically indeterminate problems are those where statics alone can not determine the internal forces or reactions. Consider a structure comprising two cylindrical rods made of steel and brass. These rods are joined at point B and restrained by rigid supports at points A and C. Now, the reactions at points A and C and the deflection at point B are to be determined. This rod structure is classified as statically indeterminate as the structure has more supports than are necessary for maintaining...

Temperature Dependent Deformation

Temperature Dependent Deformation

In a nonhomogeneous rod made up of steel and brass, restrained at both ends and subjected to a temperature change, several steps are involved in calculating the stress and compressive load. Due to the problem's static indeterminacy, one end support is disconnected, allowing the rod to experience the temperature change freely. Next, an unknown force is applied at the free end, triggering deformations in the rod's steel and brass portions. These deformations are then calculated and added...

Deformations in a Symmetric Member in Bending

Deformations in a Symmetric Member in Bending

When analyzing the deformation of a symmetric prismatic member subjected to bending by equal and opposite couples, it becomes clear that as the member bends, the originally straight lines on its wider faces curve into circular arcs, with a constant radius centered at a point known as Point C. This phenomenon helps to understand the stress and strain distribution within the member more clearly.
When the member is segmented into tiny cubic elements, it is observed that the primary stress...

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Related Experiment Video

Updated: Dec 8, 2025

Flexural Rigidity Measurements of Biopolymers Using Gliding Assays

Flexural Rigidity Measurements of Biopolymers Using Gliding Assays

Published on: November 9, 2012

Infinitely long isotropic Kirchhoff rods with helical centerlines cannot be stable.

Andy Borum¹, Timothy Bretl²

¹Department of Mathematics, Cornell University, Ithaca, New York 14853, USA.

Physical Review. E

|September 18, 2020

Summary

Planar elastic rods with constant curvature are always stable. However, nonplanar helical rods become unstable at a finite length, with critical length depending on curvature, torsion, and twist.

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Area of Science:

Solid Mechanics
Elasticity Theory
Optimal Control

Background:

Planar elastic rods with constant curvature and clamped ends are known to be stable regardless of length.
The stability of nonplanar elastic rods, particularly helical configurations, remains less understood.

Purpose of the Study:

To investigate the stability of nonplanar elastic rods with helical centerlines.
To determine if helical rods with constant nonzero curvature can maintain stable equilibrium at any length.

Main Methods:

Application of optimal control theory to analyze the stability of Kirchhoff rods.
Derivation of coordinates for critical length computation, independent of stiffness parameters.
Development of a scaling relationship for instability length.

Main Results:

Demonstration that nonplanar helical rods with constant nonzero curvature become unstable at a finite length.
Identification of a critical length independent of bending and torsional stiffness.
Establishment of a scaling law linking instability length to curvature, torsion, and twist.

Conclusions:

Unlike their planar counterparts, nonplanar helical rods are not universally stable.
The derived critical length and scaling relationship provide key insights into the stability limits of helical elastic rods.