# What is stress stiffening in Finite Element Analysis?

In a Finite Element Analysis, if we turn ON the geometric non-linearity, it also takes into account the stress stiffening if any structure is undergoing it. That is what I have read in the literature.

But what is stress stiffening? When does it occur? It is related to what? Why does it happen?

• Is there an entry for “stress stiffening@ in the literature? Aug 13, 2021 at 7:28
• An alternative name for this is the material stiffness matrix (which is similar to the standard stiffness matrix in linear finite elements). Aug 13, 2021 at 10:41
• I believe it may be something related to a behavior similar of work hardening for metals. Probably it needs you to specify some more parameters in order to change the slope of the elastic modulus due to the stiffening happening Aug 13, 2021 at 11:05
• @BiswajitBanerjee "material stiffness matrix" seems a very poor name for the stress stiffness, because it does not depend on the elastic properties of the material. (But the fact that it is a poor name doesn't mean that nobody has ever used it, of course...) Aug 13, 2021 at 16:03
• @alephzero: The total stiffness matrix is the sum of two terms: the geometric term and the material term. If you look at the rate form equations, you'll notice that both terms contain the stress. However, the first contains the [velocity gradient times the stress] while the second contains the [deformation gradient times stress rate] (which is replaced by the material law). People sometime use "stress stiffness" for the geometric term. That's the reason for the confusion. Aug 13, 2021 at 19:56

My undestanding is that stress stiffening is the additional stiffness in structures when there is an applied prestressed state.

The most common example, -I know of- is how the guitar string (or a beam) changes it fundamental frequency as pretension of the guitar string increases. When you increase the tension of the string (and therefore its stress), then the fundamental frequency changes.

Parenthesis: Difference between fundamental vibration frequency of structures and strings

The fundamental frequency in the dynamics of a structure is affected by its mass and a spring constant.

In strings, Vincenzo Galilei, discovered that the velocity of propagation of a wave in a string $$v$$ is :

$$v=\sqrt {\frac{T}{\mu}}$$

where:

• $$T$$ the force of tension of the string
• $$\mu$$ the linear density of the string

which can be used to obtain the Mersenne law for the fundamental frequency: $$f = \frac{1}{2L}\sqrt {\frac{T}{\mu}}$$ where:

• $$T$$ is the tension (in Newtons),
• $$\mu$$ is the linear density (that is, the mass per unit length),
• $$L$$ is the length of the vibrating part of the string.

What happens -in the above case - is that there is coupling between in-plane stress and transverse stiffness, which changes the vibrational behavior of the structure. This coupling is most pronounced in thin, highly stressed structures, such as cables or membranes.

In FEA, one possible implementation to account for stress stiffening, is that an additional stiffness matrix is calculated based on the previous state of the solution (for dynamic transient problem). This additional stress stiffness matrix is added to the structure stiffness matrix (which depends on geometric parameters) and the solution is updated.

Stress stiffening is the work done to displace the structure that is caused by its internal stress.

The simplest formulation of FEA (and classical theories like Euler-Timoshenko beam theory) assume that displacements and strains can be approximated to first order (i.e. "engineering strains") which means that the stress stiffening terms are second order (proportional to displacement squared) and therefore ignored.

If we assume that the strains are small but displacements can be arbitrarily large, we need different measures of stress and strain (e.g. Green's strain and Piola-Kirchhoff stress) where for example the strain components are all zero for arbitrary, large, rigid body rotations of the body. In that case, the stress stiffening terms are first order in the displacements, and also proportional to the initial stress in the structure.

The example of a stretched string in another answer is a good simple example. If you pull a stretched string transversely at a point, there is a restoring force returning it to its original straight position, and you have to do work against that force. The work depends on the geometry of the string (e.g. its length) and its internal stress (i.e. the tension). It does not depend on the material properties of the string itself.

In the simplest model of a stretched string, the elastic stiffness caused by the material properties is ignored. A more accurate model includes both the elastic stiffness and the stress stiffness, and the frequencies of the harmonics of the string are then not in an exact 1:2:3:4:... ratio. Google for "inharmonicity" for more information about how this affects musical instruments like pianos and guitars, for example.