Extra Energy Stability criterion

The Extra Energy Stability criterion is a method for calculating the static stability of non-conservative massless systems, which was proposed by Ingerle in 2013.^[1] The classical Euler- and the Energy Stability criterion relate to deformed, closely spaced non-trivial equilibrium positions of systems. Since non-conservative systems often have no non-trivial equilibrium positions, these criteria can not be applied to non-conservative static systems. The Extra Energy Stability criterion, as proposed by Ingerle, refers to still undeformed conservative and non-conservative static systems and is argued to be applicable to both types of systems.

Historical background

Since the beginning of the 20th century the notion that non-conservative elastic systems without non-trivial states of equilibrium do not have static stability limits predominates in the scientific discourse on the topic. The commonly employed Euler- and the energy criterion are applicable for cases where non-trivial equilibrium states exist. In 1969 Ingerle ^[2] demonstrated for the first time that such systems have static stability limits. In the decade after also El Naschie came to a similar conclusion.^[3] However, the topic continued to be discussed controversially for the remainder of the 20th century. In 2005 the ongoing discussion surrounding the topic motivated I. Elishakoff to publish a review paper on the subject, listing more than 200 references.^[4] With the presentation of a novel static stability criterion by Ingerle in 2013^[1] the existence of static stability for non-conservative elastic systems without non-trivial states of equilibrium is now largely accepted.

Background and Assumptions

To deform an undeformed, still stable stability system, extra energy E_ex – in form of extra forces Z_i – must be applied. The energy E_ex - necessary for a defined deformation of the system – allows the following conclusions: For E_ex ≥ 0, the system is stable and for E_ex = 0 a critical load p_c is reached. An infinite number of extra forces Z_i applied to a system corresponds to an infinite number of deformations with an infinite number of critical loads p_c. To find the smallest critical load p_cs, all deformations caused by extra forces must be investigated.

Three important steps are necessary for the analysis:

In order to deform an undeformed elastic stability system, extra forces $Z_{i}$ are required. All possible deformations with extra forces have to be considered.
Energy $E_{e x}$ of all extra forces $Z_{i}$ must be $0$ (or the energy of all loads on the system $E_{p}$ - without extra forces - corresponds to the energy stored in the system $E_{S}, E_{e x} + E_{p} + E_{S} = 0$ ).
In order to find the smallest critical load, all possible deformations caused by extra forces have to be investigated (extreme value investigation).

The “extra energy method” ( $E_{e x} = 0$ ) takes into account also all possible extra forces. The equilibrium equations and the energy equation of the extra forces $E_{e x} = 0$ are established. In contrast, the “energy method” ( $E_{P} = E_{S}$ ) refers to all possible deformations and must not consider extra forces. In the analysis the energy of the loads and of the system is determined.

The “Extra Energy Stability” criterion - applicable to conservative and non-conservative systems – can be formulated in a very short form:

The smallest stability load of an undeformed stability system, for which the deformation energy - generated by extra forces - is zero ( $E_{e x, m i n} = 0$ ) needs to be found.

Application of the extra energy method and the energy method in absence of non-trivial equilibrium positions (Example of Ziegler's Column)

For a better understanding the calculation of the stability of Ziegler’s column is shown. Ziegler's column is a simple example for massless non-conservative static systems.

Ziegler’s column – loaded with a tangential load P – consists of two rigid rods connected by spring hinges with constant stiffness $c$ (figure a in the depiction of Ziegler's column). Figure b depicts the column deformed with two extra forces $Z_{1}$ and $Z_{2}$ . Deformations and the extra forces are of infinitesimal size while the load $P$ is of finite dimension.

The non-conservative Ziegler’s column is analysed using the “extra energy method”. This method must take into account all possible extra forces $Z_{i}$ such that all extra forces can be represented by horizontal force $Z_{1, h}$ applied to the middle hinge and the horizontal force $Z_{2, h}$ on the tip of the system. Possible vertical infinitesimal extra forces can be ignored since they are applied in axial direction and are negligible compared to the finite load $P$ (see figure c below). The equilibrium equations and the equation for the extra forces $E_{x} = 0$ lead to the infinite number of critical loads $p_{c}$ . Extreme value analysis leads subsequently to the smallest critical load $p_{c s}$ _:

File:Figure 1 - Ziegler's column-.jpg

Ziegler Column

$c φ_{1} = Z_{1} l + Z_{2} 2 l - P l φ_{2}; c φ_{2} = Z_{2} l$

$E_{e x} = 0 = \frac{Z_{1} l φ_{1}}{2} + \frac{Z_{2} l (2 φ_{1} + φ_{2})}{2}$

$\frac{φ_{1}}{φ_{2}} = x, p_{c} = - \frac{(x^{2} + 1)}{x}$

$\frac{\partial p_{c}}{\partial x} = - x^{2} + 1 = 0; p_{c} = \frac{P_{c} l}{c}$

$(x = - 1; Z_{1} = - Z; Z_{2} = + Z); p_{c s} = 2.0$

To compare the "Energy Method" $(E_{S} = E_{P})$ with the proposed Extra Energy Method, Ziegler's column is investigated once more. The follower load $P$ is split into the finite vertical component $P_{V} = P [1 - \frac{(φ_{1} + φ_{2})^{2}}{2}] \approx P$ and into an infinitesimal horizontal component $P_{H} = P (φ_{1} + φ_{2})$ . In this case the Energy Method leads to the same smallest critical load as the Extra Energy method.

$E_{s} = c (φ_{1}^{2} + φ_{2}^{2}) / 2; E_{s} / φ_{2}^{2} = c (x^{2} + 1) / 2$

$E_{p} = P l [φ_{1}^{2} + (φ_{1} + φ_{2})^{2} - (2 φ_{1} + φ_{2}) (φ_{1} + φ_{2})] / 2; E_{p} / φ_{2}^{2} = - P l x / 2$

$E_{s} = E_{p}; p_{c} = - (x^{2} + 1) / x; \frac{\partial p_{c}}{\partial x} = 0;$

$x^{2} - 1 = 0; x = - 1; p_{c s} = 2.0$

Supplementary Reading

In his 2018 book on non-conservative systems Ingerle expands on the ideas proposed in his 2013 article.^[5]

References

↑ ^1.0 ^1.1 Ingerle, Kurt (2013). "Stability of massless non-conservative elastic systems". Journal of Sound and Vibration. 332, 11: 4529–4540.
↑ Ingerle, Kurt (1969). "Die Verwendung des statischen Stabilitätskriteriums für Probleme der Elastostatik bei Fehlen nichttrivialer Gleichgewichtslagen". Forschung im Ingenieurwesen. 35: 196–197.
↑ El Naschie, M. S. (1977). "Some Remarks on the Beck Problem". AIAA Journal. 15: 1200–1201.
↑ Elishakoff, Isaac (2005). "Controversy Associated With the So-Called "Follower Forces": Critical Overview". Applied Mechanics Reviews. 58 (2): 117–142.
↑ Ingerle, Kurt (2018). Non-conservative systems - New static and dynamic stability criteria. Boca Raton: CRC Press, Taylor & Francis Group. ISBN 978-1-138-30584-7. Search this book on

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[:1-1] 1.0 ^1.1 Ingerle, Kurt (2013). "Stability of massless non-conservative elastic systems". Journal of Sound and Vibration. 332, 11: 4529–4540.

[2] Ingerle, Kurt (1969). "Die Verwendung des statischen Stabilitätskriteriums für Probleme der Elastostatik bei Fehlen nichttrivialer Gleichgewichtslagen". Forschung im Ingenieurwesen. 35: 196–197.

[3] El Naschie, M. S. (1977). "Some Remarks on the Beck Problem". AIAA Journal. 15: 1200–1201.

[:0-4] Elishakoff, Isaac (2005). "Controversy Associated With the So-Called "Follower Forces": Critical Overview". Applied Mechanics Reviews. 58 (2): 117–142.

[5] Ingerle, Kurt (2018). Non-conservative systems - New static and dynamic stability criteria. Boca Raton: CRC Press, Taylor & Francis Group. ISBN 978-1-138-30584-7. Search this book on

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