8.2: Trefftz Condition for Stability
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In 1933 the German scientist Erich Trefftz proposed the energy criterion for the determination of the stability of elastic structures. We shall explain this criterion on a simple example of a one-degree-of-freedom structure. Consider a rigid column free at one end and hinged at the other. There is a torsional spring mounted at the hinge. Upon rotation by an angle \(\theta\), a bending moment develops at the hinge, resisting the motion
\[M = K \theta\]
where \(K\) is the constant of the rotational spring. The column is initially in the vertical position and is loaded by a compressive load \(P\), Figure (\(\PageIndex{1}\)). In the deformed configuration, the force \(P\) exerts a work on the displacement \(u\)
\[u = l(1 − \cos \theta) \cong l \frac{\theta^2}{2} \label{8.2.2}\]
The total potential energy of the system is
\[\prod = \frac{1}{2} M \theta − P u = \frac{1}{2} K \theta^2 − \frac{1}{2} Pl \theta^2 \]
Upon load application the column is of course rigid and remains straight up to the critical point \(P = P_c\). The path \(\theta = 0\) is called the primary equilibrium path. If the column were elastic rather than rigid, there would be only axial compression along that path. This stage is often referred to as a pre-buckling configuration. At the critical load \(P_c\) the structure has two choices. It can continue resisting the force \(P > P_c\) and remain straight. Or it can bifurcate to the neighboring configuration and continue to rotate at a constant force. The bifurcation point is the buckling point. The structure is said to buckle from the purely compressive stage to the stage of a combined compression and bending.
The above analysis have shown that consideration of the equilibrium with nonlinear geometrical terms, Equation \ref{8.2.2} predicts two distinct equilibrium paths and a bifurcation (buckling) point. Let’s now explore a bit further the notion of stability and calculate the second variation of the total potential energy
\[\delta^2 \prod = (K − Pl) \delta \theta \delta \theta \]
The plot of the normalized second variation \(\delta^2 \prod / \delta \theta \delta \theta \) is shown in Figure (\(\PageIndex{1}\)).
It is seen that in the range \(0 < P < P_c\), the second variation of the total potential energy is positive. In the range \(P > P_c\), that function is negative. A transition from the stable to unstable behavior occurs at \(\delta^2 \prod = 0\). Therefore, vanishing of the second variation of the total potential energy identifies the point of structural instability or buckling.
Physically, the test for stability looks like this. We bring the compressive force to the value \(P^*\), still below the critical load. We then apply a small rotation \(\pm \delta\theta\) in either direction of the buckling plane. The product \(\delta\theta\delta\theta\) is always non-negative.
For equilibrium the first variation of the total potential energy should banish, \(\delta\theta = 0\), which gives
\[(K − Pl) \theta \delta \theta = 0 \]
There are two solutions of the above equation, which corresponds to two distinct equilibrium paths:
- \(\theta = 0 - \text{ primary equilibrium path}\)
- \(P = P_c = \frac{K}{l} - \text{ secondary equilibrium path}\)
And so is the second variation of the total potential energy (length AB in Figure (\(\PageIndex{1}\))). When the lateral load needed to displace the column by \(\delta\theta\) is released, the spring system will return to the undeformed, straight position.
We repeat the same test under the compressive force \(p^{**} > P_c\). The application of the infinitesimal rotation \(\delta\theta\) will make the function \(\delta^2/\delta\theta\delta\theta\) negative. This is a range of unstable behavior. Upon releasing of the transverse force, the column will not returned to the vertical position, but it will stay in the deformed configuration. It should be pointed up that the foregoing analysis pertains to the problem of stability of the primary equilibrium path. The secondary equilibrium path is stable, as will be shown below.
To expression for total potential, it can be constructed with the exact equation for the displacement \(u\) rather than the first two-term expansion, Equation \ref{8.2.2}
\[\prod = \frac{1}{2} K\theta^2 − lP(1 − \cos \theta) \]
The secondary equilibrium path obtained from \(\delta \prod = 0\) is
\[\frac{P}{P_c} = \frac{\theta}{\sin \theta}\]
The plot of the above function is shown is Figure (\(\PageIndex{4}\)).
For small values of the column rotation, the force \(P\) is almost constant, as predicted by the two-term expansion of the cosine function. For larger rotations, the column resistance increases with the angle \(\theta\). Such a behavior is inherently stable. The force is monotonically increasing and reach infinity at \(\theta \rightarrow \pi\).
