Abstract
The linearized theory of buckling is illustrated here, aimed to determine the critical load of prestressed elastic structures. Under the fundamental hypothesis of negligibility of the precritical strains, a quadratic expression is determined for the total potential energy. It differs from that of the linear theory for the presence of an additional peculiar term, representing the second-order work of the prestress. The variational procedure leads to a linear eigenvalue problem in the unknown critical load. Reference is made both to discrete and continuous systems. The eigenvalue problem for the Cauchy continuum is derived as an example. In view of studying nonconservative systems, an alternative formulation is described, grounded on the virtual work principle. Finally, a direct derivation of the equations is also illustrated, useful to interpret the equilibrium of forces in the adjacent configuration. In the last part of the chapter, imperfect systems, with a single or several degrees of freedom, are addressed, whose linearized response to incremental loads is determined. The solution suggests the definition of an amplification factor of the linear response, which accounts for the geometric effects not included in the linear theory. An example of imperfect two degrees of freedom system is worked out.
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Notes
- 1.
In this book, the terms “linearized theory” and “linear theory” are used with different meanings. The former accounts for prestress; the latter ignores it.
- 2.
The term should be understood as opposed to the linear theory, or of the first order, classic of the elementary textbooks of mechanics of solids and structures, in which the effects of prestress are ignored, by virtue of the “superposition principle.”
- 3.
Here, it is assumed that the work is linear in the Lagrangian parameters, as it is often possible to obtain with a suitable choice of them (an example is given in the Sect. 6.6); otherwise it is necessary to add some nonlinear terms (as it will be seen in the Sect. 9.7, dealing with forces applied to rigid cross-sections of thin-walled beams).
- 4.
In this book, the term “congruence equation” is used, instead of the more common (and pedestrian) “strain-displacement relationship.” The locution “compatibility condition,” also improperly used in literature, is instead reserved to the integrability conditions of the kinematic problem.
- 5.
The geometric stiffness matrix cannot be written by matrix symbolism, but indicial. By letting \(\boldsymbol {\sigma }_{0}=\left (\sigma _{k}^{0}\right ),\) \({\mathbf {d}}_{2}\left (\mathbf {q}\right )=\left (\sum _{i}\sum _{j}d_{kij}q_{i}q_{j}\right )\), with dkij = dkji, it follows:
$$\displaystyle \begin{aligned} \boldsymbol{\sigma}_{0}^{T}{\mathbf{d}}_{2}\left(\mathbf{q}\right)=\sum_{i}\sum_{j}\left(\sum_{k}d_{kij}\sigma_{k}^{0}\right)\,q_{i}q_{j} \end{aligned}$$so the coefficient \(\left (i,j\right )\) of the matrix Kg is \(K_{ij}^{g}=\left (\sum _{k}d_{kij}\sigma _{k}^{0}\right )=K_{ji}^{g}\).
- 6.
Here and in the following, the column matrices ε, σ list the independent components of strain and stress. For example, for the Cauchy continuum, the six independent components of strain and stress tensors; for the Kirchhoff plate, the three curvatures (two flexural and one torsional) and the three moments (two bending and one torsional); and so on.
- 7.
It should be remembered that the elastic tensor has weak symmetries Eijhk = Ejihk = Eijkh = Ejikh (consequent to the symmetry of σij and εhk), as well as strong symmetries Eijhk = Ehkij, referred to the existence of an elastic energy.
- 8.
The symmetry of T has been exploited, and the indices i and h in the second term have been exchanged.
- 9.
This follows from Green’s formula in space:
$$\displaystyle \begin{aligned} \underset{\mathcal{V}}{\int}f\:g_{,j}\mathrm{d}V=-\underset{\mathcal{V}}{\int}f_{,j}\:g\mathrm{d}V+\underset{\mathcal{S}}{\int}f\:g\:n_{j}\mathrm{d}S \end{aligned}$$where \(\mathbf {n}=\left (n_{i}\right )\) is the outward normal.
- 10.
The discussion, for the sake of brevity, is limited to the field equation, by ignoring the boundary conditions. These latter, however, can be thought as incorporated in the operator.
- 11.
Since \(\boldsymbol {\mathcal {E}}^{*}\) is a two-dimensional matrix and u a column matrix, \(\boldsymbol {\mathcal {E}}_{0,\mathbf {u}}^{*}\) is a three-dimensional matrix.
- 12.
- 13.
The analogy with the response of the harmonic oscillator to a harmonic excitation should be noticed again.
- 14.
Indeed, the second-order TPE, \(\Pi =\frac {1}{2}\left (k_{e}-\mu k_{g}\right )q^{2}-\left (k_{e}q_{0}+\tilde {f}\right )q\), has second derivative \(\frac {\partial ^{2}\Pi }{\partial q^{2}}=k_{e}-\mu k_{g}>0\) when μ < μc, and second derivative \(\frac {\partial ^{2}\Pi }{\partial q^{2}}<0\) when μ > μc.
- 15.
This result is due to the fact that the stability limit curve on the bifurcation diagram, i.e., the locus on which the second derivative of the energy vanishes (e.g., the dash-dotted curve in Fig. 5.3), crushes on the line μ = μc, due to the truncation of the energy. Therefore, all the super-critical states are (erroneously) unstable in the linearized theory.
- 16.
Here, the index c is suppressed on μck and uck, since, in the linearized context, no notational confusion can arise with the series expansions of load and displacements.
- 17.
Equations 6.45 are proved as follows. Written the eigenvalue problem for two distinct k and h, premultiplying for uh and uk respectively, one has:
$$\displaystyle \begin{aligned}{\mathbf{u}}_{h}^{T}\left({\mathbf{K}}_{e}+\mu_{k}{\mathbf{K}}_{g}\right){\mathbf{u}}_{k}=\mathbf{0}\\ {\mathbf{u}}_{k}^{T}\left({\mathbf{K}}_{e}+\mu_{h}{\mathbf{K}}_{g}\right){\mathbf{u}}_{h}=\mathbf{0} \end{aligned}$$By subtracting these equations member to member, and exploiting the symmetry of the two matrices, it follows:
$$\displaystyle \begin{aligned} \left(\mu_{k}-\mu_{h}\right){\mathbf{u}}_{h}^{T}{\mathbf{K}}_{g}{\mathbf{u}}_{k}=0 \end{aligned}$$from which \({\mathbf {u}}_{h}^{T}{\mathbf {K}}_{g}{\mathbf {u}}_{k}=0\) for h ≠ k (orthogonality property); hence, \({\mathbf {u}}_{h}^{T}{\mathbf {K}}_{e}{\mathbf {u}}_{k}=0\) for h ≠ k. On the other hand, if h = k, given the arbitrariness of the length of the eigenvector, it is possible to take \({\mathbf {u}}_{k}^{T}{\mathbf {K}}_{g}{\mathbf {u}}_{k}=1\) (normalization condition), and consequently \({\mathbf {u}}_{k}^{T}{\mathbf {K}}_{e}{\mathbf {u}}_{k}=-\mu _{k}\).
- 18.
The imperfections are assumed to be of generic form. For particular forms, close to the higher modes, the contribution of these latter may be relevant. However, when μ → μ1 the first mode prevails, whatever the imperfection.
- 19.
For example, when an Euler beam, compressed by a force P smaller than the critical load Pc, is subject to transverse loads, the relevant abscissa-dependent bending moment is \(M^{+}\left (x\right )=\frac {1}{1-\frac {P}{P_{c}}}M_{I}\left (x\right )\), where \(M_{I}\left (x\right )\) is the law established by the linear theory, in which the effects of the axial force are ignored. This example will be addressed in Sect. 7.4.
- 20.
Differently from what was done in the Sect. 5.6, here two Lagrangian parameters are adopted, which reproduce circumstances that will be systematically encountered in the analysis of continuous systems. In particular, these parameters allow to express the external work as linear in the displacements.
- 21.
The reasons for the different degree of truncation will be explained immediately.
- 22.
For this reason, in Eqs. 6.53, θ has been truncated at the first order and Δℓ at the second order.
References
Bolotin, V.V.: Nonconservative problems of the theory of elastic stability. Macmillan, London (1963)
Como, M.: Theory of elastic stability (in Italian). Liguori, Napoli (1967)
Pignataro, M., Rizzi, N., Luongo, A.: Stability, bifurcation and postcritical behaviour of elastic structures. Elsevier, Amsterdam (1990)
Timoshenko, S.P., Gere, J.M.: Theory of elastic stability. McGraw-Hill, New York (1961)
van Der Heijden, A.M.A.: W.T. Koiter’s elastic stability of solids and structures. Cambridge University Press, Cambridge (2009)
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Luongo, A., Ferretti, M., Di Nino, S. (2023). Linearized Theory of Buckling. In: Stability and Bifurcation of Structures. Springer, Cham. https://doi.org/10.1007/978-3-031-27572-2_6
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