Encyclopedia of Continuum Mechanics

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| Editors: Holm Altenbach, Andreas Öchsner

Analytical Stability Considerations in Lightweight Design

  • Melanie TodtEmail author
  • Franz G. Rammerstorfer
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DOI: https://doi.org/10.1007/978-3-662-53605-6_7-1

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Lightweight structures are thin walled and/or slender; hence, they are prone to the occurrence of structural instabilities under various loading conditions. In this chapter analytical equations for estimating the critical loads of some important structures, such as beams, plates, and cylindrical shells, are presented. All equations discussed are based on a static stability criterion and limited to linear elastic material behavior.

Stability Criteria and Post-buckling Behavior

Depending on the stiffness of the structure and the loading conditions, different forms for the loss of structural stability can be observed involving bifurcation of equilibria, snap-through, snapback, and flutter instabilities. Reaching the plastic limit load can also be considered as an occurrence of a structural instability but is not further discussed within this chapter.

Bifurcation of equilibrium is characterized by a critical state at which the system switches from a current load-displacement path to an energetically more favorable one. Buckling of beams, plates, or cylindrical shells are examples for equilibrium bifurcations. Snap-through occurs if the structural stiffness of the system with respect to the applied load decreases with an increase in load. When the critical load is reached, the structural stiffness has completely vanished, and the structure snaps through dynamically. This can be observed, e.g., for flat von Mises frames (see, e.g., Bažant and Cedolin 2010, pp. 228) or flat arches and caps. A turning point with a vertical tangent within the load-displacement path indicates a so-called snapback. Snapback can be observed, e.g., for so-called Lee frames (Lee et al., 1968). Characteristic load-displacement diagrams for the different forms of structural instabilities are shown in Fig. 1. Flutter instabilities are characterized by a dynamic swing-up of alternating deformations for loads beyond the critical load and may occur for structures subjected to non-conservative follower loads. If a flutter instability appears, the critical load can only be detected with a dynamic stability criterion.
Fig. 1

Different forms of structural instabilities: (a) bifurcation of equilibrium, (b) snap-through, (c) snapback (under the assumption of a displacement-controlled process)

Herein, all considerations are limited to conservative loads, and hence, the occurrence of flutter instabilities is excluded.

Depending on the geometry of the structure and its local stiffness properties, different global and/or local instabilities may occur; see Fig. 2. Strictly speaking, the overall critical load of the structure is determined by the lowest buckling load obtained under consideration of all possible local and global instabilities. This means, the critical load is the load at which an instability appears first. This does not necessarily mean total collapse of the structure. However, the stiffness of the structure changes, and, consequently, stresses are redistributed having an influence on the structural behavior under further load increase.
Fig. 2

Different forms of local (ae) and global (f) instabilities involving (a) local Euler buckling of beams, (b) buckling of cylindrical shells, (c) local buckling of thin-walled profiles, (d) wrinkling of the face layers or shear buckling of the core of a sandwich, (e) dimpling of sandwich face layers on a honeycomb core, and (f) global buckling of the beam. ((ae) taken from Rammerstorfer and Daxner (2009) with permission)

Within this chapter only bifurcation type instabilities are considered. The bifurcation point within the load-displacement diagram, i.e., the critical load, is detected using a static stability criterion. The characteristic of the bifurcation point, i.e., the nature of the post-buckling path, determines the imperfection sensitivity of the structure; see Fig. 3. For structures with a stable bifurcation point, such as beams or plates, the load can be further increased beyond the critical point. Such structures are not imperfection sensitive, but imperfections lead to a change from a stability problem to a pure stress problem. For structures, which possess an unstable bifurcation point, the load-carrying capacity is significantly reduced beyond the critical point. In this case, small imperfections turn the bifurcation type instability into a snap-through type instability accompanied by a significantly lower critical load as estimated for the perfect structure. Hence, an unstable bifurcation point indicates imperfection sensitivity of the structure. Accessing the post-buckling behavior of a structure requires to investigate the nonlinear problem, which is beyond the scope of this chapter. In the following, the focus is on determining the critical load of structures which behave linearly with negligible small deformations in the pre-buckling regime.
Fig. 3

Load-displacement diagrams for structures with stable (a) and unstable (b and c) post-buckling behavior. The parameters λ and w are measures for the applied load and the resulting characteristic deformations, respectively

General Concept of Static Analytic Stability Considerations

The general concept for estimating the critical load of a structure is exemplified for the simply supported Bernoulli-Euler beam shown in Fig. 4. The beam is assumed to be slender and has a constant cross section along its length. The cross section is symmetric in y- and z-direction, and the compressive load P acts in the cross section’s center of gravity. Furthermore, the beam axis is assumed to be inextensible.
Fig. 4

Simply supported beam subjected to a compressive axial force (a) and dependence of the buckling length lB on the boundary conditions (b)

For estimating the critical load by using a static stability criterion, first, the equilibrium conditions are formulated on the deformed structure (second-order theory); see Fig. 4a. This leads to the fourth-order differential equation
$$\displaystyle \begin{aligned} \frac{\partial}{\partial x^2}\left(EJ\frac{\partial^2 w}{\partial x^2}\right)+P\frac{\partial^2 w}{\partial x^2}=0 \end{aligned} $$
(1)
where w = w(x) denotes the deflection of the beam and E and J are the Young’s modulus and the cross section’s moment of inertia, respectively. Since J(x) is assumed to be constant, Eq. (1) simplifies to
$$\displaystyle \begin{aligned} \frac{\partial^2 w}{\partial x^2}+\lambda^2w=0,\quad \mathrm{with}\quad\lambda = \sqrt{\frac{P}{EJ}}. \end{aligned} $$
(2)
Second, with the Eulerian ansatz w(x) = eαx, the solution of Eq. (2) for the special case of a simply supported beam resulting in \(\alpha _{1,2}=\pm \sqrt {-\lambda ^2}\) is determined as
$$\displaystyle \begin{aligned} w=A\cos{}(\lambda x)+B\sin{}(\lambda x). \end{aligned} $$
(3)
Third, the coefficients A and B as well as the parameter λ are estimated such that Eq. (3) fulfills the kinematic boundary conditions and that a nontrivial solution, i.e., w≢0, is obtained. This leads to an eigenvalue problem with λ as eigenvalue and the corresponding solution w as eigenform. For the case of a simply supported beam with w(x = 0) = 0 and w(x = l) = 0, this leads to A = 0 and \(B\sin {}(\lambda l)=0\). A nontrivial solution requires B≠0 and \(\sin {}(\lambda l)=0\) being fulfilled for \(\lambda _n=\frac {n\pi }{l}\), where n = 1, 2, 3… is the half-wave number of the eigenform. With \(\lambda = \sqrt {\frac {P}{EJ}}\), this leads to
$$\displaystyle \begin{aligned} P_n = \frac{\pi^2n^2EJ}{l^2} , \end{aligned} $$
(4)
where the smallest buckling load, i.e., the critical load P, is obtained for n = 1. It can be shown that for different boundary conditions, Eq. (4) can be stated as
$$\displaystyle \begin{aligned} P^* = \frac{\pi^2EJ}{l_{\mathrm{B}}^2}. \end{aligned} $$
(5)
where the buckling length lB depends on the specific boundary conditions; see Fig. 4b. For the simply supported case lB = l, Eq. (5) is also known as Euler’s equation of beam buckling.
So far infinite transverse shear stiffness has been assumed (Bernoulli’s hypothesis). When taking a finite shear stiffness into account, a modified form of Eq. (2) is obtained as
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{\partial^2 w}{\partial x^2}+\lambda_{\mathrm{S}}^2w&\displaystyle =0,\quad \mathrm{with}\notag\\ \lambda_{\mathrm{S}} &\displaystyle = \sqrt{\frac{P}{EJ}/(1-\frac{P}{GA_{\mathrm{S}}})}. \end{array} \end{aligned} $$
(6)
The quantity G denotes the shear modulus, and AS = γA is the “shear area,” where A is the area of the cross section and γ ∈ [0, 1] is a correction factor dependent on the shape of the cross section; see, e.g., (Ziegler, 1998) p. 231. The critical load \(P^*_{\mathrm {S}}\) under consideration of a finite transverse shear stiffness is obtained as
$$\displaystyle \begin{aligned} P^*_{\mathrm{S}} = \frac{P^*}{1+\frac{P^*}{GA_{\mathrm{S}}}} \end{aligned} $$
(7)
with P defined according to Eq. (5).

Beams

Euler buckling introduced in the previous section is a bending type instability which is not the only form of structural instability that can occur for slender beams subjected to axial compression. In general, bending-buckling, twisting-buckling, or a combination of bending- and twisting-buckling is possible. In the following transverse shear deformations are neglected, and the axial load is assumed to act in the cross section’s center of gravity. The generalized stability problem is then described by three coupled differential equations

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle EJ_z \frac{\partial^4 v}{\partial x^4}+P\frac{\partial^2 v}{\partial x^2}+Pz_{\mathrm{M}}\frac{\partial^2 \chi}{\partial x^2}=0\notag\\ &\displaystyle &\displaystyle \quad EJ_y \frac{\partial^4 w}{\partial x^4} +P\frac{\partial^2 w}{\partial x^2}-Py_{\mathrm{M}}\frac{\partial^2 \chi}{\partial x^2}=0\notag\\ &\displaystyle &\displaystyle \quad Pz_{\mathrm{M}}\frac{\partial^2 v}{\partial x^2}-Py_{\mathrm{M}} \frac{\partial^2 w}{\partial x^2}+EC_{\mathrm{W}}\frac{\partial^4 \chi}{\partial x^4}\notag\\ &\displaystyle &\displaystyle \quad -GJ_{\mathrm{T}}\frac{\partial^2 \chi}{\partial x^2}+Pi_{\mathrm{M}}^2\frac{\partial^2 \chi}{\partial x^2} = 0 \end{array} \end{aligned} $$
(8)
where \(i_{\mathrm {M}}^2=i_{\mathrm {P}}^2+r_{\mathrm {M}}^2\) with \(r_{\mathrm {M}}^2=y_{\mathrm {M}}^2+z_{\mathrm {M}}^2\) and \(J_{\mathrm {P}}=Ai_{\mathrm {P}}^2\). The quantities v = v(x) and w = w(x) denote the deflections in y- and z-direction, respectively, and χ = χ(x) is the twist of the cross section. The parameters A , JT, and JP are the area, the torsional, and polar moments of inertia of the cross section, and CW denotes the warping resistance. The distances yM and zM between the center of gravity and the shear center of the cross section are illustrated in Fig. 5.
Fig. 5

Location of the shear center for different beam cross sections

For obtaining a solution of the set of differential equations defined in Eq. (8), an appropriate ansatz is required, which takes the applied boundary conditions into account. For a beam with, e.g., v = 0, w = 0, χ = 0, v″ = 0, w″ = 0, and χ″ = 0 at x = 0 and x = l, this ansatz reads
$$\displaystyle \begin{aligned} \begin{array}{rcl} v &\displaystyle =&\displaystyle V\sin\left(\frac{\pi x}{l}\right),\ w=W\sin\left(\frac{\pi x}{l}\right),\notag\\ \chi &\displaystyle =&\displaystyle X\sin\left(\frac{\pi x}{l}\right), \end{array} \end{aligned} $$
(9)
where for the half-wave number n = 1 is used, as it shows to give the smallest buckling load for Euler buckling; see Eq. (4). This ansatz leads to a system of three coupled algebraic equations given in matrix notation as
$$\displaystyle \begin{aligned} \underbrace{ \left[\begin{array}{ccc} EJ_z(\frac{\pi}{l})^2-P & 0 & -Pz_{\mathrm{M}}\\ 0 & EJ_y(\frac{\pi}{l})^2-P & Py_{\mathrm{M}}\\ -Pz_{\mathrm{M}} &Py_{\mathrm{M}} & EC_{\mathrm{W}}(\frac{\pi}{l})^2+GJ_{\mathrm{T}}-Pi_{\mathrm{M}}^2 \end{array}\right]}_{\boldsymbol{M}} \left[\begin{array}{c} V\\ W\\ X \end{array}\right] = \left[\begin{array}{c} 0\\ 0\\ 0 \end{array}\right]. \end{aligned} $$
(10)
A nontrivial solution vector [V, W, X]T can only be obtained if det(M) = 0 with M being the coefficient matrix in Eq. (10). For double-symmetric cross sections with Open image in new window, the individual equations in Eq. (10) become uncoupled, leading to
$$\displaystyle \begin{aligned} P_y^*&=\frac{\pi^2EJ_y}{l^2},\ P_z^*=\frac{\pi^2EJ_z}{l^2},\notag\\ P_\chi^* &= \frac{1}{i_{\mathrm{M}}^2}[EC_{\mathrm{W}}(\frac{\pi}{l})^2+GJ_{T}],\end{aligned} $$
(11)
where \(P_y^*\) and \(P_z^*\) are the critical loads for bending-buckling around the y- and z-axis, respectively, and \(P_\chi ^*\) is the critical load for twisting-type buckling. The overall critical load for the beam subjected to axial compression P is obtained as \(P^*=\min (P_y^*,P_z^*,P_\chi ^*)\).
For arbitrary cross sections, the condition det(M) = 0 leads to a cubic equation for the critical load P
$$\displaystyle \begin{aligned} &\left(1-\frac{P_z^*}{P^*}\right)\left(1-\frac{P_y^*}{P^*}\right)\left(1-\frac{P_\chi^*}{P^*} \right)\notag\\ &\quad -\left(1-\frac {P_z^*}{P^*}\right)\left(\frac{y_{\mathrm{M}}}{i_{\mathrm{M}}}\right)^2 \notag\\ &\quad - \left(1-\frac {P_y^*}{P^*}\right)\left(\frac{z_{\mathrm{M}}}{i_{\mathrm{M}}}\right)^2 =0, \end{aligned} $$
(12)
with \(P_y^*\), \(P_z^*\), and \(P_\chi ^*\) defined in Eq. (11). In general, roots \(P_i^*\) of Eq. (12) include coupled bending-twisting buckling loads. The critical load P is found as \(P^*=\min (P_1^*,P_2^*,P_3^*)\).
For beams with Jy ≫ Jz and comparably small JT, transverse shear loading as well as a bending moment around the y-axis can lead to lateral torsional buckling as exemplified in Fig. 6. For a double-symmetric cross section the problem of lateral torsional buckling is governed by two coupled differential equations,
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle EJ_z\frac{\partial^4 v}{\partial x^4}+P\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 (M_y\chi)}{\partial x^2}=0\\ &\displaystyle &\displaystyle \quad M_y \frac{\partial^2 v}{\partial x^2}+EC_{\mathrm{W}}\frac{\partial^4 \chi}{\partial x^4}\notag\\ &\displaystyle &\displaystyle \quad -(GJ_{\mathrm{T}}-Pi_{\mathrm{P}}^2)\frac{\partial^2 \chi}{\partial x^2}=0,\qquad\end{array} \end{aligned} $$
(13)
with χ(x) as angle of rotation of the cross section and v(x) as displacement in y-direction (Fig. 6). The bending moment My depends on the applied loading comprising external bending moments Mext as well as moments resulting from transverse shear loading Qext and q, respectively. The solution of Eqs. (13) with respect to the critical loading state strongly depends on the boundary and loading conditions. Critical loading states for different loading scenarios are derived, e.g., in Bažant and Cedolin (2010) (pp. 384).
Fig. 6

Lateral torsional buckling of a beam with Jy ≫ Jz and comparably small JT subjected to an applied axial compressive force Pext, to transverse shear loading Qext and q, and to an external bending moment Mext

In the given form of Eqs. (13), it is assumed that all transverse shear loads are applied along the beam axis (z = 0) as shown in Fig. 6. However, it should be noted that the z-coordinates of the points of application of the transverse shear loads have an influence on the critical loading state. Hence, for the case of transverse shear loads not being applied along the beam axis but, e.g., on the top or bottom surface of the beam, Eqs. (13) lead only to an approximation of the critical loading state.

Plates

Plates can buckle under several in-plane loading conditions involving compression, bending, and shear; see Fig. 7. In analogy to Eq. (1), the critical loading state is obtained using the differential equation for plate bending, where the equilibrium conditions are formulated for the deformed plate. Under the assumptions that the membrane forces Nxx and Nyy are positive for compression and that there is no out-of-plane loading of the plate, the corresponding differential equation according to second-order theory reads
$$\displaystyle \begin{aligned} K\varDelta\varDelta w +N_{xx} \frac{\partial^2 w}{\partial x^2} + N_{yy} \frac{\partial^2 w}{\partial y^2}+2N_{xy} \frac{\partial^2 w}{\partial x \partial y}=0; \end{aligned} $$
(14)
see, e.g., Ziegler (1998) p. 414. In Eq. (14), \(K=\frac {Et^3}{12(1-\mu ^2)}\) denotes the bending stiffness, and w = w(x, y) is the deflection of the plate. The parameters E, μ, and t are the Young’s modulus, the Poisson’s ratio, and the plate’s thickness, respectively. Plate buckling is associated with a non-zero deflection under membrane loading. The question for the smallest load \(N_{ij}^*\), corresponding to one of the load cases shown in Fig. 7, for which such a nontrivial equilibrium state occurs, leads to an eigenvalue problem. For the case of uniform compressive loading in either x- or y-direction, the solution of this eigenvalue problem is found as
$$\displaystyle \begin{aligned} \sigma^* = \kappa \pi^2 K \frac{1}{t b^2}=\kappa \frac{\pi^2}{12(1-\mu^2)}E(\frac{t}{b})^2, \end{aligned} $$
(15)
with b being the shorter edge of the plate. The parameter κ depends on the half-wave numbers n and m in directions parallel and normal to the loaded edge, respectively, as well as on the boundary conditions. For a simply supported plate, \(\displaystyle\kappa _{xx}(m,n=1)=[\frac {mb}{a}+\frac {a}{mb}]^2\) is illustrated in Fig. 8 in dependence of the plate’s aspect ratio ab. Although, Eq. (15) is derived for uniaxial loading, it can be shown that the typical form of this equation is also valid for other in-plane loading conditions, such as bending or shear loading. Different loading and boundary conditions can be accounted for with the parameter \(k_{ij}=\kappa _{ij} \frac {\pi ^2}{12(1-\mu ^2)}\) leading to
$$\displaystyle \begin{aligned}{}[\sigma_{xx}^*,\sigma_{yy}^*,\sigma_{\mathrm{B}}^*,\tau^*]=[k_{xx},k_{yy}, k_{\mathrm{B}},k_{\tau}]\ E(\frac{t}{b})^2 \end{aligned} $$
(16)
with kxx, kyy , kB, and kτ being the buckling factors for uniform compressive loading in x- or y-direction, bending, and shear loading. The stresses \(\sigma _{xx}^*\), \(\sigma _{yy}^*\), \(\sigma _{\mathrm {B}}^*\), and τ are the corresponding critical stresses. Values for buckling factors kij (for μ = 0.3) for different boundary and loading conditions can be found, e.g., in Hertel (1986) pp. 65–68 and Rammerstorfer and Daxner (2009) p. 27.
Fig. 7

Plates subjected to (a) compressive membrane loading, (b) bending, and (c) shear loading

Fig. 8

Buckling factor κxx for a simply supported uniaxially loaded plate depending on the plate dimensions for the case n = 1

Plates are often subjected to combinations of different loading conditions. This can be accounted for by interaction equations
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &&\mbox{compression}\\mbox{and}\\mbox{shear:}\quad\big(\frac{\sigma_{xx}}{\sigma_{xx}^*}\big)+\big(\frac{\tau}{\tau^*}\big)^2\le 1 \end{array} \end{aligned} $$
(17)
$$\displaystyle \begin{aligned} \begin{array}{rcl} && \mbox{bending}\\mbox{and}\\mbox{shear:}\quad \big(\frac{\sigma_{\mathrm{B}}}{\sigma_{\mathrm{B}}^*}\big)^2+\big(\frac{\tau}{\tau^*} \big)^2\le1{} \end{array} \end{aligned} $$
(18)
$$\displaystyle \begin{aligned} \begin{array}{rcl} && \mbox{bi-axial}\\mbox{compression:}\notag\\ &&\quad \left\{\begin{array}{ll} \frac{\sigma_{yy}}{\sigma_{yy}^*}\le 1 & \mathrm{for}\ \frac{\sigma_{xx}}{\sigma_{xx}^*}\le0.5 \\ \frac{1}{4}\big(\frac{\sigma_{yy}}{\sigma_{yy}^*}\big)\big(\frac{\sigma_{xx}}{\sigma_{xx}^*} \big)^{-1}{+}\big(\frac{\sigma_{xx}}{\sigma_{xx}^*}\big){\le} 1 & \mathrm{for}\ \frac{\sigma_{xx}}{\sigma_{xx}^*}>0.5 \end{array} \right. {} \end{array} \end{aligned} $$
(19)
with \(\sigma _{xx}^*\), \(\sigma _{yy}^*\), \(\sigma _{\mathrm {B}}^*\), and τ each calculated separately according to Eq. (16) and σxx, σyy, σB, and τ being the present stresses due to the applied loading. If the respective inequality (Eqs. 17, 18, and 19) is fulfilled, buckling can be excluded as long as the assumption of linear elastic material behavior holds true.

Cylindrical Shells

Thin-walled cylindrical shells can buckle when subjected to axial compression, an external excess pressure, torsion, bending, and combinations of these loading scenarios. All considerations made in this section are only valid if the obtained critical stresses are far smaller than the yield stress of the material. For determining the critical state of a cylinder with radius R, height l, and wall thickness t subjected to axial compressive membrane force Nxx (positive for compression) and an external pressure p, the corresponding differential equation reads
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} K\varDelta^4 w &\displaystyle +&\displaystyle \frac{D}{R^2}(1-\mu^2)\frac{\partial^4 w}{\partial x ^4} + N_{xx}\varDelta^2 \frac{\partial^2 w}{\partial x ^2} \notag\\ &\displaystyle +&\displaystyle \frac{p}{R}\varDelta^2 \frac{\partial^2 w}{\partial \varphi^2}=0. \end{array} \end{aligned} $$
(20)
In Eq. (20) cylindrical coordinates x, r, and φ with fixed radial coordinate r = R, i.e., cylinder shell coordinates, are used; see Fig. 9. The parameters \(\displaystyle K=\frac {Et^3}{12 (1-\mu ^2)}\) and \(\displaystyle D=\frac {Et}{1-\mu ^2}\) denote the bending and membrane stiffness of the shell, respectively, and \(\displaystyle\varDelta = \frac {\partial ^2}{\partial x^2}+\frac {1}{R^2}\frac {\partial ^2 }{\partial \varphi ^2}\) is the Laplacian operator in cylindrical coordinates. Equation (20) is obtained from Donnell’s equations (Donnell, 1932) by adding the term for the external pressure p.
Fig. 9

Buckling coefficient of a cylinder subjected to axial compressive loading for different parameters \(\beta = \frac {t^2}{12 R^2}\). The parameters m and n denote the half-wave number in axial direction and the wave number in circumferential direction, respectively. Donnell’s solution is (Eq. 22) given by the dashed line and Flügge’s solution (Eq. 23) by the solid lines

For p = 0 and radially and tangentially fixed edges, the critical axial membrane stress is obtained as (see Pflüger (1975) p. 262)
$$\displaystyle \begin{aligned} \sigma^*=\frac{N_{xx}^*}{t}= \kappa_{\mathrm{A}}\frac{E}{1-\mu^2}. \end{aligned} $$
(21)
Therein,
$$\displaystyle \begin{aligned} \kappa_{\mathrm{A}} &= \frac{(1-\mu^2)\lambda^4+\beta(\lambda^2+n^2)^4}{ \lambda^2(\lambda^2+n^2)^2},\ \mathrm{with}\notag\\ \beta &= \frac{t^2}{12R^2}\ \mathrm{and}\ \lambda=m\pi\frac{R}{l}, \end{aligned} $$
(22)
is the buckling coefficient, where m is the half-wave number in axial direction and n is the wave number in circumferential direction. This solution is often referred to as Donnell’s solution of cylinder buckling under axial compression. A more generalized solution can be found in Flügge (1932), where the structure of Eq. (21) is the same but the buckling factor is obtained as
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \kappa_{\mathrm{A}} &\displaystyle =&\displaystyle \frac{1}{\lambda^2(\lambda^2+n^2)^2+\lambda^2n^2}\notag\\ &\displaystyle &\displaystyle \{ (1-\mu^2)\lambda^4+\beta[(\lambda^2+n^2)^4\notag\\ &\displaystyle &\displaystyle -2(\mu\lambda^6+3\lambda^4n^2 + (4-\mu)\lambda^2n^4+n^6)\notag\\ &\displaystyle &\displaystyle +2(2-\mu)\lambda^2n^2+n^4]\}; \end{array} \end{aligned} $$
(23)
see also Pflüger (1975) p. 488. The buckling factors κA, as defined by Eqs. (22) and (23), are depicted in Fig. 9. Cylinders subjected to axial compression can be categorized in three groups depending on their normalized length l∕(mR), viz., short, middle, and long. As can be seen from Fig. 9, this classification also depends on the parameter β.
For a cylinder of middle length, Donnell’s solution (Eq. 22) is in good agreement with Flügge’s solution (Eq. 23), and the buckling coefficient is almost independent of the wave number n. Hence, a more simplified solution can be obtained from Eq. (22) when only the connection line of the minima of the corresponding curve in Fig. 9 is considered. The wave number n, relevant for buckling, can then be obtained from the condition \(\frac {\partial \kappa }{\partial n} =0\), leading to an expression n = f(l∕(mR)), which describes the relation between n and the normalized length of the cylinder. Although only natural numbers are admissible for n, the relation n = f(l∕(mR)) can be inserted into Eq. (22) allowing to eliminate its dependency on n and leading to
$$\displaystyle \begin{aligned} \sigma^*=\frac{E}{\sqrt{3(1-\mu^2)}}\frac{t}{R}. \end{aligned} $$
(24)
The value σ is only a theoretical value as middle-long cylinders possess an unstable post-buckling behavior and, therefore, show a high imperfection sensitivity; see Fig. 3. This imperfection sensitivity has to be considered by a knockdown factor αA which is, e.g., suggested as \(\alpha _{\mathrm {A}}=\left (1/\sqrt {1+\frac {R}{100\ t}}\right )\) in Pflüger (1975) for geometrical imperfections in the range of R∕200. An internal excessive pressure reduces the imperfection sensitivity being considered by \(\bar {\alpha }_{\mathrm {A}} = \alpha _{\mathrm {A}} + \varDelta \alpha _{\mathrm {A}}\). The parameter ΔαA depends on the internal pressure pi, the Young’s modulus of the cylinder material, and the geometry of the cylinder. Values for ΔαA can be found in engineering standards for cylinder buckling. Hence, the relevant critical stress \(\sigma ^*_{\mathrm {pr}}\) is obtained as
$$\displaystyle \begin{aligned} \sigma^*_{\mathrm{pr}} = (\alpha_{\mathrm{A}}+\varDelta\alpha_{\mathrm{A}})\frac{E}{\sqrt{3(1-\mu^2)}}\frac{t}{R }, \end{aligned} $$
(25)
with ΔαA = 0 for pi = 0.
The asymptotic solution for Open image in new window (Open image in new window) for the critical axial membrane stress is obtained from Eq. (23) as
$$\displaystyle \begin{aligned} \sigma^*_{S} = \frac{\pi^2}{12(1-\mu^2)}E(\frac{t}{l})^2. \end{aligned} $$
(26)
The critical stress obtained by Eq. (26) is equal to the critical stress of a simply supported plate subjected to uniaxial compressive loading, where the loaded edge has an infinite width; compare Eq. (15). In practice, the critical axial stress of a short cylinder is estimated as \(\sigma ^* = \max (\sigma ^*_{\mathrm {pr}},\sigma ^*_{S})\).
For lR → (λ → 0) and n = 1, Eq. (23) leads to
$$\displaystyle \begin{aligned} \sigma_{B}^* = \frac{\pi^2}{2}E\left(\frac{r}{l}\right)^2, \end{aligned} $$
(27)
being the equation of beam buckling for a simply supported beam with a circular ring cross section with t ≪ R. From Fig. 9 it can be seen that for long cylinders, also the case of cylinder buckling with n = 2 is of importance leading to
$$\displaystyle \begin{aligned} \sigma_{\mathrm{L}}^* = \frac{\sqrt{12}}{10\sqrt{1-\mu^2}}E\frac{t}{R}. \end{aligned} $$
(28)
In practice, the critical axial stress for long cylinders is estimated as \(\sigma ^*=\min (\sigma _{B}^*,\alpha _{\mathrm {A}}\sigma _{\mathrm {L}}^*)\), where the imperfection sensitivity of cylinder buckling is accounted for by the parameter αA.

Note: If not both ends of the cylinder are simply supported, l in Eqs. (26) and (27) is to be replaced by lB according to Fig. 4.

For practical applications, the theoretical critical stress for cylinder buckling under axial membrane loading can be calculated as
$$\displaystyle \begin{aligned} \sigma^*=kE\frac{t}{R}. \end{aligned} $$
(29)
The parameter k is given in a diagram similar to those in Fig. 9, but where the dependency on the parameter β has been eliminated; see, e.g., (Rammerstorfer and Daxner (2009) p. 31). This allows a classification of cylinders in long, middle, and short independent of β. Of course, expressing k as function of the cylinder length leads to the same relations as given by Eqs. (24), (26), and (28). Hence, for long and short cylinders, the same considerations as made above apply. The imperfection sensitivity also needs to be accounted for by the knockdown factor αA.

If the axial stress is not the result of axial compressive loading but of bending, the critical stress can be estimated in the same way as discussed for axial membrane loading. However, a slightly higher value for αA can be used.

For buckling under external excess pressure p with Nxx = 0, the solution obtained by Flügge (1932) reads (see, e.g., Pflüger (1975) p. 148)
$$\displaystyle \begin{aligned} p^* = \kappa_{\mathrm{P}} \frac{E}{1-\mu^2}\frac{t}{R} \end{aligned} $$
(30)
with κP being the buckling factor given as
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \kappa_{\mathrm{P}} &\displaystyle =&\displaystyle \frac{1}{n^2(\lambda^2+n^2)^2-(3\lambda^2n^2+n^4)}\notag\\ &\displaystyle &\displaystyle \{(1-\mu^2)\lambda^4+\beta[(\lambda^2+n^2)^4\notag\\ &\displaystyle &\displaystyle -2(\mu\lambda^6+3\lambda^4n^2\notag\\ &\displaystyle &\displaystyle +(4-\mu)\lambda^2n^4+n^6)\notag\\ &\displaystyle &\displaystyle +2(2-\mu)\lambda^2n^2+n^4]\} \end{array} \end{aligned} $$
(31)
where a constant pressure along the cylinder height l is assumed. The parameters β and λ are defined according to Eq. (23). Equation (31) gives the buckling factor κP for a cylinder with radially and tangentially fixed but otherwise simply supported ends. According to Eq. (31), the critical pressure is always obtained for a half-wave number m = 1 in direction of the cylinder axis, see also Fig. 10. This is in contrast to buckling under axial membrane loading where m depends on the cylinder geometry. The fact that p is independent of m leads to a strong dependency of p on the cylinder length and the boundary conditions. This dependency can be accounted for by a factor cB, which describes the relation of the buckling factor κP as given by Eq. (31) to the buckling factor obtained for the respective boundary conditions. Furthermore, the imperfection sensitivity has to be considered using a knockdown factor αP. Cylinders subjected to external pressure are less imperfection sensitive than cylinders subjected to axial membrane loading. Hence, the respective knockdown factor αP is larger than αA. A typical value is αP = 0.7. For practical applications, the critical pressure \(p^*_{\mathrm {pr}}\) for arbitrary boundary conditions can be estimated from the critical pressure p of the perfect simply supported cylinder as
$$\displaystyle \begin{aligned} p^*_{\mathrm{pr}}=\alpha_{\mathrm{P}} c_{\mathrm{B}}p^*. \end{aligned} $$
(32)
Values for the knockdown factor αP and the factor cB can be found in engineering standards for cylinder buckling.
Fig. 10

Buckling coefficient of a cylinder subjected to an external excess pressure for different parameters \(\beta = \frac {t^2}{12 R^2}\). The parameters m and n denote the half-wave number in axial direction and the wave number in circumferential direction, respectively. Black solid lines correspond to a half-wave number m = 1

For very long cylinders (lR →), the buckling pressure \(p^*_{\mathrm {L}}\) is obtained as
$$\displaystyle \begin{aligned} p^*_{\mathrm{L}}= \frac{E}{4(1-\mu^2)}(\frac{t}{R})^3 \end{aligned} $$
(33)
being the equation for buckling of a circular ring (with E replaced by E∕(1 − μ2)) under radial loading taking the follower load character of p into account. It is independent of the cylinder length and, consequently, also independent of the applied boundary conditions (cB = 1).
The critical shear stress τ for torsional loading can be estimated as
$$\displaystyle \begin{aligned} \tau^* = \kappa_{\mathrm{T}}\frac{E}{1-\mu^2} \end{aligned} $$
(34)
with κT as corresponding buckling factor. The factor κT strongly depends on the geometry of the cylinder and on the applied boundary conditions. Values for the buckling factor can be found, e.g., in Pflüger (1975), pp 492–493 where, e.g., for a cylinder with radially immoveable edges, κT is given as
$$\displaystyle \begin{aligned} \begin{array}{rcl} \kappa_{\mathrm{T}} &\displaystyle =&\displaystyle \frac{1}{4(\lambda^5+7\lambda^3+12\lambda)}\lbrace (1-\mu^2)\lambda^4 \\ &\displaystyle +&\displaystyle \beta[ \lambda^8+2(8-\mu)\lambda^6+72\lambda^4\notag\\ &\displaystyle +&\displaystyle 24(6+\mu)\lambda^2+144] \rbrace.\qquad\end{array} \end{aligned} $$
(35)
The critical shear stress obtained by Eq. (34) is again a theoretical value. The imperfection sensitivity of the cylinder has to be considered by a corresponding knockdown factor αT, leading to \(\tau ^*_{\mathrm {pr}}=\alpha _{\mathrm {T}}\tau ^*\). For the parameter αT, values according to local engineering standards and regulations have to be used (Fig. 10).

Equation (34) can also be used to estimate the critical shear stress \(\tau ^*_{\mathrm {Q}}\) for transverse shear loading. This is rather a conservative estimation, as it does not account for the nonuniform shear stress distribution within the cross section.

When a cylinder is subjected to a combination of different loadings, this can be accounted for by corresponding interaction equations:
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle &\displaystyle \mbox{axial}\\mbox{compression}\\mbox{+}\\mbox{ext.}\\mbox{pressure:}\notag\\ &\displaystyle &\displaystyle \quad \frac{p}{p^*_{\mathrm{pr}}}+\frac{\sigma_{xx}}{\sigma^*_{\mathrm{pr}}}\le1 \end{array} \end{aligned} $$
(36)
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \mbox{axial}\\mbox{compression}\\mbox{+}\\mbox{torsion:}\notag\\ &\displaystyle &\displaystyle \quad \left(\frac{\tau}{\tau^*_{\mathrm{pr}}}\right)^2+\frac{\sigma_{xx}}{\sigma^*_{\mathrm{pr}}}\le 1{} \end{array} \end{aligned} $$
(37)
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \mbox{bending}\\mbox{+}\\mbox{transverse}\\mbox{shear:} \notag\\ &\displaystyle &\displaystyle \quad \max\left(\frac{\sigma_{xx}}{\sigma^*_{\mathrm{pr}}},\frac{\tau}{\tau^*_{\text{Q, pr}}} \right)\le \frac{\sqrt{2}}{2}{}. \end{array} \end{aligned} $$
(38)
If the respective inequality (Eqs. (36), (37), and (38)) is fulfilled, cylinder buckling can be excluded.

Cross-References

References

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Lightweight Design and Structural BiomechanicsTU WienViennaAustria

Section editors and affiliations

  • Franz G. Rammerstorfer
    • 1
  • Melanie Todt
    • 2
  • Isabella C. Skrna-Jakl
    • 3
  1. 1.Institut für Leichtbau und Struktur-BiomechanikTU WienWienAustria
  2. 2.Institut für Leichtbau und Struktur-BiomechanikTU WienWienAustria
  3. 3.Institut für Leichtbau und Struktur-BiomechanikTU WienWienAustria