# Analytical Stability Considerations in Lightweight Design

**DOI:**https://doi.org/10.1007/978-3-662-53605-6_7-1

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## Synonyms

## Definitions

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.

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

## General Concept of Static Analytic Stability Considerations

*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.

*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

*w*(

*x*) = e

^{αx}, 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

*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

*P*

^{∗}, is obtained for

*n*= 1. It can be shown that for different boundary conditions, Eq. (4) can be stated as

*l*

_{B}depends on the specific boundary conditions; see Fig. 4b. For the simply supported case

*l*

_{B}=

*l*, Eq. (5) is also known as Euler’s equation of beam buckling.

*G*denotes the shear modulus, and

*A*

_{S}=

*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

*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

*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*,

*J*

_{T}, and

*J*

_{P}are the area, the torsional, and polar moments of inertia of the cross section, and

*C*

_{W}denotes the warping resistance. The distances

*y*

_{M}and

*z*

_{M}between the center of gravity and the shear center of the cross section are illustrated in Fig. 5.

*v*= 0,

*w*= 0,

*χ*= 0,

*v*″ = 0,

*w*″ = 0, and

*χ*″ = 0 at

*x*= 0 and

*x*=

*l*, this ansatz reads

*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

*V*,

*W*,

*X*]

^{T}can only be obtained if det(

*) = 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*

**M***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 \(P^*=\min (P_y^*,P_z^*,P_\chi ^*)\).

*) = 0 leads to a cubic equation for the critical load*

**M***P*

^{∗}

*P*

^{∗}is found as \(P^*=\min (P_1^*,P_2^*,P_3^*)\).

*J*

_{y}≫

*J*

_{z}and comparably small

*J*

_{T}, 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,

*χ*(

*x*) as angle of rotation of the cross section and

*v*(

*x*) as displacement in

*y*-direction (Fig. 6). The bending moment

*M*

_{y}depends on the applied loading comprising external bending moments

*M*

_{ext}as well as moments resulting from transverse shear loading

*Q*

_{ext}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).

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

*N*

_{xx}and

*N*

_{yy}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

*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

*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

*a*∕

*b*. 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

*k*

_{xx},

*k*

_{yy},

*k*

_{B}, and

*k*

_{τ}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

*k*

_{ij}(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.

*τ*

^{∗}each calculated separately according to Eq. (16) and

*σ*

_{xx},

*σ*

_{yy},

*σ*

_{B}, 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

*R*, height

*l*, and wall thickness

*t*subjected to axial compressive membrane force

*N*

_{xx}(positive for compression) and an external pressure

*p*, the corresponding differential equation reads

*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*.

*p*= 0 and radially and tangentially fixed edges, the critical axial membrane stress is obtained as (see Pflüger (1975) p. 262)

*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

*κ*

_{A}, 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*.

*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

*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

*α*

_{A}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

*Δα*

_{A}depends on the internal pressure

*p*

_{i}, the Young’s modulus of the cylinder material, and the geometry of the cylinder. Values for

*Δα*

_{A}can be found in engineering standards for cylinder buckling. Hence, the relevant critical stress \(\sigma ^*_{\mathrm {pr}}\) is obtained as

*Δα*

_{A}= 0 for

*p*

_{i}= 0.

*l*∕

*R*→

*∞*(

*λ*→ 0) and

*n*= 1, Eq. (23) leads to

*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

*α*

_{A}.

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

*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

*α*

_{A}.

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 *α*_{A} can be used.

*p*with

*N*

_{xx}= 0, the solution obtained by Flügge (1932) reads (see, e.g., Pflüger (1975) p. 148)

*κ*

_{P}being the buckling factor given as

*l*is assumed. The parameters

*β*and

*λ*are defined according to Eq. (23). Equation (31) gives the buckling factor

*κ*

_{P}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

*c*

_{B}, which describes the relation of the buckling factor

*κ*

_{P}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

*α*

_{P}. Cylinders subjected to external pressure are less imperfection sensitive than cylinders subjected to axial membrane loading. Hence, the respective knockdown factor

*α*

_{P}is larger than

*α*

_{A}. A typical value is

*α*

_{P}= 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

*α*

_{P}and the factor

*c*

_{B}can be found in engineering standards for cylinder buckling.

*l*∕

*R*→

*∞*), the buckling pressure \(p^*_{\mathrm {L}}\) is obtained as

*E*replaced by

*E*∕(1 −

*μ*

^{2})) 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 (

*c*

_{B}= 1).

*τ*

^{∗}for torsional loading can be estimated as

*κ*

_{T}as corresponding buckling factor. The factor

*κ*

_{T}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,

*κ*

_{T}is given as

*α*

_{T}, leading to \(\tau ^*_{\mathrm {pr}}=\alpha _{\mathrm {T}}\tau ^*\). For the parameter

*α*

_{T}, 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.

## Cross-References

## References

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