김경종
412 lines
32 KiB
Markdown
412 lines
32 KiB
Markdown
<!-- source-page: 461 -->
|
||
|
||
reference configuration defined by $u_{2}$ and $w_{2}$ . Thus the reference configuration may display an angle $\theta$ far different from its initial value $\theta = 0$ .
|
||
|
||
A correspondence with preceding equations may be recognized as follows. If displacements $u_{2}$ and $w_{2}$ in Eq. 14.4-14 are sufficiently small, the third matrix in Eq. 14.4-14, which depends quadratically on displacements, may be discarded in comparison with the other two matrices. If in addition we substitute $(AE/L)u_{2}=P$ , set $w_{2}=0$ , and presume that $u_{2}<<L$ , then there is a further simplification to
|
||
|
||
$$
|
||
\left(\frac {A E}{L} \left[ \begin{array}{l l} 1 & 0 \\ 0 & 0 \end{array} \right] + \frac {P}{L} \left[ \begin{array}{l l} 3 & 0 \\ 0 & 1 \end{array} \right]\right) \left\{ \begin{array}{l} d u _ {2} \\ d w _ {2} \end{array} \right\} = \left\{ \begin{array}{l} d R _ {x} \\ d R _ {z} \end{array} \right\} \tag {14.4-15}
|
||
$$
|
||
|
||
The second matrix in Eq. 14.4-15 is analogous to the lower right 2 by 2 submatrix in Eq. 14.4-9, except that the number 3 appears instead of the number 1. Equation 14.4-15 can be reconciled with Eq. 14.2-11 by means of the same arguments that are applied to Eq. 14.4-9; that is, by noting that in practice $AE \gg P$ . Thus the number 3 is discarded and the second matrix in Eq. 14.4-15 is recognized as $[K_{\sigma}]$ .
|
||
|
||
In summary, by discarding the quadratic stiffness terms in Eq. 14.4-14, we obtain two forms of incremental equations:
|
||
|
||
$$
|
||
([ \mathbf {K} ] + [ \mathbf {N} _ {1} ]) \{d \mathbf {D} \} = \{d \mathbf {R} \} \quad \text { and } \quad ([ \mathbf {K} ] + [ \mathbf {K} _ {\sigma} ]) \{d \mathbf {D} \} = \{d \mathbf {R} \} \tag {14.4-16}
|
||
$$
|
||
|
||
Equations 14.4-16 are symbolic statements of approximations that can be applied to linearly elastic structures in general, not only to the elastic bar. Both equations are linearized—that is, made to depend on first powers of displacements—by discarding quadratic terms. Either equation may be used to compute a buckling load. Upon buckling, infinitesimal displacements $\{dD\}$ occur, measured with reference to a prebuckling configuration $\{D\}$ . In computing a buckling load, use of $[K_{\sigma}]$ implies that prebuckling rotations in $\{D\}$ are zero, whereas use of $[N_{1}]$ implies only that prebuckling rotations are small. For initially flat plates and initially straight columns, the two formulations yield the same buckling load. In practical problems where prebuckling rotations are not negligible, use of $[N_{1}]$ seems always to yield a smaller buckling load than does use of $[K_{\sigma}]$ , but there is no guarantee that either approach will always err on the high side or the low side of the actual collapse load [14.1].
|
||
|
||
# 14.5 BIFURCATION BUCKLING
|
||
|
||
A bifurcation buckling load is the load for which a reference configuration of the structure and an infinitesimally close (buckled) configuration are both possible equilibrium configurations. As a buckling displacement $\{dD\}$ takes place from a reference configuration $\{D\}$ , the load does not change. Accordingly, from Eqs. 14.4-16,
|
||
|
||
$$
|
||
([ \mathbf {K} ] + [ \mathbf {N} _ {1} ]) \{d \mathbf {D} \} = \{\mathbf {0} \} \quad \text { or } \quad ([ \mathbf {K} ] + [ \mathbf {K} _ {\sigma} ]) \{d \mathbf {D} \} = \{\mathbf {0} \} \tag {14.5-1}
|
||
$$
|
||
|
||
We ask for the level of deformation (symbolized by $[N_{1}]$ ) or the level of stress (symbolized by $[K_{\sigma}]$ ) such that a solution $\{dD\}$ other than $\{dD\} = \{0\}$ is possible.
|
||
|
||
<!-- source-page: 462 -->
|
||
|
||
Use of $[N_{1}]$ implies that prebuckling rotations are small but are not to be ignored. Use of $[K_{\sigma}]$ implies that prebuckling rotations are either ignored or are zero. The latter is “classical” buckling analysis, as commonly used for straight columns and flat plates.
|
||
|
||
In what follows we emphasize classical buckling analysis, which uses $[K_{\sigma}]$ . One begins by applying to the structure a reference level of loading $\{R\}_{ref}$ and carrying out a standard linear static analysis to obtain membrane stresses in elements (e.g., to determine membrane stresses in a flat plate under thermal load). Hence, we generate a stress stiffness matrix $[K_{\sigma}]_{ref}$ appropriate to $\{R\}_{ref}$ . For another load level, with $\lambda$ a scalar multiplier,
|
||
|
||
$$
|
||
[ \mathbf {K} _ {\sigma} ] = \lambda [ \mathbf {K} _ {\sigma} ] _ {\text { ref }} \quad \text { when } \quad \{\mathbf {R} \} = \lambda \{\mathbf {R} \} _ {\text { ref }} \tag {14.5-2}
|
||
$$
|
||
|
||
Equations 14.5-2 imply that multiplying all loads $R_{i}$ in $\{R\}_{ref}$ by $\lambda$ also multiplies the intensity of the stress field by $\lambda$ but does not change the distribution of stresses. Then, since external loads do not change during an infinitesimal buckling displacement $\{dD\}$ ,
|
||
|
||
$$
|
||
([ \mathbf {K} ] + \lambda_ {\mathrm{cr}} [ \mathbf {K} _ {\sigma} ] _ {\text { ref }}) \{\mathbf {D} \} = ([ \mathbf {K} ] + \lambda_ {\mathrm{cr}} [ \mathbf {K} _ {\sigma} ] _ {\text { ref }}) \{\mathbf {D} + d \mathbf {D} \} = \lambda_ {\mathrm{cr}} \{\mathbf {R} \} _ {\text { ref }} \tag {14.5-3}
|
||
$$
|
||
|
||
Subtraction of the first equation from the second yields
|
||
|
||
$$
|
||
([ \mathbf {K} ] + \lambda_ {\mathrm{cr}} [ \mathbf {K} _ {\sigma} ] _ {\mathrm{ref}}) \{d \mathbf {D} \} = \{\mathbf {0} \} \tag {14.5-4}
|
||
$$
|
||
|
||
Equation 14.5-4 defines an eigenvalue problem whose lowest eigenvalue $\lambda_{cr}$ is associated with buckling. The critical or buckling load is, from Eq. 14.5-2,
|
||
|
||
$$
|
||
\{\mathbf {R} \} _ {\mathrm{cr}} = \lambda_ {\mathrm{cr}} \{\mathbf {R} \} _ {\mathrm{ref}} \tag {14.5-5}
|
||
$$
|
||
|
||
The eigenvector $\{d\mathbf{D}\}$ associated with $\lambda_{\mathrm{cr}}$ defines the buckling mode. The magnitude of $\{d\mathbf{D}\}$ is indeterminate. Therefore $\{d\mathbf{D}\}$ identifies shape but not amplitude.
|
||
|
||
A physical interpretation of Eq. 14.5-4 as follows. Terms in parentheses in Eq. 14.5-4 comprise a total or net stiffness matrix $[K_{net}]$ . Since forces $[K_{net}]\{dD\}$ are zero, one can say that membrane stresses of critical intensity reduce the stiffness of the structure to zero with respect to buckling mode $\{dD\}$ .
|
||
|
||
If the foregoing analysis were to use $[N_{1}]$ instead of $[K_{\sigma}]$ , the displacements needed to construct $[N_{1}]$ would be those obtained from static analysis under load $\{R\}_{ref}$ . Figure 14.5-1 gives an example of how well these two approaches to buckling analysis compare with the actual collapse load.
|
||
|
||
|
||
|
||
<details>
|
||
<summary>other</summary>
|
||
| Method | H | H (H = 35.0) | H (H = 3.49) |
|
||
| :--- | :--- | :--- | :--- |
|
||
| [Kσ] used | 21.35 | 0.520 | |
|
||
| [N1] used | 20.58 | 0.173 | |
|
||
| Nonlinear | 20.47 | 0.100 | |
|
||
</details>
|
||
|
||
Figure 14.5-1. An elastic bar, hinged at both ends. The linearized buckling load $P_{cr}$ is compared with the collapse load determined by a more exact nonlinear analysis [14.1].
|
||
|
||
<!-- source-page: 463 -->
|
||
|
||
Computational methods for determining $\lambda_{cr}$ are numerous (see Appendix C). Eigenvalue extraction methods used to compute natural frequencies and modes of vibration (Section 13.5) can also be applied to buckling problems. An algorithm that requires inversion of $[K_{\sigma}]$ may fail, because $[K_{\sigma}]$ may not be a positive definite matrix. If only one $\lambda_{cr}$ is required, it may be wasteful to use a method that automatically extracts several eigenvalues. However, at times one may wish to know the several lowest eigenvalues and their associated buckling modes in order to gain insight into ways of stiffening or supporting the structure so as to make buckling less likely.
|
||
|
||
By basing $[K]$ and $[K_{\sigma}]$ on the original, undeformed geometry, Eq. 14.5-4 ignores prebuckling nonlinearities that may actually be present; that is, the possible dependence of $[K]$ and $[K_{\sigma}]_{ref}$ on deformation is ignored. One way to account for such nonlinearity is to base $[K]$ and $[K_{\sigma}]_{ref}$ on the configuration just before buckling [14.1]. More specifically, one applies a trial level of load $\{R\}_{base}$ and performs a nonlinear static analysis. A result of this analysis is $[K_{t}]$ , the “tangent” stiffness matrix of the structure in its current deformed configuration. As compared with $[K]$ of the structure before loads are applied, $[K_{t}]$ is degraded in stiffness because of membrane stresses produced by $\{R\}_{base}$ . A small trial load increment $\{\Delta R\}$ is applied, and displacements produced by $[K_{t}]$ and $\{\Delta R\}$ are used to compute membrane stresses. Thus $[K_{\sigma}]_{ref}$ for the current configuration is established. The linear eigenvalue problem
|
||
|
||
$$
|
||
\left(\left[ \mathbf {K} _ {t} \right] + \Delta \lambda_ {\mathrm{cr}} \left[ \mathbf {K} _ {\sigma} \right] _ {\text {ref}}\right) \{d \mathbf {D} \} = \{\mathbf {0} \} \tag {14.5-6}
|
||
$$
|
||
|
||
is solved for $\Delta\lambda_{cr}$ . The computed value of $\Delta\lambda_{cr}$ reduces the net stiffness, which is the coefficient of $\{dD\}$ in Eq. 14.5-6, to zero with respect to the buckling mode. The predicted buckling load is
|
||
|
||
$$
|
||
\{\mathbf {R} \} _ {\mathrm{cr}} = \{\mathbf {R} \} _ {\text {base}} + \Delta \lambda_ {\mathrm{cr}} \{\Delta \mathbf {R} \} \tag {14.5-7}
|
||
$$
|
||
|
||
Equation 14.5-6 presumes that stresses change in intensity but not in distribution when the load increases an amount $\Delta\lambda_{cr}\{\Delta\mathbf{R}\}$ . This assumption becomes more nearly true as $\{R\}_{base}$ approaches $\{R\}_{cr}$ . By using a sequence of increasing loads $\{R\}_{base}$ , one can approach the correct buckling load arbitrarily closely. At convergence, $\Delta\lambda_{cr}=0$ and $\{R\}_{cr}=\{R\}_{base}$ .
|
||
|
||
Example: Classical Linear Buckling. Consider the uniform column in Fig. 14.5-2. Let the column be modeled by one beam element. We take [k] from Eqs. 2.4-3 and 4.2-5, and $[k_{\sigma}]$ from Eq. 14.2-9. By inspection, we see that the axial load throughout the bar has magnitude P. We arbitrarily choose the reference value of P as -1.0, where the negative sign indicates compression, not that the load is directed leftward. Nonzero d.o.f. are $u_{2}$ , $w_{2}$ , and $\theta_{2}$ . Thus, for a single element, Eq. 14.5-4 becomes
|
||
|
||
$$
|
||
\left(c _ {2} \left[ \begin{array}{c c c} c _ {1} / c _ {2} & 0 & 0 \\ 0 & 1 2 & - 6 L \\ 0 & - 6 L & 4 L ^ {2} \end{array} \right] + \lambda_ {\mathrm{cr}} \frac {- 1}{3 0 L} \left[ \begin{array}{c c c} 0 & 0 & 0 \\ 0 & 3 6 & - 3 L \\ 0 & - 3 L & 4 L ^ {2} \end{array} \right]\right) \left\{ \begin{array}{l} u _ {2} \\ w _ {2} \\ \theta_ {2} \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ 0 \\ 0 \end{array} \right\} \tag {14.5-8}
|
||
$$
|
||
|
||
where $c_{1} = AE/L$ and $c_{2} = EI/L^{3}$ . A solution other than $u_{2} = w_{2} = \theta_{2} = 0$ requires that the expression in parentheses have a zero determinant. Thus we write the char-
|
||
|
||
<!-- source-page: 464 -->
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
z, w
|
||
P
|
||
x, u
|
||
L
|
||
</details>
|
||
|
||
Figure 14.5-2. A uniform elastic bar, fixed at x = 0 and free at x = L. The exact $P_{cr}$ is $\pi^{2}EI/4L^{2} = 2.4674EI/L^{2}$ (in compression).
|
||
|
||
acteristic polynomial and extract its lowest root (this method is suitable for hand calculation provided there are few d.o.f.). The lowest root in the present case is
|
||
|
||
$$
|
||
\lambda_ {\mathrm{cr}} = 2. 4 8 6 0 E I / L ^ {2} \quad \text { hence } \quad P _ {\mathrm{cr}} = \lambda_ {\mathrm{cr}} (- 1. 0) = - 2. 4 8 6 0 E I / L ^ {2} \tag {14.5-9}
|
||
$$
|
||
|
||
Matrix $[\mathbf{k}_{\sigma}]$ in Eq. 14.5-8 comes from Eq. 14.2-9. If instead we use $[\mathbf{k}_{\sigma}]$ from Eq. 14.2-11, we obtain $P_{\mathrm{cr}} = -3EI / L^2$ , which is less accurate than $P_{\mathrm{cr}}$ in Eq. 14.5-9. Yet both answers are upper bounds to the correct magnitude of $P_{\mathrm{cr}}$ , which is $2.4674EI / L^2$ . Bounds are discussed further in Section 14.6.
|
||
|
||
The buckling mode can be computed from Eq. 14.5-8 by setting $\lambda = \lambda_{\mathrm{cr}}$ , choosing an arbitrary value for one of the d.o.f. (e.g., $\theta_{2} = 1$ ), and solving for the remaining d.o.f. Thus we obtain
|
||
|
||
$$
|
||
u _ {2} = 0 \quad w _ {2} = 0. 6 3 7 9 L \quad \theta_ {2} = 1 \tag {14.5-10}
|
||
$$
|
||
|
||
D.o.f. in Eq. 14.5-10 are buckling displacements $\{d\mathbf{D}\}$ , measured relative to the reference state in which $\lambda_{\mathrm{cr}}[\mathbf{K}_{\sigma}]_{\mathrm{ref}}$ is associated with initial membrane stresses. In this reference state, $u_{2} = -PL / AE$ and $w_{2} = \theta_{2} = 0$ . We see that $u_{2}$ plays no role in Eq. 14.5-8. Indeed, the buckling equation could have been written using $w_{2}$ and $\theta_{2}$ as the only d.o.f.
|
||
|
||
Example: Condensation of D.O.F. In structural dynamics, one can use condensation to reduce the size of the eigenvalue problem (see Section 13.7). The same can be done in buckling problems, with $[K_{\sigma}]$ taking the place of [M]. One might elect to eliminate rotational d.o.f. In the preceding example, to eliminate $\theta_{2}$ from Eq. 14.5-8, Eq. 13.7-3 becomes
|
||
|
||
$$
|
||
[ \mathrm{T} ] = \left[ \begin{array}{c c} 1 & 0 \\ 0 & 1 \\ 0 & 3 / 2 L \end{array} \right] \quad \text { where } \quad \frac {3}{2 L} = - \left(\frac {L}{4 E I}\right) \left(- \frac {6 E I}{L ^ {2}}\right) \tag {14.5-11}
|
||
$$
|
||
|
||
The transformations of Eq. 13.7-5 convert Eq. 14.5-8 to
|
||
|
||
$$
|
||
\left(c _ {2} \left[ \begin{array}{c c} c _ {1} / c _ {2} & 0 \\ 0 & 3 \end{array} \right] + \lambda_ {\mathrm{cr}} \frac {- 1}{3 0 L} \left[ \begin{array}{c c} 0 & 0 \\ 0 & 3 6 \end{array} \right]\right) \left\{ \begin{array}{l} u _ {2} \\ w _ {2} \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ 0 \end{array} \right\} \tag {14.5-12}
|
||
$$
|
||
|
||
from which $\lambda_{cr} = 2.5EI/L^{2}$ and $P_{cr} = -2.5EI/L^{2}$ . We see that condensation has slightly increased the magnitude of the computed buckling load.
|
||
|
||
# 14.6 REMARKS ON $[K_{\sigma}]$ AND ITS USES
|
||
|
||
Field on Which $[k_{\sigma}]$ is Based. A stress stiffness matrix is termed “consistent” if built from the same shape functions used to build the conventional stiffness matrix.
|
||
|
||
<!-- source-page: 465 -->
|
||
|
||
If the structure geometry is well modeled and if elements are compatible and not softened by low-order integration rules, then such a formulation yields an upper bound to the magnitude of the correct buckling load. The “correct” buckling load is the linear bifurcation load of the structure in its reference configuration; it is not necessarily the collapse load of the actual structure. Finite element analysis would yield the correct buckling load if $[k]$ and $[k_{\sigma}]$ were based on fields that include the buckled shape as a possible displacement mode. In the case of buckling of a perfect pin-ended column, this mode is sinusoidal rather than cubic, and the correct buckling load has magnitude $\pi^{2}EI/L^{2}$ .
|
||
|
||
We can base [k] and $[k_{\sigma}]$ on different displacement fields. We recall from the convergence requirements of Section 4.5 that if a strain energy expression involves displacement derivatives of order m, the displacement field must provide inter-element continuity of displacement derivatives of order m - 1 as the mesh is refined. Energy integrals that yield $[k_{\sigma}]$ involve first derivatives of displacement, so continuity of displacement is all that is required. Thus, for example, we would expect to be able to determine $P_{cr}$ for a pin-ended column by using the conventional beam [k] but the $[k_{\sigma}]$ of Eq. 14.2-11. This is indeed the case but, for a given accuracy, we must divide the column into more elements than when we use the consistent $[k_{\sigma}]$ of Eq. 14.2-9.
|
||
|
||
It is sometimes recommended that $[k_{\sigma}]$ for a complicated element be based on a simpler displacement field than that used to construct the conventional stiffness matrix, in order to increase computational efficiency with little loss in accuracy. The “best” $[k_{\sigma}]$ is probably intermediate to the consistent $[k_{\sigma}]$ and the simplest possible $[k_{\sigma}]$ . Numerical evidence suggests that computed buckling loads are increased when $[k_{\sigma}]$ is simplified. If $[k_{\sigma}]$ is generated by numerical integration, a simplified displacement field is effectively employed by adopting a reduced order of quadrature.
|
||
|
||
A valid $[\mathbf{k}_{\sigma}]$ must not generate nodal loads during a rigid-body translation. Nodal loads do appear when an element rotates. Indeed, from Eq. 14.5-1 one can interpret buckling as a displacement state $\{d\mathbf{D}\}$ in which pseudo-loads $[\mathbf{K}_{\sigma}]\{d\mathbf{D}\}$ are equal in magnitude to the corresponding resistances $[\mathbf{K}]\{d\mathbf{D}\}$ .
|
||
|
||
Stress stiffness matrices have been devised for many buckling problems, for example, for homogeneous and sandwich plates [14.5-14.7], torsional and torsional-flexural buckling of prismatic members [5.4,14.8], tapered bars and plates [14.6,14.9], and nonconservative problems [14.10].
|
||
|
||
Applications of $[K_{\sigma}]$ . A shell of revolution usually has a nonaxisymmetric buckling mode even if geometry, supports, material properties, and loading are all axisymmetric. The buckling mode will probably display many waves in each hoop circle. It is commonly assumed that the buckling mode varies circumferentially as a single Fourier harmonic. Thus buckling analysis is similar to the displacement analysis described in Sections 10.5 and 10.6 [14.11]. First the shell is divided into elements such as those shown in Fig. 12.4-2. Then one selects a specific number n of circumferential waves and computes the corresponding $[K]_{n}$ and $[K_{\sigma}]_{n}$ . Next one solves the eigenvalue problem to obtain $\lambda_{cr}$ for n waves. The entire procedure is repeated for $n + 1$ waves, for $n + 2$ waves, and so on. Provided that the initial n is sufficiently small, the lowest of the sequence of $\lambda_{cr}$ values can be identified as the desired buckling parameter. It is not obvious which mode will govern, and many analyses may be needed: Ref. 14.12 mentions a case where buckling is associated with 39 circumferential waves. If many waves also appear in the meridional direction, many elements are needed even if the shell has simple geometry.
|
||
|
||
<!-- source-page: 466 -->
|
||
|
||
Dynamic analysis of undamped structures with membrane forces leads to the equations
|
||
|
||
dynamic response: $[\mathbf{K} + \mathbf{K}_{\sigma}]\{\mathbf{D}\} + [\mathbf{M}]\{\ddot{\mathbf{D}}\} = \{\mathbf{R}\}$ (14.6-1)
|
||
|
||
natural frequencies: $([\mathbf{K} + \mathbf{K}_{\sigma}] - \omega^{2}[\mathbf{M}])\{\overline{\mathbf{D}}\} = \{\mathbf{0}\}$ (14.6-2)
|
||
|
||
where $ [M] = mass matrix, \{\ddot{D}\} = accelerations of nodal d.o.f., \omega = circular frequency, and \{\overline{D}\} = amplitudes of nodal d.o.f. Tensile membrane forces increase the frequencies. Compressive forces decrease them and produce the root \( \omega = 0 $ if buckling impends.
|
||
|
||
A structure may have no conventional stiffness [K]. An example is a linkage of pin-connected bars, like a chain, with each link idealized as rigid. Similarly, some elastic structures may have a [K] that offers no resistance to certain loads. Examples include straight cables and flat membranes, which have no bending stiffness with which to resist lateral loads. Static problems of this type can be analyzed by the equation $[\mathbf{K}_{\sigma}]\{\mathbf{D}\} = \{\mathbf{R}\}$ , where $\{\mathbf{D}\}$ contains d.o.f. associated with small lateral deflection. Analogous dynamic problems, such as a plucked string or a vibrating membrane, can be analyzed by Eqs. 14.6-1 and 14.6-2 with [K] = [0].
|
||
|
||
# 14.7 REMARKS ON BUCKLING AND BUCKLING ANALYSIS
|
||
|
||
A real structure may collapse at a load quite different than that predicted by a linear bifurcation buckling analysis. The following remarks, extracted largely from Refs. 14.13 to 14.15, describe types of buckling behavior and caution against oversimplification in analysis. Throughout the discussion it is assumed that the material of the structure remains linearly elastic and that loads are gradually applied.
|
||
|
||
Figure 14.7-1 illustrates some of the ways a structure may behave. Here $P$ is either the load or is representative of its magnitude, and $D$ is displacement of some d.o.f. of interest. In Fig. 14.7-1a, the primary or prebuckling path happens to be linear. At bifurcation, either of two adjacent and infinitesimally close equilibrium positions are possible. Thereafter, for $P > P_{\mathrm{cr}}$ , a real (imperfect) structure
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
P
|
||
Primary path
|
||
Limit point
|
||
Secondary path
|
||
(postbuckling)
|
||
Pcr
|
||
Bifurcation point
|
||
D
|
||
</details>
|
||
|
||
(a)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>line</summary>
|
||
|
||
| Point Type | Description | P Value |
|
||
| ----------------------- | --------------------------------- | ------- |
|
||
| Bifurcation point | Bifurcation point | P_cr |
|
||
| Limit point (on primary path) | Limit point (on primary path) | P_cr |
|
||
| Postbuckling (secondary path) | Postbuckling (secondary path) | P_cr |
|
||
| Actual (imperfect) structure | Actual (imperfect) structure | P_cr |
|
||
</details>
|
||
|
||
{b}
|
||
Figure 14.7-1. Possible load versus displacement behaviors of thin-walled structures.
|
||
|
||
<!-- source-page: 467 -->
|
||
|
||
follows the secondary path. The secondary (postbuckling) path rises, which means that the structure has postbuckling strength. In this case $P_{cr}$ characterizes a local buckling action that has little to do with overall strength. This structure finally collapses at a limit point, which is defined as a relative maximum on the P versus D curve for which there is no adjacent equilibrium position. Loose terminology may refer to the limit point load as a buckling load. The action at collapse becomes dynamic, because the slope of the curve becomes negative and the structure releases elastic energy, which is converted into kinetic energy.
|
||
|
||
A different type of behavior is depicted in Fig. 14.7-1b. Here the perfect (idealized) structure has a nonlinear primary path. The postbuckling path falls, so there is no postbuckling strength. If the primary path is close to a falling secondary path, the structure is called imperfection sensitive, which means that the collapse load of the actual structure is strongly affected by small changes in direction of loads, manner of support, or changes in geometry. The actual structure, which has imperfections, displays a limit point rather than bifurcation, as shown by the dashed line.
|
||
|
||
Figure 14.7-2 shows how the response may be affected by overall structure geometry. Figure 14.7-2c, for a deep spherical cap, also applies qualitatively to a cylindrical shell under axial compression. The deep cap and the cylindrical shell are imperfection sensitive: if the radius-to-thickness ratio is large, laboratory specimens buckle at roughly one half of the theoretical bifurcation load, even when heroic efforts are made to achieve geometric perfection.
|
||
|
||
In the absence of prior knowledge about how a structure behaves, one must anticipate that a computed bifurcation buckling load may be far above or far below the actual collapse load, that imperfections may be influential, and that prebuckling nonlinearities may be important. Nonlinearities may arise because pressure loads change direction as the structure deforms, because the deformed shape is more (or less) susceptible to instability than the undeformed shape, or because of the effects of deformation on the membrane stress distribution. Nonlinearities may be accounted for as described in connection with Eqs. 14.5-6 and 14.5-7, or by a
|
||
|
||
|
||
Figure 14.7-2. Pressure p versus center deflection D for thin-walled elastic structures. Dashed line: linear theory. Solid line: nonlinear theory and actual behavior. (a) Circular plate. (b) Shallow spherical cap. (c) Deep spherical cap.
|
||
|
||
<!-- source-page: 468 -->
|
||
|
||
nonlinear analysis that effectively plots load versus displacement and signals collapse when the total stiffness matrix of the structure in its current configuration becomes singular.
|
||
|
||
Collapse analysis should be approached with caution and with expertise. Novices are cautioned against misuse of computer programs. For example, the axially compressed cylindrical shell is an attractive test case, yet the problem is analytically difficult because several eigenvalues are clustered and correspond to quite different eigenmodes. Even a program that can negotiate the difficulties will not produce the correct result if, misled by the geometric simplicity of the problem, the user has employed so few d.o.f. that the many waves of the actual buckling mode cannot be properly modeled $[14.14]$ .
|
||
|
||
# PROBLEMS
|
||
|
||
# Section 14.1
|
||
|
||
14.1 A straight wooden column has an axial hole that fits closely but without friction around a metal rod, as shown. The rod is tensioned by tightening nuts that bear on the ends of the column. Assume that linear elasticity prevails and that the rod remains precisely centered in the column. Will the column buckle?
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
L
|
||
</details>
|
||
|
||
Problem 14.1
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
P
|
||
M₀
|
||
P
|
||
k
|
||
k
|
||
L
|
||
</details>
|
||
|
||
Problem 14.2
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
P
|
||
e
|
||
k
|
||
L
|
||
</details>
|
||
|
||
Problem 14.4
|
||
|
||
14.2 A rigid bar is supported by two springs, each of stiffness k, and loaded by horizontal forces P, as shown. Use the view that loads P do work during a small rotation of the bar (noted in the footnote in Section 14.1), and determine:
|
||
|
||
(a) the angle of rotation of the rigid bar, in terms of $M_0, k, P$ , and $L$ .
|
||
(b) the (compressive) value of $P$ for buckling (when $M_0 = 0$ ), in terms of $k$ and $L$ .
|
||
|
||
14.3 For the problem described by Fig. 14.1-1, let $P_{cr} = -\pi^{2}EI/L^{2}$ and $w_{c0}$ represent the value of $w_{c}$ produced by q alone (when P = 0). Plot $w_{c}/w_{c0}$ versus $P/P_{cr}$ as P goes from $P = P_{cr}$ (compressive) to $P = 5|P_{cr}|$ (tensile).
|
||
|
||
14.4 A rigid bar is pivoted at the lower end and held by a linear spring at the upper end, as shown. Load P is offset a distance e from the bar axis. Find $P_{cr}$ (for e = 0). Also, using small-angle approximations, express lateral deflection $\Delta$ of the top in terms of P, e, k, and L. Plot $\Delta/L$ versus $P/P_{cr}$ for e/L = 0, 0.01, and 0.02.
|
||
|
||
<!-- source-page: 469 -->
|
||
|
||
# Section 14.2
|
||
|
||
14.5 Buckling of a tapered column is to be studied. Each element of the column is tapered. In which element matrices ([k] or $[\mathbf{k}_{\sigma}]$ ) does the effect of taper appear, and how is it to be included?
|
||
|
||
14.6 Show that Eq. 14.2-9 is produced by Eq. 14.2-8 and the standard cubic beam shape functions.
|
||
|
||
14.7 (a) Show that Eq. 14.2-4 results from Eqs. 14.2-1 through 14.2-3. (b) How must $u_{,x}$ and $w_{,x}$ be related if the term in $\epsilon_x^2$ that contains $w_{,x}^4$ is to be less than $5\%$ of $u_{,x}w_{,x}^2$ ? Hence, what limiting angle of rotation is indicated if $u_{,x}$ is 0.002?
|
||
|
||
14.8 Construct a 4 by 4 matrix $[\mathbf{k}_{\sigma}]$ for a uniform beam element, analogous to Eq. 14.2-9, by using the quadratic displacement field $w = (1 - \xi)w_{1} + \xi w_{2} + (1 - \xi)\xi L(\theta_{1} - \theta_{2}) / 2$ , where $\xi = x / L$ .
|
||
|
||
14.9 Derive each of the two $[k_{\sigma}]$ matrices in Eq. 14.2-11 by imposing the appropriate restrictions on $[k_{\sigma}]$ of Eq. 14.2-9.
|
||
|
||
14.10 A member of a plane, pin-jointed truss makes an angle $\beta$ with the global x axis. Determine an expression for $[k_{\sigma}]$ , analogous to Eq. 14.2-11b, that operates on global d.o.f. $u_{1}$ , $w_{1}$ , $u_{2}$ , and $w_{2}$ . Express your answer in terms of P, L, and $\beta$ .
|
||
|
||
14.11 For the system shown, establish a set of two equations that could be solved to determine $w_{2}$ and $\theta_{2}$ in member 1–2 in terms of P, Q, E, I, L, and c. The connection at node 2 transmits no moment.
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
EI
|
||
Q
|
||
Rigid
|
||
1
|
||
2
|
||
3
|
||
P
|
||
L
|
||
c
|
||
</details>
|
||
|
||
Problem 14.11
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
z, w
|
||
L
|
||
Rigid
|
||
Q
|
||
1
|
||
2
|
||
P
|
||
x
|
||
Linear spring
|
||
k
|
||
</details>
|
||
|
||
Problem 14.12
|
||
|
||
14.12 The bar shown is hinged at node 1 and may be considered rigid and weightless. In terms of $P, Q, k,$ and $L$ , what is deflection $w_2$ if $P$ is (a) zero, (b) $0.96kL$ in tension, and (c) $0.96kL$ in compression?
|
||
|
||
14.13 Solve Problem 14.12(c), in which $P = -0.96kL$ , by the following iterative method. For $Q$ alone, $w_{2} = Q / k$ . Now, with $w_{2} > 0$ , $Q$ and $P$ exert a moment about node 1. Another analysis therefore yields a $w_{2}$ larger than before. The process repeats.
|
||
|
||
14.14 The column shown is fixed at the left end. Axial load $P$ at the simply supported end has eccentricity $e$ from the centerline. Model the column by one element.
|
||
|
||
(a) Determine the rotation $\theta_{2}$ in terms of $P, L, E, I$ , and $e$ .
|
||
|
||
(b) If $e = 0$ , what load $P$ will make $\theta_2 \neq 0$ ? What is the percentage error of this result?
|
||
|
||
<!-- source-page: 470 -->
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
z, w
|
||
P
|
||
1
|
||
2
|
||
e
|
||
x
|
||
L
|
||
</details>
|
||
|
||
Problem 14.14
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
z, w
|
||
EI = 110 N · m²
|
||
0.1 N
|
||
1
|
||
2
|
||
P
|
||
x
|
||
3.0 m
|
||
</details>
|
||
|
||
Problem 14.15
|
||
|
||
(a) Determine $w_{2}$ if $P = 0$ .
|
||
(b) Determine $w_{2}$ if $P = 30.0 \, \mathrm{N}$ in compression.
|
||
(c) Determine $w_{2}$ if P = 30.0 N in tension.
|
||
|
||
14.15 The cantilever beam shown may be subjected to either a tensile or a compressive axial force P. Model the beam by one element. Use $w_{2}$ and $\theta_{2}$ as d.o.f. and take $[k_{\sigma}]$ from Eq. 14.2-9.
|
||
14.16 Repeat Problem 14.15, but use the $[\mathbf{k}_{\sigma}]$ developed in Problem 14.8.
|
||
14.17 In Problem 14.15(b), evaluate the bending moment at $x = 0$ by the calculation $M = EI[\mathbf{B}]\{\mathbf{d}\}$ , where $[\mathbf{B}]$ is based on a cubic field and $\{\mathbf{d}\} = [0 \ 0 \ w_2 \ \theta_2]^T$ . Compare this $M$ with that obtained by statics; that is, $M = 0.1L - Pw_2$ , where $P = -30.0 \ \text{N}$ in this case.
|
||
14.18 Let the 4 by 1 displacement vector $\{\mathbf{d}\}$ represent a small rigid-body rotation of a bar or beam about its left end. Determine the nodal forces $[\mathbf{k}_{\sigma}]\{\mathbf{d}\}$ for a bar (Eq. 14.2-11) and a beam (Eq. 14.2-9). Express your answers in terms of $P$ , $L$ , and $w_{2}$ .
|
||
14.19 Equation 4.1-6 has been used to compute consistent nodal loads on a beam (e.g., see Fig. 4.3-6). If the beam also carries an axial force, are these loads still correct? Explain.
|
||
14.20 In Chapter 13 we discussed a diagonal element mass matrix [m]. Why is an analogous diagonal stress stiffness matrix $\left[\mathbf{k}_{\sigma}\right]$ unacceptable if $\{\mathbf{d}\}$ contains only translational d.o.f.?
|
||
|
||
# Section 14.4
|
||
|
||
14.21 Show that Eq. 14.4-2 yields the small-strain approximation $\epsilon = (ds^{*} - ds)/ds$ if $ds \approx ds^{*}$ .
|
||
|
||
14.22 (a) Imagine that the bar element of Fig. 14.2-1b is generalized to three dimensions. Thus the bar lies on the $x$ axis of a local coordinate system xyz that is arbitrarily oriented with respect to global coordinates. Nodal d.o.f. consist of three translations at each node (in local directions). What is the 6 by 6 matrix $[\mathbf{k}_{\sigma}]$ in local coordinates?
|
||
|
||
(b) How would you establish the $[\mathbf{k}_{\sigma}]$ that operates on translational d.o.f. in global coordinate directions?
|
||
|
||
14.23 Show that Eq. 14.3-5 results from appropriate specialization of Eq. 14.4-7 (or of Eq. 14.4-6).
|
||
|
||
14.24 Consider the eight-node trilinear (solid) isoparametric element. How many rows and columns are there in [G] of Eq. 14.4-7? Express the $G_{ij}$ in terms of shape function derivatives and coefficients $\Gamma_{ij}$ of the inverse Jacobian matrix. For convenience, let $\{d\} = \left[u_{1} \quad u_{2} \quad \ldots \quad u_{8} \quad v_{1} \quad \ldots \quad w_{8}\right]^{T}$ .
|