Elastic buckling

Fig. 25.11. When on elastomeric foam is compressed beyond the I inear region, the cell walls buckle elastically, giving the long plateau shown in Fig. 25.9.

The bulk modulus of rubber, which depends on the strength of the van der Waals forces between the molecules, is 2 GPa. Therefore, the compressive modulus of a rubber layer increases by a factor of a thousand as the shape factor increases from 0.2 (Fig. 4.3). The responses are not shown for S < 0.2 such tall, thin rubber blocks would buckle elastically (Appendix C, Section C. 1.4), rather than deforming uniformly. When laminated rubber springs are designed, Eqs (4.5) and (4.7) allow the independent manipulation of the shear and compressive stiffness. The physical size of the bearing will be determined by factors such as the load bearing ability of the abutting concrete material, or a limit on the allowable rubber shear strain to 7 < 0.5 and the compressive strain e < -0.1. 

There are three lands of buckling elastic, inelastic, and plastic. This procedure is concerned with elastic buckling only. AISC assumes that the upper limit of elastic buckling is defined by an average stress equal to one-half the yield point. 

Heil, M. 1999. Minimal liquid bridges in non-axisymmetrically buckled elastic tubes. / Fluid Mech 380 309-337. 

The other striking feature of nanotubes is their extreme stiffness and mechanical strength. Such tubes can be bent to small radii and eventually buckled into extreme shapes which in any other material would be irreversible, but here are still in the elastic domain. This phenomenon has been both imaged by electron microscopy and simulated by molecular dynamics by lijima et al. (1996). Brittle and ductile behaviour of nanotubes in tension is examined by simulation (because of the impossibility of testing directly) by Nardelli et al. (1998). Hopes of exploiting the remarkable strength of nanotubes may be defeated by the difficulty of joining them to each other and to any other material. 

The most common conditions of possible failure are elastic deflection, inelastic deformation, and fracture. During elastic deflection a product fails because the loads applied produce too large a deflection. In deformation, if it is too great it may cause other parts of an assembly to become misaligned or overstressed. Dynamic deflection can produce unacceptable vibration and noise. When a stable structure is required, the amount of deflection can set the limit for buckling loads or fractures. 

The analysis should include the modulus of soil reaction, because in a buried RTR piping system the elastic medium surrounding the pipe helps increase the pipe s resistance to buckling. The formula into which all these factors can be inserted to determine the critical buckling pressure of the pipe is called the Luscher Hoeg formula (44,56). 

In many cases, a product fails when the material begins to yield plastically. In a few cases, one may tolerate a small dimensional change and permit a static load that exceeds the yield strength. Actual fracture at the ultimate strength of the material would then constitute failure. The criterion for failure may be based on normal or shear stress in either case. Impact, creep and fatigue failures are the most common mode of failures. Other modes of failure include excessive elastic deflection or buckling. The actual failure mechanism may be quite complicated each failure theory is only an attempt to explain the failure mechanism for a given class of materials. In each case a safety factor is employed to eliminate failure. 

We present a few basic ideas of structural mechanics that are particularly relevant to the design of telescopes and to the support of related optics. This talk only touches on a very rich and complex held of work. We introduce the ideas of kinematics and kinematic mounts, then review basic elasticity and buckling. Simple and useful mles of thumb relating to structural performance are introduced. Simple conceptual ideas that are the basis of flexures are introduced along with an introduction to the bending of plates. We finish with a few thoughts on thermal issues, and list some interesting material properties. 

We have discussed the value of struts or columns in structural mechanics and described their linear elastic properties. They have another characteristic that is not quite so obvious. When columns are subject to a compressive load, they are subject to buckling. A column will compress under load until a critical load is reached. Beyond this load the column becomes unstable and lateral deformations can grow without bound. For thin columns, Euler showed that the critical force that causes a column to buckle is given by... 

Stiffness is the ability to resist bending and buckling. It is a function of the elastic modulus of the material and the shape of the cross-section of the member (the second moment of area). 

The classic example of failure due to elastic instability is the buckling of tall thin columns (struts), which is described in any elementary text on the Strength of Materials . 

Elastic buckling is the decisive criterion in the design of thin-walled vessels under external pressure. 

Two types of process vessel are likely to be subjected to external pressure those operated under vacuum, where the maximum pressure will be 1 bar (atm) and jacketed vessels, where the inner vessel will be under the jacket pressure. For jacketed vessels, the maximum pressure difference should be taken as the full jacket pressure, as a situation may arise in which the pressure in the inner vessel is lost. Thin-walled vessels subject to external pressure are liable to failure through elastic instability (buckling) and it is this mode of failure that determines the wall thickness required. 

Micromechanics theories for closed cell foams are less well advanced for than those for open cell foams. The elastic moduli of the closed-cell Kelvin foam were obtained by Finite Element Analysis (FEA) by Kraynik and co-workers (a. 14), and the high strain compressive response predicted by Mills and Zhu (a. 15). The Young s moduli predicted by the Kraynik model, which assumes the cell faces remain flat, lie above the experimental data (Figure 7), while those predicted by the Mills and Zhu model, which assumes that inplane compressive stresses will buckle faces, lie beneath the data. The experimental data is closer to the Mills and Zhu model at low densities, but closer to the Kraynik theory at high foam densities. 

Gibson and Ashby (a. 13) propose separate models for elastic collapse by cell edge buckling and plastic collapse by stretching of cell faces. The latter model gave a scaling relationship between the (initial) collapse stress a pi and the relative densities ... 

A Kelvin foam model with planar cell faces was used (a. 17) to predict the thermal expansion coefficient of LDPE foams as a function of density. The expansion of the heated gas is resisted by biaxial elastic stresses in the cell faces. However SEM shows that the cell faces are slightly wrinkled or buckled as a result of processing. This decreases the bulk modulus of the... 

Since MWNTs produced by arc discharge have nearly perfect crystallinity of graphite and are very stiff [Young s modulus being l. 8 TPa on average (36)l, almost all the MWNTs observed are straight. Elastically deformed (bent) ones are only rarely observed. Bent tubes are plastically deformed, with buckling on the concave side (37). 

Spencer and Dillon (S9) have discussed the problem of elastic recovery of polymeric melts and solutions at the exit of a tube. For the polystyrenes studied, a criterion was developed for the point at which such recovery was sufficient to cause buckling of the filament leaving the tube. 

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