Plastic collapse

Similar treatments can be used for all sorts of two-dimensional problems for calculating the plastic collapse load of structures of complex shape, and for analysing metal-working processes like forging, rolling and sheet drawing. 

We shall now examine material selection for a pressure vessel able to contain a gas at pressure p, first minimising the weight, and then the cost. We shall seek a design that will not fail by plastic collapse (i.e. general yield). But we must be cautious structures can also fail by fast fracture, by fatigue, and by corrosion superimposed on these other modes of failure. We shall discuss these in Chapters 13, 15 and 23. Here we shall assume that plastic collapse is our only problem. 

First, the pressure vessel must be safe from plastic collapse that is, the stresses must everywhere be below general yield. Second, it must not fail by fast fracture if the largest cracks it could contain have length 2a (Fig. 16.4), then the stress intensity K CTV must everywhere be less than K. Finally, it must not fail by fatigue the slow growth of a crack to the critical size at which it runs. 

Ceramics, without exception, are hard, brittle solids. When designing with metals, failure by plastic collapse and by fatigue are the primary considerations. For ceramics, plastic collapse and fatigue are seldom problems it is brittle failure, caused by direct loading or by thermal stresses, that is the overriding consideration. 

This enormous hardness is exploited in grinding wheels which are made from small particles of a high-performance engineering ceramic (Table 15.3) bonded with an adhesive or a cement. In design with ceramics it is never necessary to consider plastic collapse of the component fracture always intervenes first. The reasons for this are as follows. 

Cellular materials can collapse by another mechanism. If the cell-wall material is plastic (as many polymers are) then the foam as a whole shows plastic behaviour. The stress-strain curve still looks like Fig. 25.9, but now the plateau is caused by plastic collapse. Plastic collapse occurs when the moment exerted on the cell walls exceeds its fully plastic moment, creating plastic hinges as shown in Fig. 25.12. Then the collapse stress (7 1 of the foam is related to the yield strength Gy of the wall by... 

The explanation is almost the same as that for the transverse modulus the cell walls bend like beams, and collapse occurs when these beams reach their plastic collapse load. As with the moduli, moisture and temperature influence the crushing... 

Plastic Collapse Pressure Formula. The minimum collapse pressure for the plastic range of collapse is calculated by... 

The formula for minimum plastic collapse pressure is applicable for D/t values as shown in Table 4-151. The factors A, B, and C are given in Table 4-152. 

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 ... 

In a plot similar to Figure 6.25, the log of the relative plastic collapse stress is plotted as a function of the log of the relative density9 for nine different experimental sets of data (Figure 6.27). There is general agreement with theory yielding a 3/2 power for the relative density relationship, although the closed cells appear to support an exponent closer to 1/2. 

Since polymeric foams collapse upon compression, they do not respond to indentation by spreading laterally (displaying a Poisson s ratio of nearly 0.04). As a result, for materials with relative densities < 0.3, their indentation hardness is essentially equal to the plastic collapse stress,16 instead of the more familiar H = 3oy relation for dense solids. 

When the force on the polymer material exceeds the fully plastic moment of the weakest element of the cell structure, plastic collapse occurs (characterized by region C in Figure 6.22). In effect, the weakest elements become a plastic hinge and deform to create a collapsing cell that responds to nearly constant stress (opl) with large deformations (see Figure 6.28). 

While elastic buckling of structural walls is fully recoverable, plastic collapse of the plastic hinge sections is not. The theory of plastic collapse corresponds well for materials with a relative density of 0.3 or less materials with greater relative densities do not follow theoretical predictions because their cell partitions are too thick to buckle or hinge readily. The data represented in Figure 6.18, showing SEM from 6.17a, estimates a relative density of 0.345. 

Figure 6.27 Relative plastic collapse stress for open- and closed-cell foams.9...

Small strains are typically confined to the linear response region, after which the region of plastic collapse takes over. At high strains, the material will typically experience a densification response in which the cell walls have been fully crushed and the bulk material properties begin to be exhibited. As one would expect, the greater the relative density, the smaller the role played by the collapse of the cell walls. At somewhere near 0.3 relative density, the cells are sufficiently apart that the collapse mechanisms are insignificant (illustrated in Figure 6.31). 

British Standard BS EN ISO 3826-1 2003 Plastic collapsible containers for human blood and blood components requires that... 

Structural members—all this systematic and purely quantitative effort was transformed into two words of our opinion a plastic collapse. We synthesized the mountain of data using our background knowledge, and finally the symbolic analogy simply emerged. This symbolic analogy became obviously the equivalent of all the documentation and of all this twisted steel and in very few words perfectly described the situation. 

Plastic collapse test in lab. of optimal model frame. high (though different specimen each time) loads — clear collapse mode - fairly clear medium - high fairly high... 

Consider a limit state equation such as that for the plastic collapse of a laterally restrained, simply supported steel beam, under a uniformly distributed load. The function defining the failure boundary is... 

Tile use of mathematical models in these situations is quite different to the use of similar models in structural analysis, because the consequences of error are different. Optimising only one part of a structural scheme may lead to an inefficient design. Calibrating a code of practice using only one part of the total uncertainty may lead to misleading conclusions. Structural analysis, on the otlier hand, is concerned with one-sided accuracy, safety and as long as the solution complies with the Safe Theorem of plastic collapse it is acceptable even if it is false. The success of structural analysis in this situation does not imply that the same types of analysis will lead to success in these other far more complex problems. 

Liu, T., Deng, Z.C., and Lu, T.J. (2008) Analytical modeling and finite element simulation of the plastic collapse of sandwich beams with pin-reinforced foam cores. Interrmtioruil Journal of Solids and Structures, 45, 5127 5151. 

Extractable di-(2-ethylhexyl) phthalate, DOP, is determined in plastic collapsible containers for human blood and blood components. A sample of plastic is extracted with ethanol-water mixture having a density of 0.937 at 37°C for 60 min. The resultant solution is measured for absorbance at 272 nm and concentration of plasticizer determined from calibration curve. 

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