Elastic deflection

Boussinesq s theory/analysis can also be used to derive the elastic deflection of a homogeneous material subjected to uniform circular loading. The elastic surface deflection, A, at the centre of the loading area under the influence of a uniformly distributed load can be calculated from the following equation  

For a uniformly distributed load, there are two types of deformation of the subgrade under the loading area the uniform and non-uniform deformation. 

It should be mentioned that values of deflection obtained by the above equation and Foster and Ahlvin s nomograph are valid only when Poisson s ratio p is equal to 0.5. A supplementary work by Ahivin and Ulery (1962) allowed stresses and deflections to be calculated for any value of Poisson s ratio. The results of this work were presented in tabular form instead of diagrams. The relevant tables as well as the equations developed can be found in Ahivin and Ulery (1962) and Yoder and Witczak (1975). 

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A typical screwdriver has a shaft and blade made of a high-carbon steel, a metal. Steel is chosen because its modulus is high. The modulus measures the resistance of the material to elastic deflection or bending. If you made the shaft out of a polymer like polyethylene instead, it would twist far too much. A high modulus is one criterion in... 

Fig. 7.2. The elastic deflection of a telescope mirror, shown for simplicity as a flat-faced disc, under its own weight.

You will find the formulae for the elastic deflections of plates and beams under their own weight in standard texts on mechanics or structures (one is listed under Further Reading at the end of this chapter). We need only one formula here it is that for the deflection, 8, of the centre of a horizontal disc, simply supported at its... 

The square-section beam of length / (determined by the design of the structure, and thus fixed) and thickness t (a variable) is held rigidly at one end while a force F (the maximum service force) is applied to the other, as shown in Fig. 7.4. The same texts that list the deflection of discs give equations for the elastic deflection of beams. The formula we want is... 

Although the initial elastic and the primary creep strain cannot be neglected, they occur quickly, and they can be treated in much the way that elastic deflection is allowed for in a structure. But thereafter, the material enters steady-state, or secondary creep, and the strain increases steadily with time. In designing against creep, it is usually this steady accumulation of strain with time that concerns us most. 

If (as with body panels) elastic deflection is what counts, the logical comparison is for a panel of equal stiffness. And if, instead, it is resistance to plastic flow which counts (as with bumpers) then the proper thing to do is to compare sections with equal resistance to plastic flow. 

The allowable dimensional variation (the tolerance) of a polymer part can be larger than one made of metal - and specifying moulds with needlessly high tolerance raises costs greatly. This latitude is possible because of the low modulus the resilience of the components allows elastic deflections to accommodate misfitting parts. And the thermal expansion of polymers is almost ten times greater than metals there is no point in specifying dimensions to a tolerance which exceeds the thermal strains. 

But a polymer should never be regarded simply as an inexpensive substitute for a metal. Its properties differ in fundamental ways - most notably its modulus is far lower. Metal wheels are designed as rigid structures it is assumed that their elastic deflection under load is negligible. And - thus far - we have approached the design of a polymer wheel by assuming that it, too, should be rigid. 

The restraint coefficient, K, in the thermal stress formula is very potent. It can be varied over a wider range than any of the other parameters. If a designer can build in flexibility, and thus substitute elastic deflection for plastic flow, he can achieve a major safety factor. 

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. 

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. 

Resistance-Deflection Function. The resistance-deflection function establishes the dynamic resistance of the trial cross-section. Figure 4a shows a typical design resistance-deflection function with elastic stiffness, Kg (psi/in), elastic deflection limit, Xg (in) and ultimate resistance, r.. (psi). The stiffness is determined from a static elastic analysis using the average moment of inertia of a cracked and uncracked cross-section. (For design... 

Typical Data for Creep before, during, and after Irradiation. Curves of sample deflection vs. time before, during, and after irradiation for the three polymers used in this work are shown in Figure 3. The ordinate is presented as deflection after the first 2 seconds because this part of the elastic deflection could not be accurately measured. This normal elastic deflection was not of particular interest in the present work. 

THE ELASTIC DEFLECTION OCCURRING WITHIN TWO SECONDS AFTER STRESS APPLICATION HAS BEEN OMITTED. 

The sample deflection for each of the stress-on periods does not include the elastic deflection occurring during the first 2 seconds after stress application. The periods (48 seconds each) between the stress-on periods have been omitted from the figure to make clearer the differences that appear during the stress-on periods. 

Elastic deflection can be used both to apply the load and to sense the response of the sliding body in a friction-measuring device. This was the principle employed by Bowden and Leben in their now classical research of the late 1930 s [4]. Today their apparatus is only of historic interest. A modern application of the elastic deflection principle is exemplified by the dual cantilever beam, such as is described by Bayer al. [5] and illustrated in Fig. 7-6. Two independently acting elastic... 

Buckley, Swikert and Johnson [9] applied the principle of lever loading and elastic deflection to the measurement of friction in high vacuum (13.3 uPa, 10 torr). Their apparatus is illustrated diagram-matically in Fig. 7-10. The rotating specimen and its drive shaft are... 

In the elastic deflection formula replace 1/E by the stress-dependent creep compliance to obtain... 

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