Safe Theorem

Heyman [48] discusses in some detail a second report on the dome by Poleni. Poleni s method is one which would have been reproduced almost exactly by a modern analyst using the safe theorem of plasticity. He sliced the dome into 50 portions approximating half spherical lunes (orange slices) and worked on the premise that if each lune would stand, then so would the dome. The thrust line was determined experimentally by loading a flexible string and was found to lie within the thickness of the dome. He thus observed that the cracking was not critical but he agreed with the three mathematicians that further ties should be provided. 

The third part of a structural analysis, the analysis of safety, is normal carried out using only very simple techniques, as we shall see in Chapter 4. Ii recent series of papers on the analysis of masonry arches, Heyman [14] 1 pointed out the more general use of the plastic theorems of collapse and, particular, the Safe Theorem. In fact, tliis theorem effectively states that i. reasonable system of forces can be found to satisfy both the equilibrii... 

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. 

The application of G-spinor basis sets can be illustrated most conveniently by constructing the matrix operators needed for DCB calculations. The DCB equations can be derived from a variational principle along familiar nonrelativistic lines [7], [8, Chapter 3]. It has usually been assumed that the absence of a global lower bound to the Dirac spectrum invalidates this procedure it has now been established [16] that the upper spectrum has a lower bound when the trial functions lie in an appropriate domain. This theorem covers the variational derivation of G-spinor matrix DCB equations. Sucher s repeated assertions [17] that the DCB Hamiltonian is fatally diseased and that the operators must be surrounded with energy projection operators can be safely forgotten. 

A fundamental theorem of classical mechanics called the equipartition theorem (which we shall not derive here) states that the average energy of each degree of freedom of a molecule in a sample at a temperature T is equal to kT. In this simple expression, k is the Boltzmann constant, a fundamental constant with the value 1.380 66 X 10-21 J-K l. The Boltzmann constant is related to the gas constant by R = NAk, where NA is the Avogadro constant. The equipartition theorem is a result from classical mechanics, so we can use it for translational and rotational motion of molecules at room temperature and above, where quantization is unimportant, but we cannot use it safely for vibrational motion, except at high temperatures. The following remarks therefore apply only to translational and rotational motion. 

General.—The equations (7) which constitute the substance of my Heat Theorem, at once suggested a new experimental problem, namely, the determination of specific heats down to temperatures as low as possible. It is only if the specific heats are known that we can quite safely calculate the quantities A and U at low temperatures by means of the two already known laws, and so test the new Heat Theorem for these quantities are, for many reasons, practically always inaccessible to direct measurement at low temperatures. 

This reasoning is affected, however, by various more or less hypothetical assumptions up till now we have been moving, we may say, on absolutely safe ground, and we shall not leave it if we content ourselves, in this and the following chapter, with an indirect application of the Heat Theorem. This is possible if we apply the Theorem to condensed systems and operate, as far as the gaseous phase is concerned, solely with classical thermodynamics. A simple example may at once show us how we have to proceed. 

Now, if the heat capacity vanishes at low temperatures, the application of the Heat Theorem must also be regarded as safe hence the considerations previously developed (Chapter VII) concerning the impossibility of attaining the absolute zero are to be transferred to the present case. We thus arrive of necessity at the third fundamental assumption ... 

Consider two sets of orbitals, transforming as the irreps Fa and Fi, respectively, each occupied by one electron. A two-electron wavefunction with electron 1 in the Ya component of the first set, and electron 2 in the yj, component of the second set is written as a simple product function FaYai l)) rbYb 2)). Clearly, since the one-electron function spaces are invariants of the group, their product space is invariant, too. Now the question is to determine the symmetry of this new space. The recipe to find this symmetry can safely be based on the character theorem first determine the character string for the product basis, and then carry out the reduction according to the character theorem. Symmetry operators are all-electron operators affecting all particles together hence, the effect of a symmetry operation on a ket product is to transform both kets simultaneously. 

Bernoulli s Theorem. When friction is neghgible and there are no hydraulic machines, the conservation of energy principle is reduced to Bernoulli s equation, which has many applications in pressurized flow and open-channel flow when it is safe to neglect the losses. 

The second effect which has been neglected is that due to low-symmetry crystal fields. As the discussion in this chapter has been confined to octahedral MLg complexes one might assume that low-symmetry ligand fields could safely be ignored, but this is not so. A theorem due to Jahn and Teller states that any non-linear ion or molecule which is in an orbitally degenerate term will distort to relieve this degeneracy. This means that all Ey, and T2g terms of d" configurations, in principle, are unstable with respect to some distortion which reduces the symmetry. Of course, as... 

The described approach can be applied to other response properties apart from the perturbed energies provided that the method used for their computations satisfy the Hellmann-Feynman theorem (Feynman 1939 Hellmann 1937) according to which the derivative of the total energy with respect to the field is equivalent with the expectation value of the derivative of the Hamiltonian with respect to the field. In this case, one can safely use the following relationships ... 

A critical assumption is that real-world sidechains occur in discretizable rotamers. It has been recognized already in 1987 that this is largely true (Figure 8). This is underscored by a detailed recent study of sidechain rotamers and the authors show that more than a third of all possible sidechain rotamer states may be safely eliminated even before applying the dead-end elimination theorem to analyzing the rest. 

In the static approach the resultant of the compressive stresses is plotted at each cross section of the structure. The line containing aU the positions of the compressive stress resultants is called by thrust line (Fig. 4a). The thrust line can be determined by analytical methods or graphical methods, such as funicular polygon (Kooharian 1952). The stmcture is safe when the thrust line is located totally inside of the geometry of structure and the equitibrium with the external loads can be found. In this conditions and according to the lower bound theorem, the applied load is less... 

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