Timoshenko

The model discussed is called the Kirchhoff model. Meantime there are other approaches to describe the behaviour of a shell. For example, it can be assumed that the fibre is not orthogonal to the mid-surface and the corresponding angle between the mid-surface and the orthogonal direction may vary. In this case the models are called Timoshenko or Reissner-Timoshenko models (see Vol mir, 1972 compare Ciarlet, Sanchez-Palencia, 1996). In particular, these approaches are used in Chapter 5. 

In this section we analyse the contact problem for a curvilinear Timoshenko rod. The plastic yield condition will depend just on the moments m. We shall prove that the solution of the problem satisfies all original boundary conditions, i.e., in contrast to the preceding section, we prove existence of the solution to the original boundary value problem. 

Kovtunenko V.A. (1996a) Numerical solution of a contact problem for the Timoshenko bar model. Izvestiya Rus. Acad. Sci. Mechanics of Solid 5, 79-84 (in Russian). 

Timoshenko S.P., Goodier J.N. (1951) Theory of elasticity. McGraw-Hill, New York. 

S. P. Timoshenko and ]. N. Goodier, Theory of Elasticity, McGraw Hill, 1970, Chap. 1. 

S. Timoshenko, D. H. Young and W. Weaver, Vibration Problems in Engineering, 4th edition, Wiley, 1974. 

Timoshenko et al (1967) recommended running a set of experiments in a CSTR on feed composition (now called feed-forward study), and then statistically correlating the discharge concentrations and rates with feed conditions by second order polynomials. In the second stage, mathematical experiments are executed on the previous empirical correlation to find the form and constants for the rate expressions. An example is presented for the dehydrogenation of butane. 

It has been shown that the ultimate tensile strength, Su, for brittle materials depends upon the size of the speeimen and will deerease with inereasing dimensions, sinee the probability of having weak spots is inereased. This is termed the size effeet. This size effeet was investigated by Weibull (1951) who suggested a statistieal fune-tion, the Weibull distribution, deseribing the number and distribution of these flaws. The relationship below models the size effeet for deterministie values of Su (Timoshenko, 1966). 

The methods used to measure residual stresses in a eomponent are performed after the manufaeturing proeess, and are broadly elassed into two types meehanieal (layer removal, eutting) and physieal (X-ray diffraetion, aeoustie, magnetie). Further referenee to the methods used ean be found in Chandra (1997), Juvinall (1967), and Timoshenko (1983). 

Timoshenko, S. P. 1966 Strength of Materials Part II - Advanced Theory and Practice, 3rd Edition. NY D. van Nostrand. 

Timoshenko, S., Strength of Materials Adranced Theory Problems, 3rd ed. Van Nostrand Reinhold Pub., 1956. 

From strength of materials one can move two ways. On the one hand, mechanical and civil engineers and applied mathematicians shift towards more elaborate situations, such as plastic shakedown in elaborate roof trusses here some transient plastic deformation is planned for. Other problems involve very complex elastic situations. This kind of continuum mechanics is a huge field with a large literature of its own (an example is the celebrated book by Timoshenko 1934), and it has essentially nothing to do with materials science or engineering because it is not specific to any material or even family of materials. 

Timoshenko, S. (1934) Introduction to the Theory of Elasticity for Engineers and Physicists (Oxford University Press, London). 

The results obtained in this way are the familiar results found by Hertz. More complete treatments of the contact mechanics of particle adhesion are given in Johnson [49] and in Timoshenko and Goodier [28]. 

Timoshenko, S. and Woinomsky-Kreiger, Theory of Plates and Shells, McGraw-Hill Book Company, Inc., New York, 1959, 2nd edition. 

The analysis to find the fiber buckling load in each mode is based on the energy method described by Timoshenko and Gere [3-31], The buckling criterion is that the change in strain energy for the fiber, AUf, and for the associated matrix material, AUf, is equated to the work done by the fiber force, AW, during deformation to a buckled state, that is,... 

For the fibers, the change in strain energy is related to the curvature of the bent fiber, v", considered as a column in the manner of Timoshenko and Gere [3-31],... 

Timoshenko, S., and MacCullough, Gleason, H., Elements of Strength of Materials, Van Nostrand, Princeton, 1957. 

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