| dc.description.abstract | From the investigations on the analysis of hollow cylinders subjected to axisymmetric boundary conditions reported in the previous five chapters, the following conclusions are summarized:
1)
In Chapter 1, an exact solution has been given through the elasticity approach for the analysis of the end problem of a semi-infinite hollow cylinder. Detailed theoretical derivations are presented when the end load consists of radially symmetric shear force and self-equilibrating normal load. Numerical results are presented only when self equilibrating end load acts and the diameter of the hole is one eighth the outer diameter.
It is concluded that St. Venant抯 principle is valid in toto and that the St. Venant zone extends to an axial distance of about one outer diameter of the semi infinite hollow cylinder. A hollow cylinder longer than twice its outer diameter can be analysed as two separate semi infinite hollow cylinders. Hollow cylinders shorter than this length are to be analysed as finite ones.
2)
In Chapter 2, the application of the problem treated in Chapter 1 to the anchorage zone stress problem in post tensioned prestressed concrete beams has been discussed. It is observed that a spalling zone is developed if the radial stress is taken as the relevant transverse stress, thus agreeing with Guyon抯 observation for rectangular beams. On the contrary, choosing hoop stress as the relevant transverse stress confirms the conclusion of Douglas and Trahair for hollow concrete cylinders that no spalling zone exists. It is indicated that the spalling zone stresses given rise to by the radial stress, though small, are to be duly accounted for in the design.
3)
In the third chapter, an elasticity solution has been presented for the analysis of a semi infinite circular cylindrical shell subjected to concentrated, uniform, circumferential, radial line load at the end. Shell theory solutions (according to the theories of Timoshenko-Donnell and Fl黦ge) are also derived. Detailed numerical work has been done for a steel shell ( = 0.3) with an inner radius 9/10 of the outer radius, and comparisons of elasticity and shell theory solutions carried out.
It is concluded that the assumptions of thin shell theory that:
(a) radial normal stress is quite small in comparison with the other stresses and hence can be neglected in the stress-strain relations; and
(b) normals to the median surface of the shell before deformation remain normals after deformation,
are justified.
Also, comparison of numerical results of the elasticity solution and the shell theory solution is given when the wall thickness of the steel shell is increased from one tenth the outer radius to one half the outer radius. Percentage deviations of the shell theory results from the elasticity solution are also given.
4)
The analysis given in Chapter 4 for a finite hollow cylinder subjected to axisymmetric end loads confirms the conclusions of the first chapter.
5)
In Chapter 5, detailed theoretical derivations are given for two mixed boundary value problems connected with long (infinite) hollow cylinders. The first problem is of laminated cylindrical shells, and the second one is the shrink fit problem of a hollow cylinder. No numerical work has been done on any of these problems. Presentation of numerical results for these problems would be useful.
It is noted here that the elasticity analysis presented for the end problem of semi infinite hollow cylinders is applicable to the analysis of thermal stresses in hollow cylinders.
From the summary, it can be observed that there is a lot of scope for further research in the field. In particular, the following problems are mentioned:
Study of the effect of variation of Poisson抯 ratio in the semi infinite hollow cylinder problem would be interesting. Similar studies on the finite hollow cylinder treated in Chapter 4 would also be useful.
For the end problem of the semi infinite hollow cylinder treated in Chapter 1, it is desirable to consider various diameters for the longitudinal hole and to give tables for the stress and displacement coefficients. Such work would be of immense use to designers.
Numerical work on the end problem of the semi infinite hollow cylinder with the presence of radially symmetric end shear, in addition to the normal end load, can be taken up. Detailed theoretical derivations for such a problem are given in Chapter 1. Such an investigation would help analyse a hollow cylinder subjected to axisymmetric skew end load(s).
For the semi infinite circular cylindrical shell subjected to concentrated, uniform, circumferential radial line load analysed in Chapter 3, a study of the effect of variation of Poisson抯 ratio on the comparison between elasticity and shell theory solutions would be of interest.
Recently, Alwar has given the elastic stress analyses of interface problems. He has obtained by Fourier analysis the stress distribution in (i) an infinite compound bar, (ii) finite compound bar, (iii) joined semi infinite plates, etc. All these problems are two dimensional ones. It is noted here that these problems can easily be extended to the case of composite hollow and solid cylinders.
Regarding the shrink fit problem of the infinite hollow shaft treated in Chapter 5, an obvious extension would be to analyse a semi infinite circular cylinder stiffened by a shrink fit at the end. No evidence has been noticed in literature for the analysis of such a problem even in the solid cylinder case.
A Love function formulation for such a problem connected with a solid cylinder would be:
=A3+B( , )sin ( z)+ n=1 (Cn+Dnz)eknzJ0(knr),\phi = A_3 + B(\infty, \sigma) \sin(\sigma z) + \sum_{n=1}^{\infty} \left( C_n + D_n z \right) e^{k_n z} J_0(k_n r), =A3 +B( , )sin( z)+n=1 (Cn +Dn z)ekn zJ0 (kn r),
where knk_nkn is a root of the transcendental equation J1(kna)=0J_1(k_n a) = 0J1 (kn a)=0.
The boundary conditions of the problem are:
(a) when z=0z = 0z=0, r z=0=0\sigma_r|_{z=0} = 0 r z=0 =0;
(b) when r=ar = ar=a, r=0\sigma_r = 0 r =0 and
u= u = \deltau= for z z \leq \ellz ,
u=0u = 0u=0 for z> z > \ellz> .
Here a=a =a= radius, =\ell = = length of shrink fit, =\delta = = shrink fit.
The analysis can easily be extended to hollow cylinders by including Bessel functions of the second kind. The effect of variation of Poisson抯 ratio on the shrink fit pressure can also be ascertained.
It is hoped that the solution of the problems cited above will contribute considerably to the progress of the analysis of cylinder problems in solid mechanics. | |
