STUDY OF SHELL JUNCTIONS AND OPENINGS

Abstract

An attempt has been made to utilise Semi-Loof element models to solve the complex shell, curved beams and combined shell-beam problems for' the discontinuities resulting from junctions and openings. The effect of stress concentration for shell junctions and openings has been studied for different types of shell surfaces. A general purpose computer program has been developed for solving various problems of shell, three dimensional curved beams and their combinations. The results have been predicted in the graphical and tabular form and conclusions have been drawn. Enormous possibilities now exist for developing rational design procedure for the solution of these complex shell problems, which are common in occurrence. Cutouts and holes in structural members are the areas of weakness. Singularities due to abrupt discontinuties in stress distribution are observed near junctions and bifurcations. Combined shell-beam problems play their important role in stiffening thin shell surfaces against buckling and reinforcing the openings to reduce the stress concentration. The exact nature of stress distribution at the interface of an eccentrically attached curved beam with the shell surface, is not yet known. There are a large number of practical problems of cutouts and shell junctions which require accurate solutions. Closed form theoretical solutions of shell structures are limited to ii simple geometries, loadings and boundary conditions and these complex shell problems can only be attempted by means of numerical methods. Various shell elements are now available and the 'Semi-Loof elements offer satisfactory solutions for cutout and junction problems in shell structures. The development of the family of Semi-Loof elements is closely associated with the numerically integrated isoparametric elements using simple shape function routines, the normality hypothesis, independent description of displacements and rotations, constraint equations and reduced integration technique. Thus Semi-Loof shell and beam elements ,capable of solving difficult shell problems, have been formulated and used. The program based on Semi-Loof shell and Semi-Loof beam formulations for variable degrees of freedom per node, has been written to handle multiple right hand side vectors at a time for different load combinai;ions and can accommodate multiple boundary conditions in a single run. The program has been further developed for parametric studies of shell junctions and cutout problems. The frontal solution routines have been modified and developed which follow a suitable sequence of element assembly and thus avoiding a large number of individual problem resolutions and a set of repeatitive calculations are performed only once and used again and again for various parametric studies. This modification is also useful for solving the problems of sequential construction and excavation. iii Various test problems on cylindrical shells, doubly curved caps, surface of revolution, flat plates, highly curved shell surfaces, clamped cantilevers, beams curved in plan, built in rings under antisymmetric loads etc. have been solved to assess the accuracy and efficiency of the developed program in predicting displacements and stresses including shear effect in deep beams. The program has given excellent results. The program and the formulations are most general for the analysis of shells, 3D curved beam and combined shell-beam problems for any type of conventional loads. It is found that the present solutions with lesser elements are much more accurate as compared to the solutions of other elements. Semi-Loof shell element is flexible under point loads and corner moments because there are no slope connections at corners. A mesh refinement is therefore, necessary near the vicinity of cutouts and junctions in shell element but a coatfse mesh is sufficient for points away from singular regions. The discontinuity of geometry of intersecting and bifurcating shells at the junction interface causes stress concentration. The problem of pipe-nozzle connection has been chosen to compare the performance of Semi-Loof element with experimental results obtained by Corum at ORNL and numerical results obtained by using other shell elements. The stress distribution pattern for hoop and axial stresses are in close agreement with the experimental and numerical results. In order to predict the stress concentration near iv the shell junctions and bifurcations, the problems of square cross-section tube, pipe-nozzle junctions, cylinder-sphere junction, and pipe-bifurcations have been solved. It has been observed that the stress concentration(K) in shell junction problem is a function of diameter and thickness ratio of intersecting shells. The value of a coefficient C defined in equation K= C( Thickness/Diameter)1/ varies from 40 to 75 in different types of shell junctions and for different boundary conditions. The stress concentration is sometimes 6 to 8 times the stresses found in the plane problems with smooth surfaces and uniform loads. A stiffener element has been developed for Semi-Loof shell element and combined shell beam problems have been solved. The combined shell-beam problems have been solved by modifying the strains of the beam centroidal axis to match with the shell middle surface at the connected edge through the technique of shear centre. The analysis has been carried out successfully for a channel and a Z-section. In order to predict the stress concentration factors around the openings on shell surfaces, the parametric studies on various geometries of shells and its openings, loadings and boundary conditions have been made. Single and four symmetrical openings of circular, elliptical and rectangular shapes have been considered. Due to limitation oh computer time a very coarse mesh has been adopted to test the applicability of Semi-Loof elements. The numerical results cannot be considered as very reliable as the mesh refinement is obvious from the results, Srom the studies on cutout problems the following conclusions are drawn; The stress concentration varies with the size and shape of the hole. The concentration factors for inplane stresses go on increasing with the increase in the size of the hole, while the rate of increase of bending stress concentration factor is slow and attain nearly a constant value for circular hole. The stress concentration for rectangular hole is larger than the elliptical and circular holes of similar parameters. The principal radii of curvature and the thickness of the shell dictate the stress concentration around the openings. The maximum stress concentration varies from 15 to 20 times larger than the stress found in plane problems without discontinuity for shells of different curvatures. As the computer costs are now going down each problem can be solved with desired accuracy. In the end the important conclusions of this study are summarised along with the scope for future studies, A large number of practical problems of cutouts, junctions with stiffner elements in shells can now be solved to improve the design practices of Civil, Mechanical, Chemical and other engineering structures.