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ABSTRACT
In this investigation, the effect of four different types of directrices on the stresses in cylindrical shells to cover a given area is studied. The four directrices considered are: circular, parabolic, elliptical and inverted catenary. The membrane forces Nx, N and Nx and hence the stresses are determined using membrane theory. The loads considered are: self weight of the shell for an assumed thickness and live load as prescribed by IS 2210-1988, “CRITERIA FOR DESIGN OF REINFORCED CONCRETE SHELL STRUCTURES AND FOLDED PLATES”. It is found that to cover a given area the stresses are the least when the directrix of the cylindrical shell is in the form of inverted catenary. The highest values results when the directrixis in the form of semi ellipse.
INTRODUCTION
All the structures shaped as curved surfaces are called shells. The form of the middle surface and the thickness at every point are the two parameters required to define the geometry of the shell. The middle surface is the surface that bisects the shell thickness. A shell may have a uniform or varying thickness. Thin shell is one where the thickness is small in comparison to other dimensions and its radius of curvature.
Shells or skin roofs are preferable to plane roofs since they can be used to cover large floor spaces with economical use of materials of construction. The use of cured space roofs requires 25 to 40% less materials than that of the plane elements, structurally the shell roofs are superior since the whole cross section is uniformly stressed due to the direct forces with negligible bending effects. Due to this aspect the thickness of shells is usually very small in the range of 75 mm to 150mm.
INTRODUCTION OF CYLINDRICAL SHELLS
A cylindrical shell roof structure is a particular type of shell structure. Geometrically, this shell is a singly curved surface, domes being an example of doubly curved shell, generated by a straight line generator running along a cylindrical directrix (Fig.1). For roofs,thedirectrix can be a chord of a circler, an ellipse, a cycloid, a catenary, or a parabola. The roof is usually composed of a single shell or of multiple shells supported by traverses and/or edge beams. The traverses can be trusses, reinforced concrete diaphragms or ribs.
MEMBRENE THEORY OF CYLINDRICAL SHELLS
General discussion
In membrane theory, the loads are considered to be carried only by in-plane direct stresses Nx ,Nφ , and Nxφ (Fig.4.2), which lay in the shell middle surface. This is quite reasonable because it has a small rigidity (thin shell) and is curved, not straight like slabs, which carry loads by bending moments. If direct stresses are compressive, they can cause a buckling effect that is similar to a column under axial forces.
Equations of equilibrium
The shell problem in the membrane theory is straightforward in that it has three unknowns, two normal stresses Nx ,Nφ , and transverse shear stress Nxφ and three equations of equilibrium are enough for solving the problem. The components X ,Y, and Z denote the components of the external load applied per unit area in the x, y, and z directions. The following three equations are used to derive the stresses in different directrices.
CONCLUSIONS
Four different directrices have been selected for the analysis of cylindrical shells of
Length = 20m and chord width = 8m.
The maximum stresses obtained for a thickness of 60mm are as given below.