This document analyzes the effect of different directrix shapes (circular, parabolic, elliptical, inverted catenary) on membrane stresses in cylindrical shell roofs. Membrane theory is used to determine the normal forces (Nx, Nθ, Nxθ) and stresses under self-weight and live loads. Equations for each directrix are presented. An example problem is solved and results are shown in tables comparing stresses for each directrix. The analysis found that stresses are lowest with an inverted catenary directrix and highest with an elliptical directrix to cover the same area.

1. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 238-244 © IAEME
238
INFLUENCE OF SHAPE OF DIRECTRIX ON THE STRESSES OF
CYLINDRICAL SHELL ROOFS
Dr. N. Arunachalam1
, C. Kavitha2
1
(Professor and Dean of civil engineering, Bannari Amman Institute of Technology,
Sathyamangalam)
2
(M.E (structural Engg) II year, Bannari Amman Institute of Technology, Sathyamangalam)
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, Nφ and Nxφ 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.
Key words: Shells, Directrices, Membrane Forces, Stresses.
1. 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.
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2. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 238-244 © IAEME
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1.1 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, the directrix 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.
Fig.1
2. MEMBRANE THEORY OF CYLINDRICAL SHELLS
2.1 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.
2.2 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.
Nθ = - ZR
Zxθ = - Kx
Nθ= - ( - x2
)
Value of K for Dead and Live load
Let the dead load be g per unit area of the surface.
Y = g sinθ and Z = g cosθ,
Nθ = - g R cosθ

3. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 238-244 © IAEME
240
K = ( + Y) = 2g sinθ – g cosθ(10)
This value of K may be substituted in equations to find Nxθ and Nθ.
Next, consider a live load of p0 uniform over the horizontal projection of the shell surface .
The intensity of the vertical load = p0cosθ
Hence
Y = p0sinθcosθ and Z = p0 cos2
θ
From, N = - p0 R cos2
θ
K = ( + Y) = (3p0 sin θ cos θ – p0cos ² θ )
2.3 Stresses due to dead and live loads in circular cylindrical shell
Stresses due to dead load
Nθ = -gRcos θ , Nxθ = -2gx sin θ
Nx = - ( -x²) cos θ
Stresses due to live load
Nθ = - p0R cos² θ , Nxθ = - 1.50 p0 x sin 2θ
Nx= - 1.50 ( -x²) cos 2θ
2.4 Stresses due to dead and live loads in cylindrical shells with parabolic directrix
Stresses due to dead load
Nθ = - , Nxθ= gx sin θ
Nx = 0.50 ( - x²) cos4
θ
Stresses due to live load
Nθ = - , Nxθ =Nx = 0
2.5 Stresses due to dead and live loads in cylindrical shells with inverted catenary directrix
Stresses due to dead load
Nθ = - , Nxθ = 0 and Nx = 0

4. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 238-244 © IAEME
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Stresses due to live load
Nθ = - p0R0 ,Nxθ = - p0 x sin θ cos θ and Nx= - 0.50 ( - x²) cos 2θ cos² θ
2.6 Stresses due to dead and live loads in cylindrical shells with semiellipse directrix Stresses
due to dead load
Nθ = -g Nxθ = - gx[ 2 + sin θ
Nx = - ( - x²) [ + ( cos ² θ – sin²θ ) ] cos θ
Stresses due to live load
Nθ = -p0 Nxθ = -3p0x ( )
Nx = - p0( - x²) [ ]
We can solve the problem and findout the stresses in different directrices by using the above
equations.
3. EXAMPLE PROBLEM
To determine membrane stress resultants Nx, NθandNxθof a circular cylindrical shell, the
following dimensions are assumed.
Span L= 20 m
Radius of curvature R = 7 m
Shell thickness d = 60 mm
Semi central angle = 40
Width of the shell = 8 m
Dead load = 2650 N/m² and
Live load = 750 N/m2
For all the directrices the span thickness and width of the shell are respectively the same. In
semi elliptical directrix the length of major axis a=4m and minor axis b=3m.

5. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 238-244 © IAEME
242
Table 1: Membrane forces at mid-span section due to dead load and live loads in a cylindrical
shell with circular directrix
x
(m)
θ Nx (N/m) Nθ(N/m) Nθx (N/m) Normal
stresses
(N/mm2
)
Due to Dead
load
Due to Live
load
Due to Dead
load
Due to Live
load
Due to
Dead
load
Due to
Live
load
0 0 -37857.14 -16071.42 -18550 -5250 0 0
x = 0.630
(comp)
σθ = 0.309
(comp)
σxθ = 0
0 5 -37713.08 -15827.26 -18479.41 -5210.12 0 0
0 10 -37282.01 -15102.20 -18268.18 -5091.69 0 0
0 15 -36567.19 -13918.26 -17917.92 -4898.31 0 0
0 20 -35574.07 -12311.42 -17431.29 -4635.86 0 0
0 25 -34310.22 -10330.51 -16812.09 -4312.31 0 0
0 30 -32785.46 -8035.71 -16064.77 -3937.5 0 0
0 35 -31010.75 -5496.75 -15195.27 -3522.80 0 0
0 40 -30943.74 -2790.77 -14702.45 -3080.82 0 0
The stresses are: x= (N/mm2
); σθ= (N/mm2
)
0
0.2
0.4
0.6
0.8
0 10 20 30 40
Stress(N/mm2)
values of θº
σx (N/mm²)
σθ (N/mm²)
Fig 2: variation of x and σθ with θ in a cylindrical shell with circular directrix and parabolic
directrixat (x =0)
N.B: calculations show that x and σθ have greatest values at the mid-span section of the shell. Hence
the values obtained at other sections are not given here.
Table 2: Membrane forces due to dead load and live load in cylindrical shell with parabolic
directrix
x(m) θ Nx (N/m) Nθ (N/m) Nxθ(N/m) Normal
stresses
(N/mm2
)
Due to Dead
load
Due to
Live
load
Due to Dead
load
Due to Live
load
Due to
Dead
load
Due to
Live
load
0 0 16562.5 0 -21200 -6000 0 0
x =
0.276
(comp)
σθ = 0.602
(comp)
σxθ = 0
0 5 16311.83 0 -21362.27 -6022.91 0 0
0 10 15578.71 0 -21859.13 -6092.55 0 0
0 15 14417.86 0 -22722.09 -6211.65 0 0
0 20 12914.25 0 -24008.45 -6385.06 0 0
0 25 11174.52 0 -25809.78 -6620.26 0 0
0 30 9316.41 0 -28266.67 -6928.20 0 0
0 35 7457.34 0 -31594.63 -7324.64 0 0
0 40 5703.50 0 -36126.66 -7832.44 0 0

6. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 238-244 © IAEME
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Fig 3: variation of x and σθwith θ in cylindrical shell with inverted catenary directrix and semi
elliptical directrix (x =0)
Table 3: Membrane forces due to dead load and live load in inverted catenary directrix
X
(m)
θ Nx(N/m) Nθ (N/m) Nθx (N/m) Normal
stresses
(N/mm2
)Due to
Dead
load
Due to Live
load
Due to Dead
load
Due to Live
load
Due to
Dead
load
Due to
Live load
0 0 0 -5357.14 -10600 -3000 0 -3000
x = 0
σθ = 0.230
(comp)
σxθ = 0
0 5 0 -5235.67 -10640.49 -3000 0 -3000
0 10 0 -4882.27 -10763.52 -3000 0 -3000
0 15 0 -4328.63 -10973.93 -3000 0 -3000
0 20 0 -3623.75 -11280.28 -3000 0 -3000
0 25 0 -2828.47 -11695.80 -3000 0 -3000
0 30 0 -2008.93 -12239.83 -3000 0 -3000
0 35 0 -1229.46 -12940.21 -3000 0 -3000
0 40 0 -545.89 -13837.32 -3000 0 -3000
Table 4: Membrane forces due to loads in semi elliptical directrix
4. 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.
θ Nx(N/m) Nθ (N/m) Nxθ (N/m) Normal stresses (N/mm2
)
0 640902.5 14133.33 0 x = 10.68 (comp)
σθ = 0.235 (comp)
σxθ = 0
5 291765.0 11456.35 0
10 108625 9628.55 0
15 109759.04 8302.29 0
20 82337.54 7294.38 0
25 69915.22 6494.49 0
30 56954.11 5833.79 0
35 43841.82 5266.33 0
40 31430.79 4759.88 0

7. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 238-244 © IAEME
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Table 5: stresses at mid-span section
Sl.no Shape of
directrices
Max x (N/mm²) Max θ (N/mm²) xθ
(N/mm²)
1 Circular 0.630 0.309 0
2 parabola 0.276 0.353 0
3 Inverted
cetanary
0.087 0.05 0
4 Semi elliptical 10.68 0.235 0
From that table the normal stress xmaximum in semi elliptical directrix
The maximum stress can occurs at mid span section. The shear stress xθis zero at
mid-span section.
The normal stresses x θ , are maximum at crown point.
List of symbols
a Length of Major axis in semi elliptical directrix
b Length of Minor axis in semi elliptical directrix
R Radius of curvature at a point
Nx Force acting in x direction per unit length
Nθ Force acting in θ direction per unit length
Nxθ Force acting in x face in θ direction
x Stress in x direction
θ Stress in θ direction
g Dead load applied per unit surface area
p0 Live load applied per unit horizontally projected area
5. REFERENCES
1. Billington, D.P., Thin Shell Concrete Structures, 2nd Edition, McGraw-Hill Book Co., New
York, 1982, 373 pp.
2. “Design and construction of concrete shell roofs” by G.S. Ramaswamy.
3. Dr. K.S. Satyanarayanan and T.V. Srinivas Murthy, “Assessment and Rehabilitation of an
Existing Roofing System Subjected to Cyclonic Wind Loads”, International Journal of Civil
Engineering & Technology (IJCIET), Volume 3, Issue 2, 2012, pp. 144 - 153, ISSN Print:
0976 – 6308, ISSN Online: 0976 – 6316.
4. R. Goyal, A.K. Gupta and A.K. Ahuja, “Variation of Wind Load Distribution on Gable Roof
Building with Varying Length of Attached Canopy”, International Journal of Civil
Engineering & Technology (IJCIET), Volume 1, Issue 1, 2010, pp. 68 - 75, ISSN Print:
0976 – 6308, ISSN Online: 0976 – 6316.