design aids for fixed support reinforced concrete

24
MultiCraft International Journal of Engineering, Science and Technology Vol. 1, No. 1, 2009, pp. 148-171 INTERNATIONAL JOURNAL OF ENGINEERING, SCIENCE AND TECHNOLOGY www.ijest-ng.com © 2009 MultiCraft Limited. All rights reserved Design aids for fixed support reinforced concrete cylindrical shells under uniformly distributed loads Srinivasan Chandrasekaran 1* , S.K.Gupta 2 , Federico Carannante 3 1* Dept. of Ocean Engineering, Indian Institute of Technology Madras, Chennai 600 036, India 2 Dept. of Civil Engineering, Institute of Technology, Banaras Hindu University, Varanasi, India 3 Dept. of Structural Engineering, University of Naples Federico II, Naples, Italy * Corresponding Author: e-mail:[email protected]; Fax: +91-44-22574802 Abstract Shells are objects considered as materialization of the curved surface. Despite structural advantages and architectural aesthetics possessed by shells, relative degree of unacquaintance with shell behavior and design is high. Thin shells are examples of strength through form as opposed to strength through mass; their thin cross-section makes them economical due to low consumption of cement and steel as compared to other roof coverings such as slabs. Current study presents design curves for reinforced concrete open barrel cylindrical shells for different geometric parameters. The analysis is done in two parts namely: i) RC shell subjected to uniformly distributed load that remain constant along its length and curvature of the shell surface; and ii) RC shell subjected to uniformly distributed load varying sinusoidally along its length in addition to different symmetric edge loads present along its longitudinal boundaries. Design charts are proposed for easier solution of shell constants after due verification of results obtained from finite element analysis. Expressions for stress resultants proposed in closed form make the design more simple and straightforward; stress resultants plotted at closer intervals of φ can be useful for detailing of reinforcement layout in RC shells. Axial force-bending moment yield interaction studied on shells under uniformly distributed loads show compression failure, initiating crushing of concrete. Keywords: cylindrical shells; design curves; open barrel; reinforced concrete; stress resultants 1. Introduction Shells are skin structures by virtue of their geometry and shell action is essentially more towards transmitting the load by direct stresses with relatively small bending stresses. Presence of significant shear makes their behavior different from other roof covering structures; they have remarkable reserve strength with a greater degree-of-freedom in structure layout, shape and architecture, making them virtually impossible to collapse though their supporting structure may collapse (Rericha 1996; Ha-Wong Song et al. 2002). A number of procedures are proposed in the literature (see, for example, Flugge 1967) for analyzing the various types of thin shells for ultimate load but experimental evidences, however are not abundant. In many of these procedures, bending stresses are stated to be negligibly small justifying the application of membrane theory for analyzing thin shells, where normal forces are not necessarily tensile. Though behavior of shells to external loads is generally quite complex, thanks to second order equations developed that are sufficient to model the behavior of most civil engineering shell structures (Jacques, 1977). Forces in thin shells are, in a certain sense, statically determinant as they can be determined without referring to elastic property of material used. Linear elastic behavior is applied to provide a direct relationship between stress and strains by which the equilibrium of stress resultant and stress couples could be established. However mathematical models of thin RC shells developed on the interpretation of their physical behavior are difficult to present rationally as the rigorous analysis is extraordinarily complex, resolving to simplification (see, for example, Billington, 1965). These simplifications need extra caution for the design to become conservative. Also, interestingly, load carrying capacity of thin curved shells often exceeds the prediction of even most-refined available analysis. It is therefore commonly accepted that shells (reinforced concrete shells, in particular) are designed on the basis of approximate analysis only, which of course exhibits defects. For example, aircraft impact on containment structures is studied

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Page 1: Design aids for fixed support reinforced concrete

MultiCraft

International Journal of Engineering, Science and Technology

Vol. 1, No. 1, 2009, pp. 148-171

INTERNATIONAL JOURNAL OF

ENGINEERING, SCIENCE AND TECHNOLOGY

www.ijest-ng.com

© 2009 MultiCraft Limited. All rights reserved

Design aids for fixed support reinforced concrete cylindrical shells under uniformly distributed loads

Srinivasan Chandrasekaran1*, S.K.Gupta2, Federico Carannante3

1* Dept. of Ocean Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

2 Dept. of Civil Engineering, Institute of Technology, Banaras Hindu University, Varanasi, India 3 Dept. of Structural Engineering, University of Naples Federico II, Naples, Italy *Corresponding Author: e-mail:[email protected]; Fax: +91-44-22574802

Abstract Shells are objects considered as materialization of the curved surface. Despite structural advantages and architectural aesthetics possessed by shells, relative degree of unacquaintance with shell behavior and design is high. Thin shells are examples of strength through form as opposed to strength through mass; their thin cross-section makes them economical due to low consumption of cement and steel as compared to other roof coverings such as slabs. Current study presents design curves for reinforced concrete open barrel cylindrical shells for different geometric parameters. The analysis is done in two parts namely: i) RC shell subjected to uniformly distributed load that remain constant along its length and curvature of the shell surface; and ii) RC shell subjected to uniformly distributed load varying sinusoidally along its length in addition to different symmetric edge loads present along its longitudinal boundaries. Design charts are proposed for easier solution of shell constants after due verification of results obtained from finite element analysis. Expressions for stress resultants proposed in closed form make the design more simple and straightforward; stress resultants plotted at closer intervals of φ can be useful for detailing of reinforcement layout in RC shells. Axial force-bending moment yield interaction studied on shells under uniformly distributed loads show compression failure, initiating crushing of concrete. Keywords: cylindrical shells; design curves; open barrel; reinforced concrete; stress resultants 1. Introduction Shells are skin structures by virtue of their geometry and shell action is essentially more towards transmitting the load by direct stresses with relatively small bending stresses. Presence of significant shear makes their behavior different from other roof covering structures; they have remarkable reserve strength with a greater degree-of-freedom in structure layout, shape and architecture, making them virtually impossible to collapse though their supporting structure may collapse (Rericha 1996; Ha-Wong Song et al. 2002). A number of procedures are proposed in the literature (see, for example, Flugge 1967) for analyzing the various types of thin shells for ultimate load but experimental evidences, however are not abundant. In many of these procedures, bending stresses are stated to be negligibly small justifying the application of membrane theory for analyzing thin shells, where normal forces are not necessarily tensile. Though behavior of shells to external loads is generally quite complex, thanks to second order equations developed that are sufficient to model the behavior of most civil engineering shell structures (Jacques, 1977). Forces in thin shells are, in a certain sense, statically determinant as they can be determined without referring to elastic property of material used. Linear elastic behavior is applied to provide a direct relationship between stress and strains by which the equilibrium of stress resultant and stress couples could be established. However mathematical models of thin RC shells developed on the interpretation of their physical behavior are difficult to present rationally as the rigorous analysis is extraordinarily complex, resolving to simplification (see, for example, Billington, 1965). These simplifications need extra caution for the design to become conservative. Also, interestingly, load carrying capacity of thin curved shells often exceeds the prediction of even most-refined available analysis. It is therefore commonly accepted that shells (reinforced concrete shells, in particular) are designed on the basis of approximate analysis only, which of course exhibits defects. For example, aircraft impact on containment structures is studied

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149

by employing modal analysis to estimate their elastic response (Prabhakar, 2003). The study approximates shell with uniform thickness in longitudinal and circumferential directions and orthotropic properties are obtained by scaling the Young’s modulus values. Wheen and Wheen (2003) studied design and construction principles of a pre-stressed conical concrete tent roof, idealizing it as thin pre-tensioned radial circles suspended from a temporary central tower and anchored at the lower ends to a pre-stressed ring beam in a catenary form. They showed that the curved concrete surface, usually a felicitous combination of sound structural form and method of construction leading also to low cost, does not crack when suspended because of radial pre-stressing. Concrete domes, in particular, possess limited membrane stress variations due to soundness in structural form and therefore ideal as all stresses are compressive during construction and in final stages as well (Wheen and Anatharaman, 1998). Example structures presented above for designing/constructing RC shell structures prove the confidence level of researchers in employing relevant approximate theories for analyzing shell structures. This leads to the desire of simplifying the analysis procedure and adding to architectural aesthetics and structural advantages shell possess; the objective is to simplify analysis procedure by means of ready to use design curves and tables. This shall not only promote more construction of such advantageous and safe structures but also improve the confidence level in design offices. However, other studies conducted by the authors emphasize verification of stress resultants at closer intervals of geometric angles at boundaries, in particular, for appropriate detailing of reinforcement in RC shells (see, for example, Chandrasekara et al. 2005; Chandrasekaran and Srivastava, 2006). Ramaswamy (1968) discussed limitations and precision of various theories on shell structures and concluded that Flügge’s theory is acceptable and can be treated as a bench mark to assess the accuracy of other theories. 2. Objectives and structural idealization Based on the critical review of literature, main objective of the current study is to develop design curves based on the classical flexural theory for single barrel open cylindrical shells with different geometric parameters; the outcome is to derive the stress resultant estimates that can be readily determined without undergoing detailed computations. Though theory employed is not new but the attempt made to simplify the analysis procedure is relatively new and the analytical solutions proposed in the paper are not present in the literature. As the solution procedure of generic case of shell geometry shall involve complex mathematics, objective of simplifying the design procedure through design curves shall not be fulfilled; hence a particular geometric case of cylindrical shells is only addressed in the present study. Please note that cylindrical shells are commonly used geometry in structural/civil engg applications. With the use of proposed design curves, stress resultants and stress couples can easily be readily determined at the valley, crown or at any desired section for various type of edge loads. This shall be helpful to be employed in design offices while encouraging such new RC shell structures in future and also enhances the confidence of the structural designer. The study employs few structural idealizations namely: i) stress normal to the surface is neglected as thin shell is a curved slab whose thickness is relatively small compared to its other dimensions and radius of curvature; ii) deflections under load are small enough so that its static equilibrium remain affected due to changes in geometry, vide small deflection theorem; iii) linear elastic behavior is assumed in the analysis enabling a direct relationship between stress and strain; iv) for all kinematic relations, any point on the mid surface is considered as unaffected by the shell deformation; as well as v) all points lying normal to the middle surface before deformation lie on the normal after deformation.

3. Mathematical development Let us consider a cylindrical coordinate system ( )r,,x ϕ and a cylindrical shell with thickness t. The cylindrical shell has the directive along the x-axis and we denote ru,u,ux ϕ as the displacements along longitudinal (x), circumferential (φ) and radial directions (r), respectively. It is important to note that the displacements, strains and stress functions depend only on two variables namely ( x,ϕ ). Figure.1 shows the overall geometry of the shell considered for the study while Figure.2 shows the differential element considered for the analysis. The three strain components ϕεε ,x and ϕγ x of the middle surface and the three curvature changes ϕχχ ,x and ϕχ x can be expressed in terms of the displacement components as given below:

⎟⎟⎠

⎞⎜⎜⎝

ϕ∂∂∂

+∂

∂=φ⎟

⎟⎠

⎞⎜⎜⎝

ϕ∂

∂+

ϕ∂

∂=φ

∂=φ

∂+

ϕ∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

ϕ∂

∂=

∂∂

=

ϕϕ

ϕϕ

ϕϕ

ϕϕ

xu

xu

R1,uu

R1,

xu

,x

uuR1γ,u

uR1ε,

xuε

r2

x2r

2

22r

2

x

xxr

xx

(1)

Corresponding quantities associated with the strain components and curvatures are the membrane resultant forces ϕϕ xx N,N,N , and resultant moments ϕϕ xx M,M,M , respectively. In addition, resultant shear forces Qx, Qφ are also present. The shell is constituted by isotropic material characterized by two elastic constants namely: i) modulus of elasticity, E and ii) poisons modulusν .

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150

t

O

xdxL

d

k

zu

yuxu

u

ru

R

z,

y,x,

k

Figure 1. Analytical and FEM Model

M∂M∂+ d

M

Mx

d+∂ xM∂

xMxM

Mx dxx∂Mx∂+

xM

Mx dxx∂Mx∂+

Qx+∂ xQ∂x dx

Q d∂Q∂+

Q

xQ

r

x

rpxp

p

Nx +∂ xN∂x dx

x d∂N∂+Nx

Nx

xN

+∂N∂ dN

dxx∂Nx∂+

Nx

xN

N

x

r

Figure 2. Membrane stresses, moment and shear forces acting on infinitesimal element of shell Relationships between static quantities and strain/curvature given by the Hooke’s law are shown below:

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ϕϕϕϕϕϕ

ϕϕϕϕϕ

+=−=+

−−=+

−−=

+=+

−=+

−=

x

3

xxx2

3

x2

3

x

xxx2x2x

χν112

tEMM,νχχν112

tEM,νχχν112

tEM

,γν12

tEN,ενεν1tEN,ενε

ν1tEN

(2)

By substituting Eq. (1) in Eq. (2), force-displacement relationships are derived as given below:

( ) ( ) ( )

( ) ( )

( ) xr

23

x

2r

2

22r

2

2

3

2r

2

22r

2

2

3

x

xxr

x2r

x2x

Mx

ux

uRν112

tEM

,uuR1

xuν

ν112tEM,uu

Rv

xu

ν112tEM

,uR1

xu

ν12tEN,u

ux

uRνν1R

tEN,uu

νx

uRν1R

tEN

ϕϕ

ϕ

ϕϕ

ϕ

ϕϕ

ϕϕ

ϕ

−=⎟⎟⎠

⎞⎜⎜⎝

ϕ∂∂∂

+∂

+=

⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

ϕ∂

∂+

ϕ∂

∂+

−−=

⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

ϕ∂

∂+

ϕ∂

∂+

−=

⎟⎟⎠

⎞⎜⎜⎝

ϕ∂∂

+∂

+=⎟⎟

⎞⎜⎜⎝

⎛−

ϕ∂

∂+

∂∂

−=

⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

ϕ∂

∂+

∂∂

−=

(3)

For a general case in three dimensions, we can write three equation of equilibrium along x,y and z direction and three equations of moments with respect to the x,y, and z axes. In total there are six equations of equilibrium available for solving eight unknown

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151

functions. If ϕϕ xx N,N,N are small in comparison with their critical values at which lateral buckling of the shell may occur, their effect on bending becomes negligible; all terms containing the products of the resultant forces or resultant moments with the derivatives of the small displacements can be neglected in the equilibrium equations (Timoshenko and Krieger, 1959). With this hypothesis, equilibrium equations are now reduced to five as given below:

0RxQM

Rx

M

0RQM

Rx

M

0RpNQ

Rx

Q

0RpQN

Rx

N

0RpN

Rx

N

xx

x

rx

x

xxx

=+ϕ∂

∂−

∂∂

=ϕ+ϕ∂

∂−

=++ϕ∂

∂+

∂∂

=+−ϕ∂

∂+

=+ϕ∂

∂+

∂∂

ϕ

ϕϕ

ϕϕ

ϕϕϕϕ

ϕ

(4)

where rx ppp ,, ϕ are the load for unit surface along x-axis, circumferential and radial direction, respectively. Solving last two equations with respect to ϕQ , xQ and substituting in remain three equations, we finally obtain the three following relationship:

0RpNM

R1

xM

RxM

xM

0RpM

R1-

xM

Rx

NN

0RpN

Rx

N

r2

2x

2

2x

2x

2

xx

xxx

=++ϕ∂

∂+

ϕ∂∂

∂−

∂+

ϕ∂∂

=+ϕ∂

∂+

∂+

ϕ∂

=+ϕ∂

∂+

∂∂

ϕϕϕϕ

ϕϕϕϕϕ

ϕ

(5)

The shear functions have the following expression:

( ) ( )

( ) ⎟⎟

⎜⎜

∂∂

+ϕ∂∂

∂+

ϕ∂∂

−−=

⎥⎥⎦

⎢⎢⎣

⎟⎟

⎜⎜

∂−+

ϕ∂∂∂

+ϕ∂

∂+

ϕ∂

−−=ϕ

ϕ

ϕϕ

3r

32

2r

32

22

3

2

2

2r

32

3r

3

2

2

23

3

xuR

xu

xu

ν1R12tE

xQ

xu

ν1x

uRuuν1R12

tEQ

(6)

Substituting the equations (3) in equations (5), we obtain the equilibrium equation in terms of the unknown displacement functions rx u,u,u ϕ as given below:

( )

( )

⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪

=⎟⎟⎠

⎞⎜⎜⎝

⎛ ν+

⎥⎥⎦

⎢⎢⎣

ϕ∂

∂+

ϕ∂∂

∂−+

ϕ∂

∂+

ϕ∂∂

∂+

∂−−

ϕ∂

∂+

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛ ν+

⎥⎥⎦

⎢⎢⎣

ϕ∂

∂+

ϕ∂∂

∂+

ϕ∂

∂+

∂−+⎟

⎜⎜

ϕ∂∂

−ϕ∂

∂+

∂⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

ϕ∂∂∂+

+∂∂ν

−ϕ∂∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛ ν++

ϕ∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛ ν−+

ϕϕϕ

ϕϕϕϕϕ

ϕ

0tE

-1Rpu

R1

xu

Rν2u

R1

xu

R2

xuR

12t

Ruu

R1

xuν

0tE

-1RpuR1

xuu

R1

xu

ν1R12

tuuR1

xu

R2ν1

xu

2ν1

0tE-1p

xu

Rxu

R21u

R21

xu

2

r3

3

32

3

4r

4

322r

4

4r

42rx

2

3r

3

22r

3

2

2

22

22r

2

2

2

2x

2

2xr

2

2x

2

2x

2

(7)

3.1 Analytical solution in closed form for shells under uniform distributed load Let us consider an open barrel cylindrical shell loaded by uniform distributed load, as seen in Figure 3. The boundaries of the shell along its length are considered as fixed support in the analysis. Stress functions will depend only on one variable, namelyϕ leading to the following differential equations system:

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152

k=45° u == 0= k2

for

R/t=100Rck=25N/mm2

p

Surface load

0

0=ux

ru = 0

∂∂x 0=

ur

z = constant

fixed edge

fixed edge

k-pz

R

p

pr

Figure 3. Shell under uniform distributed load

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

=⎟⎟

⎜⎜

ϕ∂∂

∂+

∂⇒=

=⎟⎟

⎜⎜

∂ϕ∂

∂+

∂ϕ∂

∂+

∂⇒=

=⎟⎟

⎜⎜

∂ϕ∂

∂+

∂ϕ∂

∂+

∂⇒=

∂∂

=∂ϕ∂

∂+

∂⇒=

=∂∂

−∂ϕ∂

∂+

∂⇒=

=⎟⎟

⎜⎜

∂∂

−∂ϕ∂

∂+

∂⇒=

∂∂

ϕϕ

ϕϕ

ϕ

ϕϕ

ϕϕ

ϕ

00x

xx0

x

0xx

0x

x0

x

0xx

0x

xx0

x

r2

r2

r

r2

r

22

r22

r22

2

3

2x

2

3

23

3

2

3

23

3x

x2

x

2x

2xx

xu

x

uM

0uu

R1

xuν

M

uu

Rv

xuM

0uR1

x

uN

uu

xuRν

N

0uuν

xuRN

⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪⎪⎪⎪

=⎟⎟⎠

⎞⎜⎜⎝

⎛ϕ∂

∂+

=⎟⎟⎠

⎞⎜⎜⎝

⎛ϕ∂

∂+

ϕ∂∂∂

=∂

=⎟⎟⎠

⎞⎜⎜⎝

∂+

ϕ∂∂

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

ϕ∂

∂∂

=∂

ϕ

ϕ

ϕ

ϕ

0uux

0,uux

0xu

0x

uRu

x

0uu

x

0xu

r2

2

r

3r

3

x

r

2x

2

(8)

From the first and fourth equations of the above set of equations, we can derive the following: ( ) ( ) ( )( ) ( ) ( ) ( ) 2

210r

10x

xqxqqx,u

xppx,u

ϕ+ϕ+ϕ=ϕ

ϕ+ϕ=ϕ (9)

where ( ) ( ) ( ) ( ) ( )ϕϕϕϕϕ 21010 qqqpp ,,,, are unknown functions of variable ϕ . By substituting Eq. (9) in the last equation of Eq. (8), we obtain the function ( )ϕϕ x,u as given below:

( ) ( ) ( ) ( ) 2210 x

dqdxhhx,uϕϕ

−ϕ+ϕ=ϕϕ (10)

where ( ) ( )ϕϕ 10 hh , are unknown functions of variable ϕ . By substituting Eqs (9-10) in Eq. (8), we obtain the following relationships:

( )( )( )( ) ( )( ) ( ) 6211

87340

5431

212

341

CsinCcosCr2pCCsinCcosCrp

CsinCcosChsinCcosCqsinCcosCq

+ϕ+ϕ=ϕϕ++ϕ−ϕ=ϕ

+ϕ+ϕ=ϕϕ+ϕ=ϕϕ−ϕ=ϕ

(11)

By substituting in Eqs. (9-10), the displacements functions are obtained as given below:

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153

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )xCCxsinCxCxcosqx,u

CxCCxcosCxCxsinhx,ux2CCrsinx2CCrcosCxCCx,u

23410r

523410

2314867x

−ϕ−+ϕ+ϕ=ϕ

+−ϕ++ϕ+ϕ=ϕ−ϕ−+ϕ+ϕ++=ϕ

ϕ (12)

The uniform distributed load applied on surface of the shell is resolved into components along ( )r,,x ϕ as given below:

( )( )⎪

⎪⎨

ϕ−ϕ=

ϕ−ϕ−==

ϕ

kr

k

x

cosppsinpp

0p (13)

where kϕ is total shell angle in radians. By substituting Eq.(12) in equilibrium Eq. (7), the first condition is verified. Further, by substituting the function ( )ϕ0q with its first derivative, the third equation of Eq. (12) becomes:

( ) ( ) ( ) ( )xCCxsinCxCxcosd

qdx,u 23410

r −ϕ−+ϕ+ϕϕ

=ϕ (14)

Now, the second and third equations of Eq. (7) become: ( ) ( ) ( ) ( )

( )( ) ( ) ( ) 0sinν112pRsinCcosCt12Rtν2ER

dqd12R

dhdt12Rt

dqdtE

k24

12222

20

22

20

2222

40

4

=ϕ−ϕ−−ϕ−ϕ++

⎥⎥⎦

⎢⎢⎣

ϕ

ϕ−

ϕ

ϕ++

ϕ

ϕ

(15)

( ) ( ) ( ) ( )

( )( )[ ] ( ) ( ) 0cosν112pRsinCcosCt12R6RCtνR2E

dqd

dhdt

dhd

dqdtE12R

k24

1222

62

50

5

30

32002

=ϕ−ϕ−−ϕ−ϕ++−

⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

ϕ

ϕ+

ϕ

ϕ+

ϕϕ

−ϕϕ

(16)

By integrating Eq. (15) twice we get:

( ) ( ) ( ) ( )( )

( ) ( ) ( )1092

02

20

222

22k

24

122

0

CCdqdtq12R

t12R1

t12RtEsinν1R12psinCcosCν2Rh

+ϕ+⎥⎥⎦

⎢⎢⎣

ϕ

ϕ−ϕ

++

+

ϕ−ϕ−+ϕ−ϕ=ϕ

(17)

By substituting in Eq. (16) we get the following differential equation in ( )ϕ0q :

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] 0cos1pR2RCCtEtR12dqd

dqd2

dqdtE k

2269

225

05

30

303 =ϕ−ϕν−+ν++−

⎥⎥⎦

⎢⎢⎣

ϕ

ϕ+

ϕ

ϕ+

ϕϕ

(18)

By solving, function ( )ϕ0q is obtained and is given as below:

( ) ( )( ) ( )[ ] ( )( ) ( )

( )( )( )[ ] ( ) 151413123k

22222

212141169

22

3k

22222

0

CcosCCCt8E

sinsin8cos112ν1t12RpRt

sinCCCνRCCt12Rt8E

cossin112cos81νt12RpRq

+ϕϕ−−+ϕϕϕ−ϕ−ϕ−+

+

ϕϕ+++ϕ+++

ϕϕ−ϕ+ϕϕ−+=ϕ

(19)

Now the hypothesis is verified to satisfy the equilibrium and compatibility conditions. By substituting Eq. (19) and Eq. (17) in Eq. (12), displacement functions rx u,u,u ϕ are obtained in a closed form. The integration constants 14,....,2,1iC = (the constant 15C is not present in displacement function as the function ru is considered as a derivative of is present the derivate of function ( )ϕ0q given by Eq. (14)) are determined by imposing the boundary conditions at fixed supports and using polynomial identity rule. The following sets of algebraic equations are obtained.

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154

( ) ( )( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )⎪⎪⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪⎪⎪

=ϕ=ϕϕ

=ϕ=ϕϕ

=ϕ=ϕϕ

⇒∀=ϕ=ϕϕ

==ϕϕ

==ϕϕ

==ϕϕ

⇒∀==ϕϕ

=ϕ=ϕ=ϕ=ϕ=ϕ=ϕ⇒∀=ϕ=ϕ==ϕ==ϕ==ϕ⇒∀==ϕ

ϕ=ϕ=ϕ=ϕ=ϕ=ϕ⇒∀=ϕ=ϕ

=ϕ=ϕ

==ϕ==ϕ⇒∀==ϕ

=ϕ=ϕ=ϕ=ϕ⇒∀=ϕ=ϕ==ϕ==ϕ⇒∀==ϕ

ϕ

ϕ

02ddq 0;2

ddq0;)2(

ddq x02x,

dud

00ddq 0;0

ddq0;0)(

ddq x00x,

dud

02q 0;2q0;)2(q x02x,u00q 0;0q0;0)(q x00x,u

0d

2qd 0;2h0;)2(h x02x,u

0d

0qd 0;0h0;0)(h x00x,u

02p0;)2(p x02x,u00p0;0)(p x00x,u

k2

k1

k0

kr

210r

k2k1k0kr

210r

k2k1k0k

210

k1k0kx

10x

(20)

It can be easily seen from the above that there are only 14 independent equations (as same as the integration constants to be determined); the remaining are linearly dependent ones.

( ) ( )( ) ( )( )[ ]( )22

kkk22

k2k

22k

222

91,2,...8i t12Rsin4cos3t4R3cos1812R38tRtβ48C0,C

+

ϕϕ−ϕ++ϕ−ϕ+−ϕ=== (21)

( ) ( ) ( )( ) ( )( )[ ]

( )22kkk

4224k

2k

22k

222k

2

10 t12Rcos3sin4tt16R48R3cos1812R38tt12RRβ48C

+

ϕ−ϕϕ++−ϕ−ϕ+−ϕ+ϕ−= (22)

( ) ( )( )( )( )⎥⎥⎦

⎢⎢⎣

−−++ϕϕϕ+ϕ+

−ϕ+−ϕ+ϕϕβ−=

224422kkkk

2

2k

222k

42k

4k

11t72R240Rtt12Rcos2sincos312R

928t2R52224Rt2cos6C (23)

( ) ( )( )( ) ( ) ( ) ⎥⎥

⎢⎢⎢

⎟⎠⎞⎜

⎝⎛ +−ϕ++ϕ+ϕ+

ϕ−ϕϕ+ϕ+β=

224k

2222k

222k

kk2kk

222

12t5R36R6cos25t36R6Rt12R4sin

cos3cos41t12R4C (24)

( ) ( )( ) ( ) ( ) ⎥⎦

⎤⎢⎣⎡ ϕ++⎟

⎠⎞⎜

⎝⎛ +−ϕ+ϕ−

ϕ−ϕ++ϕϕ−=

k2222222

k222

k

k2k

222kk13

sin3t320R6Rt320R18Rt12Rsinβ12

3cos283t12Rcosβ2C (25)

( ) ( ) ( )( ) ( ) kk

222k22

2k

22k

2222kk14 sinsin25t36Rβ96Rcos2

t12R12t5212Rt12Rsinβ8C ϕϕ++

⎥⎥⎦

⎢⎢⎣

⎡ϕ−

+

−ϕ+−ϕ+ϕϕ= (26)

where ( )( ) ( )[ ]kk

22k

42k

22k

23

22

sin2t12Rcos224R124Rt2t16Eν1pRβ

ϕϕ++ϕ+−ϕ+ϕ

−=

Substituting the integration constants in the displacement function, we get the following displacement functions:

0ux = (27)

( ) ( )( )[ ] ( ) ( )

( ) ( ) ( )[ ]

( ) ( )[ ] ( )( )223

k222224422

22

22121411

21411

2

3k

24

221413

222212

2

22

910

t12RtE8sint12R2t216R1584Rt1νpR

t12Rsint12RCCCtCC12R

tEcos1ν12pR

t12RcosCCt12R12RtC

tt12RCCu

+

ϕ−ϕ+ϕ+−−−+

+

ϕ+ϕ+−+++

ϕ−ϕϕ−+

+

ϕϕ+++−−⎟

⎟⎠

⎞⎜⎜⎝

⎛ +ϕ+=ϕ

(28)

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155

( )( )( )[ ]

( ) ( )( )[ ] ( ) ( ) ϕϕ++ϕϕ++ϕ−ϕ+ϕϕϕ−+⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

ϕϕϕ−ϕ−ϕ−+=

sinCCcosCCsin32cos4sin1νpRCt

t12R

tE8cossin4cos321νt12RpRu

141312112

k22

92

22

3k

22222

r

(29)

The stress functions are given by:

( ) ( ) ( ) ( )( )( ) ( )( )

( ) ( )

( ) ( ) ( )( ) ( ) ( )( )[ ]( ) ( ) ( ) ( )( )

( ) ( )( )⎥⎥⎦⎤

⎢⎢⎣

ϕϕ+−ϕ−ϕ−

ϕ−+ϕϕ+ϕν−+ϕ+ϕω−=

ϕϕ++ϕ+ϕ−ϕ+−ϕϕ+ϕ−+

⎥⎦⎤

⎢⎣⎡ ϕ−ϕ++=

⎥⎥⎦

⎢⎢⎣

ϕϕ−+ϕϕ++

ϕϕ+−ϕ−ϕ−+ϕ−ϕ=

======

ϕ

ϕ

ϕ

ϕϕϕϕϕ

sintR122costR12sinsintR12costR122cos

1pRsinCcosCtE4Q

sint12R2cost4R3cossint4R3cost12R2sin1νRωp

sinCcosCt24Rt12RCR6

tEωM

sinsint12Rcost12R2sint12R2cost12Rcos

1νRωpsinCcosCtEω4N

MνM,NνN0,Q0,N0,M0,M

2222k

2222k22

14123

2222k

2222k

23

121422222

9

k2222

2222k22

12143

xxxxxx

(30)

where ( )( )1tR12R21

222 −ν+=ω

3.2 Analytical solution in closed form for shells under edge loads The open barrel RC cylindrical shell in now analyzed under uniformly distributed surface load varying sinusoidal in addition to the edge loads. Bellington (1965) presented the general equations of equilibrium under certain assumptions to simplify the stress functions. Poisson’s modulus is neglected in this treatment to get the following relationships:

⎟⎟⎠

⎞⎜⎜⎝

⎛ϕ∂

∂+

∂=⎟⎟

⎞⎜⎜⎝

⎛−

ϕ∂

∂=

∂∂

= ϕϕ

ϕϕ

xx

xx

uR1

xu

2tEN,u

uR

tEN,x

uRtEN r (31a)

⎟⎟⎠

⎞⎜⎜⎝

⎛ϕ∂

∂+

∂∂

=−=⎟⎟⎠

⎞⎜⎜⎝

ϕ∂∂

+ϕ∂

∂−=

∂∂

= ϕϕϕϕ

ϕr

3

xx2r

2

2

3

2r

23

xuu

xR12tEMM,uu

12RtEM,

xu

12tEM (31b)

⎥⎥⎦

⎢⎢⎣

∂+⎟⎟

⎞⎜⎜⎝

⎛ϕ∂

∂+

ϕ∂∂

∂∂

−=⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

ϕ∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛ϕ∂

∂+

ϕ∂∂

−=ϕ ϕϕϕ 2r

22r

2

3r2r

3

3

xuRuu

R12tE

xQ,uuRuuR12tEQ

xx2

2

2

2 (31c)

In equations (31b), terms of radial displacement ϕu are neglected and ϕQ terms in Eq. (4) are also dropped. By utilizing namely: i) equilibrium equations given by Eq. (4); ii) relationships given by Eq. (31); as well as the compatibility conditions between the displacement components and stresses forces, the elastic problem is now reduced in to a single eighth order partial differential equation with displacement along the radial direction as the only unknown. The modified differential equation is given by:

⎟⎟

⎜⎜

ϕ∂

∂+

ϕ∂∂

−∂∂

∂+∇=

∂⎟⎠

⎞⎜⎝

⎛+∇⎟

⎟⎠

⎞⎜⎜⎝

⎛ ϕϕ2

2

22

2

22x

3

3z4

4r

4

2r8

3 p

r1

x

p

r1

φxp

r1p

xu

rtEu

12tE

(32)

Where the differential operator r8u∇ and r

4p∇ are given by the following expressions:

4r

4

422r

4

24r

4

r4

8r

8

862r

8

644r

8

426r

8

28r

8

r8

pr1

xp

r2

xpp

ur1

xu

r4

xu

r6

xu

r4

xuu

ϕ∂

∂+

ϕ∂∂

∂+

∂=∇

ϕ∂

∂+

ϕ∂∂

∂+

ϕ∂∂

∂+

ϕ∂∂

∂+

∂=∇

(33)

Let the shell be subjected to surface load ‘p’ that is distributed uniformly over the shell surface and also varying sinusoidal along its length. The load can be resolved into its components along longitudinal, circumferential and radial directions as given below:

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156

( )

( )⎪⎪⎪⎪

⎪⎪⎪⎪

⎟⎠

⎞⎜⎝

⎛ϕ−ϕ=

⎟⎠

⎞⎜⎝

⎛ϕ−ϕ−=

=

ϕ

Lxπsincospp

Lxπsinsinpp

0p

kr

k

x

(34)

where, p = load per unit surface area of the shell surface and L is the span of shell. Equilibrium equations given by Eq. (4) can be satisfied by a particular integral of the form:

( ) ( ) ⎟⎠⎞

⎜⎝⎛ϕ−ϕ=ϕ

LxπsincosCx,u k0r (35)

By substituting, the value of constant C0 that depends on the geometric parameters of the shell is determined as: ( )

( ) ⎥⎦⎤

⎢⎣⎡ ++

++=

64442222

44222444

0RπL12RπLttE

RπRπL4L2RpL12C (36)

Further by substituting C0 at any desired section, Eq. (35) gives the displacement in terms of load and geometric characteristics of the reinforced concrete shell. However, particular integral of Eq. (35) does not provide the complete solution and hence complimentary function of Eq. (32) is now considered. Equating the RHS of Eq. (32) to zero, the complimentary function assumes the following form:

( ) ∑ ⎟⎠⎞

⎜⎝⎛=ϕ

=1,3,5..n

φMmr L

xπsineAx,u (37)

where, Am represents the eight arbitrary constants depending on the longitudinal boundary conditions and M (in eMΦ ) represents the corresponding roots. Variation of ( )ϕx,ur along the longitudinal boundary can be described by Fourier series considering only the first term of the series. By substituting Eq. (37) in homogenous form of Eq. (32), roots M are determined. On substitution, we get:

0LtR

π12kRM

422

442

2

2=+⎟

⎟⎠

⎞⎜⎜⎝

⎛− (38)

where L

k π= . Eq. (38) in now expressed as:

( ) 0Q4RkM 84222 =+− (39) where,

43

t3r

LQ, π== 2

648

tRk3Q (40)

By introducing the constantQRk 22

=γ , Eq. (39) is rewritten as:

( )i2γQM 22 ±= (41) By applying the Euler’s equation, Eq. (41) becomes:

( )i1γQM 22 ±±= (42) By solving, we obtain the follows roots:

)β(αM,)β(αM 222111 ±±=±±= (43) where,

,2

γ)(11γ)(1Qβ,

2γ)(11γ)(1

,2

γ)(11γ)(1Qα,

2γ)(11γ)(1

2

2

2

1

2

2

2

1

−++−=

+−++=

−−+−=

++++=

(44)

Now, the complimentary function of Eq. (32) is given by:

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157

⎟⎠⎞

⎜⎝⎛

⎥⎥⎦

⎢⎢⎣

++++

++++=ϕ

−−+−−−+−

−+−+

Lxπsin

eBeAeBeA

eBeAeBeA)(x,u

)φiβ(α4

)φiβ(α4

)φiβ(α3

)φiβ(α3

)φiβ(α2

)φiβ(α2

)φiβ(α1

)φiβ(α1

r22221111

22221111

(45)

For ( )ϕx,ur to be real, the integration constants 1,2,3,4)(iB,A ii = must be complex; by recalling the integration constants, the displacement function ( )ϕx,ur is written as:

( ) ( )( ) ( )

⎟⎠⎞

⎜⎝⎛

⎥⎥⎦

⎢⎢⎣

ϕ+ϕ+ϕ+ϕ+

ϕ+ϕ+ϕ+ϕ=

ϕ−ϕ

ϕ−ϕ

Lxπsin

esinβDcosβDesinβDcosβD

esinβDcosβDesinβDcosβDu

22

11

α2827

α2625

α1413

α1211

r (46)

where 6,7,8)1,2,3,4,5,(iDi = are integration constants. The final solution is the sum of particular integral given by Eq. (35) and complementary function given by Eq. (46). By substituting in Eq. (31) and utilizing the equilibrium conditions given by Eq. (4), we obtain the stress functions as follows:

RpuR1

xu

R2

xuR

12tEN z4

r4

322r

4

4r

43−⎟

⎟⎠

⎞⎜⎜⎝

ϕ∂

∂+

ϕ∂∂

∂+

∂=ϕ (47)

∫⎥⎥⎦

⎢⎢⎣

⎡−

ϕ∂∂

+⎟⎟⎠

⎞⎜⎜⎝

ϕ∂

∂+

ϕ∂∂

∂+

ϕ∂∂

∂−= ϕϕ dx

prx pu

R1

xu

R2

xu

12tEN 5

r4

432r

5

24r

53 (48)

∫ ∫⎥⎥⎦

⎢⎢⎣

∂∂

−ϕ∂

∂+

ϕ∂

∂−⎟

⎟⎠

⎞⎜⎜⎝

ϕ∂

∂+

ϕ∂∂

∂+

ϕ∂∂

∂= ϕ dxdx

xRp

Rp

Rpu

R1

xu

R2

xu

12RtEN x

2r

2

6r

6

442r

6

224r

63

x (49)

⎟⎟⎠

⎞⎜⎜⎝

ϕ∂∂

∂+

∂−= 2

r3

23r

33

x xu

R1

xu

12EtQ (50)

⎟⎟⎠

⎞⎜⎜⎝

ϕ∂∂

∂+

ϕ∂

∂−=ϕ 2

r3

3r

3

2

3

xuu

R1

R12EtQ (51)

By introducing the following vector, displacement function can be written in more compact form:

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

=

ϕ

ϕ

ϕ

ϕ

2

2

1

1

α-2

α2

α-1

α1

ω where ( )( )

( )( )⎥⎦

⎤⎢⎣

⎡ϕϕ

=⎥⎦

⎤⎢⎣

⎡ϕϕ

=2

22

1

11 βsin

βcosω,

βsinβcos

ω (52)

( ) ( )[ ] ⎟⎠⎞

⎜⎝⎛⋅+ϕ−ϕ=ϕ

LxπsinDωcosCx,u k0r (53)

where [ ]T821 D,.....,D,DD = . In explicit form the stress functions are given by:

( ) [ ]( )[ ]

( ) [ ] [ ]( )[ ]

( ) [ ]( )[ ]

( ) [ ] [ ]( )[ ] ⎟⎠⎞

⎜⎝⎛⋅⋅⋅+ϕ−ϕ=

⎟⎠⎞

⎜⎝⎛⋅⋅+ϕ−ϕ=

⎟⎠⎞

⎜⎝⎛⋅⋅⋅+ϕ−ϕ=

⎟⎠⎞

⎜⎝⎛⋅⋅+ϕ−ϕ=

ϕ

ϕϕ

ϕϕ

LxπsinDλΓωsinH

12EtQ

LxπcosDΓωcosH

12EtQ

LxπcosDλΓωsinH

12EtN,M

LxπsinDΓωcosH

12EtN,N,M,M

k0

3

k0

3

x

k0

3

xx

k0

3

xx

(54)

where H0 assumes different values for different stress functions as given below:

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158

( )

( )

( )

( )

( )⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

+

+

−+

−+

π−

+

=

ϕ

ϕ

ϕ

ϕ

ϕ

Q forRL

CRπL

Q forRL

CRπLπ

NfortELπ

24pRLπ

CLRπ

MforLR

πC

NforEt

12pRRL

CRπL

NforEt24pL

RπLCRπL

MforRC

MforLCπ

H

320

222

x230

222

x33430

2222

x0

3340

2222

x3

2

5220

2222

20

x20

2

0

R2

(55)

The matrices [ ] [ ]λ,Γ are given by:

[ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] ⎥⎥

⎥⎥⎥

⎢⎢⎢⎢⎢

T(2)

(2)

T(1)

(1)

Γ0000Γ0000Γ0000Γ

where ⎥⎥⎦

⎢⎢⎣

−=

⎥⎥⎦

⎢⎢⎣

−= (2)

11(2)12

(2)12

(2)11(2)

(1)11

(1)12

(1)12

(1)11(1)

ΓΓΓΓ][Γ,

ΓΓΓΓ][Γ (56)

[ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]⎥⎥

⎥⎥⎥

⎢⎢⎢⎢⎢

=

(0)

(0)

(0)

(0)

λ0000λ0000λ0000λ

λ where ⎥⎦

⎤⎢⎣

⎡−

=10

01][λ (0) (57)

( ) ( )

( ) ( ) ( )[ ]

( )[ ] ( )[ ]

( ) ( ) ( )[ ]

( )[ ] ( )[ ]

( ) ( )[ ] ( )[ ]

( ) ( )[ ] ( )[ ] ϕ

ϕ

⎪⎪⎭

⎪⎪⎬

+−−=

+−++=

+−−=

+−++=

⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪

π

α−β+β−α−=

π

++−−+−+=

π

α−β+β−α−=

π

++−−+−+=

−==−==

====

Nfor

RLRπαβLβ4αΓ,

RLββ6ααLα-βRπ2LRπΓ

,RL

RπαβLβ4αΓ,RL

ββ6ααLα-βRπ2LRπΓ

Nfor

RLLRπL-Rπβ2αΓ

,RL

ββαβ15ααLββ6ααRπ2Lα-βRπΓ

,RL

LRπL-Rπβ2αΓ

,RL

ββαβ15ααLββ6ααRπ2Lα-βRπΓ

M forRβα2Γ,

Rα-βΓ,

Rβα2Γ,

Rα-βΓ

M for0ΓΓ,LπΓΓ

34

2222

22

222(2)

1234

42

22

22

42

422

22

22244(2)11

34

2221

21

211(1)

1234

41

21

21

41

421

21

22244(1)11

522

22

22

22222

22

22222(2)

12

522

62

42

22

22

42

62

442

22

22

42

22222

22

44(2)11

522

21

21

22221

21

22211(1)

12

522

61

41

21

21

41

61

441

21

21

41

22221

21

44(1)11

222(2)

122

22

22(2)

11211(1)

122

21

21(1)

11

x(2)12

(1)122

2(2)11

(1)11

x

33

15

33

15

(58)

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159

( ) ( )[ ]

( ) ( )[ ]

( ) ( )[ ]

( ) ( )[ ]

ϕ

ϕ

⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪

+−+−+=

+−+−−=

+−+−+=

+−+−−=

−=−=−=−=

x

43

42

22

22

42

422

22

222442(2)

12

43

42

22

22

42

422

22

222442(2)

11

43

41

21

21

41

421

21

222441(1)

12

43

41

21

21

41

421

21

222441(1)

11

x2(2)

122(2)

111(1)

121(1)

11

Nfor

πRLββ10α5αL3αβRπ2LRπβΓ

πRL5ββ10ααL3βαRπ2LRπαΓ

πRLββ10α5αL3αβRπ2LRπβΓ

πRL5ββ10ααL3βαRπ2LRπαΓ

MforRLπβΓ,

RLπαΓ,

RLπβΓ,

RLπαΓ

(59)

( )[ ]

( )[ ]

( )[ ] ( )[ ]

( )[ ] ( )[ ] ϕ

⎪⎪⎭

⎪⎪⎬

−+=

−−=

−+=

−−=

⎪⎪⎭

⎪⎪⎬

−=−+

=

−=

−+=

Qfor

RL3αβLRπβΓ,

RL3βαLRπαΓ

RL3αβLRπβΓ,

RL3βαLRπαΓ

Qfor

LRπβ2αΓ,

RLαβLRππΓ

LRπβ2αΓ,

RLαβLRππΓ

32

22

22

2222(2)

1232

22

22

2222(2)

11

32

21

21

2221(1)

1232

21

21

2221(1)

11

x

222(2)

1223

22

22

222(2)11

211(1)

1223

21

21

222(1)11

(60)

An open barrel reinforced concrete cylindrical shell with geometric properties given in Table 1 is analyzed under uniform surface load varying sinusoidally in addition to the sinusoidal symmetric edge loads namely: i) fixed unitary moment; ii) unit axial edge load; and iii) unit shear edge load, considered to act one-by-one successively along the longitudinal boundaries. Figures 4-5 show the stress resultants obtained from the analysis using proposed close form expressions, for different values of φ. It can be seen from the figures that stress resultants are symmetric with respect to the crown of the shell and qualitatively similar under different edge loads. It is also seen that the obtained results on the basis of proposed expressions closely agree with those obtained from the detailed FEM analysis based on the procedures discussed by Stanley and Huges (1984). The stress resultants obtained from detailed FEM analysis are shown in Figure 6, for comparison.

Table 1 Geometric properties of the RC shell (case i) Description values Φk 45º L 26.67m R 8m t 80mm Concrete Mix and steel grade M25 and Fe 415 steel Dead load 25 x 0.08 = 2.0 kN/m2 Live load for inspection 0.5 kN/m2 Load due to finishes 0.75 kN/m2 Total load 3.25 kN/m2 Young’s modulus of concrete, E 25 x 106 kN/m2 Poisson’s ratio of concrete, ν 0.20

Page 13: Design aids for fixed support reinforced concrete

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160

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0 10 20 30 40 50 60 70 80 90

Analytical FEM

-1.00

-0.50

0.00

0.50

1.00

1.50

0 10 20 30 40 50 60 70 80 90

-6.00

-5.00

-4.00

-3.00

-2.00

-1.00

0.000 10 20 30 40 50 60 70 80 90

-35.00

-30.00

-25.00

-20.00

-15.00

-10.00

-5.00

0.00 0 10 20 30 40 50 60 70 80 90

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

0 10 20 30 40 50 60 70 80 90

-1.5

-1.0

-0.5

0.0

0.5

1.0

0 10 20 30 40 50 60 70 80 90

Analytical FEM

Analytical FEM

Analytical FEM

Analytical FEM

Analytical FEM

(deg)

(deg)

(deg)

(deg)

(deg)

Mx

(kN

-m/m

)

M(k

N-m

/m)

(deg)

Nx

(kN

/m)

N(k

N/m

)

Q(k

N/m

)

u z(m

m)

Figure 4. Stress resultants for RC shell at φ = 45º

Page 14: Design aids for fixed support reinforced concrete

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161

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0 10 20 30 40 50 60 70

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 10 20 30 40 50 60 70

-6.60

-6.40

-6.20

-6.00

-5.80

-5.60

-5.40

-5.200 10 20 30 40 50 60 70

-41.00

-40.00

-39.00

-38.00

-37.00

-36.00

-35.00

-34.00

-33.000 10 20 30 40 50 60 70

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

0 10 20 30 40 50 60 70

-0.00030

-0.00025

-0.00020

-0.00015

-0.00010

-0.00005

0.00000

0.00005

0 10 20 30 40 50 60 70

Analytical FEM

Analytical FEM

Analytical FEM

Analytical FEM

Analytical FEM

Analytical FEM

(deg) (deg)

(deg)

(deg)

(deg)

(deg)

Mx

(kN

-m/m

)N

x(k

N/m

)Q

(kN

/m)

M(k

N-m

/m)

N(k

N/m

)u z

(mm

)

Figure 5. Stress resultants for RC shell at φ = 35º

Page 15: Design aids for fixed support reinforced concrete

Chandrasekaran et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 148-171

162

Nx (kN/m) Nφ (kN/m) Mx (kN-m/m)

Mφ (kN-m/m) Qφ (kN/m) Displacement uz (mm)

Figure 6. Stress resultants and displacement of RC shell obtained from FEM at φ = 45º

3.3 Design charts After satisfactory comparison of stress resultants and displacement of RC shells obtained by employing the proposed expressions with that of detailed FEM, RC shell is now analyzed for different R/L ratios and angles of curvature to obtain Design charts for closer intervals of φ while Table 2 shows the stress resultants for edge loads. Based on classical flexural theory, using the proposed expressions in close form, Design charts are developed for computing the stress resultants in open cylindrical shells with varying geometric parameters. Figures 7-9 show the proposed Design charts plotted for different R/L ratios for practical cases of R/t and Φ angles those can be readily used for analyzing the shells without undergoing detailed computations. The expressions are also proposed in closed form for analysis of RC shells under sinusoidal surface load in addition to unit edge loads of different types. It is seen from the design curves that they follow the same trend for all edge load cases namely: i) fixed moment couple along longitudinal edge; ii) axial load along the longitudinal edge; as well as iii) shear edge load. The design curves can be readily used to obtain the stress resultants for RC open barrel cylindrical shell for varying R/L values. The stress resultants can also be obtained for any desired values of φ that can be necessary for detailing of reinforcement, particularly near the crown and the valley. The numerical problem solved using the design charts show computation of final stress resultants at closer intervals of Φ. Results obtained closely agree with that of the results obtained through finite element analysis. The proposed design curves shall be readily used in design offices that do not have access to high end software. Further, they shall also serve as an easy checking tool for assessing the correctness of the results given by the software. The proposed design curves shall enable the design engineers to understand the qualitative range of deviation, if any, from the results given by the software.

Page 16: Design aids for fixed support reinforced concrete

Table 2 Stress resultants of RC shell under sinusoidal surface load and different edge loads (case ii)

Shells under uniform distribued load Surface load + moment edge load Surface load + axial edge load Surface load + shear edge loadf N N M Q

(deg) (kN/m) (kN/m) (kN-m/m) (kN-m/m) (kN/m) (kN/m) (kN/m) (kN-m/m) (kN-m/m) (kN/m) (kN/m) (kN/m) (kN-m/m) (kN-m/m) (kN/m) (kN/m) (kN/m) (kN-m/m) (kN-m/m) (kN/m)

0 -4.988 -31.177 0.231 1.447 -2.298 1283.62 0.000 2.237 1.000 0.000 1372.76 1.00 2.52 0.00 0.00 1482.95 0.00 2.98 0.00 -1.00

10 -4.560 -28.499 -0.070 -0.439 -0.566 181.59 -8.464 1.506 -0.445 -2.947 176.91 -8.13 1.67 -1.44 -3.09 153.08 -9.82 1.93 -2.50 -3.87

20 -4.199 -26.247 -0.088 -0.549 0.278 -218.88 -24.066 0.795 -4.875 -4.042 -241.31 -24.44 0.86 -5.99 -4.19 -277.95 -26.46 0.96 -7.76 -4.66

30 -3.940 -24.625 0.004 0.023 0.449 -261.97 -35.258 0.216 -9.511 -3.164 -271.41 -35.80 0.22 -10.73 -3.25 -274.24 -37.25 0.20 -12.83 -3.44

40 -3.804 -23.778 0.081 0.507 0.197 -211.41 -40.322 -0.110 -12.285 -1.164 -206.95 -40.77 -0.14 -13.55 -1.19 -180.18 -41.57 -0.22 -15.74 -1.23

50 -3.804 -23.778 0.081 0.507 -0.197 -211.41 -40.322 -0.110 -12.285 1.164 -206.95 -40.77 -0.14 -13.55 1.19 -180.18 -41.57 -0.22 -15.74 1.23

60 -3.940 -24.625 0.004 0.023 -0.449 -261.97 -35.258 0.216 -9.511 3.164 -271.41 -35.80 0.22 -10.73 3.25 -274.24 -37.25 0.20 -12.83 3.44

70 -4.199 -26.247 -0.088 -0.549 -0.278 -218.88 -24.066 0.795 -4.875 4.042 -241.31 -24.44 0.86 -5.99 4.19 -277.95 -26.46 0.96 -7.76 4.66

80 -4.560 -28.499 -0.070 -0.439 0.566 181.59 -8.464 1.506 -0.445 2.947 176.91 -8.13 1.67 -1.44 3.09 153.08 -9.82 1.93 -2.50 3.87

90 -4.988 -31.176 0.231 1.447 2.298 1283.62 0.000 2.237 1.000 0.000 1372.76 1.00 2.52 0.00 0.00 1482.95 0.00 2.98 0.00 1.00

fx Mx f f N N M Qfx Mx f f N N M Qfx Mx f f N N M Qfx Mx f f

k=45°

p(x) sin ( xL

(p.=

M

M

k=45°k=45°

NN

p(x) sin ( xL

(p.=

QQ

p(x) sin ( xL

(p.=

k=45°

pz = constant

fixed edge

fixed edge

R

Page 17: Design aids for fixed support reinforced concrete

90

0.5

1.0

1.5

2.0

2.5

-12

-10

-8

-6

-4

-2

-4

-2

2

4

(deg)

Mx(

kN-m

/m)

M(k

N-m

/m)

-0.5

0.0

-40

-30

-20

-10

2

0

-14

(deg)

Q(k

N/m

)N

(kN

/m)

Nx

(kN

/m)

(deg)

(deg)

3

1

0

-1

-3

-5

-15

0

-25

-35

-45

500

1000

1500

0

-500

1250

750

250

-250

(deg)

10 80706050403020

9010 80706050403020

9010 80706050403020

9010 80706050403020

-5

R/L

R/L =0.3,0.35,0.4,0.45,0.5

R/L=0.3

R/L=0.35

R/L=0.4

R/L=0.45

R/L=0.5

R/L=0.3

R/L=0.35

R/L=0.5

R/L=0.3

R/L=0.35

R/L=0.4

R/L=0.45

R/L=0.5

R/L=0.35R/L=

0.4R/L=0.45

R/L=0.5

R/L=0.3

k=45°M sin ( x

L

(1.=

= 0= k2

N 0=Q 0=N 0=x

forR/t=100Rck=25N/mm2

p(x) sin ( xL

(p.=

Surface load

Edge load

LOAD CASE=Surface load + Edge load

M

M

Figure 7. Design charts for RC shells under surface load and moment edge loads

Page 18: Design aids for fixed support reinforced concrete

Chandrasekaran et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 148-171

165

45

0.5

1.0

1.5

2.0

2.5

- 14

- 12

- 10

- 8

- 6

- 4

- 2

-4

-2

2

4

-0.5

-1

-3

0

1

3

3.0

0.0

-1.0

Mx(

kN-m

/m)

Q(k

N/m

)

M(k

N-m

/m)

500

1000

1500

-40

-30

-20

-10

5

-5

-500

0

(deg)

(deg)

(deg)

-5

-15

-25

-35

-45

N(k

N/m

)

(deg)

(deg)9010 80706050403020

9010 80706050403020

9010 80706050403020

9010 80706050403020

9010 80706050403020

R/L

R/L =0.3,0.35,0.4,0.45,0.5

R/L=0.3

R/L=0.35

R/L=0.4

R/L=0.45

R/L=0.5

R/L=0.35R/L=0.4R/L=0.45R/L=0.5

R/L=0.3

R/L=0.3

R/L=0.35

R/L=0.5

R/L=0.3

R/L=0.35

R/L=0.4

R/L=0.45

R/L=0.5

k=45° Msin ( x

L

(1.=

= 0= k2

N0

=Q 0=N 0=x

forR/t=100Rck=25N/mm2

NN

Nx

(kN

/m)

p(x) sin ( xL

(p.=

Surface load

Edge load

LOAD CASE=Surface load + Edge load

Figure 8. Design charts for RC shells under surface load and axial edge loads

Page 19: Design aids for fixed support reinforced concrete

Chandrasekaran et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 148-171

166

1.0

2.0

3.0

-15.0

-10.0

-5.0

-4

-2

2

4

Mx

(kN

-m/m

)

0.5

1.5

2.5

0.0

-0.5

3.5

M(k

N-m

/m)

-1

-3

1

3

0

5

-5

0

-2.5

-7.5

-12.5

-17.5

Q(k

N/m

) 500

1000

1500

-40

-30

-20

-10

-5

-15

-25

-35

-45

(deg)

(deg)

(deg)

(deg)

(deg)

-500

1250

750

250

0

-250

-1.0

9010 80706050403020

9010 80706050403020

9080706050403020

9010 80706050403020

9010 80706050403020

R/L

R/L =0.3,0.35,0.4,0.45,0.5

R/L=0.3

R/L=0.35

R/L=0.4

R/L=0.45

R/L=0.5

R/L=0.35R/L=0.4R/L=0.45R/L=0.5

R/L=0.3

R/L=0.3

R/L=0.35

R/L=0.5

R/L=0.3

R/L=0.35

R/L=0.4

R/L=0.45

R/L=0.5

k=45°

M

sin ( xL

(1.

== 0= k2

N 0=Q =N 0=x

for

R/t=100Rck=25N/mm2

0

Nx

(kN

/m)

p(x) sin ( xL

(p.=

LOAD CASE=Surface load + Edge loadSurface load

Edge loadQ

Q

Figure 9. Design charts for RC shells under surface load and shear edge loads

Page 20: Design aids for fixed support reinforced concrete

Chandrasekaran et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 148-171

167

4. Axial force-bending moment interaction

Concrete under multi-axial compressive stress state exhibits significant nonlinearity that can be successfully represented by nonlinear constitutive models capable of handling inelastic deformations and cyclic loading. Further, it is necessary to describe a suitable failure criterion for complete description of ultimate strength. A reinforced concrete element of rectangular cross-section is studied for axial force-bending moment yield interaction behaviour for different percentage of tension and compression reinforcements (Chandrasekaran et al. 2009). Bernoulli hypothesis of linear strain over the cross-section, both for elastic and elastic-plastic responses of the element, under bending moment combined with axial force was assumed. The interaction behaviour becomes critical when one of the following conditions apply namely: i) strain in reinforcing steel in tension reaches ultimate limit; ii) strain in concrete in extreme compression fibre reaches ultimate limit; as well as iii) maximum strain in concrete in compression reaches elastic limit under only axial compression. Axial force-bending moment limit domain consisting of six sub-domains are described; collapse in sub-domains (1) and (2) is caused by yielding of steel whereas for sub-domains (3) to (6), the collapse is caused by crushing of concrete. Table 3 gives the resume of expressions for various sub-domains, as reported by the researchers; in general terms, axial force-bending moment interaction can be expressed as below:

( ) ( )( )

( ) ( )( ) ( )⎪⎪⎩

⎪⎪⎨

⎟⎠⎞

⎜⎝⎛ −++⎟

⎠⎞

⎜⎝⎛ −=

+−=

d2DAσAσdyy

2DyεσbM

AσAσdyyεσbN

scscststccu

scscststccu

c

c

Ac

Ac

x

x

ϕ

ϕ

(61)

where Ac is the area of concrete in compression; if the section is in full tension, the integral vanishes as the neutral axis is negative. For the neutral axis greater than the depth of the section, say in this case, thickness of the shell, the integral shall extend for the total thickness of the shell. In general, as the shell remains in compression, the depth of neutral axis is always greater than the thickness of the shell. Figure 10 shows the cross-section element of the shell; axial force normal to the section, Nφ and moment about the section, Mφ are now considered for studying the interaction. The stress resultants (Nφe, Mφe) are determined from the proposed equations (case i) presented above and the values are plotted in the P-M domain. Points A to E correspond to shells with different values of φk. It can be seen that the stress resultants lie in the sub-domain (3) for all values of φk indicating a compression failure initiated by crushing of concrete. For the failure point on the P-M boundary, for example, A′ , following equation holds good:

0)(xMMN

)(xN cue

ecu =− ϕ

ϕ

ϕϕ (62)

By iteration, Eq. (62) is solved for xc and further by substitution, (Nφu, Mφu) are determined using Eq. (30). Depths of neutral axis thus obtained for different angles of curvature and the corresponding points on the P-M boundary namely: A’ to E’ are shown in Figure 9. Factor of safety (F.S) is obtained from the following relationship:

AOOAF.S′

= (63)

Using the above equation, factor of safety for shells with different φk are computed and shown in Figure 10. It can be seen from the figure that factor of safety reduces with increase in angle of curvature of the shell. Also, strain in concrete, for all cases, reaches ultimate limit prompting a compression failure while those in tension and compression steel are within the yield limits. The failure points interpreted on the P-M domain show compression failure of shell section initiating crushing of concrete.

Page 21: Design aids for fixed support reinforced concrete

Chandrasekaran et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 148-171

168

Table 3. Resume of expressions for P-M interaction (Chandrasekaran et. al. 2009)

sub-

cx ( )cxq ( )cmax,c xε ( )cst xε ( )csc xε ( )cu xP ( )cu xM

(1) ,] ∞− 0 dxDxc

csu−−

εsuε ⎟⎟⎠

⎞⎜⎜⎝⎛

−−−ε

dxDdx

c

csu st,uP st,uM

(2a) ] 0 dxDxc

csu−−

εsuε ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−εdxD

dxc

csu ( )0qPP con,1ust,u =+ ( )0qMM con,1ust,u =+

(2b) ] ( )dxDx csu0c

c −−εε− dxD

xc

csu−−

εsuε ⎟⎟⎠

⎞⎜⎜⎝⎛

−−−ε

dxDdx

c

csu con,1ust,u PP + con,1ust,u MM +

(3) ccu

0ccu xεε−ε

cuε ( )c

ccux

dxD −−ε ( )cccu

xdx −ε

con,1ust,u PP + con,1ust,u MM +

(4) ccu

0ccu xεε−ε

cuε ( )c

ccux

dxD −−ε ( )cccu

xdx −ε

con,1ust,u PP + con,1ust,u MM +

(5) ] ccu

0ccu xεε−ε

cuε ( )c

ccux

dxD −−ε ( )cccu

xdx −ε

con,1ust,u PP + con,1ust,u MM +

(6) Dcu

0ccuεε−ε

( )0ccuccu

c0ccuDx

xε−ε−ε

εε ( )( )0ccuccu

c0ccuDx

dxDε−ε−ε−−εε ( )

( )0ccuccu

c0ccuDx

dxε−ε−ε

−εεcon,2ucon,1ust,u PPP ++ con,2ucon,1ust,u MMM ++

]

x'c

0

0 ,[

[ x 'c ,x"

c

][ x"c ,x "'

c

][x"'c,D-d

[D-d,D

[D, ∞+ [

dom

ain

( ) ( )0y,xx cccmax,c =ε=ε ( )( )sttsccststscscst,u ppdDbAAP σ−σ−=σ−σ= ( ) ( )( ) ⎟

⎠⎞

⎜⎝⎛ −σ−σ−=⎟

⎠⎞

⎜⎝⎛ −σ−σ= d

2DppdDbd

2DAAM sttsccststscscst,u ( ) ( )[ ]cmax,cc0c2

0c2c

2cmax,c0c

0ccon,1u xqx3x3

xqbbqP −ε+ε

ε

−εσ+σ=

( ) ( ) ( ) ( )( )[ ]D2q3xxqD3x2q4x2x12

xqbqD

2bq

M ccmax,ccc0c20c

2c

2cmax,c0c0c

con,1u −+−ε+−+εε

−εσ−−

σ=

( )[ ]( )20c

2c

2ccmax,cc0cmax,c0c

con,2ux3

xDxDx3bP

ε

−−ε+εεσ−=

( ) ( ) ( )[ ]20c

2c

2c

2max,ccc0c

2cmax,c0c

con,2ux12

xDx2Dx2xDbM

ε

−ε++ε−εσ=

Page 22: Design aids for fixed support reinforced concrete

Point N M F.S.(deg) (kN/m) (kN-m/m) (kN/m) (kN-m/m)

(m) (kN/m )

A 30 87.162 0.628 862.85 6.22 9.90

0.079 0.003500 11023 -0.00083 -174581 0.0026 360870

B 35 89.881 1.477 631.24 10.38 7.02

0.059 0.003500 11023 0.00006 12243 0.0023 360870

C 40 92.801 2.708 421.08 12.29 4.54

0.044 0.003500 11023 0.00127 266762 0.0019 360870

D 45 95.928 4.451 239.47 11.11 2.50

0.030 0.003500 11023 0.00339 360870 0.0012 252626

E 50 99.236 6.849 132.09 9.12 1.33

0.023 0.003500 11023 0.00549 360870 0.0005 105996

N Mf ff k u ue e f f

xc ec,max s c,max est s st s scesc

-250

0

250

500

750

1000

1250

-12 -10 -8 -6 -4 -2 2 4 6 8 10 12

A'

B'

C'

D'E'

2a

2b

3

4

5

6

Mu (kN-m)

Pu (kN-m)

Point

A'B'C'D'E'

k=45°

pz = constant

fixed edgefixed edge

R

2 (kN/m )2 (kN/m )2

Point

A'B'C'D'E'

A B C DE

1.00m

80mm

pz= 3.25 kN/m2

pt = pc = 0.5%Rck= 25N/mm2

fy= 415N/mm2

R/L=R/t=

0.3100

R=8m Nf

Mf

Figure 10. Axial force-bending moment interaction of RC shells under uniformly distributed surface loads (P-M interaction)

Page 23: Design aids for fixed support reinforced concrete

5. Conclusions Design curves for stress resultants based on the classical flexural theory for single barrel open cylindrical shells are presented with analytical expressions in closed form. The effect of variation of angle of curvature of shell geometry on the final stress resultants is demonstrated using the proposed design curves. Plots of axial force-bending moment values on the P-M domain are

illustrated to show the compression failure in RC shells initiated by crushing of concrete. As the behavior of the plots for various values of R/L is nonlinear, interpolations for still closer values of fraction of R/L is not recommended. Common theories adopted for analysis of cylindrical shells interpret the design behavior in different ways. Therefore every design procedure developed without undergoing rigorous analysis leads to simple and close form solution but of course with some defects. Despite shell structures possess inherent advantages gained by its structural form, they are not very common due to their analysis complexities. Current study attempts to make shell analysis easy, popular and develops confidence in design offices, encouraging the designer to handle a complex problem through a simple graphical tool. With the use of proposed Design charts, stress resultants can be readily determined at the valley, crown or at any desired section for various type of edge loads. In the present context of design offices highly influenced by use of software, this study is an attempt in the direction of developing simple graphical tools based on well established theories so that the designer is sure to know the error, if committed both qualitatively and quantitatively.

Nomenclature

R radius (m) L length (m) kϕ angle of the shell (deg) t thickness of the shell (mm)

E Young’s modulus of concrete (kN/m2) ν Poisson’s ratio of concrete ϕ generic angle in shell (deg)

xM Bending moment in plane x-z ϕM Bending moment in plane r-ϕ

ϕxM Torque moment in plane x-z xMϕ Torque moment in plane r-ϕ

xN Axial force in x-direction ϕN Axial force in ϕ -direction

ϕxN Shear force in x –direction

xQ Shear force in radial direction along longitudinal edges

ϕQ Shear force in radial direction along transversal edges

xε normal strain in x-direction ϕε normal strain in ϕ -direction

ϕγ x shear strain in plane x-ϕ xφ curvature in plane x-z

ϕφ curvature in plane r-ϕ ϕφ x curvature in plane x-z

xp surface load on the shell in x-direction (kN/m2) ϕp surface load in ϕ -direction (kN/m2)

rp surface load on the shell in radial direction (kN/m2) zp surface load on the shell in z-direction (kN/m2)

xu displacement component in x-direction ϕu displacement component in ϕ -direction

ru displacement component in radial direction zu displacement component in z- direction 8∇ differential operator of eight order 4∇ differential operator of fourth order

d effective cover of the section (m)

cx depth of neutral axis measure from extreme compression fibre (mm)

"'c

"c

'c

0c x,x,x,x limit position neutral axis (mm) q depth of plastic kernel of concrete (mm)

y depth of generic fibre of concrete measured from extreme compression fibre (mm)

εc strain in generic fibre of concrete εc,max maximum strain in concrete

εc0 elastic limit strain in concrete εcu ultimate limit strain in concrete

εst strain in tensile reinforcement εsc strain in compression reinforcement

εs0 elastic limit strain in reinforcement εsu ultimate limit strain in reinforcement

Es modulus of elasticity in steel (kN/m2) Rck compressive cube strength of concrete (kN/m2)

σc stress in generic fibre of concrete (kN/m2) σc,max maximum stress in concrete (kN/m2)

σc0 design ultimate stress in concrete in compression (kN/m2)

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Chandrasekaran et al. / International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 148-171

171

σy yield strength of steel (kN/m2) σs0 design ultimate stress in steel (kN/m2)

σst stress in tensile reinforcement (kN/m2) σsc stress in compression reinforcement (kN/m2)

pt percentage of tensile reinforcement pc percentage of compression reinforcement

Ast area of tension reinforcement (m2) Asc area of compression reinforcement (m2)

cA area of concrete in compression F.S. safety of safety in P-M interaction

eϕN Axial force in ϕ -direction in elastic range (kN) uNϕ Axial force in ϕ -direction at collapse (kN)

eϕM Bending moment in plane R-ϕ in elastic range (kN-m) uϕM Moment in plane R-ϕ at collapse (kN-m) References Billington David P. 1965. Thin shell concrete structures. Mc-Graw Hill Inc. Chandrasekaran, S., Ashutosh Srivastava, Parijat Naha. 2005. Computational tools for shell structures. Proc. of Intl. conf. on

structures and road transport (START-2005), IIT-Kharagpur, India, pp. 167-175. Chandrasekaran, S., Srivastava, A. 2006. Design aids for multi-barrel RC cylindrical shells. J. Struct. Engrg. SERC, Vol. 33, No.

4, pp. 287-296. Chandrasekaran S., Luciano Nunziante, Giorgio Serino, Federico Carannante 2009. Seismic design aids for nonlinear analysis of

reinforced concrete structures. CRC Press, Taylor and Francis, Florida, USA, 246pp. Flugge Wilhelm. 1967. Stress in shells. Springer Verlag, New York. Ha-Wong Song, Sang-Hyo Shim, Keun-Joo Byun, Koichi Maekawa. 2002. Failure analysis of reinforced concrete shell structures

using layered shell element with pressure node. J. Struct. Engrg, ASCE, Vol. 128, No. 5, pp. 655-664. Jacques H., 1977. Equilibrium of shell structures. Clarendon press, Oxford. Prabhakar G. 2003. Analysis of aircraft impact on containment structures. Proc. of 5th Asia-Pacific conf. on Shocks and Impact

loads on structures, Hunan, China, pp. 315-322. Ramaswamy G.S. 1968. Design and construction of concrete shell roof. First Edition, Mc-Graw Hill. Rericha P. 1996. Local impact on RC shells and beams. Proc. of International conf. on Structures under Shock and Impact loads

(SUSI 96), Udine, Italy, No. 4, Computational Mechanics Publications, Southampton, pp. 341-349. Stanley G.M., Huges T.T.R. 1984. Finite element procedures applicable to nonlinear analysis of RC shell structures. Report ADA

146796, Defence Technology Information Center, p. 68. Timoshenko, S.P., Woinowsky-Krieger, S. 1959. Theory of Plates and Shells. 2d ed., McGraw-Hill Book Company, New York. Wheen R.J, Anathraman. 1998. Concrete origami in Geodesic Domes, Conical tent and other Spatial Applications. Proc. of IASS

International Symposium on innovative applications of shells and spatial forms. Bangalore, India, Nov. 21-25, Vol. 1. Wheen R.J., Wheen P.R. 2003. A unique conical tent roof design and construction. Proc. of 2nd Specialty conference on the

conceptual approach to structural design, Milan, Italy, pp.855-861. Biographical notes Prof. Srinivasan Chandrasekaran is Associate Professor in Department of Ocean Engineering, Indian Institute of Technology Madras, India. He has engaged in teaching and research activities since the last 20 years. His field of specialization is Structural dynamics and offshore structures. He has published several papers in various national, international conferences and journals and also recently published a book on seismic design aids, addressing latest concepts in seismic resistant design of RC structures. He is member of ASCE and several other national and international societies and reviewer of several international journals. Prof. S.K.Gupta is working as Assistant Professor in Civil Engineering Department, Institute of Technology, Banaras Hindu University, India. He has done his research in the field of fluidized motion conveying of particulate solids. He has been in teaching and research for last 13 years and published over 15 papers in refereed international/ national journals and conferences. His research interest relates to the experimentation and modeling in the areas of multiphase flow, hydraulics, solar energy systems, and use of optimization techniques as well. He is a reviewer for an international journal published by Taylor and Francis and few other national journals. Dr. Federico Carannante is a Lecturer on contract with Dept. of Structural Engineering, University of Naples Federico II, Naples, Italy. His research interest vests in functionally graded materials and their applications in structural health monitoring. He has about fifteen publications in refereed international journals and many conference publications to his credit. Received October 2009 Accepted November 2009 Final acceptance in revised form December 2009