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\title{Application of Initial Value Model in Circular Cylindrical Shell Analysis}
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             \author[1]{Mbanusi E.  C.}

             \author[2]{Obodoh D.  A.}

             \author[3]{Ezeokonkwo J.  U}

             \author[4]{Ogunoh P.  E}

             \author[5]{Okolie K.  C}

             \affil[1]{  NNAMDI AZIKIWE UNIVERSITY}

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\date{\small \em Received: 13 April 2016 Accepted: 30 April 2016 Published: 15 May 2016}

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\begin{abstract}
        


This study simulated underground circular cylindrical shell structure to serve as water reservoir under known soil characteristics and conditions. Cognizance of prevailing acute scarcity of portable water supply, more often than not, during each year, this structural facility was also simulated for study under two distinct situations namely: when it is empty and when it is full of water. Structural analysis of the facility was carried out using the initial value model whereas the classical model served to establish validity. The study sought the stress effect arising from: empty condition and in full of water, on this facility. The investigation result revealed: Both classical and initial value models led to identical results. Effect of stress which resulted from the two different conditions did not diminish the structural integrity of this facility.

\end{abstract}


\keywords{asymptotic integration, axi-symmetric, cylindrical shell, hoop tension, thin-walled structures.}

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\let\tabcellsep& 	 	 		 
\section[{I. Introduction}]{I. Introduction}\par
hell \hyperref[b0]{[1]} is applied to bodies bounded by two curved surfaces, where, the distance between the surfaces is small in comparison with other body dimensions. The centre of points lying at equal distances from these two curved surfaces defines the middle surface of the shell. The lengths of the segment, which is perpendicular to the curved surfaces, is known as the thickness of the shell and is denoted by h. Shells have all the characteristics of plates, along with an additional one, which is curvature. Mindful of intrinsic, functional essence of shells, \hyperref[b1]{[2]} presents shells as 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. In line with this functional essence of the shell, \hyperref[b2]{[3]}, \hyperref[b14]{[15]}, shells are spatially curved surface structures which support applied external loads or forces. Shell structures \hyperref[b2]{[3]}, \hyperref[b4]{[5]}, \hyperref[b6]{[7]} can be referred to as "form resistant structures". This implies a surface structure whose strength is derived from this shape, and which resists loads by developing stresses in its own plane \hyperref[b2]{[3]}, \hyperref[b4]{[5]}, \hyperref[b6]{[7]}.\par
An early form of shell construction \hyperref[b5]{[6]} was the dome known to Romans thousands of years ago. With shell concrete construction, \hyperref[b5]{[6]} it becomes quite possible to produce satisfactory domes which weigh only a fraction of the weights of the much earlier massive domes.\par
Properties of shells \hyperref[b0]{[1]}, \hyperref[b4]{[5]} which are of particular importance in structural usage and which also earn wide application of shell structures in engineering are the following: (i) Efficiency of Load carrying behavior; (ii) High degree of reserved strength and structural integrity; (iii) High strength versus weight ratio; (iv) Very high stiffness; (v) Containment of space.\par
Areas where shell structures \hyperref[b0]{[1]}, \hyperref[b3]{[4]} are used in Building and Civil Engineering are:-(i) Large-span roofs; (ii) Liquid retaining structures and water tanks; (iii) Containment shells of nuclear power plants; and (iv) Concrete arch domes. Shell forms in Mechanical Engineering \hyperref[b0]{[1]}, \hyperref[b3]{[4]} are used in: (i) Piping systems; (ii) turbine disks; (iii) Pressure vessels technology. The use of shells \hyperref[b0]{[1]}, \hyperref[b3]{[4]} in aeronautical and marine engineering are in the following forms:-(i) aircrafts; (ii) missiles; (iii) rockets; (iv) ships; and (v) submarines. Shells \hyperref[b7]{[8]} found in various biological forms such as the eye, the skull and the egg, represent another application of shell engineering, this time, in the field of biomechanics. An egg, as a natural thin-walled structure, can be considered as one of the most beautiful structural shapes. It combines extreme fitness for its purpose with an economy of material and cleanliness of design. This account depicts only a small list of shell forms in engineering and nature.\par
There are \hyperref[b0]{[1]}, \hyperref[b7]{[8]} two different classes of shells: thin and thick shells. Shells are said to be thin when the ratio of their thickness, h, to the radius of curvature R of the middle surface is less than or equal to 1/20, i.e. h/R ? 1/20. For a large number of practical applications \hyperref[b0]{[1]}, the thickness of thin shells lies within the range: 1/100 ? h/R ? 1/20. Hence, shells for which these h/R stipulations do not lie within the stated range, belong to thick shells \hyperref[b0]{[1]}.\par
Thin plates and thin shells belong to a category of structures known as Thin-Walled Structures. Thinwalled structures \hyperref[b8]{[9]} possess the following three characteristics: (i) two dimensions are much longer than the third, its thickness; (ii) They have a great strength as a result of their spatial character of working under the action of external loads; (iii) They make use of a minimal quantity of material.\par
Depending on the curvature of the surface \hyperref[b0]{[1]}, \hyperref[b2]{[3]}, shells are divided into (i) cylindrical, comprising of noncircular and circular; (ii) conical; (iii) spherical (iv) ellipsoidal; (v) paraboloidal; (vi) toroidal; and (vii) hyperbolic paraboloidal shells.\par
The usual theory of thin shells utilizes the main suppositions of the theory of thin plates. Nonetheless, thin plate and thin shell have a substantial difference in behaviour under external loadings. The static equilibrium of a plate element under lateral load \hyperref[b1]{[2]}, \hyperref[b7]{[8]} is only possible by the action of bending and torsional or twisting moments, usually accompanied by shearing forces. Conversely, a shell, in general, \hyperref[b9]{[10]} is capable of transmitting the surface load by "membrane" stresses uniformly distributed over its thickness. This property of shells \hyperref[b7]{[8]}, \hyperref[b9]{[10]} makes them to be not only more economical but also more rigid than plates and other types of construction under the same conditions. Apart from these obvious advantages over other systems \hyperref[b0]{[1]}, \hyperref[b7]{[8]}, shell structures are very well known and used for their performance, strength against accidental damage, resistance to fire and low upkeep cost as well as their aesthetic appearance.\par
The economy and/or feasibility of many modern constructions necessitate light weight, a property which thin-walled structures are replete with. Strictly speaking, the aim in structural engineering has always been to lower, as much as possible, the cost and thus the quantity of material used without, compromising the structural integrity of the system. Thin-walled structures meet this requirement.\par
The deliberate effort in the analysis and design of shells \hyperref[b2]{[3]} is to make the shell as thin as practical requirements would permit, so the dead weight is reduced and the structure functions as a membrane free from the large bending stresses.\par
Thin shell concrete structures \hyperref[b2]{[3]} are pure compression structures formed from inverse catenary shapes. The inverse catenary is a pure compression scenario. Pure compression is ideal for concrete, as concrete has high compressive strength and very low tensile strength. These shapes maximize the effectiveness of concrete, allowing it to form thin light spans \hyperref[b2]{[3]}.\par
This paper presents the application of initial value model in analysis of underground circular cylindrical shell structure subject to axi-symmetrical loads of hydrostatic pressure but considered under two conditions: (i) when the tank is empty; (ii) when the tank is full. In addition, the paper seeks validity through the classical model.\par
Aim in this study is to investigate stress effect on the underground cylindrical shell structure when full of water as well as when empty. The study intends to achieve the aim through the following objectives:\par
2) To evaluate the five internal stresses of the underground cylindrical shell structure when full of water, using the two models named above.\par
3) To determine the five internal stresses of the underground cylindrical shell when empty, also using the same two models. 
\section[{II. Previous Works}]{II. Previous Works}\par
The analysis and design of shells attracted many researchers. Among them, perhaps the best known, are: Love, U. F., Pasternak, P. L. and Timoshenko, S. P. \hyperref[b10]{[11]}. 
\section[{a) The Membrane Theory}]{a) The Membrane Theory}\par
Modern shell construction \hyperref[b10]{[11]} has its origin in the work of Lame and Clopeyron who, in 1826, proposed the membrane analogy. This theory suggests that a shell is capable of resisting external loads by direct stresses called membrane stresses without bending. Hence, when membrane theory is applied for shell design, torsional or twisting, bending moments, and shear forces in the cross-section are neglected. This is only possible if torsion and bending stresses were small compared to stresses of normal or axial, and shear forces. Membrane theory fails to represent the true stresses in those portions close to the edges, since the edge conditions usually cannot be completely satisfied by considering only membrane stresses. It is expected \hyperref[b11]{[12]} that membrane theory gives an approximate picture of stress distribution in the case of shells not long, say L ? 2R, where R is radius of the shell and L is its length. For longer shells a satisfactory solution can be obtained only by considering bending as well as membrane stresses \hyperref[b11]{[12]}. 
\section[{b) The Moment Theory}]{b) The Moment Theory}\par
Mathematical conceptions \hyperref[b5]{[6]}, developed during the 19 th century, made possible a more accurate analysis than could be achieved using any membrane theory. Aron, H., who derived an expression for potential energy of a shell as well as equations for shell equilibrium and strains, was the first to consider the new theory, evolved from those mathematical conceptions, which made room for both membrane and bending stresses \hyperref[b10]{[11]}.\par
Love \hyperref[b12]{[13]} developed a detailed derivation of equilibrium equations and equations of motion of shells with correction to a number of slips in Aron's original treatment as analogous to the theory of plates of Kirchhoff \hyperref[b13]{[14]}, and was based on identical assumptions. In each particular case, the moment theory involves the solution of a system of three differential equations, which is very complicated. 
\section[{c) The Semi-Moment Theory}]{c) The Semi-Moment Theory}\par
In the 20 th Century, \hyperref[b14]{[15]} suggested a simplified method of analyzing and designing cylindrical shells using the theory of semi-moment. This theory, on the basis of experimental data concerning medium length cylindrical shells, length/diameter = 2 -8, neglects the effect of longitudinal bending moment, shear forces and torques, and introduces geometrical hypothesis. This method has the advantage to be simpler than the moment theory and gives accurate solution when a distributed load is applied. This paper admires this method for the simple reason it appears to be the most appropriate for circular cylindrical shell loaded symmetrically with respect to its axis -the axi-symmetric loading scenario. Detailed information on general thin-walled structure theory were given in many general treatises such as: \hyperref[b8]{[9]}, \hyperref[b7]{[8]} and \hyperref[b15]{[16]}, as well as monographs: \hyperref[b9]{[10]}, \hyperref[b16]{[17]}, \hyperref[b17]{[18]}, and \hyperref[b18]{[19]}. These consider, in a great extent, the mathematical theory of thin-walled structures and the derivation of differential equations pertaining to them. Furthermore, the possibility of solving various shell-theory problems using analytic methods, has also been discussed in: \hyperref[b24]{[25]}, \hyperref[b25]{[26]}, \hyperref[b26]{[27]}, \hyperref[b27]{[28]}, \hyperref[b28]{[29]}, \hyperref[b29]{[30]}, \hyperref[b30]{[31]}, \hyperref[b31]{[32]}, and \hyperref[b32]{[33]}. 
\section[{d) Methods of Solution}]{d) Methods of Solution}\par
As far as techniques of solving the derived differential equations are concerned, difficulties involved in realizing mathematically rigorous methods, led to approximate techniques of integrating equilibrium equations. One of such methods, known as asymptotic integration, consists in replacing a given differential equation by another with specially selected coefficients different from those in the exact equation, and whose solution can be obtained by strict method and expressed in elementary functions. Blumenthal \hyperref[b19]{[20]} suggested the method in 1912. In 1913, Timoshenko \hyperref[b20]{[21]} applied it to shell equations. Shtayerman, Novozhilov \hyperref[b7]{[8]}, and Gol'denveiser \hyperref[b21]{[22]} perfected the same asymptotic integration method.\par
Another widely accepted method was a version of the membrane theory which takes boundary effects into cognizance. Geckeler's equations \hyperref[b22]{[23]} replaced exact equations owing to the presence of boundary effect. Generally speaking, methods for obtaining solutions for the shell differential equations \hyperref[b7]{[8]} can be classified as follows: (i) Exact analytical methods, also known as classical solution; (ii) methods using variational calculus; (iii) numerical methods such as: finite difference, finite element, finite strip, to mention but a few; (iv) approximate methods based upon exact equilibrium of the problem.\par
Among others \hyperref[b0]{[1]}, \hyperref[b7]{[8]}, the classical solution, the finite difference, the finite element, the finite strip, the boundary element, the boundary collocation, and the boundary value methods are some of the well-known models used for solution of shell problems.\par
The classical solution gives accurate results but fails to capture the boundary conditions. The finite difference model considers the boundary conditions but does not give room to further optimization for the simple reason solutions are obtained only for some selected nodes. Finite element generates very large matrices for considerable accuracy, handling of those large matrices being only suitable for use of computer. Besides, they are numerical models. Numerical models are more inclined to approximate solutions than exact solutions. Those models connoting boundary in their names, sound though they may be, involve a level of mathematics beyond the scope of an average engineer!\par
The semi-moment theory is as suitable for circular cylindrical shells under hydrostatic pressure and uniform gas pressure as it is amenable to application of initial value model. This study adopts the initial value model for among the suggested analytical solutions. Initial value model gives the least number of unknowns enabling such manual handling as to imbue the analyst with the ability and privilege to solve a problem in so systemic a manner that represents a mandatorily profound understanding in the-every-step-of-the-way of the problem for the analyst. ( ) 0 3 2 2 0 2 0 1 h 4 (Z)W y (Z) W ? ? + = Z y Eh R Z y ? ? ? - ( ) 0 4 2 4 Q Z y Eh R ?????????????.. (2) ( ) ( ) ( ) ( ) 0 4 2 3 0 1 4 4 4 M Z y Eh R Z y Wh Z y Z h ? ? ? ? ? + ? = - ( ) 0 4 2 2 4 Q Z y Eh R ? ?????????????.. (3) ( ) ( ) ( ) ( ) 0 1 0 4 3 3 0 3 2 2 M Z y Z Y R Eh W Z y R Eh Z M h + + = ? ? ? ??????????????.. (4) . ( ) ( ) ( ) 0 3 2 2 0 2 2 ? ? ? Z Y R Eh W Z y R Eh Z Q h + = ( ) ( ) 0 1 0 4 4 Q Z y M Z y + ? ?????????????.. (5) ( ) ( ) ( ) 0 2 0 1 ? ? ? ? ? ? ? ? + ? ? ? ? ? ? = Z y R Eh W Z y R Eh Z N h ( ) ( ) 0 4 0 3 2 4 4 Q Z y M Z Ry ? ? ? ?????????????.. (6)\par
ii. The Initial Value Particular Solution Consider the following: d Q x -x dx x 1 1 1 x Q 0 A B C ? Z ? 1 Z Z ? ? 1 dZ ? 1 Z\par
IV. Analysis a) The Governing Equation of Equilibrium \hyperref[b0]{[1]} and \hyperref[b23]{[24]} derived the governing differential equation of equilibrium as:D W dx W d x ? ? = + 4 4 4 4 ????? (1)\par
Using, for the derivation, the following: i. Forces acting on the shell; ii. Moments acting on the shell;\par
iii. Equilibrium equations of stress; iv. Equations of forces and moments displacements. 
\section[{b) Initial Value Solution of}]{b) Initial Value Solution of}\par
The Governing Differential Equation of Equilibrium for Circular Cylindrical Shell Structure i. In Summary, initial value homogenous solution becomes:  {\ref 1}) is applicable only to circular cylindrical shell structure subjected to axi-symmetric loading. This means the loading or forces and moments do not vary along the circumferential section. In other words, the loads are radially symmetrical loads.0 2 Q Z y ? + Equation (\par
For radially symmetrical loads, the governing differential equation of circular cylindrical shell structure, the equation of equilibrium, is equivalent to that of Beam on an elastic Winkler Foundation \hyperref[b33]{[34]}.\par
Understanding an origin transformation leads to finding the initial value particular solution. The origin that was previously at A is shifted to B, while introducing, at the same time, a new variable x 1 .? ? 1 1 1 1 Z Let dZ dx x = ? =\par
He distributed load at the new origin, B, is given by:( ) ? ? 1 1 Z Z Z q w ? = ?\par
The elemental force can be expressed as:( ) ( ) 1 2 1 1 1 dZ Z Z dZ Z q dQ ? ? ? ? = = ???? (7)\par
In summary, the initial value particular solution would be:( ) [ ] Z y Z Eh R W p 2 2 ? = ? ? ?????? (8) ( ) [ ] Z y Eh R p 1 2 1 ? = ? ? ???.. (9) ( ) Z y M p 4 3 ? ? ? = ???.. (10) ( ) Z y p 3 2 ? ? ? ?????. (11) ?????. (11) ( ) [ ] Z y Z R N p 2 ? = ? ? ???.. (12)\par
Obviously, general solution = homogeneous solution + particular solution. 
\section[{c) Classical Model Homogeneous Solution}]{c) Classical Model Homogeneous Solution}\par
Basically, in summary, the homogenous solution would be as follows:\par
( )   {\ref 17}) Equation \hyperref[b12]{[13]} can be expressed as:x i x x i x i x x x i x h( ) ( )( ) ( )( ) ( )( ) ( )( ) x iSin x Cos x Sinh x Cosh C x iSin x Cos x Sinh x Cosh C x iSin x Cos x Sinh x Cosh C x iSin x Cos x Sinh x Cosh C x W h ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? ? + ? + + + + = 4 3 2 1\par
Expanding the above equation gives:\par
( )x xSin Cosh A x xCos Cosh A x W h ? ? ? ? 2 1 + = x xSin Sinh A x xCos Sinh A ? ? ? ? 4 3 
\section[{+ + ???????????? (18)}]{+ + ???????????? (18)}\par
Where:A 1 = C 1 + C 2 + C 3 + C 4 A 2 = (C 1 -C 2 -C 3 + C 4 ) i A 3 = C 1 + C 2 -C 3 -C 4 A 4 = (C 1 -C 2 + C 3 -C 4 ) i d) 
\section[{Classical Model Particular Solution}]{Classical Model Particular Solution}\par
Recall the static equation of equilibrium:x D W W IV ? ? = + 4 4 ????????????????????????..(19)\par
Assuming the particular solution is of the form: W p (x) = ax ???????????????????????? (20) 
\section[{Global Journal of Researches in Engineering ( ) Volume XVI Issue III Version I}]{Global Journal of Researches in Engineering ( ) Volume XVI Issue III Version I}\par
It follows that:( ) 0 = x W IV p ???????????????????????.. (21)\par
Making use of equations \hyperref[b19]{[20]} and \hyperref[b20]{[21]} in eqn \hyperref[b18]{[19]} gives: Therefore, the classical model particular solution would be:    A comparison of co-efficient in equation \hyperref[b21]{[22]} yields:    ( ) x D x W p 4 4 ? ? = ??????????????????????? (24) 
\section[{b) Internal Stresses}]{b) Internal Stresses}\par
The deflection curve, for the hydrostatic loading, fig.  {\ref 2}, has a parabolic like shape with a speak of 0.854 mm at x = 3.2m from the top. Zero deflection is obtained at both top and bottom ends of the reservoir, justifying zero deflection at the supports. Also, tables 3 and 4 refer.\par
Under hydrostatic pressure, fig.  {\ref 2}, tables 3 and 4, the reservoir presented a maximum rotation of 0.078 x 10 -3 radians at its top end.\par
For the hydrostatic loading, fig.  {\ref 3}, tables 3 and 4, the bending moment varies from zero, at the top, to 20.5KMN at the bottom, describing a concave parabolic-like curve due to the cantilever action.\par
In the case of hydrostatic loading, the shearing force varies from 0.472KN, at the top, through 42.503KN, at the bottom, fig.  {\ref 5}, tables 3 and 4.\par
The graphs, fig. 6, have the same shape as those obtained for direct deflection. For real, deflection and hoop tension are directly proportional. For the hydrostatic pressure, a peak of 180.729 KN is reached at x = 2.67m, fig. 6, tables 3 and 4.\par
The tables 3 and 4 show values for deflection and rotation at the bottom, x = 4m, not exactly equal to zero, but very close to zero in the extent they can be taken as zero with sufficient accuracy. Table \hyperref[tab_4]{5} reveals: (i) percentage difference for applied pressure on tank wall, when the tank is empty versus when the tank is full of water, is equal to: 36.572; (ii) Overall average percentage difference value for results of internal stresses for, when the tank is empty versus when the tank is full of water, is equal to: 36.572. 
\section[{c) Conclusion and Recommendation}]{c) Conclusion and Recommendation}\par
The half-moment theory, otherwise known as semi-moment theory appears to have proven to be one of the most accurate theories in shell analysis. It has the advantage to be more realistic than the membrane theory which does not consider the bending effect. The semi-moment theory is simpler than the moment theory which generates heavy equations. In terms of capturing the boundary conditions of systems, the initial value model is more equipped than the classical model.\par
Sequel to results revealing that: percentage difference for applied pressure on tank wall: when the tank is empty versus when the tank is full of water is equal to 36.572\%; again the overall average percentage difference value for results of internal stresses with respect to: when the tank is empty versus when the tank is full of water came to 36.572\% as well, stress effect on the water reservoir which resulted from the two conditions would not diminish the fundamental structural integrity of the underground circular cylindrical shell structure, if all other sources of stress such as from weather elements are under control. \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Global}\end{figure}
      \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.035416666666666666\textwidth}P{0.18888888888888888\textwidth}P{0.050173611111111106\textwidth}P{0.14314236111111112\textwidth}P{0.14609375\textwidth}P{0.15494791666666666\textwidth}P{0.13133680555555555\textwidth}}
X(m)\tabcellsep Z\tabcellsep Deflection\tabcellsep Slope ?( z)\tabcellsep Bending\tabcellsep Shear Force\tabcellsep Hoop Tension\\
\tabcellsep \tabcellsep W( z )(mm)\tabcellsep radians\tabcellsep Moment\tabcellsep Q( z ) (KN)\tabcellsep N( z )(KN)\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep M( z ) (KNM)\tabcellsep \tabcellsep \\
0\tabcellsep 0\tabcellsep 0\tabcellsep 0.000122536\tabcellsep 0\tabcellsep 0.743859755\tabcellsep 0\\
0.8\tabcellsep \multicolumn{2}{l}{1.036008026 0.000052788}\tabcellsep 0.000123922\tabcellsep -0.4857128716\tabcellsep -1.183174309\tabcellsep 91.03938151\\
1.6\tabcellsep \multicolumn{2}{l}{2.072016051 0.000165917}\tabcellsep 0.000129602\tabcellsep -1.516548554\tabcellsep -3.019258682\tabcellsep 170.0885245\\
2.4\tabcellsep \multicolumn{2}{l}{3.108024077 0.000473429}\tabcellsep 0.000145863\tabcellsep -4.247252695\tabcellsep -8.416752018\tabcellsep 226.0594408\\
2.67\tabcellsep \multicolumn{2}{l}{3.453360086 0.000670475}\tabcellsep 0.000156135\tabcellsep -5.962376315\tabcellsep -11.88682999\tabcellsep 284.9352786\\
3.2\tabcellsep \multicolumn{2}{l}{4.144032102 0.000134648}\tabcellsep 0.000190643\tabcellsep -11.73091094\tabcellsep -23.73505572\tabcellsep 211.3360398\\
4.0\tabcellsep 5.180040128\tabcellsep 3.55 x 10 -11\tabcellsep -4.4 x 10 -11\tabcellsep -32.31896214\tabcellsep -67.01017443\tabcellsep 2.7 x 10 -7\end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.035416666666666666\textwidth}P{0.17118055555555556\textwidth}P{0.06345486111111111\textwidth}P{0.14756944444444445\textwidth}P{0.14609375\textwidth}P{0.15494791666666666\textwidth}P{0.13133680555555555\textwidth}}
X(m)\tabcellsep Z\tabcellsep Deflection W(Z)(mm)\tabcellsep Slope ( ) Z ? radians\tabcellsep Bending M(Z) (KNM) Moment\tabcellsep Shear Force Q(Z) (KN)\tabcellsep Hoop Tension N(Z)(KN)\\
0\tabcellsep 0\tabcellsep 0\tabcellsep 0.000122486\tabcellsep 0\tabcellsep -0.743859731\tabcellsep 0\\
0.8\tabcellsep \multicolumn{2}{l}{1.036008026 0.000052764}\tabcellsep 0.000123834\tabcellsep -0.48571286i8\tabcellsep -1.183174342\tabcellsep 91.03938184\\
1.6\tabcellsep \multicolumn{2}{l}{2.072016051 0.000165855}\tabcellsep 0.000129585\tabcellsep -1.516548484\tabcellsep -3.019258654\tabcellsep 170.0885258\\
2.4\tabcellsep \multicolumn{2}{l}{3.108024077 0.000473396}\tabcellsep 0.000145792\tabcellsep -4.247252615\tabcellsep -8.416752001\tabcellsep 226.0594463\\
2.67\tabcellsep \multicolumn{2}{l}{3.453360086 0.000670493}\tabcellsep 0.000156193\tabcellsep -5.962376293\tabcellsep -11.88682675\tabcellsep 284.9352765\\
3.2\tabcellsep 4.144032102\tabcellsep 0.00134592\tabcellsep 0.000190589\tabcellsep -11.73091124\tabcellsep -23.73505498\tabcellsep 211.33360358\\
4.0\tabcellsep 5.180040128\tabcellsep 4.27 x 10 -12\tabcellsep 3.84 x 10 -13\tabcellsep -32.31896187\tabcellsep -67.01017413\tabcellsep 2.3 x 10 -8\end{longtable} \par
 
\caption{\label{tab_1}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{P{0.10552523874488404\textwidth}P{0.07073669849931787\textwidth}P{0.08465211459754433\textwidth}P{0.10784447476125511\textwidth}P{0.12407912687585267\textwidth}P{0.12060027285129604\textwidth}P{0.1275579809004093\textwidth}P{0.10900409276944066\textwidth}}
Year 2016\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
36\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
Global Journal of Researches in Engineering ( ) Volume XVI Issue III Version I E\tabcellsep X(m) 0 0.8 1.6 2.4 2.67\tabcellsep Z 0 1.036008026 2.072016051 3.108024077 3.453360086\tabcellsep Deflection W(Z)(m) 0 0.000033482 0.000105238 0.000300286 0.000425269\tabcellsep Slope ( ) Z ? (radians) 0.000077722 0.000078601 0.000082204 0.000092518 0.000099033\tabcellsep Bending Moment M(Z) (KNM) 0 -0.3085154309 -0.9619133775 -2.693938928 -3.781804103\tabcellsep Shear Force Q(Z) (KN) -0.471813873 -0.750461426 -1.915049339 -5.338560581 -7.539554672\tabcellsep Hoop Tension N(Z)(KN) 0 57.7444589 107.8828632 143.3830003 180.7287485\\
\tabcellsep 3.2\tabcellsep 4.144032102\tabcellsep 0.000854047\tabcellsep 0.000012092\tabcellsep -7.44065868\tabcellsep -15.05462351\tabcellsep 134.0462233\\
\tabcellsep 4.0\tabcellsep 5.180040128\tabcellsep -2.53 x 10 -11\tabcellsep -3.4 x 10 -11\tabcellsep -20.49920654\tabcellsep -42.50307896\tabcellsep -1.14 x 10 -7\\
\tabcellsep \multicolumn{2}{l}{© 2016 Global Journals Inc. (US)}\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_2}Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.035233160621761656\textwidth}P{0.18791018998272882\textwidth}P{0.04844559585492228\textwidth}P{0.14974093264248703\textwidth}P{0.14533678756476684\textwidth}P{0.15414507772020725\textwidth}P{0.12918825561312608\textwidth}}
X(m)\tabcellsep Z\tabcellsep Deflection W(Z)(m)\tabcellsep Slope ( ) Z ? (radians)\tabcellsep Bending M(Z) (KNM) Moment\tabcellsep Shear Force Q(Z) (KN)\tabcellsep Hoop Tension N(Z)(KN)\\
0\tabcellsep 0\tabcellsep 0\tabcellsep 0.000177694\tabcellsep 0\tabcellsep -0.471813792\tabcellsep 0\\
0.8\tabcellsep \multicolumn{2}{l}{1.036008026 0.000033471}\tabcellsep 0.000078595\tabcellsep -0.3085154293\tabcellsep -0.750461386\tabcellsep 57.7444536\\
1.6\tabcellsep \multicolumn{2}{l}{2.072016051 0.000105168}\tabcellsep 0.000082193\tabcellsep -0.9619133724\tabcellsep -1.915049295\tabcellsep 107.8828584\\
2.4\tabcellsep \multicolumn{2}{l}{3.108024077 0.000300245}\tabcellsep 0.000092512\tabcellsep -2.693938892\tabcellsep -5.338560528\tabcellsep 143.3830001\\
2.67\tabcellsep \multicolumn{2}{l}{3.453360086 0.000425218}\tabcellsep 0.000099021\tabcellsep -3.781804084\tabcellsep -7.539554622\tabcellsep 180.7287423\\
3.2\tabcellsep \multicolumn{2}{l}{4.144032102 0.000854014}\tabcellsep 0.000012024\tabcellsep -7.44065818\tabcellsep -15.05462316\tabcellsep 134.0462194\\
4.0\tabcellsep 5.180040128\tabcellsep -3.62 x 10 -12\tabcellsep 2.94 x 10 -13\tabcellsep -20.49920612\tabcellsep -42.50307854\tabcellsep 1.21 x 10 -8\end{longtable} \par
 
\caption{\label{tab_3}Table 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{P{0.85\textwidth}}
tank is full = 36.572\\
Fig. 2 : Deflection Diagram\end{longtable} \par
 
\caption{\label{tab_4}Table 5 :}\end{figure}
 			\footnote{Year 2016 E © 2016 Global Journals Inc. (US)} 			\footnote{© 2016 Global Journals Inc. (US) Application of Initial Value Model in Circular Cylindrical Shell Analysis} 			\footnote{© 2016 Global Journals Inc. (US)This page is intentionally left blank} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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\end{document}
