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1. i) Analytical Program Solution according to ASCE Code no 31(cont…)
1. Vertical line load at edge must be equal to the vertical component of the
membrane transverse force.
2. Horizontal displacement at edge must be zero.
3. Shearing line load at edge must be equal to the membrane shearing force
4. Rotation of edge must be zero.
Four Boundary Condition:
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2. i) Analytical Program Solution according to ASCE Code no 31(cont…)
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Fig: Code Developed in Matlab for Solution of Interior Barrel of Multiple Barrel Shell
3. i) Analytical Program Solution According to ASCE Code no 31(Cont…)
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Tx 𝑇𝜑
S 𝑀𝜑
Output
From
Matlab
4. ii) FEM Solution Using SAP2000 V20
(Step1)
Initialize of Model in SAP2000 ie
𝐿 = Length of Shell=20m
𝑁𝐴 = 𝑁𝑜. 𝑜𝑓 𝐷𝑖𝑣𝑖𝑠𝑖𝑜𝑛𝑠, 𝐴𝑥𝑖𝑎𝑙=40
𝑇 = 𝑅𝑜𝑙𝑙 𝐷𝑜𝑤𝑛 𝐴𝑛𝑔𝑙𝑒 = 400
𝑁𝐺 = 𝑁𝑜. 𝑜𝑓 𝐷𝑖𝑣𝑖𝑠𝑖𝑜𝑛𝑠, 𝐴𝑛𝑔𝑢𝑙𝑎𝑟 =
16
𝑅 = 𝑅𝑎𝑑𝑖𝑢𝑠 = 10𝑚
Type of Shell= Multi Bay Cylindrical
Shell
Number of Bays, Y=3
(Step2)
Initialization of Shell Material (M20
Concrete)
𝑃𝑤𝑑 = 𝑈𝑛𝑖𝑡 𝑊𝑒𝑖𝑔ℎ𝑡 = 25𝑘𝑁/𝑚3
𝐸 = 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝐸𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦 =
2.236 × 107
𝑘𝑁/𝑚2
𝑈 = 𝑃𝑜𝑖𝑠𝑠𝑜𝑛 = 0.2
𝐺 = 𝑆ℎ𝑒𝑎𝑟 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 = 9.316 ×
106
𝑘𝑁/𝑚3
(Step3)
Initialization of Shell Section Data
𝑆ℎ𝑒𝑙𝑙 𝑇𝑦𝑝𝑒 = 𝑆ℎ𝑒𝑙𝑙 − 𝑇ℎ𝑖𝑛
𝑇ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑎𝑡 𝑀𝑒𝑚𝑏𝑟𝑎𝑛𝑒 = 0.1𝑚
𝑇ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑎𝑡 𝐵𝑒𝑛𝑑𝑖𝑛𝑔 = 0.1𝑚
𝑀𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝐴𝑠𝑠𝑖𝑔𝑛𝑒𝑑 = 𝑀20
𝐿𝑖𝑣𝑒 𝐿𝑜𝑎𝑑 𝐴𝑠𝑠𝑖𝑔𝑛𝑒𝑑 = 3𝑘𝑁/𝑚2
(Step4)
Make Load Combination: Live+Dead
and Run the Analysis to find the final
Forces
𝐹11 ≡ 𝑇𝑥
𝐹22 ≡ 𝑇φ
𝐹12 ≡ 𝑆
𝑀22 ≡ 𝑀φ
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5. ii) FEM Solution Using SAP2000 V20 (Cont…)
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Fig: Multi Barrel Cylindrical Shell
(Initialization in SAP2000) Table: Final Forces Computation (Interior Panel) using SAP2000
Maximum Force
Angle in Degree
0 10 20 30 40
Tx 443.00 -81.98 -279.61 -201.47 -120.20
T 𝛗 46.56 -1.58 -36.17 -65.36 -75.63
S -1.30 -66.99 -4.62 -1.54 0.00
M 𝛗 -8.41 5.13 3.70 -2.23 -4.86
Force in
Intersecting Line
Distance x in meter
0 5 10 15 20
Tx -2410.00 243.53 445.05 243.53
-
2410.00
T 𝛗 -2211.12 31.52 46.74 31.52
-
2211.12
S 1.30 0.57 0.00 -0.57 -1.30
M 𝛗 34.41 -0.77 -8.40 -0.77 34.41
6. iii) Comparison of Forces From Program and SAP2000 Solution
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Comparision of Maximum Forces
-400.00
-300.00
-200.00
-100.00
0.00
100.00
200.00
300.00
400.00
500.00
600.00
700.00
0 10 20 30 40
Longitudinal
Force
in
kN/m
Angle (𝛗)
Tx Comparision at x=l/2
Software Solution
Program Solution
-100.00
-80.00
-60.00
-40.00
-20.00
0.00
20.00
40.00
60.00
0 5 10 15 20 25 30 35 40
Longitudinal
Force
in
kN/m
Angle (𝛗)
T𝛗 Comparision at x=l/2
Software Solution
Program Solution
7. iii) Comparison of Forces From Program and SAP2000 Solution
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Comparison of Maximum Forces
On further increasing the complexity of the simply supported shell by making it multiple barrels, the FEM solution is nearer to
the value of analytical solution. This is because intermediate support has been provided on the shell.
The maximum shear force value at the edges in the FEM solution has also matched with the analytical solution in this case. It was
observed deviating in the case of simply supported and continuous shells. So, the intermediate support has decreased the
discrepancy in the FEM solution.
Observations:
-120.00
-100.00
-80.00
-60.00
-40.00
-20.00
0.00
0 10 20 30 40
Longitudinal
Force
in
kN/m
Angle (𝛗)
S Comparision at x=0
Software Solution
Program Solution
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
8.00
0 10 20 30 40
Longitudinal
Force
in
kN/m
Angle (𝛗)
M𝛗 Comparision at x=l/2
Software Solution
Program Solution
8. iii) Comparison of Forces From Program and SAP2000 Solution
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Comparison of Forces at Intersecting Line
-3000.00
-2500.00
-2000.00
-1500.00
-1000.00
-500.00
0.00
500.00
1000.00
0 5 10 15 20
Longitudinal
Force
in
kN/m
x in meter
Tx Comparision at intersecting line
Analytical Program Solution
Software Solution
-2500.00
-2000.00
-1500.00
-1000.00
-500.00
0.00
500.00
0 5 10 15 20
Longitudinal
Force
in
kN/m
x in meter
T𝛗 Comparision at intersecting line
Analytical Program Solution
Software Solution
9. iii) Comparison of Forces From Program and SAP2000 Solution
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Comparison of Forces at Intersecting Line
• In the case of intersecting lines as in the continuous shell, the solution in the FEM solution has highly deviated from
the analytical solution
Observation:
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
0 5 10 15 20
Longitudinal
Force
in
kN/m
x in meter
S Comparision at intersecting line
Analytical Program Solution
Software Solution
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
0 5 10 15 20
Longitudinal
Force
in
kN/m
x in meter
M𝛗 Comparision at intersecting line
Analytical Program Solution
Software Solution
10. 9) Parametric Analysis
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• Parametric Analysis is performed on the interior barrel of the multiple barrel long cylindrical shell.
• Varying parameters are width and height.
• Analysis is done by the program developed for Case study III where variables can be changed easily.
Constant input Parameters for all models under analysis:
Length of shell L=30m
Thickness of shell t=100mm
Boundary condition: simply supported at both interior and
exterior barrels.
Uniform Loading on the shell= 2kN/m2
Dead Weight Loading on the shell = 25 kN/m3
11. 9) Parametric Analysis (Cont…)
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Varying Input Parameters:
Width of barrels, B=6m, 8m, 10m, 12m, 14m, 16m, 18m,20m,22m
Height of barrels, h=2m, 2.5m, 3m, 3.5m
Solution of the barrel shell with all possible combination of B and h has been done.
Dependent Parameters:
Dependent parameters are radius of shell R, subtended angle 𝛗k. R/L ratios and R/t ratios.
Output Parameters Considered:
Transverse moment M𝛗 (kNm/m) at transverse mid-section.
Longitudinal normal force Tx (kN/m) at transverse mid-section.
13. i) Variation of M𝛗 for Constant Height
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-15.000
-10.000
-5.000
0.000
5.000
10.000
15.000
20.000
0.0 20.0 40.0 60.0 80.0 100.0 120.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for h=2m
B=6m B=8m B=10m B=12m B=14m
B=16m B=18m B=20m B=22m
-15.000
-10.000
-5.000
0.000
5.000
10.000
15.000
20.000
0.0 20.0 40.0 60.0 80.0 100.0 120.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for h=2.5m
B=6m B=8m B=10m B=12m B=14m
B=16m B=18m B=20m B=22m
-15.000
-10.000
-5.000
0.000
5.000
10.000
15.000
0.0 20.0 40.0 60.0 80.0 100.0 120.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for h=3m
B=6m B=8m B=10m B=12m B=14m
B=16m B=18m B=20m B=22m
-12
-10
-8
-6
-4
-2
0
2
4
6
0.0 20.0 40.0 60.0 80.0 100.0 120.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for h=3.5m
B=6m B=8m B=10m B=12m B=14m
B=16m B=18m B=20m B=22m
14. ii) Variation of M𝛗 for Constant Breadth
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-10.0
-5.0
0.0
5.0
10.0
15.0
20.0
0.0 20.0 40.0 60.0 80.0 100.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for B=6m
h=2m h=2.5m h=3m h=3.5m
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
0.0 20.0 40.0 60.0 80.0 100.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for B=8m
h=2m h=2.5m h=3m h=3.5m
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
0.0 20.0 40.0 60.0 80.0 100.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for B=10m
h=2m h=2.5m h=3m h=3.5m
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
0.0 20.0 40.0 60.0 80.0 100.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for B=12m
h=2m h=2.5m h=3m h=3.5m
-8.000
-6.000
-4.000
-2.000
0.000
2.000
4.000
0.0 20.0 40.0 60.0 80.0 100.0 120.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for B=14m
h=2m h=2.5m h=3m h=3.5m
-10.000
-8.000
-6.000
-4.000
-2.000
0.000
2.000
4.000
6.000
0.0 20.0 40.0 60.0 80.0 100.0 120.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for B=16m
h=2m h=2.5m h=3m h=3.5m
15. ii) Variation of M𝛗 for Constant Breadth (Cont…)
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-12.000
-10.000
-8.000
-6.000
-4.000
-2.000
0.000
2.000
4.000
6.000
0.0 20.0 40.0 60.0 80.0 100.0 120.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for B=18m
h=2m h=2.5m h=3m h=3.5m
-15.000
-10.000
-5.000
0.000
5.000
10.000
0.0 20.0 40.0 60.0 80.0 100.0 120.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for B=20m
h=2m h=2.5m h=3m h=3.5m
-15.000
-10.000
-5.000
0.000
5.000
10.000
0.0 20.0 40.0 60.0 80.0 100.0 120.0
M𝛗
in
kNm/m
% of 𝛗k
Variation of M𝛗 for B=22m
h=2m h=2.5m h=3m h=3.5m
16. iii) Influence of M𝛗 (Observations)
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For constant values of h, Values of M𝛗 decreases at the edges but it becomes
constant at the crown when B increases.
For constant values of h, values of M𝛗 changes its sign moving from edge towards
the crown.
Lesser the value of B for constant values of h, M𝛗 becomes positive at the edge.
For constant value of B, M𝛗 decreases at the edge when h decreases.
Lowering the values of both B and h, M𝛗 becomes dominant at the intersecting
lines.
For higher value of breadth, M𝛗 is constant at every section ie from edge to the
crown irrespective of height.
17. iv) Variation of Tx for Constant Height
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-600.00
-400.00
-200.00
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for h=2m
B=6m B=8m B=10m B=12m B=14m
B=16m B=18m B=20m B=22m
-1000.00
-500.00
0.00
500.00
1000.00
1500.00
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for h=2.5m
B=6m B=8m B=10m B=12m B=14m
B=16m B=18m B=20m B=22m
-600.00
-400.00
-200.00
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for h=3m
B=6m B=8m B=10m B=12m B=14m
B=16m B=18m B=20m B=22m
-600
-400
-200
0
200
400
600
800
1000
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for h=3.5m
B=6m B=8m B=10m B=12m B=14m
B=16m B=18m B=20m B=22m
18. v) Variation of Tx for Constant Breadth
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-600.00
-400.00
-200.00
0.00
200.00
400.00
600.00
800.00
1000.00
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for B=6m
h=2m h=2.5m h=3m
-600.00
-400.00
-200.00
0.00
200.00
400.00
600.00
800.00
1000.00
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for B=8m
h=2m h=2.5m h=3m h=3.5m
-600.00
-400.00
-200.00
0.00
200.00
400.00
600.00
800.00
1000.00
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for B=10m
h=2m h=2.5m h=3m h=3.5m
-600.00
-400.00
-200.00
0.00
200.00
400.00
600.00
800.00
1000.00
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for B=12m
h=2m h=2.5m h=3m h=3.5m
-600.00
-400.00
-200.00
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for B=14m
h=2m h=2.5m h=3m h=3.5m
-600.00
-400.00
-200.00
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for B=16m
h=2m h=2.5m h=3m h=3.5m
19. v) Variation of Tx for Constant Breadth
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-600.00
-400.00
-200.00
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for B=18m
h=2m h=2.5m h=3m h=3.5m
-600.00
-400.00
-200.00
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for B=20m
h=2m h=2.5m h=3m h=3.5m
-1000.00
-500.00
0.00
500.00
1000.00
1500.00
0.0 20.0 40.0 60.0 80.0 100.0
Tx
in
kN/m
% of 𝛗k
Variation of Tx for B=22m
h=2m h=2.5m h=3m h=3.5m
20. vi) Influence of Tx (Observations)
20/64
For constant values of h, Values of Tx decreases at the edges but it becomes constant at the crown
when B increases.
For constant values of h, values of Tx change its sign moving from edge towards the crown.
For constant value of h, there is no significant difference in the value of Tx at crown on increasing the
value of B.
Decreasing both B and h, value of Tx increases at edge.
At 40% to 45% of 𝛗k, Tx always changes its sign moving from edge towards the crown for both
constant values of h and B.
21. 10) Conclusions
21/64
From the case studies of the long barrel thin cylindrical shells and the parametric study, following are the key points observed and
conclusions made.
● In interior panels of multiple barrel cylindrical shells of higher breadths, transverse moment has similar value at every
section of the barrel irrespective of height. So similar reinforcements can be provided in the design of those barrels.
● Huge amount of longitudinal compressive force exists in the interior panels of multiple barrel cylindrical shell which becomes
tensile with almost half its magnitude moving from edge towards the crown. The force changes its sign at 40%-45% of the
percentage of the 𝛗k. So special consideration must be done in design of that section.
● With the increase in complexity of simply supported shell ie by making it continuous or multiple barrel, FEM solution deviates
at the intersecting lines of the models and is observed to have a high percentage of error there. So, an analytical based program
considering the continuity of equations of curves in complex structures leads to the nearest results.
● In FEM analysis, when the structure is provided with intermediate support conditions, the accuracy of the analysis
increases.
● FEM doesn’t account for curvature in shell analysis as FEM solution is seen deviating at the crown of cylindrical shells.
Hence, quality of meshing also determines the accuracy of FEM solution.
22. 11) Limitation of the Study
22/64
The classical shell theory develops the higher order differential equations to solve the problems of arbitrary geometries.
Those arbitrary geometries can only be solved approximated by using the finite element method or the numerical evaluation
of infinite series. So, analytical solutions of complex geometries exist only for limited number complexities. But, those
solutions offer vital function in the evaluation of FEM or modern FEM based software. However, to analyze shells with
arbitrary geometry that interact with various supports and edge beams for static and dynamic solution of shells, the practical
approach is only provided by FEM.
23. 12) Further Research Topic for the Researchers
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From this paper, it is seen that with the increase in complexity of the cylindrical shell structure, FEM based solution highly
deviates at the intersecting line. The stiffness matrix used by the FEM solution has been derived in Chapter 2 of this paper.
So, researchers can develop FEM based program and observe why FEM solution lacks proper solution at the intersecting
lines.
24. 13) References:
1. ASCE MANUAL NO-31: Design of cylindrical concrete shell roofs, prepared by the Committee on Masonry and
Reinforced Concrete of the Structural Division, through its Subcommittee on Thin Shell Design. American Society
of Civil Engineers. New York, N.Y.: The Society, 1952
2. IS: 2210–1988, CRITERIA FOR DESIGN OF REINFORCED CONCRETE SHELL STRUCTURES AND
FOLDED PLATES (First Revision)
3. Daryl L. Logan (2015) “First Course in the Finite Element Method” University of Wisconsin–Platteville
4. “Parametric study on the structural forces and the moments of cylindrical shell roof using ANSYS” by Ashique
Jose , Ramadass S and Jayasree Ramanujan
5. Kaushalkumar M. Kansara (2004) “Development of Membrane, Plate and Flat Shell Elements in Java”
6. Nawfal Hasaine (www.mathworks.com) “Satic Structural Analysis of Shell Roof Structure”
7. http://nptel.iitm.ac.in/
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