Effect of Boundary Conditions on the Behavior of Stiffened and UnStiffened Cylindrical Shells
Abstract
The effect of boundary conditions is very important in the analysis of cylindrical shells, and is rarely studied in the literature due to its difficult experimental simulation. For large structures such as shell roofs, the type of boundary supports is among the major factors that can minimize the stresses and deflections. In this study, experimental and numerical investigations of the effect of different boundary supports for stiffened and unstiffened cylindrical shells were conducted. Two different models of the stiffened and unstiffened cylindrical shells with different boundary conditions, “pinned and with rigid diaphragms”, were studied. It was shown that by using rigid diaphragms for cylindrical shells, the deflections are minimized by 80%, and by (45–50) % for the stiffened cylindrical shells. From the experimental investigations and the numerical results obtained, the efficiency of the proposed boundary support types for cylindrical shells is confirmed, which can result in economic benefits.
Keywords
Cylindrical shell Shell element Stiffeners Rigid diaphragms Boundary supports1 Introduction
The boundary conditions of shell structures have an important effect on the state of stresses and the values of displacements. The rigid clamping of the edges of shell structures induces bending stresses at least over a narrow zone near the boundaries, and also prevents the structure from undergoing extensional deformations. The analysis of cylindrical shells with different boundary conditions is infrequently studied in the literature; this is mostly due to experimental difficulties. This problem also exists in many marine, aerospace and automotive engineering applications. The three main approaches involved in structural identification of behavior are the theoretical analysis, numerical simulation, and experimental investigation.
In a classical research, Flügge (1934) derived a set of cylindrical shell equations which included bending terms up to the second order. He did not solve the problem in its most general form, but suggested a solution for a simply supported cylindrical shell, in the form of trigonometric functions which satisfied the boundary conditions. This is certainly the reason this approach is not feasible after the advent of highspeed digital computers. Although the method requires numerical computation, the results are exact in the same sense that the numerical solution to the transcendental frequency equation for a beam yields an exact solution. Another study published by Sobel (1964) is on the closely related area of stability of cylindrical shells. The results of these two independent studies lead to the same conclusions regarding the importance of the various boundary conditions of cylindrical shells. Forsberg (1964) studied the influence of boundary conditions on the model characteristics of thin cylindrical shells; his research related to Flugge’s and Sobel’s studies, and his approach provides a powerful tool for examining a wide variety of boundary conditions and their influence on the modal behavior of cylindrical shells. The results of this study clearly indicate that care must be taken in any approximate analysis to use appropriate boundary conditions. An axisymmetric and an unsymmetrical analysis of conical and cylindrical shells with various boundary conditions were conducted by Wilkins et al. (1970). Chebili (1991) studied the problem of deformation of shells and found that the behavior is governed by both the geometry of the shell and its boundary supports. Skukis et al. (2013) studied the assessment of the effect of boundary conditions on cylindrical shell modal responses. In his study, a circular cylindrical shell employing arbitrary boundary conditions has been fabricated and physically tested, with several boundary conditions being used during the experimental setup. A numerical verification with the finite element code ANSYS has been performed in parallel in order to demonstrate the accuracy of the current solutions. Marchuk and Gnidash (2016) proposed two approaches for the analysis of the thickwalled cylindrical shells with different boundary conditions under local loads. It is shown that the effect of the boundary conditions on the stress–strain state is very weak for shells of high curvature and strong for shells of low curvature.
The present research is focusing on an assessment of boundary conditions and edge beam effects on the vertical and horizontal displacement of cylindrical shells. For this purpose; five semi cylindrical shell models with diameters of 32 cm are fabricated from stainless steel 304 grade, two of them with stiffeners. The deflection measurements have been performed by means of 50C9842 ADVANTEST 9. Two different boundary conditions were used during the experimental investigation: Pinned at four points and fixed by two rigid diaphragms. The numerical analysis is performed by a flat shell finite element called “ACMRSBE5” developed by Hamadi et al. (2015) and the “S4R, C3D8IH” developed by ABAQUS (2014). The modal characteristics and the vertical displacements are evaluated and the effect of various boundary conditions is discussed.
2 Analysis Approach
2.1 Geometry and Mechanical Properties of the Cylindrical Shell Models
The specimens have been produced by rolling of thin stainless steel sheet of 304 grade (t = 1.2 mm), to form the semi cylindrical shell structure. Five semi cylindrical models were used, two with stiffeners, one with stiffeners reposed on edge beams and two without stiffeners. Two different boundary conditions were considered; the first reposed on 4 points “pinned”, and the second reposed on two rigid diaphragms “fixed”. We proposed that the rigid diaphragms be welded to the semi cylinder to facilitate the experimental work. The dimensions of the semi cylinder are; the diameter is D = 320 mm and the length L = 900 mm, the thickness is the same for all specimens, the material properties are: the Young’s modulus E = 190,000 N/mm^{2}, and the Poisson ratio v = 0.265. A concentrated load is applied at the center of the top of the shell for all models.
2.2 Finite Element Study
2.2.1 Description of “ACMRSBE5” Element
2.2.2 Description of S4R ABAQUS Element
The S4R is a 4node doubly curved element used for thin and thick shells. It has 6 DOF at each node, and its stiffness matrix is calculated using a reduced integration and hourglass control.
2.2.3 Description of C3D8IH ABAQUS Element
2.3 Experimental Tests
3 Effect of Boundary Conditions on the Behavior of Cylindrical Shells

Cylindrical shell supported on 4 points “pinned” CS4P.

Cylindrical shell supported on two ends “Rigid Diaphragms” CSRD.
The deflection diminution percentage using ACMRSBE5 element, ABAQUS element and the experimental results for the CS4P and CSRD at Point 1
Load (N)  ACMRSBE5  S4R ABAQUS  Experimental solution  

Deflection (mm)  Percentage (%)  Deflection (mm)  Percentage (%)  Deflection (mm)  Percentage (%)  
CS4P  CSRD  CS4P  CSRD  CS4P  CSRD  
775  5.061  1.064  78.97649  5.08  1.087  80.7165358  4.649  1.438  69.06862 
800  5.224  1.098  78.981623  5.241  1.122  80.70979  5.098  1.727  66.12397 
825  5.387  1.133  78.96789  5.403  1.157  80.69591  5.606  1.831  67.338566 
850  5.551  1.167  78.976761  5.564  1.192  80.69734  6.060  1.936  68.052805 
875  5.714  1.201  78.981449  5.726  1.227  80.684597  6.422  2.013  68.654625 
900  5.877  1.236  78.96886  5.888  1.262  0.216033  6.822  2.103  69.173263 
The results obtained for the cylindrical shell model supported on two ends “Rigid Diaphragms” are presented in Table 1. In this case the finite element model slightly overpredicted the displacements since it assumed a fully rigid behavior for the end diaphragm, while a small level of deformations was observed in the tests.
3.1 Comparison of Deflection Results Between CS4P and CSRD Using ACMRSBE5 Element, ABAQUS Code and Experimental Results
The vertical displacements at the top of the cylindrical shell models with no end diaphragms and the cylindrical shell with end Rigid Diaphragms with different loadings, and the percentage of reduction of the deflection by using ACMRSBE5 element, ABAQUS element and the experimental results are also presented in Table 1.
Table 1 shows that the deflection diminution percentage observed from the experimental results in the presence of the rigid diaphragm and is almost 68%; that means that the rigid diaphragm minimized the vertical displacement at point 1 by 68%, which is an excellent contribution.
The deflection diminution percentage is almost 80% according to both the ACMRSBE5 and ABAQUS results, indicating both models resulted in reasonable simulation of the behavior.
For the cylindrical shell model supported on four points, the load applied at the top of the cylinder spreads out on the skin and goes towards the supports “4 points”; but for the cylindrical shell supported by the rigid diaphragms, the load goes from the skin to the curved boundaries of the cylinder then to the rigid diaphragms reducing the vertical displacements.
4 Effectiveness of Boundary Conditions Supports on the “Cylindrical Shell with Stiffeners”
For the study of the effectiveness of rigid diaphragms on cylindrical shell structures with stiffeners, especially for deflections, experimental tests were performed on two models and the percentage of the vertical displacements between the stiffened cylindrical shell models reposed on four points “pinned” SCS4P and the stiffened cylindrical shell supported on two ends by Rigid Diaphragms SCSRD are compared.
4.1 Comparison of Deflection Results Between SCS4P and SCSRD Using Experimental and ABAQUS Analysis Results
The deflection diminution percentage using ABAQUS element and the experimental results for the SCS4P and SCSRD at point 3
Load (N)  S4R ABAQUS  Experimental solution  

Vertical displacement (mm)  Percentage (%)  Vertical displacement (mm)  Percentage (%)  
SCS4P  SCSRD  SCS4P  SCSRD  
900  2.595  1.447  44.238921  3.453  1.591  53.924124 
950  2.737  1.527  44.208988  3.529  1.755  50.269198 
1000  2.991  1.607  46.27215  3.806  1.607  57.777194 
1050  3.023  1.688  44.161429  3.829  2.138  44.162967 
1100  3.165  1.768  44.139021  3.966  2.355  40.620272 
4.2 Comparison of Deflection Diminution Between “SCS4P” and “SCSRD” Models with ABAQUS Analysis
The percentage of deflection reduction for cylindrical shell models “SCS4P” and “SCSRD”, from both the experimental and ABAQUS results are presented in Table 2.
From Table 2, the percentage of deflection diminution according to the experimental results ranges between 40 and 50%. So, the use of a rigid diaphragm on the stiffened cylindrical shell minimized the vertical displacement at the top by around 45%, representing an excellent contribution.
From Table 2, the percentage of deflection diminution according to the ABAQUS results is almost 45% indicating that the finite element model produced satisfactory results.
Also, from Fig. 12, it can be seen that the diminution of deflection using the rigid diaphragm is between 40 and 50% for the experimental results and 44% for the ABAQUS model. The changing of boundary supports from the pinned conditions to the rigid diaphragm reduces the vertical displacement of the stiffened cylindrical shell by about 40%. In this case, by using ring stiffeners, the load is distributed from the skin to the stiffeners resulting in smaller deformations. The load is then transferred equally to the four supports. As a comparison, the stiffeners reduce the vertical displacement by about 70%, and the addition of the rigid diaphragm reduces the vertical displacement by about 40%. This is considered an excellent diminution to the deflection, and represents a great solution to improve the performance of the structure.
5 Effectiveness of Edge Beams on the Stiffened Cylindrical Shells

Stiffened cylindrical shell supported on two ends “Rigid Diaphragms” “SCD”.

Stiffened cylindrical shell supported on two ends “Rigid Diaphragms” with two edge beams (Stringers) “SCDS”.
The vertical displacement W_{1} at point 1 (experimental and ABAQUS results with meshes of 45 × 25) for the Cylindrical shell with two end diaphragms and two stiffeners “SCD”
Loads (N)  Displacement W_{1} at point 1 (mm)  

Experimental solution  C3D8IH ABAQUS  
800  1.229  1.286 
850  1.398  1.366 
900  1.591  1.447 
950  1.795  1.527 
1000  1.966  1.607 
1050  2.138  1.688 
The vertical displacement W_{1} at point 1 (experimental and ABAQUS results with meshes of 45 × 25) for the Cylindrical shell with two end diaphragms and two stiffeners resting on longitudinal beams “stringers” SCDS
Loads (N)  Displacement W_{1} at point 1 (mm)  

Experimental solution  C3D8IH ABAQUS  
800  0.883  1.280 
850  0.958  1.360 
900  1.043  1.440 
950  1.143  1.520 
1000  1.229  1.600 
1050  1.482  1.681 
1100  1.686  1.761 
1150  1.913  1.841 
1200  2.162  1.921 
1250  2.469  2.001 
1300  2.722  2.081 
Percentage of vertical displacements diminution with experimental results for the SCD and SCDS, at point 1
Load (N)  Displacement for the SCD (mm)  Displacement for the SCDS (mm)  Diminution percentage (%) 

800  1.229  0.883  28.15 
850  1.398  0.958  31.47 
900  1.591  1.043  34.44 
950  1.795  1.143  36.32 
1000  1.966  1.229  37.49 
1050  2.138  1.482  30.68 
1100  2.355  1.686  28.41 
1150  2.575  1.913  25.71 
1200  2.851  2.162  24.17 
1250  3.146  2.469  21.52 
1300  3.378  2.722  19.42 
5.1 Cylindrical Shell with Two end Diaphragms and Two Stiffeners (t = 1.2 mm)
Table 3 presents the vertical displacement W_{1} at point 1 (Experimental and ABAQUS results with meshes of 45 × 25).
5.2 Cylindrical Shell with Two end Diaphragms and Two Stiffeners Resting on Longitudinal Beams “Stringers” (t = 1.2 mm)
Table 4 presents the vertical displacement W_{1} at point 1 (Experimental and ABAQUS results with meshes of 45 × 25).
5.3 Percentage of Vertical Deflection Diminution Between “SCD” and “SCDS” Models with Experimental Results and Numerical Analysis
Table 5 presents the percentage of vertical displacements diminution from the experimental results for the SCD and SCDS, at point 1.
For the experimental results shown in Table 5, the effect of longitudinal beams “Stringers” added to the cylindrical shell with two end diaphragms and two stiffeners is very high; the vertical displacement reduction is about 20–38%.
5.4 Percentage of Horizontal Displacements Diminution at Point 2 Using the Experimental and Numerical Results
Percentage of horizontal displacements diminution with experimental results for the SCD and SCDS, at point 2
Load (N)  Displacement for the SCD (mm)  Displacement for the SCDS (mm)  Diminution percentage (%) 

1000  0.24  0.09  91.67 
1100  0.28  0.109  85.71 
1200  0.33  0.119  75.76 
1300  0.39  0.129  66.67 
1400  0.47  0.139  59.57 
Percentage of horizontal displacements diminution with ABAQUS results for the SCD and SCDS, at point 2
Load (N)  Displacement for the SCD (mm)  Displacement for the SCDS (mm)  Diminution percentage (%) 

1000  0.147  0.09  38.77 
1100  0.162  0.109  32.72 
1200  0.177  0.119  32.77 
1300  0.192  0.129  32.81 
1400  0.207  0.139  32.85 
From the results obtained in Table 6, the percentage of horizontal displacements diminution at point 2 varied between 59 and 91%; so the longitudinal beams presented a good contribution to the behavior in this case. Meanwhile; the percentage obtained by ABAQUS element is almost stable at 32% for all applied loads.
5.5 Percentage of Horizontal Displacements Diminution at Point 3 Using the Experimental and Numerical Results
Percentage of horizontal displacements diminution with experimental results for the SCD and SCDS, at point 3
Load (N)  Displacement for the SCD (mm)  Displacement for the SCDS (mm)  Diminution percentage (%) 

1000  0.47  0.21  55.32 
1100  0.57  0.29  49.12 
1200  0.70  0.39  44.29 
1300  0.85  0.52  38.82 
1400  1.03  0.67  34.95 
1500  1.25  0.87  30.40 
Percentage of horizontal displacements diminution with ABAQUS results for the SCD and SCDS, at point 3
Load (N)  Displacement for the SCD (mm)  Displacement for the SCDS (mm)  Diminution percentage (%) 

1000  0.651  0.394  39.48 
1100  0.717  0.434  39.47 
1200  0.783  0.475  39.34 
1300  0.848  0.515  39.27 
1400  0.914  0.555  39.28 
1500  0.980  0.596  39.18 
The same previous comments for point 2 can be given for the diminution of horizontal displacement for point 3 (Tables 8, 9), and the average is around 40% for both numerical and experimental results in the presence of longitudinal beams “Stringers”.
These results confirm that a very good diminution of the horizontal displacements, especially at points 2 and 3, is observed. In this case, when the load is applied at the top of the cylinder, the stringers obstruct the tendency of the straight borders to deform reducing the horizontal displacements.
6 Conclusions
From the results obtained by the experimental investigation and numerical analysis presented above, the following points can be drawn:
 1.
A significant effect on deflections can be obtained by the Rigid Diaphragms; the percentage of displacement reduction is close to 68% from the experimental observations, and 80% from the ACMRSBE5 element and the ABAQUS code. That means the vertical displacement at the top of cylinder “the point of applied the load” is minimized by 68% when using the Rigid Diaphragms. So a high percentage of deflection reduction can be achieved with Rigid Diaphragms. This is due to the fact that when using rigid diaphragms, the effect goes from the skin to the curved boundaries of the cylinder then to the rigid diaphragms. So, the cylinder with rigid diaphragms can support much higher loads with smaller deformations.
 2.
The same previous comment can be concluded for the stiffened cylindrical shell supported on 4 points and the stiffened cylindrical shell model with two end Rigid Diaphragms; good results can be obtained when using Rigid Diaphragms, the percentage of deflection reduction is close to 50% from the experimental results and 45% from the ABAQUS analysis. That means that the rigid diaphragms minimized the deflection to the half. In this case, by using ring stiffeners, the load pressure is distributed from the skin to the stiffeners resulting in smaller deformations. The load is then transferred equally to the four supports.
 3.
The stiffeners have an important effect on the deflection of cylindrical shell structures; but the efficiency of boundary conditions is more significant than the stiffeners, especially for the locations of stiffeners adopted in the experimental tests presented in this work.
 4.
Both rigid diaphragms and stiffeners play an important role in minimizing the deflections of shell structures; that means their presence can result in good design and reduce the economic cost of the structure.
 5.
The effectiveness of edge beams is very important as concluded from the experiment. The reduction is about 20–38% for the vertical displacement of point 1, varied between 59 and 91% for the horizontal displacements of point 2, and is 40% for the horizontal displacement at point 3. When using stringers, they obstruct the tendency of the straight borders to deform reducing the horizontal displacements.
 6.
The numerical modeling approach used in this work proves its efficiency compared to the experimental results for the case of cylindrical shells with and without stiffeners.
Notes
Acknowledgements
The authors would like to thank all people involved in this work at the Structural Engineering laboratory of City, University of London. Special thanks are due to Dr. Brett McKinley, the laboratory manager for his invaluable advices regarding the test rig setup, Mr. JN Hooker the laboratory technician for the preparation of models, as well as Mr. S Gendy, Mr. R. Mohammad, and Mr. D Das for their help to construct the models.
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