Generally, the buildings are closed form structures which are supposed to cover the open spaces in order to make a safe place to stay inside. Such need was usually fulfilled by the covering system and supporting system. For a system of covering roof slab were constructed while the support for that covering is provided by the system of beam and column. But for the reinforced concrete shell structure both of these systems are compiled as one. So that the structure covers the large free spaces between columns, providing more free spaces.
The structure whose thickness is very less compared to the length and breadth of the structure are considered as the thin shell structures. For these types of structures the aim is to reduce its self weight by making the structure as thin as possible, permitted by the practical requirements. Which now will behave as a membrane which is free from the large bending stresses. By which optimum structural benefit is obtained in more economic ways, with the use of less material.
Figure: Roof shell structure (Dekker, 2001)(“Introduction To The General Shell Theory” 2001)
Nowadays, mostly used in commercial buildings, storage facilities, residential buildings, well matched for constructing complex curves and so on.
As the shell structures have more strength with respect to its self weight, it is gaining the popularity in the areas where there is a risk of earthquake and hurricane. There are many places on earth where there is a risk of these natural disasters.
It is therefore necessary to analyze the shell structures to ensure the best and safe design method for future, to avoid the potential accidents. The basic mechanics of the 3-D structure are different then the 2-D structure, therefore there is not any necessary that the model combination method which yield the correct result for the 2-D structure should be valid for 3-D structure as well. The fact that the less research done so far in the dynamics of shell structure is startling. Moreover there were various limitations on the research done. On the basis of available research done so far , it is difficult to conclude anything about the shell structure. Therefore there is a need for more research in the area to come to one conclusion. (Shadi Ostavari Dailamani, 2008). This research paper examines the different model combination methods for shell structures and checks their accuracy for shell structures.
High strength with respect to its self weight. (Major criteria to measure efficiency )
Very High Stiffness
Large space covered
Lowers the construction cost
Aesthetic value
Concrete being the porous material may lead to the problem of seepage
Due to its geometry, it is not possible to add another floor above it. Basically it is the roof covering structure.
Dome structure
Cylindrical Shell
Hyperbolic paraboloid Shell
Toroidal Shell
In real life practice cylindrical shells are often selected for covering massive open spaces, as it requires less number of internal supports. We see the common examples of cylindrical shells used for covering the roofs near us, such as power stations, garages, large supermarket, warehouses etc.
The main aim of the research is to come up with the result which shows the accuracy of the different model combination methods in shell structure with the aid of deeper knowledge and understanding about the dynamic response of shell structures under earthquake loading.
To understand the development of the shell theories in the course of time.
To acquire the knowledge of dynamic response and model combination for a shell structure.
To understand the mathematical and mechanical theory underpinning numerical modelling techniques for the response of structures subjected to earthquake loading.
To verify the FEM programme by cross checking the frequency, displacement, stress etc.
To be able to develop and analyze the 3-D civil engineering problems by the use of available finite element program (ABAQUS).
To interpret the response spectra of various earthquakes.
To share the findings and knowledge of the shell structure to the global community.
Documentation of a literature review and methodologies developed.
A model is created
Verification the FEM program (ABAQUS)
Accuracy of model combination for shell structure is known
Presentation of research outcomes and suggestions
Preparation of a professional draft
Many theories have been put forward in past years. These theories were based on the linear elasticity concepts. Generally it predicts the stresses and deformation for thin shells subjected to the minutest deformation. That means the Hooke’s law can be applied to the shell structure. 2D analysis of shell structures leads to the wrong direction, therefore to get the correct result 3-D analysis should be done. Then the analysis is done with the aid of the theory of linear elasticity. Due to the complexity of the analysis of this structure it is further deduced into the analysis of deformation of the middle surface only, by the adoption of some hypothesis. i.e the structure can be considered as the 2D. (Dekker, 2001)
Love was the pioneer of the shell theory devised a shell theory which is also known as first- order approximation shell theory. This theory was based in the classical linear elasticity. Several drawbacks were seen in the Loves theory which needs some corrections.
E. Resistor later develops the first order approximation theory which eliminated some deficiency in the Loves theory. The equilibrium equation, strain-displacement relation and stress resultant equation were developed from by using the same Love-Kirchhoff hypothesis by neglecting the higher order z/Ri , (Where Ri i= 1,2) are the radii of curvature of the middle surface) which yeilds the small values.
Timoshenko’s theory is somewhat like the Love theory which adopts the Kirchhoff-Love hypothesis and ignored the ratio of z/Ri near to the one.
Higher order approximation theory is separately developed by Lur’ye, Flugge, and Byrne. This theory combines the assumptions of Kirchhoff hypothesis and small deflection to give the general solution. Special consideration is taken to omit the values of z/Ri .
As this theory seems to be not elegant, various other theories were proposed then after. Stress conditions, geometry, range of deformation and loading were the basic factors that affect the different shell theories. Novozhilov, Vlasov, E. Reissner, Leissa and many more proposed the shell theory which was based on the basic principle of Love- Kirchhoff’s assumptions. Gol’denveizer was the first man to formulate the conditions for compatibility of strain components and his contribution is considered as a precious in the history of shell theories.
During the last century, the membrane theory established by Beltrami and Lecornu was getting popularity. Sokolovskii modified the membrane theory and introduced the various characteristic properties.The usefulness of the Airy’s stress function for solving the membrane problem was discovered by Pucher.
Shallow shells are the next example of specialized shell theories developed by Donnel, Vlasov and Mustari. The thin shell which has generally low rise compared to the span and used as roof covering are known as the shallow shells. Due to their comfort and ease the governing equation of this idea were found to be amazingly easy in solving the problems. Additional assumptions were made for simplification of strain-displacement relation, equilibrium and compatibility equations along with the classical Kirchhoff-Love hypothesis to come up with a new equation.
According to the D’Alambert principle the dynamic equation of motion can be obtained by the sum of inertial force and body force and body moment produced by the ground acceleration of the mass. At first the equation was developed by Love and then various improvements were proposed in the course of time by different researchers like Kraus, Flugge and many others. (Dekker, 2001)
The basic theoretical analysis is described by Anooshiravan Farshidianfar on his paper “free vibration analysis of circular cylindrical shells”. (Anooshiravan Farshidianfar, 2012). A cylindrical shell was considered with the constant thickness(h), axial length(L), mean radius(R), poission’s ratio(Ï…) density(Ï) and the Young’s modulus of elasticity(E) as shown by the figure.
Figure: cylindrical shell with coordinate system (Anooshiravan Farshidianfar, 2012)
The equation of motion can be expressed by the following equations
In which Lij (i,j=1, 2,3) is differential operators with respect to x, Ó© and t. Various solutions are provided by the various personal, but in this paper Love-Timoshenko method is used to calculate the natural frequencies and corresponding mode shapes.
Assumption of the synchronous motion is made in order to solve the equation 1
Where,
f(t) represents the scalar mode coordinate respective to mode shapes U(x, Ó©), V(x, Ó©) and W(x, Ó©).
The spatial dependence of the model shape between the longitudinal and circumferential direction is carried out by the varying separation method. Hence the deformation in three dimensions (u,v,w) is obtained by
Where,
λm is the axial wave number
And n is the circumferential wave parameter.
ω is the circular frequency of the natural vibration
And A, B and C are the constants
Substituting the equation (3) in the previous equation (1) and solving it by using theory gives the homogenous matrix solution.
Where,
| Cij | (i,j=1,2,3) are the function of λm , n and a frequency parameter Ω.
According to Love-Timoshenko The coefficient matrix, | Cij | can be obtained by
For nontrival solution the determinant of the coefficient matrix in equation (4) must be zero.
det |Cij | = 0; i,j=1,2,3 (7)
On solving the equation the following two eigenvalue problems are obtained
For the known value of λm there exist one or more proper values for ω so the equation (7) vanishes.
Similarly, for the given value of ω one or multiple values exist so the equation (7) vanishes.
The cubic equation in terms of the non dimensional frequency parameter Ω2 is obtained by solving the equation (7). The three positive roots obtained are the natural frequencies of the cylindrical shells classified as axial, tangential and radial. A low frequency usually associated with the radial motion.
The basic mode superposition methodology, that is restricted to linear elastic analysis, produces the entire time history response of member forces and displacements. Within the past there are two major drawbacks by the use of this approach. First, the tactic produces an oversized quantity of output data that may require a big quantity of process effort to conduct all doable style checks as a operate of your time. Second, the analysis should be recurrent for many different earthquake motions so as to assure that every one frequencies are excited, since a response spectrum for one earthquake during a such that direction isn’t a swish function.
The response spectrum method has computational benefits on predicting the displacement and member forces in any structural analysis. The strategy involves the calculation of solely the utmost values of the displacements and member forces in every mode of response spectra that which is the common of many earthquake motion. A recent development in the field of technology and computer has made easy and fast to analyze the lots of response spectra in less time than it used to take before.
The typical model response is given by the following equation
………………..(8)
The participation factors can be calculated as
where i represent to three orthogonal direction x, y or z. In order to obtain the approximate response spectrum one needs to pass the basic two hurdles. First, for the known ground motion the highest possible optimum displacements and forces must be estimated. Second, after the reaction for the three orthogonal guidelines is fixed it is necessary to calculate the highest possible reaction due to the three elements of earthquake movement performing simultaneously. (Wilson, n.d.)
Typical examples of response spectrahttp://eqtap.edm.bosai.go.jp/useful_outputs/report/fujino/images/figure_4.gif
Fig: 1 (a)
Fig: 1 (b)
Fig: 1 (a) Acceleration response spectrum. (F. Yozo, 2003)
Fig :1(b) Displacement response spectrum (Spyrakos, 2012)
As the maximum response of any structure cannot be taken simply the maximum of different modes, this problem gives rise to the model combination methods for estimating the optimum response. Combination methods include the following:
Absolute – peak values are added together
Square root of the sum of the squares (SRSS)
Complete quadratic combination (CQC) – a method that is an improvement on SRSS for closely spaced modes
Complex complete quadratic combination method (CCQC)
Impulsive mass contribution (IMP )
Convective mass contribution(CONV)
The absolute maxima of the model responses plus the square root of the sum of the squares of the remaining model responses (NRL)
The average of the absolute maxima of the model responses plus the square root of the sum of the squares of the model responses (CFE)
The most traditional technique that is used to calculate an optimum value of displacement or force within a framework is to use the sum of the overall of the modal response values. This strategy represents that the highest possible modal principles, for all ways, occur simultaneously.
(Alberto Lopez, 1996)
Other alternative common method to estimate the maximum model displacement and force, is to use the Square Root of the Sum of the Squares, SRSS. The SRSS technique represents that all of the highest possible modal values are statistically independent.
(Alberto Lopez, 1996)
For three dimensional structures, in which a large number of frequencies are almost identical, SRSS assumption is not justified. Previously carried out studies indicates the SRSS method is suited to most of the two dimensional structures. Good facts and the reliability with this SRSS technique containing the great outcomes on simple two dimensional structure, it’s been used for dynamic analysis of the three dimensional structure, with no proof. (E.L Wilson, 1981)
As the various studies were done then after it was confirmed that the SRSS method is good for analyzing the two dimensional structure with well separated model frequencies. This drawback leads to the new method of modal combination based on the random vibration theories, Complete Quadratic Combination, CQC, a method that was first put out in 1981.
(Alberto Lopez, 1996)
It was widely accepted by engineers and has made place in the many latest seismic analysis computer programs too. It’s observed which regarding nonclassical damped linear method, which has complicated eigenvalues as well as eigenvectors, the particular seismal responses rely on not merely the modal shift response but also the modal velocity reaction.
A similar procedure is followed to dectuct the CQC method. Since deduction associated with CQC algorithm regarding characteristically damped linear system, any closed-up formula associated with response-spectrum complex mode superposition calculation regarding highest seismal outcomes at virtually just about almost every place of a linear elastic construction having intricate modal features is actually deduced, as well as a brand new modal displacement-velocity correlation coefficient are participating form long-familiar modal displacement correlation throughout typical CQC formula. The new criteria for calculation of highest seismal outcomes of structures with nonclassical damping complex complete quadratic combination (CCQC) method is developed. (Xi-Yuan ZHOU1, 2004) .
(Xi-Yuan ZHOU1, 2004)
Others seems to have be not importance, however they are calculated as
The average of the absolute maxima of the model responses plus the square root of the sum of the squares of the model responses (CFE)
(Alberto Lopez, 1996)
And the absolute maxima of the model responses plus the square root of the sum of the squares of the remaining model responses (NRL)
(Alberto Lopez, 1996)
Where,
Sj is the maximum of i | Si |
Various issues in using the classical SRSS method on dynamic analysis was illustrated by E.L Wilson and his colleagues in the year 1981. A symmetrical four storey building was used to illustrate the result. Center of mass of the model building was located 25 inches from the geometric center on each floor of the building. Earthquake motion was applied from one direction and first 12 natural frequencies are obtained. Out of which some are closely spaced, like 13.869 and 13.931 are first and second, 108.32 and 108.80 are eighth and ninth respectively. These closely spaced frequencies show that the three dimensional building is designed to resist earthquake equally.
A response spectrum analysis was conducted by the use of 12 frequencies, the building was subjected to one component of Taft, 1952. The result shows that the SRSS method produces the values of base shear by less than 30 percent approx, while the sum of the absolute values overestimated all values. On the other hand use of CQC method results were realistic and come into agreement with the exact time history solution. The CQC method also gives the direction of the model base shear which indicates the elimination of SRSS method error. (E.L Wilson, 1981)
This paper showed that the CQC method is better than the SRSS method in model analysis. As it has a sound theoretical basis and accepted by many experts. While it does not indicate any signs that CQC method is relevant to the shell structures as well.
Three prototype multi storey pre-cast building with varying connections (hinged, hybrid and moment resisting) was studied. The purpose of the study was to show the importance of higher modes with varying structural connection. Three prototype namely model 1 (frame system with connection), model 2 (frame system with emulative connection only on the top floor) and model 3 (frame system with emulative connection) were subjected to pseudo-dynamic test at the European Laboratory for Structural Assessment of the joint research center of the European Community (ELSA). For experimental analysis the east-west component of the modified Tolmezzo accelerogram was used for the investigation. The results predicted by the model analysis is found to be lower than the accurate nonlinear analysis. The result showed the nature and values of displacement is in agreement while the storey forces did not. Thus, the paper also demonstrates that it is not a good idea to rely on the conservative approach of seismic design depending on modal analysis with response spectrum. It also explains that the difference in the results is due to the role of higher vibration modes. The force along the height is redistributed due to the effect of opposing vibration. (Fabio Biondini, n.d.)
According to this paper the traditional method of structural analysis for multi storey building is found to be unreliable. While there is not any evidence that it holds the same for shell structure.
This paper compares the simplified Mass Spring Model (MSM) proposed by Malhotra and his colleague, (Malhotra, 2000) method for determining the base shear, overturning moment and Maximum Sloshing Wave Height (MSWH) with the Results of Finite Element Modelling (FEM) under the action of an earthquake. The analysis was conducted on three different models of cylindrical tank which has the various heights to the radius (H/R= 2.6,1.0, 0.3). Common cause of failure is pointed out to be the shell bucking towards the bottom. Three models were studied by the excitation of five available earthquakes whose peak ground acceleration ranges from 2.9 to 8.3 m/s2 .The time history analysis was performed and the results were obtained by using SUM (summation of the maximum computed values), SRSS (Square Root of Sum of Squares), IMP (Impulsive mass contribution) and CONV ( Convective mass contribution). The period obtained by the use of convective and impulsive and FEM shows an agreement while it also explains that the accuracy of the MSM in predicting natural period is directly proportional to sloshing mode. SRSS method is found to be better than SUM. However it is stated that the simplified MSM may provide the error up to 70% with respect top FEM results. (M.A. Goudarzi, 2009)
This paper provides that the simplified MSM method cannot be trusted as the results may vary immensely that may mislead the structure design. However the method of analyzing the roof shells may vary.
Two steel distillation towers have been studied due to its high slenderness ratio and the risk of installation it in the seismic zone. The result was compared between five different model combination method to show the effect of choosing various methods. 1. Absolute sum of the model responses (ABS) 2. The square root of the sum of the squares of the model response (SRSS) 3. The absolute maxima of the model responses plus the square root of the sum of the squares of the remaining model responses (NRL) 4. The average of the absolute maxima of the model responses plus the square root of the sum of the squares of the model responses (CFE) 5. The complete quadratic combination (CQC). The material property of the steel is A-516 steel, grade 70, modulus of elasticity 187.3 GPA, and Yield stress 254.9 MPa. Soil influence is neglected, mass distribution and load were symmetrical except the wind load. Only the stress is compared with various model combination methods and it illustrated that ABS gives the upper bound response while the CFE and NFL rule exhibits the similar nature and CQC and SRSS almost come up with the same magnitude. As the structure was to be built on high seismic risk area CFE method was followed and it suggests that the CQC can be used in case of economic profitability. Moreover it also states that the even a small variation on damping may influence significant change in response. (Alberto Lopez, 1996)
They do not take the influence of soil while the soil may be the integral factor in designing in the place of high seismic risk, the paper did not do any time history nonlinear analysis and low damping was considered. In my personal view the findings may not be effective to use the results for roof shell structure.
The research paper has shown that the classical SRSS method is good for estimating the response of two dimensional structure, but fails to hold the reputation for three dimensional or multi storey structures with closely spaced frequency. That error is minimized by the use of CQC method. Simplified MSM cannot be also trusted, but the interesting result is shown by the paper written by Fabio Biondini and his colleagues, that even the response spectrum may not always produce the correct result. But none of them mentioned that the theory holds the same for roof shells as well. So there is a huge need of research needed to know the real gap and understanding in roof shells.
The primary factor that governs the design of the building is its self weight and the live load that it is expected to carry. Generally the small vibration and the Rayleigh movement in the vertical direction created by the small tremors are tolerated by the most buildings. Those buildings which are situated on the lateral side of the epicenter are prone to more damages as it induces the to and fro motion. Structural failure may be of various types, Some basic mode of failures are (Khan, 2007)
Foundation settlement
It may occur due to the difference in foundation settlement. As the additional moments/ stresses will be developed, it will cause the serious cracking of walls, sagging of the floor and distress to the structural frame.
Shear Movement of structural columns
Shear Movement of the Column due to Cold Joint
Collapse of Block-Masonry Cladding
Alligator Cracking of the Brick-Masonry
Buckling of main Reinforcement of Columns
Fig b: Shear movement of column due to cold joint
Fig b: Cracks due to non-uniform
foundation settlement
Fig a, Fig b,. Different modes of failure (Khan, 2007)
Fig c: Chilean Earthquake Damage, Credit: U.S. Geological Survey, Photo: Walter Mooney
Conceptual
Soft-storey
Short column effect
Irregularity of plan stiffness
Detailing
Poor is detailing
Lack of ties
Inadequate ductility
Construction
Poor construction
Poor material quality
Others (A.W.Charleson, 2001)
The methodology is based on quantitative research, which includes an immense experiment of the shell structures, statistical and correlation analysis of model combination. In the very beginning of the project the information is collected about the shell structure and screening of information is carried out so that only the required information remains. Then after the literature review is carried out to get the knowledge of the shell structure and the finite element program (ABAQUS). The shell structure is modelled in the ABAQUS and the structure is subjected to several available earthquake data for dynamic analysis. The various model combination method is then applied to find the response of the structure and compared with the accurate response to find the appropriate modal combination for shell structures. If the results obtained is unsatisfactory then the model is updated and reanalysis is carried out. The conclusion is drawn from the results obtained and finally the report is drafted. The summary of the methodology is shown in the block diagram below.
Information Collection
Different earthquake input
Preparation of Report
yes
no
Litrature Review
Conclusion and Discussion.
Result:
Happy?
Different model combination
Modelling Shell Structure in FEM
The task schedule of the project is shown in the Gant chart below. The proposed project runs for a year. The detailed work description carried out during each task is described below
Task 1: The induction program is managed to introduce the researcher, supervisor and the university to familiarize with each other . The software and hardware is installed and learned. Information is collected for the research.
Task 2: literature review is carried out by the with the help of collecting information
Deliverables: Documentation of the literature review is done and the methodology is developed.
Task 3: A preliminary calculation of the shell structure is carried out and the model is created
Task 4: A same model is created in ABAQUS and the result is verified with the known results.
Deliverables: Finite Element Program is verified and the limitation of the program is known.
Task 5: The real model of the project is tested subjected to several earthquake data and the results are collected. Moreover the response of the structure is calculated by the use of the different modal method
Deliverables: A sound model is created.
Task 6: careful analysis of the result is carried out during this phase. Comparison is made with the accurate data.
Task 7: If the results obtained does not show any agreement with the accurate result, then there might be a problem in modeling the structure. Therefore, the model is updated again and re analysis is done.
Task 8: Conclusion is made with the help of obtaining results and an appropriate method is suggested for the study of dynamic response of shell structure.
Deliverables: Accuracy of the modal combination is known for the shell structure.
Task 9: Final document will be presented including the methodologies, implementation and results of case studies.
Deliverables: presentation about the research findings and the future suggestions in investigating a shell structure.
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