Dynamic response of concrete funicular shells with rectangular base under. impulse loads

Transcription

1 Dynamic response of concrete funicular shells with rectangular base under impulse loads Hadi Sabermahany a, Abolhassan Vafai b, Massood Mofid b,* a Graduated student at Sharif University of Technology, Tehran, Iran. Mobile: address: hadi.saberm@ymail.com b Professor of Civil Engineering, Sharif University of Technology, Tehran, Iran. addresses: vafai@sharif.edu (A. vafai), mofid@sharif.edu (M. Mofid) * Corresponding author. Postal Address: Room 516, Civil Engineering Department, Sharif University of Technology, Azadi Av, Tehran, Iran. Mobile: Tel: +98 (1) , Ext Fax: +98 (1) address: mofid@sharif.edu (M. Mofid) Abstract Funicular shells are thin doubly curved shallow shells which are in compression under dead weight due to their shape. In this study, an analytical approach is employed to consider forced linear vibration of concrete funicular shells with rectangular base under impulse loads based on shallow shells theory. Two boundary conditions simply supported and clamped, both are considered. The solution is obtained by Lagrangian approach. Accuracy of the results has been considered by comparing the results with those of finite element method. The results indicate that under impulse loads, stresses in funicular shells are not only compressive, but also tensile stresses are formed. Keywords: Forced Linear Vibration; Doubly Curved Shallow Shells; Funicular Shells; Lagrange Equations; Finite Element Method

2 1. Introduction Reinforced concrete shells are frequently used as roofing elements. Shell structures carry load through their shape rather than material strength. Funicular shells are a special type of shells that their shape is obtained so that in their membrane state, they carry a specific load by pure compression (this load is the shell s dead weight). Concrete is an appropriate material for construction of funicular shells for two reasons. First, funicular shells are primarily subected to compression and concrete compressive strength is good. Second, concrete has the flexibility to form into any shape which is necessary for obtaining funicular shell geometry. The performance of funicular shells subected to dead weight is compressive; nevertheless, other types of stresses may be generated under other loads. Therefore, investigating the performance of funicular shells subected to dynamic loads appears essential. Many researchers have proposed the analysis of funicular shells under various static loads. On the other hand, for dynamic loads, no research is available. Although funicular shells with a rectangular plan are mostly shallow and analysis of doubly curved shallow shells under dynamic loads has been presented. For instance, Bhimaraddi [1] investigated free vibration of homogenous and laminated doubly curved shallow shells over rectangular plan form. Using three-dimensional elasticity equations alongside with the assumption that the ratio of shell thickness to its middle surface radius is negligible, as compared to unity, the governing equilibrium equations have been reduced to differential equations with constant coefficients. In fact, the complex mathematical manipulations can be avoided by reducing the governing equations to those with constant coefficients and retaining 3-D characteristics of the problem. This is as opposed to reducing the governing equations to -D cases. Nonlinear free vibration behavior of single/doubly curved composite shallow shell panels was studied by Singh and Panda []. A general mathematical model which includes the nonlinear

3 higher order terms was developed. The governing equations were determined by Hamilton s principle and analyzed by using nonlinear FEM steps. The effect of changing constraint conditions upon the frequencies of shallow shells with rectangular boundaries was carried out by Qatu and Leissa [3]. Three edges were completely free and the attention was focused upon a single edge with clamped, simply supported, and free edge conditions. The Ritz method was employed in order to obtain accurate results. It was found that releasing the constraint of the u- displacement component (displacement component in x direction) has the largest effect on the fundamental frequencies. Abe et al. [4] presented nonlinear vibration characteristics of clamped laminated shallow shells. Moreover, both first-order shear deformation theory and classical shell theory were used. Nonlinear equations of motion were obtained by Hamilton s principle and analyzed by using Galerkin s procedure. It was shown that the second mode responses are very dependent on the first mode. Amabili [5] studied geometrically nonlinear vibration of shallow shells subected to harmonic excitation. Simply supported boundary conditions were considered. Furthermore, the nonlinear equations of motion were generated by Lagrangian approach and were solved with numerical techniques. Large-amplitude free vibration of magneto-electroelastic curved panels was studied by Shooshtari and Razavi [6]. Electrostatics and magnetostatics were considered by Gauss s laws and equation of motions were obtained by Donnell shell theory. Displacements and rotations were presented by trial functions. Governing nonlinear partial differential equations were changed to nonlinear ordinary differential equations by using Galerkin method, and thereafter they were solved by perturbation method. Analysis of funicular shells due to various static loads was presented in many researches. Vafai et al. [7] compared experimental values of membrane stresses and vertical deflections with finite element method results where good agreement was obtained. Forty-five concrete funicular shells (square bases

4 supported at four edges) with different rises and types of reinforcement were loaded up to failure point with a concentrated central load. It was observed that the crack and failure loads are significantly related to the rise of shells. Weber et al. [8] investigated ultimate load for concrete funicular shells by testing ten models with different geometric features. It was found that the ultimate load depends on the rise and thickness of the shell; with increasing the rise parameter (square of the ratio of rise to thickness), the ultimate load will also increase. Also, Vafai and Farshad [9] indicated that the failure load of funicular shells is affected by the age of concrete shell along with the amount of reinforcement. Elangovan [10] used finite element method in order to analyze funicular shells with clamped boundaries loaded with a uniformly distributed load. The eight-node isoparametric elements along with five degrees of freedom for each node were used. The analysis defined the zone in which tension is generated and reinforcement is required. Raasekaran and Suatha [11] studied deep funicular shells using Boundary Integral Element Method (BIEM). Due to the governing equation for deep funicular shells being nonlinear; it is difficult to get a closed form solution in order to determine deep funicular shells configuration. In addition, an incremental iterative technique along with BIE was applied to solve the nonlinear differential equation. Lakshmikandhan et al. [1] presented the effect of span to rise ratio on the performance of concrete funicular shells by using finite element method. The results indicated that reduction in the span to rise ratio caused reduction in deflection, maximum compression and maximum edge beam tension with improved stiffness. Similar to the previous study, Sivakumar et al. [13] considered the behavior of concrete funicular shells with rectangular base under uniformly distributed load. Reduction in membrane stresses and deflections when the rise and thickness increase, was discovered. Siddesh el al. [14] compared the performance of concrete funicular shells under concentrated load with the slabs. Analysis was performed via

5 finite element method. Six funicular shell units with a rectangular plan of m, rise of 5 and 10 cm, and thickness of 5, 4, and cm were considered. Each shell unit was compared with a slab of same thickness and dimension. Deflection of the shell models were founded to be 34% to 83% less compared to the slab models. Sachithanantham [15] studied concrete funicular shells over square plan with 0% to 16% openings under concentrated load using finite element method. It was concluded that with an increase in percentage of openings, the deflection, membrane stress and bending stress of concrete funicular shells will also increase up to 800%, 6% and 60%, respectively. Also, some other researches considered the ultimate load along with deflection of concrete funicular shells over rectangular plan of different dimensions under concentrated static load [16-19]. In this paper, a closed-form analytical solution to forced linear vibration of concrete funicular shells with rectangular base under impulse loads is investigated. Two boundary conditions simply supported and clamped, both are considered. Step pulse, triangular pulse and sine pulse are considered as impulse load types which are applied on a rectangular area. The analysis is based on the expansion of each displacement component in a double Fourier series which satisfies the boundary conditions. Strain-displacement relationships, from shallow shells theory, are used to compute elastic strain energy. After computing kinetic energy, elastic strain energy and the virtual work completed by external forces in terms of displacement components, the equations of motion are obtained by Lagrangian approach. Moreover, analytical solutions of equations of motion are developed via modal analysis technique. The efficiency of the analysis has been examined by comparing the results with those of finite element method. At the same time, the results indicate that under impulse loads, stresses in funicular shells are not only compressive, but also tensile stresses are formed. On the other hand, displacements and stresses,

6 especially tensile stresses, under dynamic impulse loads are negligible up to an amplitude of the load which is computable. Furthermore, the effect of rise and span of funicular shell pertaining to the time response of the shell has been shown.. Funicular shell surface over rectangular ground plan Surface equation of a shallow funicular shell of double curvature which carries dead weight in its membrane state by pure compression, could be given as [0]: Z Z g x y N (1) where Z=f(x,y) is the surface equation of the funicular shell, N is the desired compressive stress and g is the dead weight of the shell. The following equation, which is an approximate solution of equation (1), is used to define the surface of funicular shell over a rectangular ground plan [1]: ˆ 5 g 1 ˆ y b ˆ ˆ ˆ ˆ x a Z a x a b y b H 1 1 ˆ ˆ 8 N aˆ b aˆ b () where â =a/, ˆ b =b/ and a, b are the lengths of edges in x and y directions, respectively. H is also the rise of funicular shell. Moreover, Figure 1 shows the surface that is generated by Eq. (). Displacements of an arbitrary point on the middle surface in x, y and z directions are u, v and w respectively; w is taken positive inwards. 3. Kinetic energy, strain energy and virtual work done by external loads 3.1. Kinetic energy

7 The kinetic energy T of the shell, by neglecting rotary inertia, is given by: ab 1 T h u v w dxdy (3) 00 where ρ is the mass density and h is the thickness of the shell. The over dot means a time derivative. 3.. Strain energy According to shallow shells theory presented by the work of Velasov, the relationships between middle surface strains and changes in curvatures with middle surface displacement components are [0]: x u x rw (4) y v x tw (5) xy u v sw y x (6) x w x (7) y w y (8) xy w x y (9)

8 where Z Z Z r, t, s. x y x y (10) ε x, ε y and γ xy are the middle surface strains and χ x, χ y and χ xy are changes in curvatures and twist of the middle surface. The stress components (σ x, σ y and τ xy ) are linearly distributed across the thickness of the elastic shell. In addition, the stress resultants alongside with the stress couples of the middle surface that are also named internal forces (N x, N y and N xy ) and moments (M x, M y and M xy ) are obtained through the integration of the stress distribution over the shell thickness. The relationships between stress components and internal forces, as well as, moments are [1]: z N x 1M x x 3 z h h (11) y 1 z N M y y 3 z h h (1) z 1 6 xy N xy N yx 3 M xy M yx z h h (13) where z is the distance from the middle surface. The stress resultant-strain and stress couple-curvature relations are [0]: Eh N x x y 1 Eh N y y x 1 (14) (15)

9 N N Eh 1 1 xy yx xy (16) x x y M D (17) y y x M D (18) M D 1 (19) xy xy where E is the Youngʼ s Modulus, ν is the poissonʼ s ratio and D=Eh 3 /[1(1- ν )]. If the displacement components are calculated, the middle surface strains and changes in curvatures by Eqs. (4) to (10) and then the internal forces along with the internal moments and stress components by Eqs. (11) to (19) can be obtained. The following equation presented the strain energy with reference to the middle surface strains and changes in curvatures [1]: ab 1 x y x y xy Eh U dxdy 1 Eh ab x y x y xy dxdy (0) The first integral represents the membrane strain energy, whereas the second term indicates the bending strain energy. Using Eqs. (4) to (10), the strain energy can be presented in terms of displacement components:

10 ab Eh u u v v u v U rw r w tw t w 1 x x y y x y 00 u v 1 u v u v u tw rw rtw 4 s w 4 sw x y y x y x y 4 dxdy 3 ab v Eh sw dxdy x 4(1 ) w w w w 1 w x y x y x y 00 (1) 3.3. Virtual work done by external loads The virtual work W completed by external forces is obtained as: ab x y z () 00 W q u q v q w dxdy where q x, q y and q z are the distributed forces per unit area in x, y and z directions, respectively. In the present study, the applied impulsive load is distributed over a rectangular area and considered to be in the z direction. The applied load area, which its center corresponds to the center of the shell, is shown in Figure. Furthermore, Eq. () can be rewritten in the following form: W 0.6a0.6b q wdxdy (3) z 0.4a0.4b The external, normal impulsive load q z is considered in three types, comprised of, step pulse, triangular pulse and sine pulse. In general: A. step pulse q, 0 z t F0 t t qz t 0, t td B. triangular pulse d

11 0, q t F 1, 0 0 t t t t qz t t td z d d C. sine pulse 0, q t F sin, 0 0 t t t t qz t t td z d d Where F 0 is the magnitude of the force and t d is the time duration of applying impulsive load. 4. Boundary conditions and Fourier series of displacement components The following boundary conditions are considered in this study: Model A. simply supported conditions For simply supported conditions where shell edges rest on diaphragms that are rigid in their own plane and flexible out of the plane, the boundary conditions are given by: v w N M 0, at x 0, a (4) x x u w N M 0, at y 0, b (5) y y Where N is the normal force and M is the bending moment per unit length. The displacements u, v and w can be presented in the following double Fourier series which satisfy the boundary conditions: M N mx ny u x, y, t u mn, t cos sin (6) m1n1 a b

12 M N mx ny v x, y, t v mn, t sin cos (7) m1n1 a b M N mx ny w x, y, t w mn, t sin sin (8) m1n1 a b where m and n are the number of expressions used in the Fourier series in x and y directions, respectively and t is time. In addition, u m,n, v m,n and w m,n are the generalized coordinates that are unknown functions of t. Convergence of the solution can be considered by using different number of terms in Eqs. (6), (7) and (8). Model B. clamped edge conditions w u v w 0, M x c, at x 0, a x (9) w u v w 0, M y c, at y 0, b y (30) where c is the stiffness per unit length of the elastic and distributed rotational springs placed at four edges. Model B has been developed in Ref. [] and provides fixed edge in-plane with free rotation by c=0 and a perfectly clamped condition ( w/ x=0 and w/ y=0) obtained for c. The displacements u, v and w can be presented in the following double Fourier series which satisfy the boundary conditions: M N mx ny u x, y, t u mn, t sin sin (31) m1n1 a b M N mx ny v x, y, t v mn, t sin sin (3) m1n1 a b

13 M N mx ny w x, y, t w mn, t sin sin (33) m1n1 a b When c is not zero, an additional potential energy is generated in the rotational springs that must be added to the elastic strain energy. This potential energy U R is given by: b 1 a w w 1 w w U R c dy c dx x 0 x 0 x x a y y 0 y 0 y b (34) By substitution w from Eq. (33) and assuming that c is constant: U R M N m b n a cw m, n m1n1 a b (35) For both boundary conditions, through the substitution of double Fourier series of displacement components in Eqs. (3), (1) and (3), the kinetic energy, the strain energy and the virtual work performed by external loads will be obtained with regard to the generalized coordinates (u m,n, v m,n and w m,n ) which is a suitable form for them to use in Lagrange equations of motion. After substitution displacement components; the kinetic energy for both boundary conditions is obtained as: 1 ab T h v w 4 M N u m, n m, n m, n (36) m 1n 1 Also the strain energy for each of the boundary conditions is obtained as: Simply supported conditions

14 M N U 1 u m, n v m, n 3 w m, n 4 u m, nv m, n 5 u m, nw m, n 6 v m, nw m, n m1n1 M N m 1nanb 1 M N mi, m 1 n 1 7 w m, naw m, nb 8 u m, naw m, nb 9 v m, naw m, nb 10 w mi, nw m, n 11 v mi, nw m, n 1 u mi, nw m, n M N 13 w mi, naw m, nb 14 u mi, naw m, nb 15 v mi, naw m, nb mi m 1nanb 1 (37) Clamped edge conditions M N M N 1 m, n m, n 3 m, n 7 m, na m, nb 16 m, na m, nb m 1n 1 m 1 na, nb 1 U u v w w w v w M N M N w w u w w w 10 mi, n m, n 17 mi, n m, n 13 mi, na m, nb mi, m 1 n 1 mi m 1na nb 1 u v u w 18 mi, na m, nb 19 mi, na v w U m, nb 0 mi, na m, nb R (38) Basically, U R is the additional potential energy that is generated in the rotational springs. The coefficients α 1 to α 0, are presented in the appendix. And for both boundary conditions, W is: M N ab W q w m m n n mn z m, n t cos 0.4 cos 0.6 cos 0.4 cos 0.6 (39) m1n1 5. Lagrange equations of motion The Lagrange equations of motion are: d T T U Q, =1, dofs (40) dt q q q

15 where q={u m,n, v m,n, w m,n } T, m= 1, M, n= 1, N and dofs=m N. Moreover, the generalized forces Q are presented in the following equation by assuming viscous type for the nonconservative damping forces: Q F q W q (41) Nonconservative damping forces of viscous type are presented as: ab M N 1 1ab F c m, n m, n m, n m, n u v w dxdy c u v w (4) 4 00 m1n1 and mn, mn, (43) c m, n m, n where ξ m,n, μ m,n and ω m,n are modal damping ratio, natural frequency and modal mass of mode (m,n). Damping forces have insignificant effect upon the responses of impact load; therefore, one can neglect damping forces. For both boundary conditions, the kinetic energy was determined through Eq. (36). Also, the virtual work completed by external forces was determined through Eq. (39). Therefore, three terms of the Lagrange equations are the same for both boundary conditions. In particular: d T ab h q dt q 4 (44) T q 0 (45)

16 Q F W ab c q q q 4 0 if q u m, n, v m, n ab q cos 0.4m cos 0.6m cos 0.4n cos 0.6 n if q w mn z m, n (46) The strain energy was determined through Eqs. (37) and (38) for simply supported and clamped edge conditions, respectively. Therefore, one can write for each boundary condition: U dofs q ˆ k, k (47) q k 1 where coefficients can be determined with regard to coefficients α 1 to α Results and discussion of numerical analysis The equations of motion that are obtained through the Lagrangian approach and neglecting damping forces, can be presented in the following matrix form: q k q F m (48) and T 1 M, N (49) q q q q The system presented above for differential equations can be solved by modal technique. In Eq. (48), [m] and [k] can be considered as mass and stiffness matrices. Thus, with these two matrices, one can determine natural frequencies of funicular shell. By solving this system of differential equations, generalized coordinates u m,n, v m,n and w m,n are obtained in terms of time. Then by using Fourier series, displacement components are determined. By determining

17 displacement components, strains and stresses generated in the shell are acquired. In order to obtain the numerical results, a computer code was written. Furthermore, as mentioned earlier, no study is available for dynamic response of funicular shells. Therefore, dynamic analysis of funicular shells is completed by finite element method in order to compare its results with those of analytical approach. In the present study, geometric and material property of the funicular shell is given as (material is concrete): a= 1 m, b= 1 m, E= 17.8 GPa, ν= 0., ρ=400 kg/m, h=0,03 m, H= 0.09 m. The magnitude of force (F 0 ) is equal to 60 kpa, the area of applied load is 0. 0.=0.04 m and the time duration of the applied load is 0.01 Sec. In addition, the number of considered degrees of freedom used in mode expansion of displacement components (m,n) is (3 15) for simply supported conditions and (4 15) for clamped edge conditions. Moreover, the number of considered modes in modal analysis technique for both methods, analytical and FEM, is 10. Evidently, Figure 3 indicates comparison of center deflection of simply supported funicular shell for two different cases. In case 1, the number of degrees of freedom used in mode expansion is (3 15) and in case is (5 0). The good agreement between the results found in Figure 3 shows convergence of Fourier series of displacement components. Table 1 shows the natural frequency of mode (1,1) that is obtained with analytical and finite element methods for each of the two different boundary conditions. The agreement between the analytical solution and FEM solution is excellent, as it is clearly shown in Table 1. For clamped edge conditions, good agreement is obtained for c= All the results of analytical method pertaining to clamped edge conditions that are presented subsequently, are obtained for c= N/rad.

18 Tables and 3 present comparison of the maximum values of center normal deflection along with the center stresses found in x direction with finite element method for different impulse loads. Figure 4 shows center normal displacement of simply supported in comparison with finite element method. Figure 5 also shows this comparison for clamped edge conditions. The good agreement between the results found in Tables and Table 3 along with Figures 4 and 5 indicate the validity of the proposed method. In Figure 5, comparison with finite element shows a phase difference between the results which could be related to the value of the stiffness c (stiffness of rotational springs). In order to simulate clamped edges, the value of the stiffness c is determined only with regard to the natural frequency of mode (1,1). Figure 6 shows comparison of center deflection of simply supported with finite element method for a harmonic force, q z =F 0 sin(ωt), where the load frequency (ω) is equal to 1. ω 1,1 (ω 1,1 is the natural frequency of fundamental mode). In regards to the harmonic load, the effect of damping should be considered. The modal damping ratio of all modes are assumed 4 percent and the magnitude of force (F 0 ) is equal to 6 kpa. Figure 6 shows good convergence between the results of two methods for harmonic load. Figures 7 and 8 show the time response of the center point displacement of funicular shell with various pulses for simply supported and clamped edge conditions, respectively. The largest deflection occurs under step pulse due to the area under the load-time curve being greater than other pulses. Tables 4 and 5 indicate the maximum and minimum values of internal forces and moments for simply supported and clamped edge conditions, respectively.

19 Tables 6 and 7 also indicate the maximum values of compressive, tensile and shear stresses found in funicular shell for two boundary conditions. Tables 4 to 7 show that under impulse loads, internal moments are formed and stresses found in funicular shells are not only compressive, but also tensile and shear stresses are formed. In this paper, for a plate with the same geometric and material features, and for the same load applied, the dynamic responses are obtained with finite element method. Tables 8 and 9 show comparison of the maximum values of center normal deflection, center compressive and tensile stresses between funicular shell and plate for simply supported and clamped edge conditions, respectively. Tables 8 and 9 indicate that, in regards to the plate, the compressive and tensile stresses are the same and it is due to the bending performance of plate. Tables 8 and 9 also indicate that the normal deflection and stresses, especially tensile stresses found in the plate, are larger when compared to funicular shell. In other words, tensile stresses found in the plate are 4 to 15 times larger than tensile stresses found in funicular shell. Essentially, as mentioned in the introduction, the rise of funicular shell is an important parameter in the performance of the funicular shell under static loads. Here, the effect of the rise of funicular shell is considered for dynamic load. With increasing the rise of the shell, the maximum value of center normal displacement and the corresponding time response will decrease (Figure 9). If the dimension of plan (a) increases, the maximum values of center normal displacement and stresses along with time response of normal displacement will increase (Table 10 and Fig 10). Table 10 indicates that for a=3 m, the center tensile stress of the funicular shell is 9.8 MPa. By

20 increasing the shell rise from H=0.15 m to H=0.7, the center tensile stress of the shell decreases to 3.3 Mpa, which shows a 67% reduction. Thus span to rise ratio significantly affects the stresses, tensile stresses in particular. 7. Summary and conclusions In this study an analytical solution in regards to forced linear vibration of concrete funicular shells on a rectangular ground plan under impulse loads for two different boundary conditions was presented based on the shallow shells theory. The results that were successfully verified against finite element technique reveal that: - The largest deflection occurs under step pulse due to the area under time-load curve being greater than the other two pulses. - The performance of funicular shell under impulse loads is much better than a rectangular flat plate. - Under impulse loads, stresses in funicular shells are not only compressive, but also tensile stresses are formed. - The deflections and stresses, especially tensile stresses, under dynamic impulse loads are negligible up to an amplitude of the load that is computable. - The effect of rise and span on the time response of the shell has been considered. From a general perspective, the smaller the ratio of span to rise, the smaller will be the displacements and stresses. In fact, by choosing an appropriate span to rise ratio, the displacements and stresses, especially tensile stresses, will decrease significantly. References

21 1. Bhimaraddi, A. Free vibration analysis of doubly curved shallow shells on rectangular planform using three-dimensional elasticity theory, Int. J. Solids Struct., 7(7), pp (1991).. Singh, V.K. and Panda, S.K. Nonlinear free vibration analysis of single/doubly curved composite shallow shell panels, Thin Walled Struct., 85, pp (014). 3. Qatu, M.S. and Leissa, A.W. Effects of edge constraint upon shallow shell frequencies, Thin Walled Struct., 14(5), pp (199). 4. Abe, A., Kobayashi, Y. and Yamada, G. Non-linear vibration characteristics of clamped laminated shallow shells, J. Sound Vib., 34(3), pp (000). 5. Amabili, M. Non-linear vibrations of doubly curved shallow shells, Int. J. Non Linear Mech., 40, pp (005). 6. Shoshtari, A. and Razavi, S. Large-amplitude free vibration of magneto-electro-static curved panels, Scientia Iranica, 3(6), pp (016). 7. Vafai, A., Mofid, M. and E.Estekanchi, H. Experimental study of prefabricated funicular shell units, Eng. Struct., 19(9), pp (1997). 8. Weber, J.W., Wu, K.C. and Vafai, A. Ultimate loads for shallow funicular concrete shells, Northwest Sci., 58(3), pp (1984). 9. Vafai, A. and Farshad, M. Theoretical and experimental study of prefabricated funicular shell units, Build. Environ., 14, pp (1979).

22 10. Elangovan, S. Analysis of funicular shells by the isoparametric finite element, Comput. Struct., 34(), pp (1990). 11. Raasekaran, S. and Suatha, P. Configuration of deep funicular shells by boundary integral element method, Comput. Struct., 44(1/), pp (199). 1. Lakshmikandhan, K.N., Sivakumar, P., Jose, L.T., Sivasubramanian, K., Balasubramanian, S.R. and Saibabu, S. Parametric study on development, testing and evaluation of concrete funicular shells, International Journal of Engineering and Innovative Technology (IJEIT), 3(1), pp (014). 13. Sivakumar, P., Manunatha, K. and Harish, B.A. Experimental and FE analysis of funicular shells, International Journal of Engineering and Innovative Technology (IJEIT), 4(9), pp (015). 14. Siddesh, T.M., Harish, B.A. and Manunatha, K. Finite element analysis of funicular shells with rectangular plan ratio 1:0.7 under concentrated load using SAP000, International Research Journal of Engineering and Technology (IRJET), 3(9), pp (016). 15. Sachithanantham, P. Study of shallow Funicular concrete shells of plan to rise ratio 1:, International Journal of Biotech Trends and Technology (IJBTT), (3), pp (01). 16. Tarunkumar, T. and Sachithanantham, P. Study on shallow Funicular concrete shells over rectangular ground plan ratio 1:0.8, International Journal of Computer Trends and Technology (IJCTT), 3(6), pp (01).

23 17. Sachithanantham, P. Study of geo-grid reinforced shallow Funicular concrete shells subected to ultimate loads, International Journal of Biotech Trends and Technology (IJBTT), (), pp (01). 18. Sachithanantham, P., Sankaran, S. and Elavenil, S. Study on shallow Funicular concrete shells over rectangular ground plan ratio 1:0.6, International Journal of Computer Trends and Technology (IJCTT), 3(6), pp (01). 19. Sachithanantham, P., Sankaran, S. and Elavenil, S. Study on shallow Funicular concrete shells over rectangular ground plan ratio 1:0.9, International Journal of Emerging Technology and Advanced Engineering (IJETAE), 4(4), pp (014). 0. Ramaswamy, G.S., Design and construction of concrete shell roofs, McGraw-Hill, New York, USA (1968). 1. Ventsel, E. and Krauthammer, T., Thin plates and shells, theory, analysis, and applications, Marcel Dekker, New York, USA (001).. Amabili, M. Effect of boundary conditions on nonlinear vibrations of circular cylindrical panels, J. Appl. Mech., 74, pp (007). Figure Captions Figure 1. Funicular surface and coordinate system. Figure. The area of applied load in plan. Figure 3. The comparison of center normal displacement of simply supported for two different cases.

24 Figure 4. Center normal displacement of simply supported conditions for sine pulse. Figure 5. Center normal displacement of clamped edge conditions for sine pulse. Figure 6. Center normal displacement of simply supported conditions for harmonic load. Figure 7. Center normal displacement of simply supported conditions for different pulses. Figure 8. Center normal displacement of clamped edge conditions for different pulses. Figure 9. Center normal displacement of simply supported conditions for sine pulse and also for different rises. Figure 10. Center normal displacement of simply supported conditions for sine pulse and also for different plan dimensions. Table Captions Table 1. Natural frequency of mode (1,1) for different boundary conditions. Table. Comparison of the maximum values of center normal deflection and center stresses of simply supported for different pulses. Table 3. Comparison of the maximum values of center normal deflection and center stresses of clamped edge for different pulses.

25 Table 4. The maximum and minimum values of internal forces and moments of simply supported for different pulses. Table 5. The maximum and minimum values of internal forces and moments of clamped edge for different pulses. Table 6. The maximum values of stresses of simply supported for different pulses. Table 7. The maximum values of stresses of clamped edge for different pulses. Table 8. Comparison of the maximum values of center normal deflection and center stresses of simply supported conditions. Table 9. Comparison of the maximum values of center normal deflection and center stresses of clamped edge conditions. Table 10. Comparison of the maximum value of center stresses of simply supported for sine pulse and also for different plan dimensions. Figures Figure 1.

26 Figure. Figure 3. Figure 4.

27 Figure 5. Figure 6.

28 Figure 7. Figure 8.

29 Figure 9. Figure 10. Tables Table 1. Boundary Conditions Natural Frequency (Hz) Analytical FEM Simply supported clamped c= edge c= (N/rad)

30 Table. Impulse Load Deflection σ x (MPa) ( 10-4 m) C * T ** Step pulse Analytical FEM Triangular Analytical pulse FEM Sine pulse Analytical FEM * Compression ** Tension Table 3. Deflection σ Impulse Load x (MPa) ( 10-4 m) C * T ** Analytical Step pulse FEM Triangular pulse Sine pulse * Compression ** Tension Analytical FEM Analytical FEM Table 4. Internal Forces and Moments Step pulse Triangular pulse Sine pulse N x (kn/m) N y (kn/m) N xy (kn/m) M x (N.M/M) M y (N.m/M) M xy (N.M/M) Max Min Max Min Max Min Max Min Max Min Max Min

31 Table 5. Internal Forces and Moments Step pulse Triangular pulse Sine pulse N x (kn/m) N y (kn/m) N xy (kn/m) M x (N.M/M) M y (N.m/M) M xy (N.M/M) Max Min Max Min Max Min Max Min Max Min Max Min Table 6. Stresses Step Triangular Sine pulse pulse pulse σ x (MPa) Tension Compression σ y (MPa) Tension Compression τ xy (MPa) Stresses Table 7. Step pulse Triangular pulse Sine pulse σ x (MPa) Tension Compression σ y (MPa) Tension Compression τ xy (MPa)

32 Impulse Load Table 8. Deflection ( 10-4 m) σ x (MPa) C * T ** Step pulse Funicular shell Plate Sine Funicular shell pulse Plate * Compression ** Tension Impulse Load Table 9. Deflection ( 10-4 m) σ x (MPa) C * T ** Step pulse Funicular shell Plate Sine Funicular shell pulse Plate * Compression ** Tension Plan Dimension (a) Table 10. σ x (MPa) Compression Tension Appendix Definitions of α 1 -α 0 : Eh m b n a 1 4a 8b 1 1

33 Eh n a m b 1 4b 8a Eh m b n a m n Eh 5g a 4b ab 1 N a b b 3b a 3a b b a a a b n 60 4m 1 4n 1 4m a a b b 1 6 m 6 n Eh mn Eh g 1 b b mb a a a b 5 5 m 1 1 N a b 1 4n a 1 4m 4m Eh g 1 a a na b b b a 6 5 n 1 1 N a b 1 4m b 1 4n 4n 1 4m Eh g 1 16 b nn a b a a 7 5 N a b na nb ab n anb na nb 16b nan b a a (1 ) 4 4 n ( na nb) 6 m a n b 3 Eh g 1 0mb n 51 an a bnan b b 8 1 N a b m na nb na nb

34 Eh g 1 0 ab n 51 an ab n b b 9 1 N a b na nb na nb 1 4n Eh g 1 16 a mim b b 10 5 N a b mi m 1 i i i (1 ) mi m mi m ba m m m m a m m b b n 0na m m 51 b am m 3 Eh g 1 i i 11 1 N a b n mi m mi m 0ba m m 51ba m Eh g 1 i 1 1 N a b mi m mi m Eh g a b mi m nanb a b mi m nanb N a b mi m na nb mi m na nb Eh g a bm nan b 14 1 N a b mi m na nb Eh g b am i m n b 15 1 N a b mi m na nb For α 7 -α 15, indices of m, (i,) and indices of n, (a,b) should be both even or odd. Otherwise, these coefficients will be zero.

35 a b 3 a b a b n a b a nb 3 3 Eh g 1 ab n n n n nn a a N a b n n 1 4m b an anb 1 na nb In the above equation, one of the indices of n, a or b should be even and the other one should be odd. Otherwise, this equation will be equal to zero. i 3 i i m i i m 3 3 Eh g 1 ba m m m m mm b b N a b m m 1 4n a bmim 1 mi m In the above equation, one of the indices of m, i or should be even and the other one should be odd. Otherwise, this equation will be equal to zero. Eh m m n n 41 1 i a b mi m na nb 18 In the above equation, one of the indices of m, i or should be even and the other one should be odd and similarly one of the indices of n, a or b should be even and the other one should be odd. Otherwise, this equation will be equal to zero. b m m n n 1 a bm m n n 3 Eh g 1 i a b i a b N a b na nb mi m mi m na nb

36 In the above equation, one of the indices of m, i or should be even and the other one should be odd and both indices of n, a and b should be even or odd. Otherwise, this equation will be equal to zero. a m m n n 1 b am m n n 3 Eh g 1 i a b i a b N a b mi m na nb na nb mi m In the above equation, both indices of m, i and should be even or odd and one of the indices of n, a or b should be even and the other one should be odd. Otherwise, this equation will be equal to zero. Biographies Hadi Sabermahany is a PhD scholar at Civil Engineering Department, University of Tehran, Tehran, Iran. He received his MSc from Sharif University of Technology, Tehran, Iran, in 015. His MSc thesis was about analysis of funicular shells. Massood Mofid is a Professor of Civil and Structural Engineering at Sharif University of Technology. He received his MS and PhD degrees from Rise University, Houston, Texas. Abolhassan Vafai is a Professor of Civil Engineering at Sharif University of Technology. He has authored/co.authored numerous papers in different field of engineering. He has also been active in the area of higher education and has delivered lectures and published papers on challenges of higher education, the future of science and technology and human resources development.