Study the Dynamic Response of the Stiffened Shallow Shell Subjected to Multiple Layers of Shock Waves

The ANSYS APDL programming and the results of calculating the stiffened shallow shell on elastic supports subjected to multiple layers of shock waves presented in the study. The program set up allows for the survey and evaluation of structural parameter and loads to the dynamic response of different types of shallow shell.


INTRODUCTION
The shell is used in many areas due to its good coverage and light weight. Some structures can be mentioned as: roof of the building, cover tunnel, engine cover, … Calculation of shell structure is influenced by different types of load are many scientists concerned. One of the types of high destructive load mentioned is the shock wave load [1,2,3,4,5]. In fact, when exposed to multiple layers of shock waves, the structural response is complex. To avoid the impact of impulse load, in some cases the elastic supports is used. In this paper, the problem of shallow shell structure with (or without) elastic supports subjected to multi layers of shock waves is investigated. The study results may give the readers a more complete vision on the response of the mentioned shell structure and may be used for reference in the its design.

II.
PROBLEM MODELING The Considering an eccentrically stiffened singly (or doubly) curved shell that is simply supported at the edges by elastic springs with stiffness k (Fig. 1). The shell is affected by multiple layers of shock waves p(t). For establishing an algorithm for the problem the following assumptions are used: -The ribs and the shell material are homogeneous and isotropic; -The shock-wave presssure is uniformly distributed on the shell surface.

III. TYPES OF ELEMENTS USED IN THE PROGRAM
To describe the bending shell, the SHELL63 element is used. The element has six degrees of freedom at each node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z-axes. Stress stiffening and large deflection capabilities are included [8].

Fig. 2: SHELL63 Geometry
To describe the stiffener, 3-D Linear Finite Strain Beam (BEAM188) is used. BEAM188 is a linear (2-node) or a quadratic beam element in 3-D. It has six or seven degrees of freedom at each node, with the number of degrees of freedom depending on the value of KEYOPT(1). When KEYOPT(1) = 0 (the default), six degrees of freedom occur at each node. The eccentricity of the stiffener is described by the SECOFFSET command [8].  program is extended to other types of viscoelastic pillows. In this case, the damping coefficient should be added when declaring the constants of the COMBIN14 element. Fig. 4: COMBIN14 Geometry [8] COMBIN14 has longitudinal or torsional capability in 1-D, 2-D, or 3-D applications. The longitudinal springdamper option is a uniaxial tension-compression element with up to three degrees of freedom at each node: translations in the nodal x, y, and z directions. No bending or torsion is considered. The torsional springdamper option is a purely rotational element with three degrees of freedom at each node: rotations about the nodal x, y, and z axes. No bending or axial loads are considered [8].

IV. GOVERNING EQUATIONS AND CALCULATION PROGRAM
The connection of the rib and the support elements into the flat shell elements is implemented by the direct stiffness method and the Skyline diagram is established by using the general algorithm of the FEM [6,7,8]. After connecting the element matrices and the element vectors in to the global ones the differential equation describing the oscillation of the stiffened shell may be written in the form: where:  Consider an eccentrically stiffened shallow cylindrical shell whose plan view is a rectangular, a = 2.0m, b = 1.0m, the radius of curvature R = 1.6m, the thickness h = 0.025m. Shell material has an elastic modulus Es = 2.110 11 N/m 2 , poisson coefficient s = 0.3, specific weight s = 7850kg/m 3 . Four edges of shell are supported by elastic springs with stiffness k = 3.10 4 kN/m. The eccentric ribs of the shell has hr = 0.03m, br = 0.01m, the ribs in the directions are 6 (6 ribs parallel to the generating line, 6 ribs perpendicular to the generating line). The ribbed material has Er = 2.510 11 N/m 2 , r = 0.3, r = 7500kg/m 3 .

International Journal of Advanced Engineering Research and Science (IJAERS)
[   Comment: With the given set of data, the maximum dynamic response values of the system reached at the time of two waves of simultaneous effects (time t = 0.021s). At center point, the stress σx max is greater than the stress σy max .

Problem 2
Considering the shallow cylindrical shell whose plan view is a rectangular, generating line's length a = 2.0m, opening angle of the shell θ = 40 o , the radius of curvature is R = 2.0m, h = 0.02m, Es = 2.210 11 N/m 2 , s = 0.31, s = 7800kg/m 3 . The eccentrically ribbed shell with hr = 0.03m, br = 0.01m, the shell with 4 ribs is parallel to the generating line, 6 ribs is perpendicular to the generating line, the ribs are equispaced. Er = 2.410 11 N/m 2 , r = 0.3, r = 7000kg/m 3 . The mentioned shell has a round hole in the middle position, with d = 0.2 m (fig. 10). The load acting and the boundary are the same as Problem 1. Vertical displacement and stress at point A, field of von mises stress of the shell and the overall transposition field of the structure at time t = 0.021s are shown in Figure 11, 12, 13, 14 and table 2.   /dx.doi.org/10.22161/ijaers.4.11.24  ISSN: 2349-6495(P) | 2456-1908(O) www.ijaers.com Page | 163 Comment: Immediately after the second wave appeared, the vertical displacement of the point A increases quite a lot, which represents the great influence of the second wave on the structure. At the time t = 0.021s, the shell appeared four symmetrical areas through the center with relatively large stress and displacement responses compared to the other positions. This is due to the reduction in the pressure applied to the position at the center of the cover.

Problem 3
Considering the doubly curved shell whose plan view is a rectangular, a = 1.5m, b = 1.0m, R1 = 2.0m, R2 = -4.0m, the thickness h = 0.005m. Shell material has Es = 2.110 11 N/m 2 , s = 0.31, s = 7800kg/m 3 ( fig. 15). The load acting is the same as Problem 1. Vertical displacement and stress at center point, field of von mises stress of the shell and the overall transposition field of the structure at time t = 0.025s are shown in Figure 16, 17, 18, 19 and table 3.

Problem 4
The parameters of the model are similar to the parameters in problem 3. The difference is that the shell has a square hole (a1 x a1) in the middle position, with a1 = 0.2 m.  is much greater, this shows that the more susceptible to damage of the structure when there is a defect on its.
VI. CONCLUSION In this study, using the ANSYS APDL programming language, a program has been established that allows for solving many different problem classes. The paper focuses on solving the problem of calculating the shell structure with one or two curvature with or without holes, which is affected by the impulse load system. The results show the complex response of the structure when multiple layers of shock wave load are applied. Solving different problem classes demonstrates the ability of the program. The results of the study may be good references when calculating, designing the same structural.