Dynamic Analysis of Isotropic Plate

Dynamic Analysis of Isotropic Plate

V. Keshav and S. N. Patel

Department of Civil Engineering BITS Pilani Campus, Pilani-333031, (Raj), India

Abstract- This paper deals with the study of simply supported isotropic plates subjected to transverse dynamic loads. Thefinite element explicit formulation (ABAQUS/Explicit finite element code) is utilised for determining the responses of the plates. The transient and steady state responses are found out. The present results are compared with the available results along with the convergence studies. Various parameters such as thickness, aspect ratio and type of loading arechanged in order to understand the impact of dependent parameters which played a key role in altering the response of the plates.

Keywords Dynamic load, finite element, isotropic load, dynamic load

1. INTRODUCTION

   When a structure is designed for static loading conditions, it is expected that all the loads that are acting on the structure either are permanent or the loads vary very slowly with time. However, it is possible that during the service life of the structure may be subjected to time dependent loads. These loads are known as dynamic loads example, earthquake, wind forces, tidal waves, vibrations from machines etc. A structure resists external forces by inertia and because the dynamic loads vary with time, the structure may not begiven time to mobilize the required amount of resistive forces. This could lead to catastrophic results even if themagnitude of the dynamic force is small. Hence it is essential to analyse any structure for dynamic loads.

   As one of the fundamental aspect of structure numerous studies have been carried out to understand the behaviour of structures under dynamic loads. But since these loads vary with time, there is always a level of uncertainty in terms of the magnitude and direction in which they act, which essentially means that there is no end to the analysis in this field. Mallikarjuna and Kant(1988) studied the effects of finite element mesh, time step, lamination scheme on the transient response of multi-layered symmetrically laminated composite plate based on higher order displacement model theory.They found that the simple C0 isoparametric formulation was efficient in predicting the transient response of composite plate.Kantet al.(1992) studied the same effects using higher order shear deformable theory of composite and sandwich plates. Ghosh and Sinha(2004)investigated the initiation and propagation of damages of laminated composite plates under forced vibrations and impact loads.

   Su and Banerjee (2015)investigated the free vibration of functionally graded Timoshenko beam by developing the dynamic stiffness method. They related the amplitudes of forces and displacements at the ends of the beam in order to

   formulate the dynamic stiffness matrix. They generated a 66 frequency dependent dynamic stiffness matrix which is used to analyse the functionally graded material which was found to be efficient and accurate. Nefovska-Danilovic and Petronijevic (2015)generated a computer program to examine the free vibration and dynamic response analysis of isotropic plates with arbitrary boundary conditions. They verified the results with the available literature and successfully concluded that the model performed accurately even though the numbers of variables were significantly reduced.

   In the current study, a simply supported square plate is analysed for different type of loading condition and the response of the plate is observed for variation in the thickness and aspect ratio of the plate.

2. THEORY AND FORMULATIONS

   The linear dynamic analysis is performed using the finite element software ABAQUS/Explicit. It solves the equation of motion(dynamic equilibrium equation), which can be written as,

   []{} + []{} + []{} = {()} ——————-(1) Where,

   {} – Global acceleration vector

   {} – Global velocity vector

   {} – Global displacement vector

   {()}- Force vector [] – Damping matrix []- Stiffness matrix [] – Mass matrix

   In the present study the damping effect is neglected, so the dynamic equation of equilibrium Eq. (1) becomes,

   []{} + []{} = {()} ——————————-(2)

   Equation (2) is solved for all analysis and the results are reported in the next section.

3. RESULTS AND DISCUSSION

   First the convergence and validation study is carried by taking some reported results from the literature. Then the proposed study is discussed and the results are reported.

   1. Convergence and Validation study:

      The analysis was carried out in ABAQUS/Explicit on a square isotropic simply supported plate(Fig.1), subjected to uniformly distributed load. The load is applied instantaneously upto 360 microseconds and the responses are

      reported upto the same time. The material properties and other parameters taken are as follows:

      a = b = 25cm; h = 5cm; = 0.25; E = 2.1 X 106 N/cm2; = 8 X 10-6 Ns2/cm4; q = 10 N/cm2; t = 5 s.

      Fig.1 Model of the plate analysed in Abaqus 6.10-1

      The dynamic analysis is performed by ABAQUS/Explicit considering different mesh sizes and the results are plotted in Fig.3.The element type is taken as S4R.S4R means a 4-node doubly curved thin or thick shell, reduced integration, hourglass control, finite membrane strains. It is seen from the figure that the results with 8×8 mesh sizes is almost converged. To validate the analysis further the results are compared with finite element results ofMallikarjuna and Kant(1988). In the reported results of Mallikarjuna and Kant(1988), it is found that the central displacements are shown in cm(w×1000). In the present study these are found to be in cm(w×10-3). The results of Mallikarjuna and Kant(1988) with cm(w×10-3) and the present results with 8×8 mesh size is plotted in Fig.2. It is found that the results are matching well.

      Fig. 2 Convergence and validation of the model

      Fig. 3 Central displacement of isotropic plate with different mesh sizes

   2. Proposed study

   The plate taken in the convergence and validation study is considered in this case with uniformly distributed load with two functions(Fig.4). The effect thickness and the aspect ratio on dynamic behaviouris studied.

   Load (N/cm2)
   Time (s)

   Load (N/cm2)

   Fig. 4(a) Rectangular loading

   Time (s)

   Fig.4(b) Triangular loading

   Effect of loading functions

   The square isotropic simply supported plate is subjected to rectangular and triangular loading (Fig.4). In order to observe the steady state response of the plate, the load is applied till 400 s and response is recorded till 4000s. The results are shown in Fig. 5 and Fig. 6.

   Fig. 5 Central displacement of isotropic plate with Rectangular loading

   Fig. 6 Central displacement of isotropic plate with Triangular loading

   It can be seen that the loading function completely alters the response of the system.

   Effect of thickness

   Further, the thickness of the plate is reduced to 2.5cm, 1.25cm and 0.625cm and subjected to rectangular loading. The load is applied till 400 s and response is recorded till 4000s. The results are plotted in Fig.7 Fig.10.

   Fig. 7 Central displacement of isotropic plate with 2.5cm thickness

   Fig. 8 Central displacement of isotropic plate with 1.25cm thickness

   Fig. 9 Central displacement of isotropic plate with 0.625cm thickness

   Fig. 10 Central displacement of isotropic plate with various thicknesses

   Combining the 4 graphs wont help much because when the thickness is halved, the response increases by a factor of approximately 8.

   3.2.3 Effect of aspect ratio

   The aspect ratio is calculated as /. is kept constant and b is increased. The thickness of the plate is 5cm and the plate is subjected to rectangular loading. The plate with aspect ratio 2,3,4,5 and 6 is considered and responses are recorded. The load is applied till 400 s and response is recorded till 4000s. The results are plotted in Fig. 11 and Fig. 12.

   Fig. 11 Central displacement of isotropic plate with various aspect ratios

   Fig. 12 Central displacement of isotropic plate with various aspect ratios

   The results of aspect ratio 1, 2, 4 and 6 are plotted in Fig. 11 and the results of aspect ratio 1, 2, 3 and 5 are plotted in Fig.

   1. Two separate plots are used as the one single graph will become overcrowded and interpreting the results is difficult. It can be seen from the two figures that the increase in the aspect ratio alters the frequency of the responses and after certain aspect ratio, the results do not vary much.

4. CONCLUSION

   The conclusions of the study can be summarised as:

   1. The linear dynamic analysis of simply supported square isotropic plate subjected touniform transverse loading is carried out using ABAQUS/Explicit, version 6.10-1.

   2. Initially the convergence and validation study is carried out, taking examples from available literature based on different mesh sizes.

   3. The change in transient response of the plate when subjected to rectangular and triangular loading is noticeable and it is also observed that the maximum displacement is very less in triangular loading compared to rectangular loading.

   4. In steady state response of the two loading conditions, it is observed that the decay in response is prominent in rectangular loading. After the load is removed, the amplitude is decaying slowly even without the consideration of damping because of the combined effects of stiffness and mass of the plate.

   5. With decrease in the thickness of the plate, the frequency of the response reduces and the amplitude of the response increases.

   6. The plate with aspect ratios 5 and 6 showed almost same response. With increase in aspect ratio, the frequency of the response reduces. The frequency of response of aspect ratio 1 is twice that of plate with aspect ratio 5 and 6.

   ACKNOWLEDGEMENT

   The authors thankfully acknowledge the Council of Scientific and Industrial Research (CSIR), New Delhi for providing the financial support for this investigation through the project No.22(0666)/14/EMR-II.

5. REFERENCES

1. Mallikarjuna and Kant, T., Dynamics of laminated composite plates with a higher order theory and finite element Discretization, Journal of Sound and Vibrations (1988) 126(3),463-475.

2. Kant, T., Arora C.P., Varaiya, J.H., Finite element transient analysis of composite and sandwich plates based on a refined theory and a mode superposition method, Composite Structures 22 (1992) 109-120.

3. Ghosh, A., Sinha, P.K., Dynamic and impact response of damaged laminated composite plates, Aircraft engineering and Aerospace Technology, vol. 76 lss 1 pp. 29-37(2004)

4. Su, H., Banerjee, J.R., Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams, Computer and structures 147 (2015) 107-116.

5. Nefovska-Danilovic, M., Petronijevic, M., In-plane free vibration and response analysis of isotropic rectangular plates using the dynamic stiffness method Computers and structures 152 (2015) 82-95.